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arXiv:0904.0538v1 [math.PR] 3 Apr 2009
Ces` aro summation for random fields
Allan Gut
Uppsala University
Ulrich Stadtm¨ uller
University of Ulm
Abstract
Various methods of summation for divergent series of real numbers have been generalized
to analogous results for sums of i.i.d. random variables.
corresponding to Ces` aro summation amounts to proving almost sure convergence of the Ces` aro
means. In the present paper we extend such results as well as weak laws and results on
complete convergence to random fields, more specifically to random variables indexed by Z2
the positive two-dimensional integer lattice points.
The natural extension of results
+,
1 Introduction
Various methods of summation for divergent series have been studied in the literature; see e.g.
[10, 21]. Several analogous results have been proved for sums of independent, identically distributed
(i.i.d.) random variables.
The most commonly studied method is Ces` aro summation, which is defined as follows: Let
{xn, n ≥ 0} be a sequence of real numbers and set, for α > −1,
Aα
n=(α + 1)(α + 2)···(α + n)
n!
,n = 1,2,...,andAα
0= 1.(1.1)
The sequence {xn, n ≥ 0} is said to be (C,α)-summable iff
1
Aα
n
n
?
k=0
Aα−1
n−kxk
converges asn → ∞.(1.2)
It is easily checked (with A−1
as ordinary convergence, and that (C,1)-convergence is the same as convergence of the arithmetic
means.
Now, let {Xk, k ≥ 1} be i.i.d. random variables with partial sums {Sn, n ≥ 1}, and let X be a
generic random variable. The following result is a natural probabilistic analog of (1.2).
n
= 0 for n ≥ 1 and A−1
0
= 1) that (C,0)-convergence is the same
Theorem 1.1 Let 0 < α ≤ 1. The sequence {Xk, k ≥ 1} is almost surely (a.s.) (C,α)-summable
iff E|X|1/α< ∞. More precisely,
1
Aα
n
n
?
k=0
Aα−1
n−kXk
a.s.
→ µasn → ∞⇐⇒E|X|1/α< ∞ and E X = µ.
For α = 1 this is, of course, the classical Kolmogorov strong law. For proofs we refer to [14]
(1
2).
2< α < 1), [1] (0 < α <1
2) and [2] (α =1
AMS 2000 subject classifications. Primary 60F15, 60G50, 60G60, 40G05; Secondary 60F05.
Keywords and phrases. Ces` aro summation, sums of i.i.d. random variables, complete convergence, convergence in
probability, almost sure convergence, strong law of large numbers.
Abbreviated title. Ces` aro summation for random fields.
Date. April 3, 2009
1
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A. Gut and U. Stadtm¨ uller
Convergence in probability for strongly integrable random variables taking their values in real
separable Banach spaces was establised in [11] under the assumption of strong integrability. In
the real valued case finite mean is not necessary; for α = 1 we obtain Feller’s weak law of large
numbers for which a tail condition is both necessary and sufficient; cf. e.g. [8], Section 6.4.1.
Next we present Theorem 2.1 of [7] where complete convergence was obtained.
Theorem 1.2 Let 0 < α ≤ 1. The sequence {Xk, k ≥ 1} converges completely to µ, i.e.,
∞
?
n=1
P????
n
?
k=0
Aα−1
n−kXk− µ
??? > Aα
nε?< ∞for every ε > 0,
if and only if
E|X|1/α< ∞,
E|X|2log+|X| < ∞,
E|X|2< ∞,
for
for
for
0 < α <1
α =1
2,
1
2< α ≤ 1,
2,
and E X = µ.
Here and in the following log+x = max{logx,1}.
The aim of the present paper is to generalize these results to random fields. For simplicity we
shall focus on random variables indexed by Z2
+, leaving the corresponding results for the index set
Zd
+, d > 2, to the readers.
