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Journ@l électronique d’Histoire des Probabilités et de la Statistique/ Electronic Journal for
History of Probability and Statistics . Vol.5, n°1. Juin/June 2009
How Paul L´
evy saw Jean Ville
and Martingales
Laurent MAZLIAK1
R´
esum´
e
Dans le pr´
esent article, nous examinons d’une part la mani`
ere dont Paul L´
evy dans les
ann´
ees 1930 a fait usage de conditions du type martingales pour ses ´
etudes de sommes de
variables al´
eatoires d´
ependantes, et d’autre part l’attitude qu’il a eue envers Jean-Andr´
e
Ville et ses travaux math´
ematiques.
Abstract
In the present paper, we consider how Paul L´
evy used martingale-type conditions for
his studies on sums of dependent random variables during the 1930s. In a second part,
we study L´
evy’s troubled relationship with Jean-Andr´
e Ville and his disdain for Ville’s
mathematical work.
Keywords and phrases : History of probability theory, martingales, dependent
random variables
AMS classication :
Primary : 01A60, 60-03
Secondary : 60G42, 60G44
Introduction
The present paper is a complement to several articles published in this issue of
the Electronic Journal for History of Probability and Statistics, devoted to the his-
tory of martingales. We will give here some extra information about some actors in
probabilistic history (Paul L´
evy (1886-1971) and Jean-Andr´
e Ville (1910-1989) in
1Laboratoire de Probabilit´
es et Mod`
eles Al´
eatoires et Institut de Math´
ematiques-Histoire des
Sciences, Universit´
e Paris VI
1
Journ@l électronique d’Histoire des Probabilités et de la Statistique/ Electronic Journal for
History of Probability and Statistics . Vol.5, n°1. Juin/June 2009
the rst place) and try to explain why they never succeeded in nding a common
basis for reection though their mathematical interest could once have ‘easily’
converged. Ville’s production is studied in several items of the present issue, and
it seems natural to expose some details about L´
evy’s work. Paul L´
evy was one of
the major gures on the probabilistic scene of the 20th Century, and his research
on limit theorems for sums of dependent variables in the middle of the 1930s had
considerable inuence on the future martingale theory. However, L´
evy was never
interested in nding an independent denition for martingales, and the martin-
gale condition always remained a technical condition for him. Added to L´
evy’s
personal mathematical disdain for Ville for which we will suggest some hints of
explanation, this disinterest also explains why L´
evy remained away from the birth
of martingale theory after World War 2.
The rst part of the paper is about L´
evy’s important research on subjects connec-
ted to the martingale property : how he grew interested in the question, how he
dealt with it, what kind of sequences of random variables satised the technical
condition he introduced. After having briey recalled the singular path followed
by L´
evy towards probability after the Great War, we will provide some informa-
tion on the kind of problems he considered and their origin. In particular, we insist
on the important question of the probabilistic study of continuous fractions which,
from the very beginning of 20th Century (especially in Borel’s studies) had been a
source of inspiration for major developments in probability. We will then describe
several works by L´
evy in which he introduced martingale-like conditions. More
precisely, we propose a detailed presentation of chapter VIII of his seminal book
[31], where L´
evy collected the results obtained in the 1930s about the extension
of limit theorems to dependent variables satisfying a martingale like condition.
We see chapter VIII as a kind of survey of the ultimate vision of martingales L´
evy
kept for the remaining of his life.
The second part of the present article focuses on L´
evy’s troubled relationship
with Ville and tries to explain his constant misunderstanding of the signicance
of his work. An unfortunate combination of circumstances, added to a clumsy pu-
blication by Ville in 1936, L´
evy’s taste for quick and nal judgments on people
and later the troubled times of the war and the Occupation, widened the gap bet-
ween the two mathematicians. L´
evy never had a real consideration for Ville and
this fact is recurrently proved by scornful comments to be found in his correspon-
dence with Maurice Fr´
echet (1878-1973). We do not know exactly to what extent
this disdain had an effect on Ville - but it probably had some. We believe that the
description of this complicated situation highlights some aspects of the creation
of the fundamental tools of modern probability theory.
2
Journ@l électronique d’Histoire des Probabilités et de la Statistique/ Electronic Journal for
History of Probability and Statistics . Vol.5, n°1. Juin/June 2009
the rst place) and try to explain why they never succeeded in nding a common
basis for reection though their mathematical interest could once have ‘easily’
converged. Ville’s production is studied in several items of the present issue, and
it seems natural to expose some details about L´
evy’s work. Paul L´
evy was one of
the major gures on the probabilistic scene of the 20th Century, and his research
on limit theorems for sums of dependent variables in the middle of the 1930s had
considerable inuence on the future martingale theory. However, L´
evy was never
interested in nding an independent denition for martingales, and the martin-
gale condition always remained a technical condition for him. Added to L´
evy’s
personal mathematical disdain for Ville for which we will suggest some hints of
explanation, this disinterest also explains why L´
evy remained away from the birth
of martingale theory after World War 2.
The rst part of the paper is about L´
evy’s important research on subjects connec-
ted to the martingale property : how he grew interested in the question, how he
dealt with it, what kind of sequences of random variables satised the technical
condition he introduced. After having briey recalled the singular path followed
by L´
evy towards probability after the Great War, we will provide some informa-
tion on the kind of problems he considered and their origin. In particular, we insist
on the important question of the probabilistic study of continuous fractions which,
from the very beginning of 20th Century (especially in Borel’s studies) had been a
source of inspiration for major developments in probability. We will then describe
several works by L´
evy in which he introduced martingale-like conditions. More
precisely, we propose a detailed presentation of chapter VIII of his seminal book
[31], where L´
evy collected the results obtained in the 1930s about the extension
of limit theorems to dependent variables satisfying a martingale like condition.
We see chapter VIII as a kind of survey of the ultimate vision of martingales L´
evy
kept for the remaining of his life.
The second part of the present article focuses on L´
evy’s troubled relationship
with Ville and tries to explain his constant misunderstanding of the signicance
of his work. An unfortunate combination of circumstances, added to a clumsy pu-
blication by Ville in 1936, L´
evy’s taste for quick and nal judgments on people
and later the troubled times of the war and the Occupation, widened the gap bet-
ween the two mathematicians. L´
evy never had a real consideration for Ville and
this fact is recurrently proved by scornful comments to be found in his correspon-
dence with Maurice Fr´
echet (1878-1973). We do not know exactly to what extent
this disdain had an effect on Ville - but it probably had some. We believe that the
description of this complicated situation highlights some aspects of the creation
of the fundamental tools of modern probability theory.
2
1 L´
evy and the martingale condition
1.1 L´
evy and his growing interest for probability
Before looking more carefully at the main topic of this paper, we want to re-
call some general information explaining why and how Paul L´
evy, who before
the Great War had never been interested in probability theory, was suddenly cap-
tivated by the subject to the point of becoming the unchallenged major French
probabilist of the inter-wars period. We shall only present a sketch of this history
here and suggest the interested reader consult other articles where the subject is
treated more deeply (see L´
evy’s comments in his autobiography [33], and secon-
dary litterature : [34], [3], [4], [37] ).
The rst encounter of L´
evy with probabilities as a professional mathematician
happened merely by chance. In 1919, Georges Humbert’s illness prevented him
from reading part of his lectures at the Ecole Polytechnique where he was pro-
fessor of mathematical analysis. L´
evy, who had been a r´
ep´
etiteur (lecturer) at the
Polytechnique since 1913 (a school where he had been himself an outstanding
student 12 years before), was asked to replace Humbert on the spot for some lec-
tures. Among them were three lectures on probability theory. We luckily have the
lecture notes on L´
evy’s rst teaching on probability. They were published in 2008
in Volume 3.1 of the Electronic Journal for History of Probability and Statistics,
along with the commentaries [4]. A regain of interest for teaching probability at
the Polytechnique resulted from the experience of the war where some basic pro-
babilistic techniques had been used at a very large scale. This is in particular the
case of the least square method used in ballistics to improve the precision of gun
ring.
L´
evy’s story with probability could have been limited to (rather basic) teaching
questions. However, at the same moment, freed at last from the military obliga-
tions (during the war, L´
evy said he had mainly worked on anti aircraft defense -
see [33] pp.54-55), he was resuming his research into potential theory. The pro-
minent gure of the probabilist somehow overshadows today that before beco-
ming a specialist in probability theory, Levy had been a brilliant follower of Vol-
terra and Hadamard’s techniques of function of lines for the potential theory of
general electric distributions. In 1911, he had defended a brilliant thesis in which
he studied Green functions as functions of lines which are solutions of integro-
differential equations. The paper [37] explains how after the war L´
evy had been
asked by Hadamard to prepare the posthumous edition of Gateaux’s papers. Young
French mathematician Gateaux (1889-1914) had been killed on the Front in Oc-
tober 1914. In the previous months, he had collected material for a thesis (also
on potential theory) where he began to construct an original theory of innite di-
mension integration. Hadamard’s request played a major role in L´
evy’s evolution,
when he realized that a probabilistic framework was well adapted to his problems.
A letter written to Fr´
echet much later (on April 1945) testies to the technology
transfer operated by L´
evy during those years between probability and potential
analysis.
3
Journ@l électronique d’Histoire des Probabilités et de la Statistique/ Electronic Journal for
History of Probability and Statistics . Vol.5, n°1. Juin/June 2009
As for myself, I learnt the rst elements of probability during the
spring of 1919 thanks to Carvallo (the director of studies at the Ecole
Polytechnique) who asked me to hold three conferences on that topic
to the students there. Besides, in three weeks, I succeded in proving
new results. And never will I claim for my work in probability a date
before 1919. I can even add, and I told M.Borel so, that I had not
really seen before 1929 how important were the new problems im-
plied by the theory of denumerable probabilities. But I was prepared
by functional calculus to the studies of functions with an innite num-
ber of variables and many of my ideas in functional analysis became
without effort ideas which could be applied in probability.
In fact, a rst trace of the probabilistic vision can be found in L´
evy-Fr´
echet’s
correspondence as early as January 1919 (so even before L´
evy really became in-
volved in probability. ..) when L´
evy wrote to Fr´
echet
For example, I think to limit the oscillations and irregularities of the
functions by bounding an integral Isuch as u2(t)dt, or at least by
considering as «less probable»the functions for which Iwould be
too large2.
