Summability of Double Independent Random Variables

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Volume 2008, Article ID 948195,12pages doi:10.1155/2008/948195

Research Article

Summability of Double Independent Random Variables

Richard F. Patterson¹ and Ekrem Savas¸²

1Department of Mathematics and Statistics, University of North Florida, 1 UNF Drive, Jacksonville, FL 32224, USA

2Department of Mathematics, Istanbul commerce University, Uskudar, 34672 Istanbul, Turkey

Correspondence should be addressed to Richard F. Patterson,[email protected] Received 21 May 2008; Accepted 1 July 2008

Recommended by Jewgeni Dshalalow

We will examine double sequence to double sequence transformation of independent identically distribution random variables with respect to four-dimensional summability matrix methods.

Copyrightq2008 R. F. Patterson and E. Savas¸. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Let Xk,lbe a factorable double sequence of independent, identically distributed random variables withE|Xk,l|<∞andEXk,l μ. LetAam,n,k,lbe a factorable double sequence to double sequence transformation defined as

Ax_m,n ^∞,∞

k,l1,1

am,n,k,lxk,l. 1.1

These factorable sequences and matrices will be used to characterize such transformations with respect to Robison and Hamilton-type conditions see 1, 2. That is,regularity conditions of the following type. The four-dimensional matrixAis RH-regular if and only if

RH₁: P-lim_m,na_m,n,k,l0 for eachkandl;

RH2: P-limm,n

k,lam,n,k,l1;

RH3: P-limm,n

k|am,n,k,l|0 for eachl;

RH₄: P-lim_m,n

l|am,n,k,l|0 for eachk;

RH₅:

k,l|am,n,k,l|is P-convergent; and

RH₆: there exist positive numbersAandBsuch that

k,l>B|am,n,k,l|< A.

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Throughout this paper, we will denote _∞,∞

k,l1,1am,n,k,lXk,l by Ym,n and examine Ym,n with respect to the Pringsheim converges. To accomplish this goal, we begin by presenting and prove the following theorem. A necessary and suﬃcient condition thatY_m,n Y˘_mY˘˘_n P-converges to μ in probability is that maxk,l|am,n,k,l| maxk,l|am,kan,l| converges to 0 in the Pringsheim sense. This theorem and other similar to it will be used in the pursuit of establishing the following. If max_k,l|am,n,k,l| max_k,l|am,ka_n,l| Om^−γ¹On^−γ²,γ₁, γ₂ > 0, then

E|X|˘ ^11/γ¹ <∞, E|X|˘˘ ^11/γ² <∞ 1.2 implies thatYm,n→μalmost sure P-convergence.

2. Definitions, notations, and preliminary results

Let us begin by presenting Pringsheim’s notions of convergence and divergence of double sequences.

Definition 2.1see3. A double sequencex xk,lhas Pringsheim limitLdenoted by P- limx Lprovided that given > 0 there existsN ∈ N such that|xk,l−L| < whenever k, l > N. We will describe such anxmore briefly as “P-convergent.”

Definition 2.2. A double sequencexis called definite divergent, if for everyarbitrarily large G >0 there exist two natural numbersn1andn2such that|xn,k|> Gforn≥n1, k≥n2.

Throughout this paper, we will also denote_∞,∞

k,l1,1 by

k,l. Using these definitions, Robison and Hamilton presented a series of concepts and matrix characterization of P-convergence. The first definition they both presented was the following. The four- dimensional matrix A is said to be RH-regular if it maps every bounded P-convergent sequence into a P-convergent sequence with the same P-limit. The assumption of bounded- ness was made because a double sequence which is P-convergent is not necessarily bounded.

They both independently presented the following Silverman-Toeplitz type characterization of RH-regularity4,5.