The definition of Ces` aro summability for arrays extends as follows:
Definition 1.1 Let α, β > 0. The array {xm,n, m,n ≥ 0} is said to be (C,α,β)-summable iff
1
Aα
mAβ
n
?
m,n
m,n
?
k,l=0
Aα−1
n−kAβ−1
n−lxk,l
converges as m,n → ∞.(1.3)
Our setup thus is the set {Xk,l, (k,l) ∈ Z2
and Marcinkiewicz-Zygmund strong law runs as follows.
+} with partial sums Sm,n, (m,n) ∈ Z2
+. The Kolmogorov
Theorem 1.3 Let 0 < r < 2, and suppose that X,{Xk, k ∈ Zd} are i.i.d. random variables with
partial sums Sn=?
then
Sn
|n|1/r
k≤nXk, n ∈ Zd. If E|X|r(log+|X|d−1) < ∞, and E X = 0 when 1 ≤ r < 2,
a.s.
→ 0as
n → ∞.
Conversely, if almost sure convergence holds as stated, then E|X|r(log+|X|d−1) < ∞, and E X = 0
when 1 ≤ r < 2.
Here |n| =?d
The theorem was proved in [18] for the case r = 1 and, generally, in [5].
For the analogous weak laws a finite moment of order r suffices (in fact, even a little less), since
convergence in probability is independent of the order of the index set.
The central object of investigation in the present paper is
k=1niand n → ∞ means inf1≤k≤dni→ ∞, that is, all coordinates tend to infinity.
1
Aα
mAβ
n
m,n
?
k,l=0
Aα−1
m−kAβ−1
n−lXk,l,(1.4)
for which we shall establish conditions for convergence in probability, almost sure convergence and
complete convergence
Let us already at this point observe that for α = β = 1 the quantity in (1.4) reduces to that
of Theorem 1.3 with r = 1, that is, to the multiindex Kolmogorov strong law obtained in [18]. A
second thought leads us to extensions of Theorem 1.3 to the case when we do not normalize the
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Ces` aro summation for random fields
3
partial sums with the product of the coordinates raised to some power, but the product of the
coordinates raised to different powers, viz., to, for example (d = 2),
Sm,n
mαnβ
for0 < α < β ≤ 1,
(where thus the case α = β = 1/r relates to Theorem 1.3). Here we only mention that some
surprises occur depending on the domain of the parameters α and β. For details concerning this
“asymmetric” Kolmogorov-Marcinkiewicz-Zygmund extension we refer to [9].
After some preliminaries we present our results for the different modes of convergence mentioned
above. A final appendix contains a collection of so-called elementary but tedious calculations.
2Preliminaries
Here we collect some facts that will be used on and off, in general without specific reference.
•The first fact we shall use is that whenever weak forms of convergence or sums of proba-
bilites are inyvolved we may equivalently compute sums “backwards”, which, in view of the i.i.d.
assumption shows that, for example
?
m,n
m,n
?
k,l=0
P(Aα−1
m−kAβ−1
n−l|Xk| > Aα
mAβ
n) < ∞ ⇐⇒
?
m,n
m,n
?
k,l=1
P(Aα−1
k
Aβ−1
l
|X| > Aα
mAβ
n) < ∞. (2.1)
In the same vein the order of the index set is irrelevant, that is, one-dimensional results and
methods remain valid.
•Secondly we recall from (1.1) that Aα
0= 1 and that.
Aα
n=(α + 1)(α + 2)···(α + n)
n!