The new probabilistic oriented mind proved especially spectacular in L´
evy’s
1922 book [26] on functional analysis, in particular in Chapter VI devoted to the
innite dimensional sphere.
1.2 Genesis of the martingale property
The genesis of a martingale type condition in L´
evy’s work had already been
presented by Cr´
epel in an unpublished and only half-developed note of a seminar
given in 1984 in Rennes. The present section closely follows Cr´
epel’s chronology.
Moreover, it will be interesting for the reader to compare several points we shall
develop in this section with the contents of the paper [15] (this issue).
As Cr´
epel mentioned, Soviet mathematician Serguey N. Berstein (1880-1968)
had studied several martingale situations during the 1920s and the beginning of
the 1930s, though he had not singled out the notion as an autonomous mathe-
matical denition. So one may ask what L´
evy exactly knew about these works
before he himself considered martingale situations. It is hard to have a denitive
answer to such a question but we nevertheless think that S.Bernstein’s inuence
on L´
evy at that moment was quite limited. First because it was often repeated by
L´
evy himself that he was not very fond of reading the works of others. Certainly
one must not take such an assertion for granted but in L´
evy’s case it seems cor-
roborated by converging information. A striking point is that S.Bernstein’s name
appears only very late in L´
evy’s correspondence with Fr´
echet (at least in the letters
which were found at the Paris Academy of Science, and published in [3]), contrary
2Our emphasis.
4
Journ@l électronique d’Histoire des Probabilités et de la Statistique/ Electronic Journal for
History of Probability and Statistics . Vol.5, n°1. Juin/June 2009
As for myself, I learnt the rst elements of probability during the
spring of 1919 thanks to Carvallo (the director of studies at the Ecole
Polytechnique) who asked me to hold three conferences on that topic
to the students there. Besides, in three weeks, I succeded in proving
new results. And never will I claim for my work in probability a date
before 1919. I can even add, and I told M.Borel so, that I had not
really seen before 1929 how important were the new problems im-
plied by the theory of denumerable probabilities. But I was prepared
by functional calculus to the studies of functions with an innite num-
ber of variables and many of my ideas in functional analysis became
without effort ideas which could be applied in probability.
In fact, a rst trace of the probabilistic vision can be found in L´
evy-Fr´
echet’s
correspondence as early as January 1919 (so even before L´
evy really became in-
volved in probability. ..) when L´
evy wrote to Fr´
echet
For example, I think to limit the oscillations and irregularities of the
functions by bounding an integral Isuch as u2(t)dt, or at least by
considering as «less probable»the functions for which Iwould be
too large2.
The new probabilistic oriented mind proved especially spectacular in L´
evy’s
1922 book [26] on functional analysis, in particular in Chapter VI devoted to the
innite dimensional sphere.
1.2 Genesis of the martingale property
The genesis of a martingale type condition in L´
evy’s work had already been
presented by Cr´
epel in an unpublished and only half-developed note of a seminar
given in 1984 in Rennes. The present section closely follows Cr´
epel’s chronology.
Moreover, it will be interesting for the reader to compare several points we shall
develop in this section with the contents of the paper [15] (this issue).
As Cr´
epel mentioned, Soviet mathematician Serguey N. Berstein (1880-1968)
had studied several martingale situations during the 1920s and the beginning of
the 1930s, though he had not singled out the notion as an autonomous mathe-
matical denition. So one may ask what L´
evy exactly knew about these works
before he himself considered martingale situations. It is hard to have a denitive
answer to such a question but we nevertheless think that S.Bernstein’s inuence
on L´
evy at that moment was quite limited. First because it was often repeated by
L´
evy himself that he was not very fond of reading the works of others. Certainly
one must not take such an assertion for granted but in L´
evy’s case it seems cor-
roborated by converging information. A striking point is that S.Bernstein’s name
appears only very late in L´
evy’s correspondence with Fr´
echet (at least in the letters
which were found at the Paris Academy of Science, and published in [3]), contrary
2Our emphasis.
4
to other Soviet scientists such as Andrei N. Kolmogorov (1903-1987) and Alek-
sandr Y.Khinchin (1894-1959). The rst mention of Berstein occured in 1942. Of
course, the correspondence is not complete and Berstein may certainly have been
quoted before. But in his letter dated 4 November 1942, L´
evy explained that he
asked Lo`
eve to give him a description of S.Bernstein’s 1932 talk at the interna-
tional congress of mathematicians in Z¨
urich, which seems to reveal that he had
at most a supercial knowledge of the paper. Cr´
epel says that L´
evy had read the
paper [6], where the Soviet mathematician obtained limit theorems - in particular
central limit theorems - for sequences of dependent random variables satisfying
martingale-type conditions. He was besides probably encouraged to read it as it
was written in French. And it is true that L´
evy wrote at the very beginning of his
paper [29] that S.Bernstein’s paper was an important step in the study of sums of
dependent variables. But one must certainly not overestimate the inuence of the
paper on L´
evy. The latter is not referred to before 1935, and maybe L´
evy was not
acquainted with it at all before someone told him that S.Berntein had dealt with
similar questions as himself. Fr´
echet, who read everything published, often played
this role of bibliographical source for L´
evy. Our hypothesis is therefore that L´
evy
had almost not been inspired by S.Bernstein’s works when he began to consider
martingales.
A rst trace of L´
evy’s observation of the martingale condition in a primitive set-
ting can be found in a paper written by L´
evy in 1929 [28] about the decomposition
of a real number in continued fractions.
Continued fractions decompositions had been studied by several analysts at the
end of the 19th Century. Let us in particular mention the important works by
Stieltj`
es (1856-1894) ([39]). In this study Stieltj`
es needed to introduce his gene-
ralization of Riemann’s integral, later extended by Lebesgue (see [22], Epilogue
pp.179 and seq). But how did continued fractions enter probability theory ? The
probabilistic study of continued fractions began with Swedish astronomer Gylden
(1841-1896) who was interested in describing the mean motion of planets around
the sun. To approximate this motion represented by a quasi-periodical function,
Gylden considered Lagrange’s techniques of approximation by continued frac-
tions (this fundamental approximation technique was developed some years later
by a student of Hermite, French mathematican Henri Pad´
e (1863-1953), is known
today as Pad´
e approximants - see [1] ). A smooth (analytical) function fcan be
represented as
f(t) = a0+tn1
a1+tn
2
a2+...
.
Gylden was therefore led to study the structure of the decomposition in conti-
nued fractions of a real number xto which he devoted three papers dated 1888
(including 2 excerpts from letters to Hermite published by the latter as notes in
the CRAS). In one of the papers, Gylden chose a probabilistic approach in which
he tried to specify the probability distribution of the quotients anfor a number x
drawn at random from [0,1]. More precisely, Gylden proved that the probability
of a value kfor anis of order 1/k.
5
Journ@l électronique d’Histoire des Probabilités et de la Statistique/ Electronic Journal for
History of Probability and Statistics . Vol.5, n°1. Juin/June 2009
In 1900, Gylden’s colleague, Lund astronomer Ander Wiman (1865-1959) consi-
dered the problem again in [43]3, applied to it Borel’s new theory of the measure
of sets, and obtained the value of the asymptotic probability for an=kunder
the form 1
ln 2 ln 1+1/k
1+1/k + 1.More details on these subjects can be found in [42],
pp.29-31.
Unfortunately, we do not know how Emile Borel (1871-1956) got acquainted
with Wiman’s work. There is no trace of a direct correspondence between Wi-
man and Borel. Nevertheless, one may suppose that Wiman sent his paper to
Borel, maybe through Mittag-Lefer (1846-1927) who had several exchanges
with Borel the same year 1900 about the interventions at the Paris International
Congress.An interesting possibility may also be another member of Borel’s Scan-
dinavian contacts, the Finnish analyst Ernst Lindel¨
of (1870-1946). On 2 January
1904, Lindel¨
of wrote to Borel the following line
One of my compatriots, M.Karl Sundman, a docent in astronomy
in our university, has been in Paris for a while and studies astro-
nomy and mathematics. He is a young man with exceptional intel-
ligence and perspicacity who will , probably, make a name in science.
Besides, he deserves already great congratulations by having dealt
with the edition of Gylden’s works which had been left uncomple-
ted. In one word, this young man wish to be a member of the Soci´
et´
e
Math´
ematique [de France] and I hope you will accept to be his spon-
sor.
We have not been able to cross-check Sundman’s meeting with Borel. But the
young Finn may have been a rsthand informer for Borel about Wiman’s works.
Anyway, in his rst publication devoted to probability in 1905 [7], Borel men-
tions that to his knowledge, Wiman’s work represents the rst attempt to apply his
measure theory of sets to a probabilistic problem.
Borel always saw the example of continued fractions as a fundamental source
of randomness. This example was particularly important in Borel’s seminal 1909
publication [8] where he presented the application of denumerable probabilities
to the decomposition of real numbers, both in decimal and in continued fractions
developments. Borel introduced in [8] the notion of almost sure convergence and
a rst version of the strong law of large numbers, thus inaugurating a way of pro-
ving existence by a probability computation which became a typical feature of the
Borelian reasoning. This reasoning was directly inherited from how he had intro-
duced the measure of sets in his thesis 15 years earlier. To prove the existence of
an arc of a circle on which a certain series was uniformly convergent, Borel proved
that he could choose the center of such an arc in the complement of a set which he
had proved to be of measure zero (see [22]). Therefore, from the very beginning
of his probabilistic life, Borel used the proof that an event has probability 1 as a
3In fact, Wiman was second in line to revise Gylden’s papers. He was preceded by another
Lund astronomer, Torsten Broden, and Wiman’s paper was a criticism and alternative approach to
Broden’s paper. See [42], p.31
6
Journ@l électronique d’Histoire des Probabilités et de la Statistique/ Electronic Journal for
History of Probability and Statistics . Vol.5, n°1. Juin/June 2009
In 1900, Gylden’s colleague, Lund astronomer Ander Wiman (1865-1959) consi-
dered the problem again in [43]3, applied to it Borel’s new theory of the measure
of sets, and obtained the value of the asymptotic probability for an=kunder
the form 1
ln 2 ln 1+1/k
1+1/k + 1.More details on these subjects can be found in [42],
pp.29-31.