Theorem 2.3. The four-dimensional matrixAis RH-regular if and only if RH1: P-limm,nam,n,k,l0 for eachkandl;

RH₂: P-lim_m,n

k,la_m,n,k,l1;

RH₃: P-lim_m,n

k|am,n,k,l|0 for eachl;

RH4: P-limm,n

l|am,n,k,l|0 for eachk;

RH₅:

k,l|am,n,k,l|is P-convergent; and

RH6: there exist positive numbersAandBsuch that

k,l>B|am,n,k,l|< A.

Following Robison and Hamilton work, Patterson in6presented the following two notions of subsequence of a double sequence.

Definition 2.4. The double sequenceyis a double subsequence of the sequencexprovided that there exist two increasing double index sequences{nj}and{kj}such that ifz_j x_n_j_,k_j,

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thenyis formed by

z1 z2 z5 z10

z4 z3 z6 — z₉ z₈ z₇ —

— — — —.

2.1

Definition 2.5 Patterson 6. A number β is called a Pringsheim limit point of the double sequencexprovided that there exists a subsequenceyofxthat has Pringsheim limit β: P-limy β.

Using these definitions, Patterson presented a series of four-dimensional matrix characterizations of such sequence spaces. Let{xk,l} be a double sequence of real numbers and, for eachn, letα_n sup_n{xk,l :k, l ≥ n}. Patterson7also extended the above notions with the presentation of the following. The Pringsheim limit superior of x is defined as follows:

1ifα ∞for each n, then P-lim supx: ∞;

2ifα <∞for somen, then P-lim supx:infn{αn}.

Similarly, letβn infn{xk,l : k, l ≥ n}. Then the Pringsheim limit inferior ofxis defined as follows:

1ifβ_n−∞for each n, then P-lim infx:−∞;

2ifβ_n>−∞for somen, then P-lim infx:sup_n{βn}.

3. Main result

The analysis of double sequences of random variables via four-dimensional matrix transformations begins with the following theorem. However, it should be noted that the relationship between our main theorem that is stated above and the next four theorems will be apparent in their statements and proofs.

Theorem 3.1. A necessary and suﬃcient condition thatY_m,nY˘_mY˘˘_nP-converges toμin probability is that maxk,l|am,n,k,l|maxk,l|am,kan,l|converges to 0 in the Pringsheim sense.

Proof. First, note that

lim˘ T→∞

TP˘ |X| ≥˘ T˘ 0, lim

T→∞˘˘

TP˘˘ |X| ≥˘˘ T˘˘ 0 3.1

becauseE|X|˘ <∞andE|X|˘˘ <∞. LetT T˘T,˘˘ Xm,n,k,l X˘_m,kX˘˘_n,l,am,n,k,lXk,l am,kX˘_kan,lX˘˘_l, and Z_m,n Z˘_mZ˘˘_n

k,lX_m,n,k,l. For suﬃciently largemandn and since max_k,l|am,n,k,l|is

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a P-null sequence, it follows from3.1that PZm,n/Ym,n≤

k,l

PXm,n,k,l/am,n,k,lXk,l

k,l

P

|X| ≥˘ 1

|am,k|;|X| ≥˘˘ 1

|an,l|

≤

k,l

|am,n,k,l|

≤M,

3.2

whereMis define by RH₆of regularity conditions. Therefore, it suﬃces to show that P-lim