,n = 1,2,...,
which behaves asymptotically as
Aα
n∼
nα
Γ(α + 1)
asn → ∞, (2.2)
where ∼ denotes that the limit as n → ∞ of the ratio between the members on either side equals
1. Combining the two relations above tells us that
?
m,n
m,n
?
k,l=0
P(Aα−1
m−kAβ−1
n−l|X| > Aα
mAβ
n) < ∞⇐⇒
?
m,n
m,n
?
k,l=1
P(kα−1lβ−1|X| > mαnβ) < ∞.(2.3)
•We shall also make abundant use of the fact that if {ak∈ R, k ≥ 1}, then
an→ 0asn → ∞=⇒
1
n
n
?
k=1
ak→ 0asn → ∞,(2.4)
that if, in addition, wk∈ R+, k ≥ 1, with Bn=?n
k=1wk, n ≥ 1, where Bnր ∞ as n → ∞, then
1
Bn
n
?
k=1
wkak→ 0 asn → ∞, (2.5)
as well as integral versions of the same.
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A. Gut and U. Stadtm¨ uller
3 Convergence in probability
We thus begin by investigating convergence in probability. We do not aim at optimal conditions,
except that, as will be seen, the weak law does not require finiteness of the mean (whereas the
strong law does so).
Theorem 3.1 Let 0 < α ≤ β ≤ 1 and suppose that {Xk,l, k,l ≥ 0} are i.i.d. random variables.
Further, set, for 0 ≤ k ≤ m, 0 ≤ l ≤ n,
Ym,n
k,l
= Aα−1
m−kAβ−1
n−lXk,lI{|Xk,l| ≤ Aα
mAβ
n},S′
m,n=
m,n
?
k,l=0
Ym,n
k,l
andµm,n= E S′
m,n.
Then
1
Aα
mAβ
n
?m,n
k,l=0
?
Aα−1
m−kAβ−1
n−lXk,l− µm,n
?
p
→ 0 asm,n → ∞(3.1)
if
nP(|X| > n) → 0 asn → ∞.(3.2)
If, in addition,
µm,n
Aα
mAβ
n
→ 0asm,n → ∞,(3.3)
then
1
Aα
mAβ
n
m,n
?
k,l=0
Aα−1
m−kAβ−1
n−lXk,l
p
→ 0 asm,n → ∞.(3.4)
Remark 3.1 Condition (3.2) is short of E|X| < ∞, i.e., the theorem extends the Kolmogorov-
Feller weak law [12], [13], and [3], Section VII.7, to a weak law for weigthed random fields for a
class of weights decaying as powers of order less than 1 in each direction.
Corollary 3.1 If, in addition, E X = 0, then (3.4) holds (and if the mean µ is not equal to zero
the limit in (3.4) equals µ).
Corollary 3.2 If, in addition, the distribution of the summands is symmetric, then (3.2) alone
suffices for (3.4) to hold.
Proof of Theorem 3.1. The proof of the theorem amounts to an application of the so-called
degenerate convergence criterion, see e.g. [8], Theorem 6.3.3.
Recalling (2.1) and (2.3) we may, equivalently, prove the theorem for the respective powers of
k and l, viz., we redefine the truncated means as
Ym,n
k,l
= kα−1lβ−1Xk,lI{kα−1lβ−1|Xk,l| ≤ mαnβ},(3.5)
with partial sums and means as
S′
m,n=
m,n
?
k,l=1
Ym,n
k,l
andµm,n= E S′
m,n,(3.6)
respectively.
In order to check the conditions of the degenerate convergence criterion we thus wish to show
that, if (3.2) is satisfied, then
m,n
?
k,l=1
P(kα−1lβ−1|X| > mαnβ) → 0asm,n → ∞,(3.7)
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Ces` aro summation for random fields
5
and that
1
mαnβ
m,n
?
k,l=1
Var?Ym,n
k,l
?→ 0as m,n → ∞.(3.8)
As for (3.7),
m,n
?
k,l=1
P(kα−1lβ−1|X| > mαnβ) =
1
mαnβ
m,n
?
k,l=1
kα−1lβ−1· mαnβk1−αl1−βP(kα−1lβ−1|X| > mαnβ),
which converges to 0 as m,n → ∞ via (2.5).
