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Statistics & Probability
NorthHolland
Letters 11 (1991) 161165
February 1991
Poisson approximation
of Bernoulli random variables
for random sums
Nikos Yannaros
Department of Mathematics, Royal Institute of Technology, S10044 Stockholm, Sweden
Received December
Revised January
1989
1990
Abstract:
and an appropriate
marked and thinned
Bounds for the total variation
Poisson distribution
point processes.
distance
are given. The results can be used to derive limit theorems
Some examples are given.
between the distribution of the sum of a random number
with rates of convergence
of Bernoulli summands
for
Keywords: Poisson convergence, random sums, total variation, point processes, thinned point processes, Hellinger integral.
1. Introduction
Poisson
value
matics, and in many other applications
following general
sequence of dependent
that each event has small probability
Then, it is of interest
events which occur
However, many of the known
not give any rates of convergence.
cannot be useful in applications.
we need explicit error bounds
tions.
Let cl, Ez,. . . , 5, be Bernoulli
with P( [, = 1) = p, = 1  P( 6, = 0), and consider
the sum S, = E, + .& + . . . t5,. In the last three
decades there has been a continued
approximation of the distribution
by a suitably chosen Poisson
effort has been made to evaluate
approximations by measuring
variation distance.
For any nonnegative
approximations
theory,
are essential
in insurance
in extreme
mathe
where the
We have a
events, such
of occurring.
in reliability,
situation appears.
or independent
to prove that the number
is approximately
limit theorems
of
Poisson.
do
Such theorems
In applications
for the approxima
random variables
interest
of the sum S,
distribution.
the error of such
the fit via the total
in the
Great
integervalued random
variables
tance d by
X and Y define the total variation dis
d(X, Y) = supIP(X~4) P(YEA)~.
A
It can be shown that
d(X, Y)=+ 5
k=O
IP(X=k)P(Y=k)l.
(1)
In the sequel we denote by U, a Poisson distribu
tion with mean A. In the case of independent
Barbour and Hall (1984) showed that
6,‘s
d(S,,, U,) < y f: P:,
k=l
(2)
where
where
gives the following
proximating
son distribution
X =p, + . . . +p,.
for any i = 1,. . , n, inequality
bound for the error
the binomial distribution
In the simplest case
(2)
p, =p
in ap
by the Pois
nk
e“p(np)k

k!
G (1  eCnP)p.
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161
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Another
d n.p was given by Romanowska
sharp bound for the variation distance
(1977):
d l7,p G tp/fi.
(4)
Let us now consider the sum S, = E, + Ez
integerval
of the
way in connec
of point process,
example
a population
to be random, and suppose
+ . . . +‘&
ued random
[,‘s. Such sums appear in a natural
tion with marking
and in extreme value theory. Another
the following. Consider
which is assumed
we want to study the subpopulation
the individuals having some property
if individual
i has the property
otherwise. Then, the number
property d is the sum S,.
In Section 2 we give bounds
between the distribution
distribution in the case of independent
summands. A new bound
two Poisson distributions
alizations and applications
cussed in Section 3.
where N is a nonnegative
variable, which is independent
and thinning
is
of size N
that
consisting
b. Let E, = 1
b, and
of individuals
of
5, = 0
with
for the distance
and the Poisson
Bernoulli
for the distance
is also obtained.
of the results are dis
of S,
between
Gener
2. Independent summands
To prove our results we need a result on Hellinger
integrals, and for properties
the reader to the book
(1987). Let F and G be two probability
and v a afinite measure
and G are absolutely continuous
and R with respect to v. The Hellinger integral (of
order t) is defined by
and details
by Jacod
we refer
and Shiryaev
measures
with respect to which F
with densities
f
P=P(F, @=/a d v.
The Hellinger
measure
integral does not depend on the
v and we have
(Ip)=sd(F,G)</~.
(5)
For any nonnegative
bles X and Y we obtain
integervalued random varia
P=E~=&(X=k)P(Y=k).
162
Let U,, and U, be Poisson
ters X and p respectively.
that
with mean parame
Freedman (1974) showed
+A, Upi,> < IAPI.
(6)
We shall first improve (6).
Theorem
2.1.
d(Kl q
<nun{ IfiJFrI, IAPI}.