Unfortunately, we do not know how Emile Borel (1871-1956) got acquainted
with Wiman’s work. There is no trace of a direct correspondence between Wi-
man and Borel. Nevertheless, one may suppose that Wiman sent his paper to
Borel, maybe through Mittag-Lefer (1846-1927) who had several exchanges
with Borel the same year 1900 about the interventions at the Paris International
Congress.An interesting possibility may also be another member of Borel’s Scan-
dinavian contacts, the Finnish analyst Ernst Lindel¨
of (1870-1946). On 2 January
1904, Lindel¨
of wrote to Borel the following line
One of my compatriots, M.Karl Sundman, a docent in astronomy
in our university, has been in Paris for a while and studies astro-
nomy and mathematics. He is a young man with exceptional intel-
ligence and perspicacity who will , probably, make a name in science.
Besides, he deserves already great congratulations by having dealt
with the edition of Gylden’s works which had been left uncomple-
ted. In one word, this young man wish to be a member of the Soci´
et´
e
Math´
ematique [de France] and I hope you will accept to be his spon-
sor.
We have not been able to cross-check Sundman’s meeting with Borel. But the
young Finn may have been a rsthand informer for Borel about Wiman’s works.
Anyway, in his rst publication devoted to probability in 1905 [7], Borel men-
tions that to his knowledge, Wiman’s work represents the rst attempt to apply his
measure theory of sets to a probabilistic problem.
Borel always saw the example of continued fractions as a fundamental source
of randomness. This example was particularly important in Borel’s seminal 1909
publication [8] where he presented the application of denumerable probabilities
to the decomposition of real numbers, both in decimal and in continued fractions
developments. Borel introduced in [8] the notion of almost sure convergence and
a rst version of the strong law of large numbers, thus inaugurating a way of pro-
ving existence by a probability computation which became a typical feature of the
Borelian reasoning. This reasoning was directly inherited from how he had intro-
duced the measure of sets in his thesis 15 years earlier. To prove the existence of
an arc of a circle on which a certain series was uniformly convergent, Borel proved
that he could choose the center of such an arc in the complement of a set which he
had proved to be of measure zero (see [22]). Therefore, from the very beginning
of his probabilistic life, Borel used the proof that an event has probability 1 as a
3In fact, Wiman was second in line to revise Gylden’s papers. He was preceded by another
Lund astronomer, Torsten Broden, and Wiman’s paper was a criticism and alternative approach to
Broden’s paper. See [42], p.31
6
proof of existence. A good example is given in section 13 of the second Chap-
ter of [8], where Borel commented on the proof that almost every real number is
absolutely normal. Let us recall that a number is said to be normal if each gure
between 0 and 9 appears with a frequency 1/10 in its decimal decomposition ; it
is absolutely normal if the same property is true with the d-basis decomposition
(with a frequence 1/d) for each integer d. Borel wrote
In the present state of science, the effective determination of an ab-
solutely normal number seems to be the most difcult problem ; it
would be interesting to solve it either by building an absolutely nor-
mal number, either by proving that, among the numbers which can be
effectively dened, none is absolutely normal. However paradoxical
may this proposition seem, it is not the least incompatible with the
fact that the probability for a number to be absolutely normal is equal
to one
This kind of strange existence proof is probably the reason why, as von Plato
observes ([42], p.57), the strong law of large numbers and denumerable probabili-
ties seem to have caught mathematicians by surprise and attracted several uncom-
prehending comments. A vigorous reaction came in 1912 from Felix Bernstein
(1878-1956) when he revisited Gylden’s approach of the problem of secular per-
turbations in his article [5] by a systematic use of the ‘measure of sets of E.Borel
and H.Lebesgue’ ([5], p.421)4. F.Bernstein contested in his paper the result ob-
tained by Borel in [8] concerning the asymptotical order of the quotients in a
continued fraction and thought he had found a contradiction with his own results.
F.Bernstein wrote
For the continued fractions, [Borel] established the following result :
if one considers only quotients which have an inuence on liman,
then their growth order is smaller than ϕ(n)with denumerable proba-
bility 1 if 1
ϕ(n)converges, and larger that ϕ(n)if 1
ϕ(n)diverges.
The last part of the theorem is contained in the second part of theo-
rem 45. On the contrary, the rst part is in contradiction with the result
obtained in theorem 4. The reason for this contradiction is of crucial
importance and we shall explain it precisely. The following fact is
true : for geometrical probabilities under consideration, the indepen-
dence of the elementary cases is not realized.
The basis of the contradiction for F.Bernstein was thus Borel’s application of
his (Borel-Cantelli) lemma to a non independent case. Several weeks later, Bo-
rel replied in a short paper published in the same journal [9]. He emphasized the
fact that F.Bernstein’s result is in no way contradictory with his own, but admit-
ted that he did not precisely write [8] for the case of dependent variables as the
quotients anare. Borel proposed thus a new proof. In [9] (p.579), he assumes that
4F.Bernstein’s interest for secular perturbations had grown from a paper published by Bohl in
1909.
5Exposed earlier in [5]
7
Journ@l électronique d’Histoire des Probabilités et de la Statistique/ Electronic Journal for
History of Probability and Statistics . Vol.5, n°1. Juin/June 2009
the conditional probability pnof the n-th event given the preceding ones satises
p
n≤pn≤p
nwhere the series p
nand p
nhave the same behavior (convergence or
divergence). Borel does not give any hint of how one may obtain the two terms p
n
and p
n. Moreover he limits the proof (of the conditional Borel(-Cantelli) lemma)
to the case when p
nand p
nare convergent series, asserting without any comment
that the proof would be the same in the divergent case (an unfortunate observation
as the result is false in the non independent divergent case !). Nevertheless, one
may detect in this proof (where Borel considers the evolution of the conditional
means) a rst use of a martingale convergence theorem. This is today used as a
common tool for obtaining the conditional version of Borel-Cantelli lemma (see
for instance [2], p.35). Moreover, it is not by mere chance that at the same mo-
ment, Borel revisited Poncar´
e’s card shufing problem in note [10] and proposed
a probabilistic proof of the convergence to the uniform distribution (ergodic theo-
rem) by consideration of the evolution of the means ; this was the rst appearance
of a probabilistic proof of convergence of a Markov chain, apart from Markov’s
original proof which remained completely unknown until much later. Besides Bo-
rel’s note also remained unnoticed, and his proof was rediscovered and extended
by L´
evy, Hadamard, Hostinsk´
y and others at the end of the 1920s (see [14] and
[36] on these subjects).
In [9], Borel underlines F.Bernstein’s confusion ; for him, F.Bernstein did not
understand that in the convergence case, with probability 1, the inequality an≥
ϕ(n)stopped being true beyond a rank nwhich changed with ω.
Still more interesting is what Borel wrote in a subsequent part, when he com-
mented on Berstein’s axiom on p.419. F.Bernstein indeed explained
When one relates the values of an experimentally measured quantity
to the scale of all the reals, one can exclude in advance from the latter
any set of measure 0. One should expect only such consequences of
the observed events which are maintained when the observed value is
changed to another one within the interval of observation.
Borel wrote ([9], pp.583-584)
I have often thought about the same kind of considerations and, as
M.Bernstein, I am convinced that the theory of measure, and espe-
cially of measure zero, is intended to play a major role in the ques-
tions of statistical mechanics.
Maybe in F.Bernstein’s text Borel found a rst formulation of what he called much
later (in [12]) the unique law of randomness ; for Borel, the signicance of pro-
bability is related to the events with small probability which are the only ones for
which probability has a practical and objective meaning : these events have to be
considered as impossible.
As said above, in his 1929 paper [28], L´
evy considered continuous fractions.
His general problem was to look for properties that the sequence of incomplete
quotients had in common with a sequence of independent random variables. On
page 190, he wrote
8
Journ@l électronique d’Histoire des Probabilités et de la Statistique/ Electronic Journal for
History of Probability and Statistics . Vol.5, n°1. Juin/June 2009
the conditional probability pnof the n-th event given the preceding ones satises
p
n≤pn≤p
nwhere the series p
nand p
nhave the same behavior (convergence or
divergence). Borel does not give any hint of how one may obtain the two terms p
n
and p
n. Moreover he limits the proof (of the conditional Borel(-Cantelli) lemma)
to the case when p
nand p
nare convergent series, asserting without any comment
that the proof would be the same in the divergent case (an unfortunate observation
as the result is false in the non independent divergent case !). Nevertheless, one
may detect in this proof (where Borel considers the evolution of the conditional
means) a rst use of a martingale convergence theorem. This is today used as a
common tool for obtaining the conditional version of Borel-Cantelli lemma (see
for instance [2], p.35). Moreover, it is not by mere chance that at the same mo-
ment, Borel revisited Poncar´
e’s card shufing problem in note [10] and proposed
a probabilistic proof of the convergence to the uniform distribution (ergodic theo-
rem) by consideration of the evolution of the means ; this was the rst appearance
of a probabilistic proof of convergence of a Markov chain, apart from Markov’s
original proof which remained completely unknown until much later. Besides Bo-
rel’s note also remained unnoticed, and his proof was rediscovered and extended
by L´
evy, Hadamard, Hostinsk´
y and others at the end of the 1920s (see [14] and
[36] on these subjects).
In [9], Borel underlines F.Bernstein’s confusion ; for him, F.Bernstein did not
understand that in the convergence case, with probability 1, the inequality an≥
ϕ(n)stopped being true beyond a rank nwhich changed with ω.
Still more interesting is what Borel wrote in a subsequent part, when he com-
mented on Berstein’s axiom on p.419. F.Bernstein indeed explained
When one relates the values of an experimentally measured quantity
to the scale of all the reals, one can exclude in advance from the latter
any set of measure 0. One should expect only such consequences of
the observed events which are maintained when the observed value is
changed to another one within the interval of observation.
Borel wrote ([9], pp.583-584)
I have often thought about the same kind of considerations and, as
M.Bernstein, I am convinced that the theory of measure, and espe-
cially of measure zero, is intended to play a major role in the ques-
tions of statistical mechanics.
Maybe in F.Bernstein’s text Borel found a rst formulation of what he called much
later (in [12]) the unique law of randomness ; for Borel, the signicance of pro-
bability is related to the events with small probability which are the only ones for
which probability has a practical and objective meaning : these events have to be
considered as impossible.
As said above, in his 1929 paper [28], L´
evy considered continuous fractions.