In order to verify (3.8) we apply the usual “slicing device” to obtain
1
m2αn2β
m,n
?
k,l=1
Var?Ym,n
k,l
?≤
m,n
?
k,l=1
1
m2αn2β
m,n
?
k,l=1
E?Ym,n
k,l
?2
≤
1
m2αn2β
E?k2(α−1)l2(β−1)X2I{kα−1lβ−1|X| ≤ mαnβ}?
=
1
m2αn2β
m,n
?
k,l=1
k2(α−1)l2(β−1)
mnβ/α
?
j=1
E?X2I{(j − 1)α< kα−1lβ−1|X| ≤ jα}?
≤
1
m2αn2β
m,n
?
k,l=1
mnβ/α
?
j=1
j2αP?(j − 1)α< kα−1lβ−1|X| ≤ jα?
≤
C
m2αn2β
m,n
?
k,l=1
mnβ/α
?
j=1
?
j
?
i=1
i2α−1?
P?(j − 1)α< kα−1lβ−1|X| ≤ jα?
≤
C
m2αn2β
m,n
?
k,l=1
mnβ/α
?
i=1
i2α−1P(|X| ≥ iαk1−αl1−β)
=
C
mαnβ
m,n
?
k,l=1
m,n → ∞,
kα−1lβ−1?
1
mαnβ
mnβ/α
?
i=1
iα−1?iαk1−αl1−βP(|X| ≥ iαk1−αl1−β)??
,
→ 0as
by applying (2.5) twice to (3.2). This completes the proof of (3.1), from which (3.4) is immediate.
2
Proof of Corollary 3.1. In order to conclude that also (3.4) holds we use the usual method to
show that the normalized trruncated means tend to zero, where w.l.o.g. we assume that E X = 0.
Then
???
1
mαnβ
m,n
?
k,l=1
E?k(α−1)l(β−1)XI{k(α−1)l(β−1)|X| ≤ mαnβ}????
??? −
1
mαnβ
k,l=1
=
1
mαnβ
m,n
?
k,l=1
E?k(α−1)l(β−1)XI{k(α−1)l(β−1)|X| > mαnβ}????
≤
m,n
?
E?k(α−1)l(β−1)|X|I{k(α−1)l(β−1)|X| > mαnβ}?→ 0asn,m → ∞.
Proof of Corollary 3.2. Immediate, since the truncated means are (also) equal to zero.
2
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A. Gut and U. Stadtm¨ uller
4Complete convergence
Theorem 4.1 Let 0 < α ≤ β ≤ 1. The field {Xk,l, k,l ≥ 0} converges completely to µ, i.e.,
?
mn
P????
m,n
?
k,l=0
Aα−1
m−kAβ−1
n−lXk,l− µ
??? > Aα
mAβ
nε?< ∞for every ε > 0,
if and only if
E|X|
1
α, for0 < α < 1/2, α < β ≤ 1,
E|X|
1
αlog+|X|,for0 < α = β <1
2,
E|X|2(log+|X|)3, forα = β =1
2,
E|X|2(log+|X|)2, forα =1
2< β ≤ 1,
E|X|2log+|X|, for
1
2< α ≤ β ≤ 1.
and E X = µ.
Proof. For the proof of the sufficiency we refer to the Appendix.
As for the necessity, we argue as in [6], p. 59. We first suppose that the distribution is symmetric.
Now, if complete convergence holds, then, using the fact that
max
0≤k,l≤m,nAα−1
m−kAβ−1
n−l|Xk,l| ≤ 2max
0≤µ,ν≤m,n
???
µ,ν
?
k,l=0
Aα−1
m−kAβ−1
n−lXk,l
???,
together with the L´ evy inequalities we must have, say,
?
m,n
P?
max
0≤k,l≤m,nAα−1
m−kAβ−1
n−l|Xk,l| > Aα
mAβ
n
?< ∞,
so that, by the first Borel-Cantelli lemma
P(Aα−1
m−kAβ−1
n−l|Xk,l| > Aα
mAβ
n
i.o. for 1 ≤ k,l ≤ m,n ;m,n ≥ 1) = 0.