m,nZ_m,nμin probability. 3.3

Observe that

EZm,n−μ

k,l

am,n,k,l

|x|<1/|a˘ m,k|x d˘ F˘

|x|<1/|a˘˘ n,l|x d˘˘ F˘˘−μ

μ

k,l

am,n,k,l−1

, 3.4

which is a P-null sequence. Since 1

T˘T˘˘

|˘x|<T˘

|x|<˘˘ T˘˘

˘

x²x˘˘²dF d˘ F˘˘ 1 T˘T˘˘

{−T˘²P|X| ≥˘ T˘·−T˘˘²P|X| ≥˘˘ T˘˘}

1 T˘T˘˘

2

_T^˘

0

˘

xP|X˘| ≥xd˘ x˘·2 _T^˘˘

0

˘˘

xP|X| ≥˘˘ xd˘˘ x˘˘

3.5

is a P-null sequence with respect toT, we have

k,l

VarX_m,n,k,l≤

k,l

|am,n,k,l|²

|x|<1/|a˘ m,k|x˘²dF˘

|x|<1/|a˘˘ n,l|x˘˘²dF˘˘ ≤

k,l

|am,n,k,l| ≤M 3.6

formandnsuﬃciently large, whereFF˘F˘˘andxx˘x. It is also clear that˘˘ E

k,lx_m,n,k,l²is finite. Thus,

k,l

VarX_m,n,k,lVar

k,l

X_m,n,k,l

3.7 is finite. The result clearly follows from the Chebyshev’s inequality. Thus, the suﬃciency is proved.

Now, let us consider the necessary part of this theorem. Similar to Pruitt’s notation8, letUk,l Xk,l−μand consider the transformationTm,n

k,lam,n,k,lUk,l. Our goal become showing thatT_m,nP-converges in probability to 0. Which imply thatT_m,nP-converges in law to 0. Let us consider the characteristic function ofT_m,n,that is,

EeûT^m,n Eeû^k,lâ^m,n,k,lÛ^k,l EΠk,leûa^m,n,k,lÛ^k,l Πk,lEeûa^m,n,k,lÛ^k,l: Πk,lguam,n,k,l. 3.8

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Observe that

P-lim

m,n{Πk,lguam,n,k,l}1. 3.9

Because

|Πk,lguam,n,k,l| ≤ |guam,n,k,l| ≤1 3.10

for allm, nwe have that

P-lim

m,nguam,n,k,l 1 3.11

for allk, l. Clearly, there existsu₀such that|guam,n,k,l|<1 for 0<|u|< u₀. Letuu₀/2M then there exists a double subsequenceam,n,km,lnsuch that

|uam,n,km,ln| ≤Mu u₀

2 . 3.12

Thus P-limm,nuam,n,km,ln 0. Therefore, clearly we can choosekm, lnsuch that

|am,n,km,ln|max

k,l |am,n,k,l|. 3.13

Theorem 3.2. If E|X|˘ ^11/γ¹ < ∞, E|X|˘˘ ^11/γ² < ∞, and maxk,l|am,n,k,l| maxk|am,k| · max_l|an,l| ≤Bm˘ ^−γ¹Bn˘˘ ^−γ²,then for every >0

m,n

P|am,n,k,lXk,l| ≥for somek, l<∞, 3.14

that is,

m,n

P|am,kX˘_k| ≥;|an,lX˘˘_l| ≥for somek, l<∞. 3.15

Proof. Let

N_m,nx N_m,nx˘x ˘˘

{k,l:1/|am,k|≤x; 1/|a˘ n,l|≤x}˘˘

|am,n,k,l|. 3.16

Notex x˘x, and observe that˘˘ Nm,nx 0 for ˘x < m^γ¹,x <˘˘ n^γ²,and _∞

0 dNm,nx

k,l|am,n,k,l| ≤M. If

Gx P|X| ≥x P|X| ≥˘ xP|˘ X| ≥˘˘ x ˘˘ GxG˘ x,˘˘ 3.17

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then xGx converges to 0 in the Pringsheim sense because EX < ∞ and recalled that T T˘T. Therefore,˘˘

k,l

P|am,n,k,lxk,l| ≥1

k,l

G 1

|am,n,k,l|

k,l

1

|am,n,k,l|G 1

|am,n,k,l|

_∞

0

xGxdNm,nx

Nm,nTTGT|^∞₀|^∞₀ − _∞

0

Nm,nxdxGx

lim

T→∞N_m,nTTGT−

_∞

0

N_m,nxdxGx

≤M

_∞

m^γ¹

_∞

n^γ²

|dxGx|

M _∞

m^γ¹

_∞

n^γ²

|dxG˘ xd˘ xG˘˘ x|.˘˘

3.18

Our goal now is to get an estimate for_∞

m^γ¹

_∞

n^γ²|dxGx|.To this end observe that, forz < y

yGy−zGz y−zGz yGz−Gy, 3.19

where y − zGz and yGz − Gy are increasing and decreasing functions of y, respectively. Thus

