# Cesàro Summation for Random Fields

**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. The natural extension of results corresponding to Cesàro summation amounts to proving almost sure convergence

of the Cesàro 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 ℤ+2, the positive two-dimensional integer lattice points.

KeywordsCesàro summation-Sums of i.i.d. random variables-Complete convergence-Convergence in probability-Almost sure convergence-Strong law of large numbers

Mathematics Subject Classification (2000)60F15-60G50-60G60-40G05-60F05

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**ABSTRACT:**The classical Marcinkiewicz-Zygmund law for i.i.d. random variables has been generalized by Gut [Gut, A., 1978. Marcinkiewicz laws and convergence rates in the law of large numbers for random variables with multidimensional indices. Ann. Probab. 6, 469-482] to random fields. Therein all indices have the same power in the normalization. Looking into some weighted means of random fields, such as Cesro summation, it is of interest to generalize these laws to the case where different indices have different powers in the normalization. In this paper we give precise moment conditions for such laws.Statistics [?] Probability Letters 04/2009; 79(8):1016-1020. · 0.53 Impact Factor

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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.

→ µ asn → ∞⇐⇒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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2

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 everyε > 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.

→ 0 as

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→ 0 asn → ∞,(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→ 0asn → ∞,(2.5)

as well as integral versions of the same.

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4

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

→ 0as m,n → ∞ (3.1)

if

nP(|X| > n) → 0asn → ∞. (3.2)

If, in addition,

µm,n

Aα

mAβ

n

→ 0as m,n → ∞,(3.3)

then

1

Aα

mAβ

n

m,n

?

k,l=0

Aα−1

m−kAβ−1

n−lXk,l

p

→ 0 as m,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) → 0 asm,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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6

A. Gut and U. Stadtm¨ uller

4 Complete 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

5 Almost 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.

→ µ asn,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

?

< ∞.

Page 8

8

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.

→ 0 as m,n → ∞,

and, hence, also that

Xm,n

mαnβ

a.s.

→ 0as m,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

6 Concluding 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 every ε > 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.

Page 9

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,forα <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

References

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[8] Gut, A. (2007). Probability: A Graduate Course, Corr. 2nd printing. Springer-Verlag, New

York.

[9] Gut, A. and Stadtm¨ uller, U. (2008). An asymmetric Marcinkiewicz-Zygmund LLN for

random fields. Report U.U.D.M. 2008:38, Uppsala University.

[10] Hardy, G.H. (1949). Divergent Series. Oxford University Press.

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Gr¨ oßen. Math. Ann. 99, 309-319.

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[15] Marcinkiewicz, J. and Zygmund, A. Sur les fonctions ind´ ependantes. Fund. Math. 29,

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[16] Moore, C.N. (1966). Summable Series and Convergence Factors. Dover, New York.

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[19] Stadtm¨ uller, U. and Thalmaier, M. (2008). Strong laws for delayed sums of random

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Page 12

12

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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