At this point we use a device from [17], p. 379. Namely, if the sums?m,n
independent, we would conclude that?
however, finiteness of the sum is only a matter of the tail probabilities, the sum is also finite in the
general case.
An application of (A.6) now tells us that the finiteness of the sum is equivalent to the moment
conditions as given in the statement of the theorem.
This proves the necessity in the symmetric case. The general case follows the standard desym-
metrization procedure, for which we use Theorem 3.1 in order to take care of the asymptotics
for the normalized medians (cf. [5], p. 472 for analogous details in the multiindex setting of the
Marcinkiewicz-Zygmund strong laws).
k,l=1Aα−1
mAβ
m−kAβ−1
n) were finite. Since,
n−lXk,lwere
m,n
?m,n
k,l=1P(Aα−1
m−kAβ−1
n−l|X| > Aα
2
5Almost sure convergence
Theorem 5.1 Let 0 < α ≤ β ≤ 1. The field {Xk,l, k,l ≥ 0} is almost surely (a.s.) (C,α,β)-
summable, that is,
1
Aα
n
k,l=0
mAβ
m,n
?
Aα−1
m−kAβ−1
n−lXk,l
a.s.
→ µasn,m → ∞
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Ces` aro summation for random fields
7
if and only if
E|X|
1
α, for0 < α < β ≤ 1,
E|X|
1
αlog+|X|,for0 < α = β ≤ 1.
and E X = µ.
Proof. Since complete convergence always implies almost sure convergence, the sufficiency follows
immediately for the case α < 1/2.
Thus, let in the following 1/2 ≤ α ≤ β ≤ 1. We first consider the symmetric case (and recall
Section 2. In analogy with [11], p. 538, the moment assumptions permit us to choose an array
{ηk,l, k,l ≥ 1} of nonincreasing reals in (0,1) converging to 0, and such that
∞
?
k,l=1
P(|Xk,l| > ηk,lkαlβ) < ∞.
Defining
Yk,l= Xk,lI{|Xk,l| ≤ ηk,lkαlβ} andS′
m,n=
m,n
?
k,l=0
Ym,n
k,l
,
it thus remains to prove the theorem for the truncated sequence.
This will be achieved via the multiindex Kolmogorov convergence criterion (see e.g [4]) and the
multiindex Kronecker lemma (cf. [16]). The first series has just been taken care of, the second one
vanishes since we are in the symmetric case, so it remains to check the third series.
Toward that end, let, for k,l ≥ 1,
Zk,l=(m − k)α−1(n − l)β−1
mαnβ
Yk,l.
Then
|Zk,l| ≤(m − k)α−1(n − l)β−1
mαnβ
kαlβηk,l≤ ηk,l≤ η00. (5.1)
Now, for any ε > 0, arbitrarily small, we may choose our η-sequence such that η00< ε, so that an
application of the (iterated) Kahane-Hoffman-Jørgensen inequality (cf. [8], Theorem 3.7.5) yields
P
????
m,n
?
k,l=0
Zk,l
??? > 3jε
?
≤Cj
?
P
????
m,n
?
k,l=0
Zk,l
??? > ε
??2j
≤Cj
??m,n
k,l=0
?(m − k)(α−1)(n − l)β−1?1/αE|X|1/α
?εmαnβ?1/α
k,l=0k(1−1/α)l(β−1)/α
mnβ/α
?2j
?2j
log m log n
nm
, for
?2j
=C′
j
??m,n
?2j
=
C′′
j
?
?
?
1
(mn)
1
α−1
, for
1
2< α < β < 1,
C′′
j
log m
nm
,for
1
2= α < β < 1,
C′′
j
?2j
1
2= α = β,
(since the usual first term in the RHS vanishes in view of (5.1)).
By choosing j sufficiently large it then follows that
?
m,n
m,n
?
k,l=0
P
????
m,n
?
k,l=0
Zk,l
??? > 3jε
?