_y_˘

˘ z

_y_˘˘

z˘˘

d|xGx| ≤y˘−zG˘ z ˘ yG˘ z˘ −Gy˘ ·y˘˘−zG˘˘ z ˘˘ yG˘˘ z˘˘ −GY˘˘. 3.20

The last inequality grant us the following:

_∞

m^γ¹

_∞

n^γ²

|dxG˘ xd˘ xG˘˘ x|˘˘

^∞,∞

i,jm,n

_i1^γ1

i^γ¹

_j1^γ2

j^γ²

≤ ^∞,∞

i,jm,n

{i1^γ¹−i^γ¹Gi^γ¹·j1^γ²−j^γ²Gj^γ²}

^∞,∞

i,jm,n

{i1^γ¹Gi^γ¹−Gi1^γ¹·j1^γ²Gj^γ²−Gj1^γ²}.

3.21

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Therefore, _∞

m^γ¹

_∞

n^γ²

≤2

∞,∞

i,jm,n

{i1^γ¹Gi^γ¹−Gi1^γ¹·j1^γ²Gj^γ²−Gj1^γ²}.

∞ m,n

P|am,n,k,lXk,l| ≥for somek, l

≤^∞

m,n

∞ k,l

P|am,n,k,lXk,l| ≥

≤2M

∞,∞

m,n1,1

∞,∞

i,jm,n

{i1^γ¹Gi^γ¹−Gi1^γ¹·j1^γ²Gj^γ²−Gj1^γ²} 2M

∞,∞

i,j1,1

{i1^γ¹Gi^γ¹−Gi1^γ¹·j1^γ²Gj^γ²−Gj1^γ²}

≤2^1γ¹2^1γ²M

|x|˘^11/γ¹|x|˘˘^11/γ²dF˘ xd˘ F˘˘ x˘˘

<∞.

3.22

Theorem 3.3. Letx andF be define as inTheorem 3.2. If E|X˘|^11/γ¹ < ∞,E|X|˘˘ ^11/γ² < ∞, and maxk,l|am,n,k,l|maxk|am,k| ·maxl|an,l| ≤Bm˘ ^−γ¹Bn˘˘ ^−γ²then forα1< γ1/2γ11andα2 < γ2/2γ2 1

m,n

P|am,n,k,lX_k,l| ≥m^α¹n^α² for at least two pairsk, l<∞, 3.23

that is,

m,n

P|am,kX˘_k| ≥m^α¹;|an,lX˘˘_l| ≥n^α² for at least two pairsk, l<∞. 3.24

Proof. By Markov’s inequality, we have the following:

m

P|am,kX˘_k| ≥m^α¹≤ |am,k|^11/γ¹E|x|˘ ^11/γ¹m^α¹^11/γ¹,

n

P|an,lX˘˘_l| ≥n^α²≤ |an,l|^11/γ²E|x|˘˘ ^11/γ²n^α²^11/γ².

3.25

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Therefore,

m,n

P|am,kX˘_k| ≥m^α¹;|an,lX˘˘_l| ≥n^α² for at least two pairsk, l

≤

i /k, j /l

P|am,iX˘_i| ≥m^α¹;|am,kX˘_k| ≥m^α¹;|an,jX˘˘_j| ≥n^α²;|an,lX˘˘_l| ≥n^α²

≤E|x|˘^11/γ¹²m^2α¹^11/γ¹

i /k

|am,i|^11/γ¹|am,k|^11/γ¹

·E|x|˘˘^11/γ²²n^2α²^11/γ²

j /l

|an,j|^11/γ²|an,l|^11/γ²

≤E|x|˘^11/γ¹²·E|x|˘˘^11/γ²²B˘^2/γ¹B˘˘

2/γ2

M⁴m^2−1α¹^11/γ¹n^2−1α²^11/γ²,

3.26

which is P-convergent when sum on n and m provided that α1 < γ1/2γ1 1 and α2 <

γ₂/2γ21.