His general problem was to look for properties that the sequence of incomplete
quotients had in common with a sequence of independent random variables. On
page 190, he wrote
8
In an unlimited series of experiments giving probabilities α1, α2, . . . , αn, . . .
to an event A, its frequency during the rst nexperiments differs from
the mean probability
α
n=α1+. . . +αn
n
by a quantity almost surely small for ninnite, that is to say that it
converges to zero, except in cases of total probability inferior to any
given positive quantity.
It must be observed that this property does not suppose the existence
of a limit for αn: it is besides of little importance that the considered
probability be independent or not ; if they form a succession, every
probability αnbeing estimated at the moment of the experiment on
the basis of the previous experiments, the theorem remains clearly
true.
As seen, L´
evy expressed himself in a rather loose way, proposing rather an as-
sertion than any proof. Only several years later did he feel necessary to provide a
complete proof, among a series of papers from 1934-1936 devoted to the studies
of limit theorems for sequences (and series) of dependent variables. In the intro-
duction of his paper [30] (pp.11-12), L´
evy explains how he interpreted his new
considerations on the strong law of large numbers as an extension of the intuition
he had had in 1929.
The idea on which this research is based, rst mentioned in 1929
about an application to the study of continued fractions, is that most
theorems related to sequences of independent random variables may
be extended to a sequence of variables in chain
u1, u2, . . . , un, . . .
if one takes care of introducing, for each of these variables un, not
its a priori probability distribution, but the a posteriori distribution
on which it depends when u1, u2, . . . , un−1are given, and which in
practice characterizes the conditions of the experience which leads to
the determination of un. It is well known that, without this precaution,
the extension of the simplest asymptotical theorems is impossible ;
when these a posteriori distributions are introduced, it becomes on
the contrary easy.
The simplest application of this observation leads to think that, under
slightly restrictive conditions, one obtains a good evaluation of the
sum
Sn=u1+u2+. . . +un
when each term uνis replaced, not by E{uν}, but by Eν−1{uν}. One
probably will object that the so-obtained approximated value is a ran-
dom variable, and does not have the practical value of an a priori
9
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History of Probability and Statistics . Vol.5, n°1. Juin/June 2009
evaluation. But in the calculus of probability, at least in a general
theory, one cannot hope more than to specify the probable relation
between the probability distribution and the result of the experiment,
between the cause and the effect ; the obtained assertions could only
lead to more precise conclusions in the special cases where one is
able to specify how the conditions of each experiment depend on the
results of the previous ones. The already mentioned application to the
study of continued fractions is sufcient to justify the interest of the
method.
In the same paper, in a footnote on page 13, L´
evy commented on the loose
presentation he provided in 1929.
If I limited myself to a statement without proof, it was partly not to
interrupt a paper devoted to continued fractions by too long a digres-
sion, and partly because, being unsure of having read all the published
works on the strong law of large numbers, I thought that so simple a
result may have been already known; since then I came to the conclu-
sion that it was a new result, and I do not think that its proof had been
published before.
Cr´
epel already mentioned that L´
evy’s explanation is reliable but insisted that
L´
evy’s lack of precision must also be understood as a proof that at that moment
(1929) he had not yet understood that he may formulate an independent property
which would guarantee the validity of the theorem.
The martingale condition was formulated in a subsequent paper ([29]), though
not at the beginning. [29] is devoted to the extension of the strong law to the case
of dependent variables. In L´
evy’s mind, such an extension was a continuation of
the theory of Markov chains.
L´
evy’s main tool for considering general sequences of random variables was to
see them as points in the innite-dimensional cube [0,1]IN equipped with the “Le-
besgue” measure. One may recognize there a direct inheritance of L´
evy’s rst pro-
babilistic consideration on the innite dimensional spaces. In [29], L´
evy proves a
version of a 0-1 law which is stated in the following way (p.88).
P(E)and Pn(E)represent respectively the probability of an event
Ebefore the determination of the xν, and after the determination of
x1, x2, . . . , xnand as a function of these known variables. This event
Edepends on the indenite sequence of the xν.
Lemma 1 If an event Ehas a probability α, the sequences realizing
this event, except in cases of probability zero, also realize the condi-
tion lim
n→+∞Pn(E) = 1.
In modern terms, one recognizes a particular case of a martingale convergence
theorem asserting that if (Fn)is a ltration such that Fn↑ F∞and zis a random
variable, then E(z/Fn)→E(z/F∞)a.s. (the theorem is considered here with
z= 1IE).
10
Journ@l électronique d’Histoire des Probabilités et de la Statistique/ Electronic Journal for
History of Probability and Statistics . Vol.5, n°1. Juin/June 2009
evaluation. But in the calculus of probability, at least in a general
theory, one cannot hope more than to specify the probable relation
between the probability distribution and the result of the experiment,
between the cause and the effect ; the obtained assertions could only
lead to more precise conclusions in the special cases where one is
able to specify how the conditions of each experiment depend on the
results of the previous ones. The already mentioned application to the
study of continued fractions is sufcient to justify the interest of the
method.
In the same paper, in a footnote on page 13, L´
evy commented on the loose
presentation he provided in 1929.
If I limited myself to a statement without proof, it was partly not to
interrupt a paper devoted to continued fractions by too long a digres-
sion, and partly because, being unsure of having read all the published
works on the strong law of large numbers, I thought that so simple a
result may have been already known; since then I came to the conclu-
sion that it was a new result, and I do not think that its proof had been
published before.
Cr´
epel already mentioned that L´
evy’s explanation is reliable but insisted that
L´
evy’s lack of precision must also be understood as a proof that at that moment
(1929) he had not yet understood that he may formulate an independent property
which would guarantee the validity of the theorem.
The martingale condition was formulated in a subsequent paper ([29]), though
not at the beginning. [29] is devoted to the extension of the strong law to the case
of dependent variables. In L´
evy’s mind, such an extension was a continuation of
the theory of Markov chains.
L´
evy’s main tool for considering general sequences of random variables was to
see them as points in the innite-dimensional cube [0,1]IN equipped with the “Le-
besgue” measure. One may recognize there a direct inheritance of L´
evy’s rst pro-
babilistic consideration on the innite dimensional spaces. In [29], L´
evy proves a
version of a 0-1 law which is stated in the following way (p.88).
P(E)and Pn(E)represent respectively the probability of an event
Ebefore the determination of the xν, and after the determination of
x1, x2, . . . , xnand as a function of these known variables. This event
Edepends on the indenite sequence of the xν.
Lemma 1 If an event Ehas a probability α, the sequences realizing
this event, except in cases of probability zero, also realize the condi-
tion lim
n→+∞Pn(E) = 1.
In modern terms, one recognizes a particular case of a martingale convergence
theorem asserting that if (Fn)is a ltration such that Fn↑ F∞and zis a random
variable, then E(z/Fn)→E(z/F∞)a.s. (the theorem is considered here with
z= 1IE).
10
Cr´
epel quotes Lo`
eve’s enthusiastic comment in [35]. For Lo`
eve, the previous
lemma is the rst convergence theorem of martingales and perhaps one of the
most beautiful results of probability theory. L´
evy also made comments later on
the result (in [33], p.93). He wrote
This theorem has an important particular case. If αnis independent
of n, and so equal to the a priori probability α=α0of the event E,
αis equal to zero or one (otherwise αn=αcould not tend towards
one of these possible limits). It is Kolmogorov’s theorem of zero-one
alternative. It is anterior to my 1934 work, but I did not know it when
I wrote this paper, which appeared in 1935.
L´
evy’s comment is conrmed by what he wrote to Fr´
echet about the same result
in January 1936, when they discussed together Kolmogorov’s measure-theoretic
proof of the 0-1 law in [23]
[Kolmogorov’s] proof is very simple and correct. One must get rid
of the impression that it is a conjuring trick. It uses the following
essential notion : the probability of the unlimited sequence of the xν
cannot be considered well dened unless it appears as the limit (in the
sense of convergence in probability) of the probability of a property of
the set of the rst nvariables - which implies the studied property with
a probability close to one, if it is realized for very large n. The desired
consequence is immediate. My own proof, I think, better highlights
these ideas. But one can feel them implicitly in Kolomogorov’s.
On Kolmogorov’s axiomatic version of probabilities, and in particular his proof
of the 0-1 law, and the connection with L´
evy’s vision, see [38].
The rst appearance of an explicit martingale condition is placed later in the
paper under the name Condition (C). It is stated on page 93 as
(C)En−1(un) = 0.
It is unclear what L´
evy had in mind with this letter ‘C’. Maybe ‘centered’, maybe
‘convergence’, maybe simply ‘condition’.
As a main use of condition (C), L´
evy proposes the following theorem which can
be seen as an extension of Kolmogorov’s theorem for the independent case.
Theorem 1 If the sequence (un)satises condition (C) and is uniformly bounded
by a number U, then unand En−1(un)2have the same nature (convergent
or divergent) with probability 1.
In Hostinsk´
y’s recension of the paper for the Zentrablatt, the Czech mathema-
tician alluded to this result under the condition that the probable value of un,
evaluated when one knows u1, u2, . . . , un−1in equal to zero.
What was the genesis of such a condition ? Unfortunately, the years when L´
evy
formulated it are precisely those when the major gap in L´
evy-Fr´
echet’s corres-
pondence is found, between 1931 and 1936 ! However, it is seen that at that time
11
Journ@l électronique d’Histoire des Probabilités et de la Statistique/ Electronic Journal for
History of Probability and Statistics . Vol.5, n°1. Juin/June 2009
L´
evy was looking for extensions of limit theorems to more general cases than in-
dependent sequences. He was therefore led to put a condition on the general term
unof the series to guarantee the convergence. The condition is stated on this ge-
neral term and was never seen by L´
evy as a property of the sequence of partial
sums Sn. L´
evy always kept this opinion and never considered a martingale-like
property as a property of a sequence of random variables (see below).
1.3 Chapter 8 of the book Th´
eorie de l’addition des variables
al´
eatoires
L´
evy’s most famous book [31] was published in 1937 and was mostly completed
during Summer 1936. It played an important role in making several fundamental
tools of modern probability theory known (such as L´
evy-Khinchin’s decomposi-
tion formula) and is now considered a classic. We may observe that L´
evy himself
was probably convinced of the particular importance of the results he had obtai-
ned between 1934 and 1936 about the behavior of the sums of random variables.