Proof. We have
=e ( x+Pw
~ =e~scx. P,
X=0
where
6(h, p)=t(X+II.)&.
It follows that
d( U,, U,) < /l  p2 = 41  e26(h.lr’
</m=lfiJTII,
which completes the proof. 0
(9)
(10)
r
Note that Ifi fil
+ fi
= lhpl/(fi
> 1. The following
+,/II.)
sim < I X  p 1 when fi
ple lemma will be useful.
Lemma
we have
2.2. If N and the .$,‘s are independent, then
d(S,,
U,) 6 f f’(N=n)d(S,,
n=O
4). 0 (11)
In the case of i.i.d. Bernoulli
we have the following
random variables
result.
Theorem
random variables with P( 6, = 1) = p, and N a non
2.3. Let E,, t2,.
. be i.i.d. Bernoulli
(7)
(8)
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STATISTICS & PROBABILITY LETTERS
February 1991
negative
independent of the .$,‘s. Then we have
integervalued random variable which is
d(%J, u,J
for any a > 0.
Proof. Put h =pa. By (11) (3), and (4)
E P(N=
n=O
n)d(S,,, U,,)
Consider
Without
A> ~1, because
Then we have
now
loss of generality
6(X, p)=S(p,
the function 6 defined
we may assume
A). Put h=p+
by (9).
that
E.
6(X, jJ)==+fa/rd_.
Now,
,/w = 1 + +E/‘P  $( e/p)’ + E,
where E is positive
6(X, p) = ~E*/P  pE, which implies that
and
1 E 1 < &(~/p)~.
Hence,
6(h, p) G $(A  ~L)*/~.
By this inequality,
we have
(lo), and since E(n) < &%,
EMU,,, UPJ) =G 2 I {F””
(14)
Inequality
inequality
(12) follows
E(d(U,,,
from (13)
u,,>>,<pEINaI.
(14) and the
0
Note that a is arbitrary
that it minimizes
special case a = EN
equality, which follows from Lemma
inequality
E 1 N  EN I < /Var( N)
and may be chosen so
given by (12). In the
we have the following
the bound
in
2.2 and the
:
pE(1  epPN)
++ P J
Var( N )
EN
min{l, 2JpEN).
(15)
If N = n is fixed, then (15) implies (3) and (4) and
is therefore a natural generalization
equalities.
In the case where the 5,‘s are independent
not identically distributed
of these in
but
we have
d(S,, U,) G E P(N=n)d(S,,,
n=O
r/,,+ +,,)
+ f P(N=n)d(U,,+
n=O
+p,, U,).
BY (2),
d(S,, UA) GE
+E
l_,(Pl+“‘+Ph )
l
N 2
Pl+ ...
+tp,&J ;;lpl
.
(16)
= i
3. Generalizations and applications
In many applications
pendent events,
tional probability
Bernoulli
and for i>2put
fling (1975) showed that
we have a sequence
such that each has small condi
given the past. Let E,, E2.. . . be
variables with
P(,$,=ljt ,,...,
of de
random
P(t, = 1) =p,,
<,_,)=p,.
Ser
@,A+
&EPA+
I=1
~E’P,.
!=I
(17)
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February 1991
where p = C:=, Ep,. Using Lemma 2.2 and (17) we
have
P,EP,I+ tE2p,
r=l
i
Thus,
d(S,, U,)<E ?E,P,EPA+
i I=1
:E’P,
1=l
N
IE c Ep,h
(18)
I,=1 /
In the special case h = E(Cr/ _, Ep,) we obtain
Let X,, . . _ , X, be independent
tegervalued random
sum by S,. Define
6, which equals 1 when
Denote by Si the sum of the 6,‘s. Serfling (1975)
showed that
nonnegative
and denote
random
in
variables
the Bernoulli
their
variable
X, = 1, and 0 otherwise.
d(.S,, S;) < c P(X, > 2).
(20)
,=l
We get
d(.S,, S;) < z P(N=n) k P(422).
1=l
(21)
n=O
From the triangular inequality it follows that
We have reduced
cial case of Bernoulli
obtain from (16),
the general
summands.
problem
For example,
to the spe
we
1 N
I
d(S,, U,)<E
I I
bh
I=1
+ E
1 _ ,(PI+
(
/ N
‘.. +PN) N
c Pf
r=*
PI + ...