< ∞.
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A. Gut and U. Stadtm¨ uller
By replacing 3jε by ε we have thus, due to the arbitrariness of ε, shown that
P?|Zk,l| > ε i.o.?= 0for any ε > 0,(5.2)
from which the desired almost sure convergence follows via the multiindex Kronecker lemma re-
ferred to above.
This proves the sufficiency in the symmetric case from which the general case follows by the
standard desymmetrization procedure hinted at in the proof of Theorem 4.1.
Finally, suppose that almost sure convergence holds as stated. It then follows that
Aα−1
0
Aβ−1
0
Aα
Xm,n
mAβ
n
a.s.
→ 0asm,n → ∞,
and, hence, also that
Xm,n
mαnβ
a.s.
→ 0asm,n → ∞,
which, in view of i.i.d. assumption and the second Borel-Cantelli lemma, tells us that
?
m,n
P(|X| > mαnβ) < ∞,
which, in turn, is equivalent to the given moment conditions.
This concludes the proof of the theorem.
2
6Concluding remarks
We close with some comments on the present and related work.
• Convergence in probability has earlier been established in [11] via approximation with in-
dicator variables, and under the assumption of finite mean. Our proof is simpler (more
elementary) and presupposes only a Feller condition.
• As pointed out above, almost sure convergence was established in three steps ([14], [1] and
[2]) with different proofs. Our proof, which also works for the case d = 1, takes care of the
whole proof in one go (since our proof also works for the case α < 1/2).
• For simplicity we have confined ourselves to the case d = 2. The same ideas can be modified
for the case d > 2 and (C,α1,α2,...,αd)-summability. However, the moment conditions
then depend on the number of α:s that are equal to the smallest one (corresponding to α < β
or α = β in the present paper); see [9] for Kolmogorov-Marcinkiewicz-Zygmund laws.
• Results on complete convergence are special cases of results on convergence rates. In this
vein our results are extendable to results concerning
?
m,n
nr−2mr−2P????
m,n
?
k,l=0
Aα−1
m−kAβ−1
n−lXk,l− µ
??? > Aα
mAβ
nε?< ∞for everyε > 0
(cf. [7] for the case d = 1). For the proofs one would need i.a. extensions of the relevant
computations in the appendix below.
A Appendix
In this appendix we collect a number of so-called elementary but tedious calculations.
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Ces` aro summation for random fields
9
First, let 0 < α ≤ β < 1. Then
?
m,n
m,n
?
k,l=1
P(kα−1lβ−1|X| > mαnβ) < ∞⇐⇒
?∞
1
?∞
1
?x
1
?y
1
P(|X| > u1−αv1−βxαyβ)dudvdxdy < ∞ ⇐⇒
?
?y
yβ
u1−αxα= z,v1−βyβ= w
?
?∞
1
?∞
1
?x
xα
?z
x
?
α
1−α?w
y
?
β
1−βP(|X| > zw)dzdwdxdy < ∞ ⇐⇒
?∞
1
?∞
1
??z1/α
z
dx
x
α
1−α
???w1/β
w
dy
y
β
1−β
?
z
α
1−αw
β
1−βP(|X| > zw)dzdw < ∞. (A.1)
In case 0 < α < β = 1 we have
?
m,n
m,n
?
k,l=1
P(kα−1|X| > mαn) < ∞ ⇐⇒
?∞
1
?∞
1
??z1/α
z
dx
x
α
1−α
?
z
α
1−αw P(|X| > zw)dzdw < ∞.(A.2)
Next we note that
?y1/γ
y
dx
1−γ∼ C
x
γ
y
logy,
1−2γ
1−γ,
1−2γ
γ(1−γ), for
for
0 < γ <1
γ =1
2,
1
2< γ < 1,
2,
y
for
(A.3)
so that
??z1/α
z
dx
x
α
1−α
???w1/β
w
dy
y
β
1−β
?