This could explain why he decided so quickly to collect them in a book. It is
not impossible that his meeting with Doeblin ( L´
evy rst met him during Spring
1936) inuenced him. It is known that Doeblin made great impression on the ra-
ther scarcely accessible L´
evy (on Doeblin’s beginnings in probability see [13] and
[36]). And in a letter to Fr´
echet ([3], 21 December 1936), L´
evy mentioned that he
prepared for 21-years-old Doeblin a copy of the manuscript.
The eighth chapter of [31] is called Various questions related to sums of va-
riables in chain. L´
evy himself presents it in a footnote as a collection of questions
studied in previous chapters for the case of independent variables and taken again
in that chapter but for ‘chained’ (dependent) variables. The chapter collects the
results obtained by L´
evy in previous years about the extension of limit theorems
to dependent variables and remained probably for him the vision of martingales
he accepted. It is therefore interesting to give a more detailed description to un-
derstand this ultimate vision. We shall now present a quick survey of Chapter VIII
of [31]. Basically, our aim is to emphasize two main ideas, already mentioned
above. First for L´
evy the (martingale) condition he introduced was nothing but a
technical condition on the general term of a series which could allow the extension
of the classical limit theorems. L´
evy never considered martingales as a property
related to the sequence itself. Second, Chapter VIII of the book [31] was probably
seen by L´
evy as a kind of conclusion to his research in the direction of the series
of random variables. And this also may explain why he did not later feel really
concerned with the way Ville and Doob began a full theory of martingales.
1.3.1 Representation of a sequence of dependent variables
L´
evy begins Chapter VIII by explaining what is for him the General problem of
chained probability (section 64, page 225). In general, ’chained probability’ is a
term covering any sequence of (dependent) random variables X1, X2, . . . , Xn, . . .
12
Journ@l électronique d’Histoire des Probabilités et de la Statistique/ Electronic Journal for
History of Probability and Statistics . Vol.5, n°1. Juin/June 2009
L´
evy was looking for extensions of limit theorems to more general cases than in-
dependent sequences. He was therefore led to put a condition on the general term
unof the series to guarantee the convergence. The condition is stated on this ge-
neral term and was never seen by L´
evy as a property of the sequence of partial
sums Sn. L´
evy always kept this opinion and never considered a martingale-like
property as a property of a sequence of random variables (see below).
1.3 Chapter 8 of the book Th´
eorie de l’addition des variables
al´
eatoires
L´
evy’s most famous book [31] was published in 1937 and was mostly completed
during Summer 1936. It played an important role in making several fundamental
tools of modern probability theory known (such as L´
evy-Khinchin’s decomposi-
tion formula) and is now considered a classic. We may observe that L´
evy himself
was probably convinced of the particular importance of the results he had obtai-
ned between 1934 and 1936 about the behavior of the sums of random variables.
This could explain why he decided so quickly to collect them in a book. It is
not impossible that his meeting with Doeblin ( L´
evy rst met him during Spring
1936) inuenced him. It is known that Doeblin made great impression on the ra-
ther scarcely accessible L´
evy (on Doeblin’s beginnings in probability see [13] and
[36]). And in a letter to Fr´
echet ([3], 21 December 1936), L´
evy mentioned that he
prepared for 21-years-old Doeblin a copy of the manuscript.
The eighth chapter of [31] is called Various questions related to sums of va-
riables in chain. L´
evy himself presents it in a footnote as a collection of questions
studied in previous chapters for the case of independent variables and taken again
in that chapter but for ‘chained’ (dependent) variables. The chapter collects the
results obtained by L´
evy in previous years about the extension of limit theorems
to dependent variables and remained probably for him the vision of martingales
he accepted. It is therefore interesting to give a more detailed description to un-
derstand this ultimate vision. We shall now present a quick survey of Chapter VIII
of [31]. Basically, our aim is to emphasize two main ideas, already mentioned
above. First for L´
evy the (martingale) condition he introduced was nothing but a
technical condition on the general term of a series which could allow the extension
of the classical limit theorems. L´
evy never considered martingales as a property
related to the sequence itself. Second, Chapter VIII of the book [31] was probably
seen by L´
evy as a kind of conclusion to his research in the direction of the series
of random variables. And this also may explain why he did not later feel really
concerned with the way Ville and Doob began a full theory of martingales.
1.3.1 Representation of a sequence of dependent variables
L´
evy begins Chapter VIII by explaining what is for him the General problem of
chained probability (section 64, page 225). In general, ’chained probability’ is a
term covering any sequence of (dependent) random variables X1, X2, . . . , Xn, . . .
12
and L´
evy wants to explain how the distribution of the sequence may be construc-
ted. The main tool, L´
evy explains, is to obtain a representation of the following
kind : Xn=Gn(Y1, Y2, . . . , Yn)where (Yn)is a sequence of independent ran-
dom variables with uniform distribution on [0,1]. The Ynmay be dened as Yn=
Fn(X1, X2, . . . , Xn)where Fn(X1, X2, . . . , Xn−1, z)is the distribution function
of the conditional distribution of Xnwhen X1, X2, . . . , Xn−1are given.
1.3.2 Markov Chains
In section 65 (p.227), L´
evy concentrates on the most important case, Markov
chains. After having presented the Chapman-Smoluchowski equations describing
the evolution of the transition probabilities, L´
evy provides interesting considera-
tions for justifying the importance of the Markovian situation. There are, L´
evy
writes, situations in Physics where one is not able to know all the parameters de-
ning the state of a system. One has to deal with the ‘apparent’ parameters and
to neglect the ‘hidden’ parameters. Of that kind are two particularly important
situations.
The rst one is when the knowledge of the past compensates for the ignorance of
the present values of the hidden parameters, and hence allows to predict the future.
This is the theory of hereditary phenomena developed by Volterra, for whom the
analytical tool is given by integro-differential equations. The second one is when
only the present value of the (apparent) parameters is known. One then cannot
do better than describe the probabilities of the future states (as a simple example,
L´
evy quotes gambling systems). For this situation, the natural analytical tool is
Markov chains for which the Huygens principle (the principle asserting that for
given times t0< t1< t2, one can equivalently determine the situation at time t2
by looking at the direct evolution from t0to t2or by looking rst at the evolution
from t0to t1and then from t1to t2) is expressed by the Chapman-Smoluchowski
equations. L´
evy’s connection between Volterra’s theory and Markov chains is a
direct interpretation of the early story of Markov chains at the end of the 1920s,
and in particular of Hostinsk´
y’s considerations. It is indeed probably from his stu-
dies on Volterra’s integro-differential equations that Hostinsk´
y was led to propose
a rst model of Markov chain with continuous state in 1928 (on Hostinsk´
y’s be-
ginnings in probability, see in particular [21]). L´
evy then develops the classical
historical model of cards shufing proposed by Hadamard for the description of
the mixing of two liquids, and subsequently studied by Poincar´
e, Borel and Hos-
tinsk´
y. It has already been mentioned that L´
evy had also considered this model in
his 1925 book, but without connecting it to a general situation (see [14] and the
letters from November 1928 in [3]). L´
evy takes advantage of his new book to de-
velop the proof of convergence towards uniform distribution of the cards (ergodic
principle) which was only sketched in [27] (L´
evy had already written down the
proof earlier on Fr´
echet’s request - see Letters 18 and 19 in [3]).
13
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1.3.3 The ‘martingale’ condition
After this long introduction about Markov chains, L´
evy presents section 66
whose title is extension of Bernoulli theorem and of Chebyshev’s method to sums
of chained variables. L´
evy begins by looking for conditions under which the va-
riance of the sum Snof centered random variables is equal to the sum of variances.
It sufces, L´
evy writes, that M(Xj)equals 0 for each i<jwhere M(Xj)is
the probable value of Xjwhen Xiis known (conditional expectation) . This is
obviously implied by the more restrictive hypothesis
(C)Mν−1(Xν) = 0, ν = 1,2,3, . . .
where Miis the probable value calculated as a function of X1, X2, . . . , Xisuppo-
sed given. And L´
evy adds : This hypothesis will play a major role in the sequel. If
Xndoes not satisfy C, one can consider the new sequence Yn=Xn−Mn−1(Xn).
In the same way, writing
Sn− M(Sn) =
n
1
(Mν(Sn)− Mν−1(Sn)),
allows to control the approximation of Snby M(Sn)with an error of order √n
when the inuence of the ν-th experiment is small on the n-th experiment when
n−νis large (for instance when
p
h=0 Mν(Xν+h)− Mν−1(Xν+h)is bounded
independently of νand p).
1.3.4 Consequences of condition (C): Central Limit theorem
Section 67 is devoted to the central limit theorem for sums of dependent va-
riables. The proof is presented as an extension of Lindeberg’s method for random
variables which are small with respect to the dispersion of their sum. Apart from
(C), L´
evy rst introduces two more hypotheses
(C1)Mν−1(X2
ν) = σ2
ν=M(X2
ν)
(C)|Xν|< εbn,where b2
n=
n
i=1
σ2
ν.
L´
evy observes that hypothesis (C1)implies that the conditional expectation of X2
ν
is not dependent on X1, X2, . . . , Xν−1. Under these hypotheses, L´
evy proves that
P(Sn
bn
< x)→1
√2πx
−∞
e−u2/2du,
along the lines of Lindeberg’s proof. In a second part of the section (p.242), he
proposes to weaken condition (C1), and to replace it by the requirement that the
probability of divergence of σ2
νbe positive.
14
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History of Probability and Statistics . Vol.5, n°1. Juin/June 2009
1.3.3 The ‘martingale’ condition
After this long introduction about Markov chains, L´
evy presents section 66
whose title is extension of Bernoulli theorem and of Chebyshev’s method to sums
of chained variables. L´
evy begins by looking for conditions under which the va-
riance of the sum Snof centered random variables is equal to the sum of variances.
It sufces, L´
evy writes, that M(Xj)equals 0 for each i<jwhere M(Xj)is
the probable value of Xjwhen Xiis known (conditional expectation) . This is
obviously implied by the more restrictive hypothesis
(C)Mν−1(Xν) = 0, ν = 1,2,3, . . .
where Miis the probable value calculated as a function of X1, X2, . . . , Xisuppo-
sed given. And L´
evy adds : This hypothesis will play a major role in the sequel. If
Xndoes not satisfy C, one can consider the new sequence Yn=Xn−Mn−1(Xn).