Theorem 3.4. LetxandFbe define as inTheorem 3.2. Ifμ 0,E|X˘|^11/γ¹ <∞,E|X|˘˘ ^11/γ² < ∞, and max_k,l|am,n,k,l|max_k|am,k| ·max_l|an,l| ≤Bm˘ ^−γ¹Bn˘˘ ^−γ²then for >0

m,n

P

k,l

|am,n,k,lXk,l| ≥

<∞, 3.27

where

k,l

am,n,k,lXk,l

{k:|am,kXk|<m^−α¹l:|an,lXl|<n^−α²}

am,n,k,lXk,l, 3.28

α1< γ1, andα2< γ2. Proof. Let

Xm,n,k,l:

⎧⎪

⎪⎨

⎪⎪

⎩

Xm,k; if|am,kXk|< m^−α¹, X_n,l; if|an,lX_l|< n^−α², 0; otherwise,

3.29

andβ_m,n,k,lEXm,n,k,l. Ifa_m,n,k,l 0,thenβ_m,n,k,lμ0 and ifa_m,n,k,l/0,then

|βm,n,k,l| μ−

|˘x|≥m^−α¹|am,k|⁻¹

|x|≥m˘˘ ^−α²|an,l|⁻¹x dF

≤

|x|≥m˘ ^−α¹B˘⁻¹m^γ¹

|x|≥n˘˘ ^−α²B˘˘⁻¹n^γ²

|x|dF.

3.30

Therefore, P-lim_m,nβ_m,n,k,l0 uniformly ink, land P-lim_m,n

k,la_m,n,k,lβ_m,n,k,l 0. Let

Z_m,n,k,lZ_m,kZ_n,lX_m,n,k,l−β_m,n,k,l, 3.31

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so thatEZm,n,k,l 0,E|Zm,k|^11/γ¹ < c1,andE|Zn,l|^11/γ² < c2 for somec1 and c2. Also

|am,kZ_m,k| ≤2m^−α¹and|an,lZ_n,l| ≤2n^−α². Observe that

k,l

a_m,n,k,lX_k,l

k,l

a_m,n,k,lX_m,n,k,l

k,l

a_m,n,k,lZ_m,n,k,l

k,l

a_m,n,k,lβ_m,n,k,l. 3.32

Note for suﬃciently largemandn

k,l

a_m,n,k,lX_k,l ≥

⊂

k,l

a_m,n,k,lZ_m,n,k,l ≥

2

. 3.33

Thus it is suﬃcient to show that

m,n

P

k,l

|am,n,k,lZ_m,n,k,l| ≥

<∞. 3.34

Letη1andη2be the least integers greater than 1/γ1and 1/γ2, respectively. Our goal now is to produce an estimate for

E

k

am,kZm,k

_2η₁

l

an,lZn,l

_2η₂

. 3.35

Observe that

E

k

a_m,kZ_m,k

_2η₁

l

a_n,lZ_n,l _2η₂

3.36

is equal to

k1,k2,...,k2p;l1,l2,...,l2q

E 2p

i1

2q j1

am,n,ki,ljZm,n,ki,lj

. 3.37

It happens to be the case thatE

ka_m,kZ_m,k^2η¹

la_n,lZ_n,l^2η²is zero ifk_i, l_i/k_j, l_jfori /j because theZ_m,n,k,l’s are independent andEZm,n,k,l 0. Let us now consider the general term. Thus

p₁of theksφ₁, . . . , p_θ₁of theksφ_θ₁, q1 of theksϕ1, . . . , qθ2 of theksϕθ2, r1 of thelsκ1, . . . , rτ1 of thelsκτ1, s₁ of thelsω₁, . . . , s_τ₂ of thelsω_τ₂,

3.38

where 2≤pi≤11/γ1,qj >11/γ1, 2≤rλ≤11/γ2,sχ >11/γ2,

θ1

i1

p_i^θ²

j1

q_j2η₁,

τ1

λ1

r_i^τ²

χ1

s_χ2η₂.