+PN
1
\
+E
(22)
Now let N be some point
and let N, be the thinnedresealed
obtained by retaining
ability
p independently
the point process, and then replacing
point at time instant
have
process on [0, cc)
point process
every point of N with prob
of the other points and of
the retained
t, by a point at pt,. Then we
N,(t) = 5, + . . . +SN(r,p) = qv(,,p)’
where the 6,‘s are i.i.d. Bernoulli
which are independent
that if N is renewal,
process N, converges
same intensity as p + 0. Intuitively,
of N are deleted independently
very close to one, then it is natural
dependence between
ing a Poisson process as limit.
Renyi’s paper
research on limit theorems
but no rates of convergence
theorems were interesting,
theory was new and theoretical
oped. However, the theorems
applications since
given. Our results
theorems for random
rates of convergence.
It follows from (12) and (15) that
random variables
of N. RCnyi (1956) showed
then the thinnedresealed
to a Poisson process with the
if the points
with a probability
to have little
points, the retained indicat
was a starting point for much
deletions
These
when
for random
were given.
particularly
tools were devel
are not so useful in
no explicit error
can be used to derive
deletions and also to give
the
bounds are
limit
d( N,(t), U,,) 4 min
’
2J1p
’ p’EN(Vdj
+ min{
K,, K2},
(23)
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February 1991
where
and
K,=/pEN(t/‘p)AtI
Var( N( t/p)) “2 1
EN(t/p)
’
p/Var(N(t/p)) 1
Theorem 3.1. Assume that
lim El N( t)/t
t+m
 h 1 = 0,
(24)
(25)
(26)
where 0 < h < 00. Then, as p + 0, the thinnedres
caled process N, converges to a stationary
process with intensity X, and we have
Poisson
d(N,(t), U,,) < a\/l/)p +EipN(t/‘~) Xtl.
(27)
Proof. Inequality
By (26)
d(N,(A),
ring generated
notes the Lebesgue
process is uniquely
son distribution
generated
result is established.
(27) follows from (23) and (24).
EIpN(t/p)XtI +O as p0.
U,,,,)0
as p+0,
by the intervals,
measure.
determined
of counts for all sets in the ring
by the intervals (see Rtnyi
Hence,
for all A in the
where
Since the Poisson
by having a Pois
1 A 1 de
(1967)) the
?
Let now N be a renewal
that 0 < p < co, where p is the mean of the inter
arrival distribution. Then,
process, and assume
N(t)/t + l/p
a.s. and
in L’. Hence, condition
l/p. It follows
process with
RCnyi’s theorem.
strengthening
the following
mation.
(26) is satisfied
N, converges
l/p as p + 0, which
As a matter
of his result because
bound for the error in the approxi
with X =
to a Poisson that
rate gives
of fact, we have a
we also have
It is clear from the proof of Theorem
the result holds for point processes on R, and also
for point processes in more general
vided that condition (26) is assumed
class of sets generating
space. Convergence to a nonhomogeneous
son process follows also easily by considering
A(t) a function of t, and also to a mixed Poisson
process when X is a random
3.1 that
spaces pro
to hold for a
the Bore1 algebra in the
Pois
X =
variable.
References
Barbour,
convergence,
Freedman,
dent events, Ann. Probab. 2, 256269.
Jacod, J. and A.N. Shiryaev
chastic Processes (Springer,
RCnyi, A. (1956), A characterization
Translated in: P. Turan,
R&yi, Vol. I (Akad.
Renyi, A. (1967), Remarks
Math. Hungar. 2, 119123.
Romanowska, M. (1977), A note on the upper bound
distribution in total variation
Poisson distribution,
Stat. Neerlandica
Sertling, R.J. (1975), A general
rem, Ann. Probnb. 3, 726731.
A.D. and P. Hall (1984),
Math. Proc. Comb. Philos. Sm. 95, 473480.
D. (1974). The Poisson
On the rate of Poisson
approximation for depen
(1987),
Berlin).
Lwmt Theorems for Sto
of the Poisson
Selected
Budapest,
on the Poisson process,
process,
ed.,
Papers of Al/&d
1976) pp. 622628.
Studio Sci.
Kiado,
for the
and the between the binomial
31, 127130.
Poisson approximation theo
165