In the same way, writing
Sn− M(Sn) =
n
1
(Mν(Sn)− Mν−1(Sn)),
allows to control the approximation of Snby M(Sn)with an error of order √n
when the inuence of the ν-th experiment is small on the n-th experiment when
n−νis large (for instance when
p
h=0 Mν(Xν+h)− Mν−1(Xν+h)is bounded
independently of νand p).
1.3.4 Consequences of condition (C): Central Limit theorem
Section 67 is devoted to the central limit theorem for sums of dependent va-
riables. The proof is presented as an extension of Lindeberg’s method for random
variables which are small with respect to the dispersion of their sum. Apart from
(C), L´
evy rst introduces two more hypotheses
(C1)Mν−1(X2
ν) = σ2
ν=M(X2
ν)
(C)|Xν|< εbn,where b2
n=
n
i=1
σ2
ν.
L´
evy observes that hypothesis (C1)implies that the conditional expectation of X2
ν
is not dependent on X1, X2, . . . , Xν−1. Under these hypotheses, L´
evy proves that
P(Sn
bn
< x)→1
√2πx
−∞
e−u2/2du,
along the lines of Lindeberg’s proof. In a second part of the section (p.242), he
proposes to weaken condition (C1), and to replace it by the requirement that the
probability of divergence of σ2
νbe positive.
14
The section 68 is devoted to the general problem of convergence of series with
non independent terms. As L´
evy stipulates, the essential hypothesis is that condi-
tion (C)is satised and the second moments of Xνare nite. L´
evy begins by
showing that Kolmogorov’s inequality can be extended to that case, which allows
him to prove that the series Xνand Mν−1(X2
ν)have the same behaviour. This
in particular proves the conditional generalization of the Borel-Cantelli lemma
(called by L´
evy the lemma of M.Borel). Sections 69 to 72 are devoted to the ex-
tension of the strong law of large numbers and of the law of the iterated logarithm.
These parts are quite technical and we shall not enter into details. Let us only note
that L´
evy’s approach is always the same : extending former results (generally
Khinchin’s and Kolmogorov’s) under condition (C).
2 L´
evy versus Ville
The second part of our paper is devoted to the complicated relationship between
L´
evy and Ville. When one has a look at the index nominum of the L´
evy-Fr´
echet
correspondence [3], it is surprising to see that Ville’s name appears many times
in the letters. It is quoted 13 times, rst in 1936 (in a letter following the afore-
mentioned letter of December 1936 where Doeblin is mentioned for the rst time)
and eventually in 1964. However, and quite impressively, when one looks at these
quotations one after the other, one can observe that Ville’s name is almost always
associated with criticisms, being even sometimes rather derogatory remarks. It is
well known that L´
evy was a scathing person who never hesitated to show disdain
for works he considered uninteresting or without originality. But in his letters to
Fr´
echet he recurrently expressed particular negativity towards Ville.
It is interesting to have rst a closer look at the last letter in which Ville is quoted.
It was written on 28 April 1964, at a moment when L´
evy had just conquered a
long desired seat at the Paris Academy of Science (at the age of 78) where he
succeeded to the almost centenarian Hadamard. The tortuous story of Fr´
echet and
L´
evy’s elections to the Academy can be followed in details in [3]. As may be
imagined, one of the most urgent tasks of a new Academician is to think about
future candidates to replace the next dead Immortal and L´
evy’s letter probably
responds to Fr´
echet’s suggestion to take into consideration a possible application
from Ville.
I have never understood Ville’s rst denition of the collectives ; Lo`
eve
and Khinchin had told me and written to me they had not understood
either. It is in 1950, in Berkeley, that I learnt from Lo`
eve that the pro-
cesses called martingales are those I had considered as early as 1935 ;
after your letter, his second denition, p.99, coincides with mine at
least by adding constants.
Naturally, I did not use a word that I did not know in 1937 in the 1954
re-edition of my 1937 book ; in order to allow the photographic re-
production, I had only corrected some mistakes and added two notes.
15
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But condition C, introduced p.238, means that the sequence of Xνis a
martingale. This condition appears in the sequel : theorems 67,1 ;67,2 ;
67,3 ;68 ; n.69 1◦and 2◦. I have therefore sketched a theory, deve-
loped afterwards by Doob, and which generalizes the sequences of
independent random variables with probable values equal to zero.
As for the theory of collectives, despite all the credits I attribute to
von Mises, I have always found it absurd, and I did not hide this from
Wald when he presented it in Geneva. I am grateful to Ville for having
helped me ghting this theory. But it is not sufcient to place him at
the same level as... say Fortet and Dugu´
e, to speak only about the
probabilists from the Sorbonne.
From the last sentence, it seems that for L´
evy anyone could have been preferable
to Ville for the election at the Academy. And the way he insists on quoting all the
theorems from Chapter VIII of [31] where the condition Cwas used is probably
a sign of irritation against what may have seem to him Ville’s undue claim of ha-
ving constructed a new mathematical concept. L´
evy’s assertion that it was only
in 1950 that he learnt about the theory of martingales is probably true (though he
was present in Lyon in 1948 and listened to Doob’s conference - but maybe the
language made difculties for him to understand it6. L´
evy had never been a great
reader and often selected only papers that were in connection with his present
research. However, as the word had been introduced by Ville in the 1930s, his
observation also sounds as a renewed proof of disinterest for Ville’s contribution.
Besides, there is irony in seeing L´
evy going astray with the denition of martin-
gale when he mentions that the sequence Xνis a martingale and not the sequence
of the partial sums. We have already observed in the previous section that L´
evy
had never considered the property otherwise than a technical property on the ge-
neral term of a series which can allow the extension of limit theorems. The small
confusion here is probably related to this fact.
L´
evy’s rst comments on Ville in his correspondence with Fr´
echet happened in
1936. The name was quoted for the rst time on 23 December, but most of the
previous letter on December 21 is devoted to demeaning comments on a note by
Ville presented to the Academy of Science by Borel on 14 December 1936 [40].
The title of the note is On the convergence of the median of the rst nresults of
an innite sequence of independent trials. It was Ville’s third note that year (all
presented by Borel) but the two other concerned Ville’s studies of collectives. It is
not clear why Ville decided to publish this relatively elementary results. That Bo-
rel presented it is not so surprising as Borel’s opinion on Ville was very positive ;
Ville had been a brilliant student at the Ecole Normale Sup´
erieure, and anyway
Borel seems never to have been very particular about the notes he transmitted to
the Academy. Besides knowing when Ville became closely associated with Bo-
rel is an interesting question. Ville claimed later he had been writing up Borel’s
lectures on games in October 1937 when Fr´
echet wanted him to go to Geneva;
6On Doob’s 1948 conference in Lyon, see Bernard Locker’s comments, along with the original
text, in this issue.
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But condition C, introduced p.238, means that the sequence of Xνis a
martingale. This condition appears in the sequel : theorems 67,1 ;67,2 ;
67,3 ;68 ; n.69 1◦and 2◦. I have therefore sketched a theory, deve-
loped afterwards by Doob, and which generalizes the sequences of
independent random variables with probable values equal to zero.
As for the theory of collectives, despite all the credits I attribute to
von Mises, I have always found it absurd, and I did not hide this from
Wald when he presented it in Geneva. I am grateful to Ville for having
helped me ghting this theory. But it is not sufcient to place him at
the same level as... say Fortet and Dugu´
e, to speak only about the
probabilists from the Sorbonne.
From the last sentence, it seems that for L´
evy anyone could have been preferable
to Ville for the election at the Academy. And the way he insists on quoting all the
theorems from Chapter VIII of [31] where the condition Cwas used is probably
a sign of irritation against what may have seem to him Ville’s undue claim of ha-
ving constructed a new mathematical concept. L´
evy’s assertion that it was only
in 1950 that he learnt about the theory of martingales is probably true (though he
was present in Lyon in 1948 and listened to Doob’s conference - but maybe the
language made difculties for him to understand it6. L´
evy had never been a great
reader and often selected only papers that were in connection with his present
research. However, as the word had been introduced by Ville in the 1930s, his
observation also sounds as a renewed proof of disinterest for Ville’s contribution.
Besides, there is irony in seeing L´
evy going astray with the denition of martin-
gale when he mentions that the sequence Xνis a martingale and not the sequence
of the partial sums. We have already observed in the previous section that L´
evy
had never considered the property otherwise than a technical property on the ge-
neral term of a series which can allow the extension of limit theorems. The small
confusion here is probably related to this fact.
L´
evy’s rst comments on Ville in his correspondence with Fr´
echet happened in
1936. The name was quoted for the rst time on 23 December, but most of the
previous letter on December 21 is devoted to demeaning comments on a note by
Ville presented to the Academy of Science by Borel on 14 December 1936 [40].
The title of the note is On the convergence of the median of the rst nresults of
an innite sequence of independent trials. It was Ville’s third note that year (all
presented by Borel) but the two other concerned Ville’s studies of collectives. It is
not clear why Ville decided to publish this relatively elementary results. That Bo-
rel presented it is not so surprising as Borel’s opinion on Ville was very positive ;
Ville had been a brilliant student at the Ecole Normale Sup´
erieure, and anyway
Borel seems never to have been very particular about the notes he transmitted to
the Academy. Besides knowing when Ville became closely associated with Bo-
rel is an interesting question. Ville claimed later he had been writing up Borel’s
lectures on games in October 1937 when Fr´
echet wanted him to go to Geneva;
6On Doob’s 1948 conference in Lyon, see Bernard Locker’s comments, along with the original
text, in this issue.
16
Doeblin went instead. Perhaps it was during the winter term of 1936-1937 that
Borel gave the course and Ville was taking notes. The lectures were published as
[11]. But one may ask whether Ville asked Fr´
echet’s opinion about his project.