z
α
1−αw
β
1−β
∼ C
z
(zw)
zwlogz logw =zw
−(logz)2− (logw)2?,
z
z
zwlogz,
zw,
1−α
α w
1−β
β ,
1−α
α ,
for
for
0 < α,β <1
0 < α = β <1
2,
2,
2
?(logzw)2
for
for
for
for
for
α = β =1
α < β =1
α <1
2< β ≤ 1,
α =1
2< β ≤ 1,
1
2< α ≤ β ≤ 1,
2,
2,
1−α
α wlogw,
1−α
α w,
Page 10
10
A. Gut and U. Stadtm¨ uller
from which it follows that
?∞
1
?∞
1
??z1/α
z
dx
x
α
1−α
???w1/β
w
dy
y
β
1−β
?
z
α
1−αx
β
1−βP(|X| > zw)dzdw
=
?
x = zw,y = z
?
=
?∞
1
?x
1x
= C?∞
?x
= C?∞
?x
=1
6
?x
= C?∞
?x
= C?∞
?x
=1
2
?x
=1
2
1−β
β y
1
α−1
β−1P(|X| > x)dydx
1x
1
α−1P(|X| > x)dx, for0 < α < β <1
2,
?∞
1
1x
1−α
α
1
yP(|X| > x)dydx
1x
1−α
α logxP(|X| > x)dx, for0 < α = β <1
2,
?∞
1
1
?1
2x(logx)2 1
?∞
1xy
y− x(log y)2
y
?P(|X| > x)dxdy
1x(logx)3P(|X| > x)dx,forα = β =1
2,
?∞
1
1
α−2(logx − logy)P(|X| > x)dydx
1x
1
α−1P(|X| > x)dx, forα < β =1
2,
?∞
1
1xy
1
α−2P(|X| > x)dydx
1x
1
α−1P(|X| > x)dx,for α <1
2< β ≤ 1,
?∞
1
1xlog y
yP(|X| > x)dydx
?∞
yP(|X| > x)dydx
?∞
1x(logx)2P(|X| > x)dx,forα =1
2< β ≤ 1,
?∞
1
1x1
1xlogxP(|X| > x)dx,for
1
2< α ≤ β ≤ 1.
(A.4)
Summarizing this we have shown that, for 0 < α ≤ β < 1,
?
m,n
m,n
?
k,l=1
P(Aα−1
k
Aβ−1
l
|X| > Aα
mAβ
n) < ∞ ⇐⇒(A.5)
E|X|
1
α,for0 < α < 1/2, α < β ≤ 1,
E|X|
1
αlog+|X|,for0 < α = β <1
2,
E|X|2(log+|X|)3,forα = β =1
2,
E|X|2(log+|X|)2,forα =1
2< β ≤ 1,
E|X|2log+|X|,for
1
2< α ≤ β ≤ 1.
(A.6)
Acknowledgement
The work on this paper has been supported by Kungliga Vetenskapssamh¨ allet i Uppsala. Their
support is gratefully acknowledged. In addition, the second author likes to thank his partner Allan
Gut for the great hospitality during two wonderful and stimulating weeks at the University of
Uppsala.
Page 11
Ces` aro summation for random fields
11
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Page 12
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A. Gut and U. Stadtm¨ uller
[21] Zygmund, A. (1968). Trigonometric Series. Cambridge University Press.
Allan Gut, Department of Mathematics, Uppsala University, Box 480,
SE-75106 Uppsala, Sweden;
Email:
allan.gut@math.uu.se
URL:
http://www.math.uu.se/~allan
Ulrich Stadtm¨ uller, Ulm University, Department of Number Theory and Probability Theory,
D-89069 Ulm, Germany;
Email
ulrich.stadtmueller@uni-ulm.de
URL:
http://www.mathematik.uni-ulm.de/matheIII/members/stadtmueller/stadtmueller.html
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