The results Ville obtained could be seen as a consequence of Glivenko-Cantelli’s
theorem on the uniform convergence of the empirical distribution functions. This
theorem had been stated and published in 1933 in an issue of the Italian journal
of actuaries (Giornale italiano degli attuari, whose director was Cantelli). Be-
sides, the issue in question contained three independent papers with the result,
by Cantelli, Glivenko and Kolmogorov (who was surprisingly forgotten when na-
ming the theorem). A striking fact is that the three papers [16], [20] and [24] had
the same title ‘On the empirical determination of a probability distribution’. In
1936, the result was well known among probabilists and statisticians. Fr´
echet de-
voted to the theorem two pages of his volume [19] published the same year. Ville
knew Fr´
echet’s book : he mentioned it as a reference for (Kolmogorov) strong
law of large numbers at the beginning of [40]. It is very likely that he did not
make the connection between his result and Glivenko-Cantelli theorem. Ville had
learnt probability with Fr´
echet at the beginning of the 1930s. It is possible that
he failed to realize that there were new topics in [19]. Besides, after two years
abroad in Berlin (1933-34) and then in Vienna (1934-35), and, back in Paris, his
interests for collectives and game theory in the years 1935-37 had marginalized
Ville in the small group dealing with Markov chains around Fr´
echet at the IHP,
where Doeblin became the leader. So, it is not obvious that Fr´
echet paid much at-
tention to what Ville was doing, and his attempt to support Ville possibly resulted
from his conscientiousness about doctorate students, and from a kind of tradition
of inter-generational solidarity at the Ecole Normale. Reading L´
evy’s letter on 21
December 1936, it seems that Fr´
echet tried to justify Ville’s submission to the
Academy but L´
evy’s reaction was rather contemptuous.
Let me come back to yesterday’s talk. It is certain that one can so-
metimes nd important and easy theorems that escaped former scien-
tists and saying that a theorem is easy does not mean condemning it.
But, when is under consideration a particular case in a general pro-
blem solved a long time ago, except in some difcult particular cases
which have been recently studied, I frankly think it would be quite ri-
diculous to look for a particular case of the classical theorem to build
that very case up. (. . .) Such is the case of the median. (. . .) The role
of the median has been elucidated for a long time ; it is an obvious
consequence of Borel and Cantelli’s results.
Fr´
echet immediately answered on December 22, probably trying once again to
milden L´
evy’s opinion. But in a new letter on December 23, extended by a kind of
post-scriptum on December 24, L´
evy drove a point home. First, he wrote a com-
plete elementary proof of Ville’s result (based on the Glivenko-Cantelli theorem
about which he referred Fr´
echet to his own book [19]). Second, he took the op-
portunity to expose his vision of mathematics and explained how different it was
from Fr´
echet’s vision in not so agreeable a tone. In the post-scriptum, he wrote
17
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In the case under consideration, I see only two fundamental ideas :
the uniform convergence, which is well known ; and the strong law of
large numbers of Borel Cantelli. Once admitted these two points, all
the theorems of Polya Glivenko Cantelli and Ville do not seem to me
to overpass what Darmois proposes to his students as an examination
test for the licence7.
The subject was closed with this letter, but it certainly convinced Ville not to
go forward in that direction, and persuaded L´
evy, who liked to have a denite
opinion on people (think about the difcult relationship he entertained with Ba-
chelier), that Ville was a dull mathematician. Let us observe moreover that Ville
was particularly unlucky with the (unexpected) confrontation with L´
evy about the
median at the precise moment when the latter was brilliantly making use of it to
prove convergence results for sums of random variables.
Though in his 1964 letter (see above) L´
evy wrote that he was grateful to Ville
for having fought von Mises collectives, there remains some doubts that he really
had read Ville’s thesis. When he was interviewed by Cr´
epel in 1984 (see [18])
[Ville said that ] Paul L´
evy had not read his thesis. ‘I don’t read’
he told Ville. Aside from his aversion to reading other mathemati-
cians, L´
evy was displeased that Ville’s thesis had been printed by the
Rendiconti del Circolo Matematico di Palermo. ‘You had your thesis
printed by the fascists’ he objected. ‘I didn’t have any money’, Ville
responded.
The last part of Ville’s remembrance must not be overinterpreted and, if it is
true, it is probably related to the particular situation in 1939 with the outbreak of
WW2 . It does not seem that in the 1920s L´
evy had harbored an open hostility
against Mussolini’s regime. Ironically, when he was in semi-clandestinity during
the war, he found, with other Jews, a relative security in the Italian occupation
zone in France.
Anyway, it is true that he never explicitly mentioned Ville’s thesis in his letters
to Fr´
echet. Only in his (late) letter from 28 April 1964 did he write that he had
never understood Ville’s rst denition of collectives - which is in Ville’s thesis -
but knowing about one denition is not a real proof of having read anything else
in the thesis. Therefore, when he wrote that Doob extended his theory of martin-
gales, L´
evy probably honestly thought that Ville had not substantially modied
the notion. However, as we have noticed before, L´
evy never considered the mar-
tingale property as an intrinsic property of a sequence of random variables. And it
is initially in Ville that Doob found his future ideas on martingales.
Ville proposed two denitions of a martingale in his thesis. The rst one stipu-
lates that a process with binary outcomes is a non-negative capital process. The se-
cond is more mathematical and concerns a sequence of functions sn(X1, . . . , Xn)
of a sequence of (dependent) random variables. It is related to L´
evy’s condition
(C)in the following way : the sequence (sn)is a martingale in Ville’s sense if
7This means for their graduation.
18
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History of Probability and Statistics . Vol.5, n°1. Juin/June 2009
In the case under consideration, I see only two fundamental ideas :
the uniform convergence, which is well known ; and the strong law of
large numbers of Borel Cantelli. Once admitted these two points, all
the theorems of Polya Glivenko Cantelli and Ville do not seem to me
to overpass what Darmois proposes to his students as an examination
test for the licence7.
The subject was closed with this letter, but it certainly convinced Ville not to
go forward in that direction, and persuaded L´
evy, who liked to have a denite
opinion on people (think about the difcult relationship he entertained with Ba-
chelier), that Ville was a dull mathematician. Let us observe moreover that Ville
was particularly unlucky with the (unexpected) confrontation with L´
evy about the
median at the precise moment when the latter was brilliantly making use of it to
prove convergence results for sums of random variables.
Though in his 1964 letter (see above) L´
evy wrote that he was grateful to Ville
for having fought von Mises collectives, there remains some doubts that he really
had read Ville’s thesis. When he was interviewed by Cr´
epel in 1984 (see [18])
[Ville said that ] Paul L´
evy had not read his thesis. ‘I don’t read’
he told Ville. Aside from his aversion to reading other mathemati-
cians, L´
evy was displeased that Ville’s thesis had been printed by the
Rendiconti del Circolo Matematico di Palermo. ‘You had your thesis
printed by the fascists’ he objected. ‘I didn’t have any money’, Ville
responded.
The last part of Ville’s remembrance must not be overinterpreted and, if it is
true, it is probably related to the particular situation in 1939 with the outbreak of
WW2 . It does not seem that in the 1920s L´
evy had harbored an open hostility
against Mussolini’s regime. Ironically, when he was in semi-clandestinity during
the war, he found, with other Jews, a relative security in the Italian occupation
zone in France.
Anyway, it is true that he never explicitly mentioned Ville’s thesis in his letters
to Fr´
echet. Only in his (late) letter from 28 April 1964 did he write that he had
never understood Ville’s rst denition of collectives - which is in Ville’s thesis -
but knowing about one denition is not a real proof of having read anything else
in the thesis. Therefore, when he wrote that Doob extended his theory of martin-
gales, L´
evy probably honestly thought that Ville had not substantially modied
the notion. However, as we have noticed before, L´
evy never considered the mar-
tingale property as an intrinsic property of a sequence of random variables. And it
is initially in Ville that Doob found his future ideas on martingales.
Ville proposed two denitions of a martingale in his thesis. The rst one stipu-
lates that a process with binary outcomes is a non-negative capital process. The se-
cond is more mathematical and concerns a sequence of functions sn(X1, . . . , Xn)
of a sequence of (dependent) random variables. It is related to L´
evy’s condition
(C)in the following way : the sequence (sn)is a martingale in Ville’s sense if
7This means for their graduation.
18
sn(X1, X2, . . . , Xn)−sn−1(X1, X2, . . . , Xn−1)satises L´
evy’s condition. It is
probably the second part of Chapter V of Ville’s thesis which caught Doob’s in-
terest above all. Here Ville generalized his second denition of martingales to
continuous time, adopting Kolmogorov’s denition of conditional expectation and
trying to prove the gambler’s ruin inequality in the framework of Doob’s 1937
paper on stochastic processes with a continuous parameter. Though Ville failed
because he tried to use as probability space the outsize set of all functions of time
instead of the topologically suitable set of continuous functions, he gave Doob a
fundamental new tool.
It is remarkable that L´
evy kept in touch with Ville during the Occupation period,
when he lived near Grenoble. Probably, if L´
evy had a bad opinion about Ville,
the latter had on the contrary a great admiration for L´
evy and wished to stay in
contact with him. However, he had a second scientic misfortune with L´
evy, this
time about the recurrence property of Brownian motion. Ville published in 1942
a note to the Comptes-Rendus on the subject ([41]) and was preparing a related
paper when he was informed by Fr´
echet that L´
evy had already published some
of his results in his great 1940 memoir to the AMS about Brownian motion [32].
Ville decided in 1943 to withdraw his own paper (maybe also because he knew
that L´
evy could not submit any paper at the time because of Vichy racial laws).
All this did not help L´
evy change his opinion on Ville as a poor mathematician,
but maybe made him feel some sympathy for the young man. He considered him
a serious and capable reader of his papers. In the long letter L´
evy wrote to Fr´
echet
on September 27, 1943, L´
evy mentioned that he would be happy to learn that Ville
would examine his new manuscript about random derivatives. L´
evy wrote
If you had the impression that I had little admiration for his works
(and actually they never seem very original to me, he is above all a
good pupil) I realize that he is very serious, has a great sense of rigor
and deeply knows the questions he deals with. I shall put my complete
trust in him.
In fact, Fr´
echet chose Lo`
eve for the work, maybe for safety reasons because he
was concerned about L´
evy’s difcult character. And, after the Liberation, L´
evy
returned to his former disdain. In the rst letter we have (12 March 1945), L´
evy
again explains to Fr´
echet that Ville’s 1936 note on the medians was not origi-
nal. However, this time, L´
evy had made a mistake, probably because he wrongly
remembered Ville’s note. A week before, he had copied on a sheet of paper a
theorem from [31] (theorem 43.2 which says that if Snis a sequence of random
variables converging in probability to S, then any converging sequence of me-
dians of Snconverges to a median of S) ; L´
evy asserted that Ville’s result was a
direct consequence of this theorem. However, this consequence was only indirect
because Ville considered empirical medians, a fact L´
evy was besides well aware
of in 1936. This was probably what Fr´
echet had replied to him. At the end of
the letter, Fr´
echet had written with a pencil : Replied on March 5 that it is a dif-
ferent theorem from Ville’s. Nevertheless, L´
evy, made the following not-so-kind
comment
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I was amazed when I received your letter. I always made a confu-
sion about Ville’s result you mentioned to me in 1936 when it was
published. I am sorry about that, but it does not change a lot my opi-
nion on the lack of originality of this note. The strong law of large
numbers (. . .) had been known for a long time (1917 or even 1909).
(. . .) Moreover, in my theorem 43.2, it is of little importance that the
distributions be theoretical probability distributions or empirical ones.
(. ..)Taking into account the strong law of large numbers, Ville’s result
appears therefore as an application of my theorem 43.2. Obviously, I
cannot blame Ville for not having known my book at the time when I
was correcting the proofs. But my theorem 43,2 has always been, in
my opinion, an obvious observation that I have explicitly stated only
because I needed it. In the same way, Ville’s theorem is for me only
an obvious corollary of the strong law of large numbers.
That was still not enough and two years later (on 20 August 1947), the subject
came back and L´
evy expressed that he was really fed up. He wrote to Fr´
echet :
Let me frankly tell you that there are details to which I cannot give as much im-
portance as you do. Later he added
I sometimes make the mistake of not making clear results which seem
obvious to me but are not for others. I have also skipped several priori-
ties which I am not in a position to claim afterwards. In the case under
consideration, the only thing I told you is that I had known Ville’s re-
sult for a long time. But, due to the fact that it is an obvious corollary
to Glivenko-Cantelli’s result, I did not claim to take any pride in it, or
to call it ‘my theorem’.
This letter seemed to have completed the discussion, and (if we consider the set
of L´
evy’s letters to Fr´
echet reasonably complete up to 1965), it was the last time
Ville was quoted in the correspondence except the 1964 letter mentioned at the
beginning of this section.
The above comments make us conclude that L´
evy had only a supercial know-
ledge of Ville’s works, including his thesis. He never set much value to the new
approach proposed by Ville and remained convinced that his Chapter VIII of [31]
was the ultimate knowledge on ‘martingales’ before Doob extended it.
Conclusion
As we wrote in the introduction, we do not know exactly how Ville faced L´
evy’s
lack of interest, but it probably played a part in his choice to leave university after
WW2 and to begin a career in industry (see Ville’s biography by Glenn Shafer in
this issue). In fact, this disinterest was only one element among others and Ville
had become a perfect outsider in the mathematical community. When he returned
from captivity in Germany, Ville mostly turned towards mathematical statistics. It
was in particular the theme of his Peccot lectures in 1942-43. Then a professorship
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I was amazed when I received your letter. I always made a confu-
sion about Ville’s result you mentioned to me in 1936 when it was
published. I am sorry about that, but it does not change a lot my opi-
nion on the lack of originality of this note. The strong law of large
numbers (. . .) had been known for a long time (1917 or even 1909).
(. . .) Moreover, in my theorem 43.2, it is of little importance that the
distributions be theoretical probability distributions or empirical ones.
(. ..)Taking into account the strong law of large numbers, Ville’s result
appears therefore as an application of my theorem 43.2. Obviously, I
cannot blame Ville for not having known my book at the time when I
was correcting the proofs. But my theorem 43,2 has always been, in
my opinion, an obvious observation that I have explicitly stated only
because I needed it. In the same way, Ville’s theorem is for me only
an obvious corollary of the strong law of large numbers.
That was still not enough and two years later (on 20 August 1947), the subject
came back and L´
evy expressed that he was really fed up. He wrote to Fr´
echet :
Let me frankly tell you that there are details to which I cannot give as much im-
portance as you do. Later he added
I sometimes make the mistake of not making clear results which seem
obvious to me but are not for others. I have also skipped several priori-
ties which I am not in a position to claim afterwards. In the case under
consideration, the only thing I told you is that I had known Ville’s re-
sult for a long time. But, due to the fact that it is an obvious corollary
to Glivenko-Cantelli’s result, I did not claim to take any pride in it, or
to call it ‘my theorem’.
This letter seemed to have completed the discussion, and (if we consider the set
of L´
evy’s letters to Fr´
echet reasonably complete up to 1965), it was the last time
Ville was quoted in the correspondence except the 1964 letter mentioned at the
beginning of this section.
The above comments make us conclude that L´
evy had only a supercial know-
ledge of Ville’s works, including his thesis. He never set much value to the new
approach proposed by Ville and remained convinced that his Chapter VIII of [31]
was the ultimate knowledge on ‘martingales’ before Doob extended it.
Conclusion
As we wrote in the introduction, we do not know exactly how Ville faced L´
evy’s
lack of interest, but it probably played a part in his choice to leave university after
WW2 and to begin a career in industry (see Ville’s biography by Glenn Shafer in
this issue). In fact, this disinterest was only one element among others and Ville
had become a perfect outsider in the mathematical community. When he returned
from captivity in Germany, Ville mostly turned towards mathematical statistics. It
was in particular the theme of his Peccot lectures in 1942-43. Then a professorship
20
of probability in Bordeaux was amazingly offered to Pisot, though Ville was the
leading probabilist of the place. A few months later, Mal´
ecot was preferred to
him for the position in Lyon. Mal´
ecot, a typical follower of Darmois’ methods in
statistics - a recycling of the British methodology (Pearson, Udny Yule) - applied
to biology (see [17], [25]) was obviously supported by the latter. A small scandal
happened because Mal´
ecot was closely related to Lyon (he was Eyraud’s son-in-
law) and the university had to face an accusation of localism. Joseph P´
er`
es wrote
an ambiguous report for the national committee judging the case, supporting Ville
but recommending to allow Lyon university to be free for the choice. The choice
of Mal´
ecot was conrmed.
A possible interpretation of Fr´
echet’s insistance on Ville in the letters with L´
evy
during and after the war is maybe precisely that he tried to obtain at least a small
support from L´
evy for Ville who needed to obtain an academic position. Anyway,
the support never came and L´
evy, after his bad judgment on Ville’s note [40] never
changed his opinion. In particular, he was not interested in Ville’s thesis and did
not pay much attention to his introduction of a new category of random processes
called martingales. L´
evy later claimed that it was only in the 1950s, when he went
to USA, that he learnt by chance from Lo`
eve that Doob had devised a theory
for this kind of processes. L´
evy’s disinterest was nevertheless not only due to
his bad opinion on Ville. A deeper reason was certainly that he was convinced
of having presented in [31] (especially, Chapter VIII with its condition (C)) a
rather complete version of how these processes could be dened and studied. L´
evy
never had the idea of considering ‘martingales’ which were not successive sums
of random variables , because his basic interest was to study extensions of the law
of large numbers and central limit theorem. It is thus true that he was not seduced
by Ville, but he was not really seduced by Doob either, though he later admitted
that Doob’s methods had proven more powerful than his own. Had L´
evy studied
with more care and attention what Ville had proposed, maybe some martingale
techniques would have arrived sooner in France after WW2 and under a different
shape. This may be a good subject for an alternate history study.
R´
ef´
erences
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Markov Chains, Chapman & Hall/CRC, 2002
[3] BARBUT, Marc, LOCKER, Bernard & MAZLIAK, Laurent : Paul L´
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Maurice Fr´
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Uber eine Anwendung der Mengenlehre auf ein aus
der Theorie des s¨
akularen St¨
orungen herr¨
uhrendes Problem, Math.Ann., 71,
417-439, 1912
[6] BERNSTEIN, Serge : Sur l’extension du th´
eor`
eme limite du calcul des pro-
babilit´
es aux sommes de quantit´
es d´
ependantes, Math.Ann, 97, 1-59, 1927
[7] BOREL, Emile : Remarques sur certaines questions de probabilit´
es, Bull.
SMF, 33, 123-128, 1905
[8] BOREL, Emile : Les probabilit´
es d´
enombrables et leurs applications
arithm´
etiques, Rend. Circ. Palermo 27 , 247–271 (1909)
[9] BOREL, Emile : Sur un probl`
eme de probabilit´
es relatif aux fractions conti-
nues, Math.Annalen, 72, 578-584, 1912
[10] BOREL, Emile : Sur le battage des cartes, CRAS, 154, 23-25, 1912
[11] BOREL, Emile : Applications aux jeux de hasard (J.Ville, r´
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e
du calcul des probabilit´
es et de ses applications par E.Borel, t.VI, fasc.II,
Gauthier-Villars ,1938
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es, Trait´
e du
calcul des probabilit´
es et leurs applications (Emile Borel, editor), Gauthier-
Villars, 1939
[13] BRU, Bernard : Doeblin’s life and work from his correspondence, in Doeblin
and Modern Probability, H.Cohn (editor), American Mathematical Society,
1-64, 1993
[14] BRU, Bernard : Souvenirs de Bologne, Jour.Soc.Fr.Stat, 144, 135-226, 2003
[15] BRU, Bernard & EID, Salah : Jessen’s theorem and L´
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pondence. Jehps, this issue. 2009
[16] CANTELLI, Francesco Paolo : Sulla determinazione empirica delle leggi di
probabilit`
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[17] CATELLIER, R´
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Preprint. 2009
[18] CREPEL, Pierre : Jean Ville’s recollections in 1984 and 1985, concerning
his work on martingales, reported by Pierre Cr´
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French by G.Shafer. Jehps, this issue.
[19] FRECHET, Maurice : Recherches th´
eoriques modernes sur le calcul des pro-
babilit´
es, Livre I. Trait´
e du calcul des probabilit´
es et de ses applications par
E.Borel, t.I, fasc.III, Gauthier-Villars ,1937
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bilit`
a, Giornale Ist.Ital.Attuari, 4, 92-99, 1933
[21] HAVLOVA, Veronika, MAZLIAK, Laurent & ˇ
SIˇ
SMA, Pavel : Le d´
ebut des
relations math´
ematiques franco-tch´
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a travers la correspon-
dance Hostinsk´
y-Fr´
echet, Electronic Journal for History of Probability and
Statistics, Vol.1, 1, 2005
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24