3.39

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Now let us consider the following expectation:

E

θ1

i1

am,φiZ_m,φ_i^pⁱ·^θ²

j1

am,ϕjZ_m,ϕ_j^q^j·^τ¹

λ1

an,κλZ_n,κ_λ^r^λ^τ²

χ1

an,ωχZ_n,ω_χ^s^χ

≤1c₁^θ¹1c₂^τ¹·^τ²

χ1

|am,φi|^pⁱ^τ¹

λ1

|an,κλ|^r^λ

·E

θ2

j1

am,ϕjZm,ϕj^q^j·^τ²

χ1

an,ωχZn,ωχ^s^χ

≤1c1^θ¹1c2^τ¹·^θ¹

i1

|am,φi|^pⁱ^τ¹

λ1

|an,κλ|^r^λ

·^θ²

j1

|am,ϕj|^11/γ¹2m^−α¹^q^j^−1−1/γ¹·^τ²

χ1

|an,ωχ|^11/γ²2n^−α²^s^χ^−1−1/γ²

≤1c₁^θ¹1c₂^τ¹·^θ¹

i1

|am,φi||am,φi|^pⁱ⁻¹

·^τ¹

λ1

|an,κλ||an,κλ|^r^λ⁻¹·^θ²

j1

|am,ϕj|^11/γ¹2m^−α¹^q^j^−1−1/γ¹

·^τ²

χ1

|an,ωχ|^11/γ²2n^−α²^s^χ^−1−1/γ²

≤1c1^θ¹1c2^τ¹·^θ¹

i1

|am,φi|^τ¹

λ1

|an,κλ|^θ²

j1

|am,ϕj|^τ²

χ1

|an,ωχ|

·Bm˘ ^−γ¹^θⁱ¹¹^pⁱ^−1θ²^/γ¹2m^−α¹^θ^j1²^q^j^−1−1/γ¹

·Bn˘˘ ^−γ²^τ^λ1¹ ^r^λ^−1τ²^/γ²2n^−α²^τ^χ1² ^s^χ^−1−1/γ²,

3.40

wherec1andc2are upper bound forE|Zm,k|andE|Zn,l|, respectively. Now let us examine the negative exponents, that is,

γ1 θ1

i1

pi−1 θ2α1 θ2

j1

qj−1− 1 γ₁

,

γ2 τ1

λ1

rλ−1 τ2α2 τ2

χ1

sχ−1− 1 γ2

.

3.41

Observe that, ifθ₂andτ₂are 1 or large, then

θ₂α₁

θ2

j1

q_j−1− 1 γ₁

≥1α₁

η₁− 1 γ₁

,

τ₂α₂

τ2

χ1

s_χ−1− 1 γ₂

≥1α₂

η₂− 1 γ₂

,

3.42

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respectively. Also isθ2τ20,then

γ₁

θ1

i1

pi−1 γ₁2η1−θ₁≥γ₁η₁≥1γ₁

η₁− 1 γ1

≥1α₁

η₁− 1 γ1

,

γ₂

τ1

λ1

rλ−1 γ₂2η2−τ₁≥γ₂η₂≥1γ₂

η₂− 1 γ2

≥1α₂

η₂− 1 γ2

.

3.43

Thus the expected value in3.40is bounded by the product of

K₁

θ1

i1

|am,φi|^θ²

j1

|am,ϕj|m^−1−α¹^η¹^−1/γ¹^, K2

τ1

λ1

|an,κλ|^τ²

χ1

|an,ωχ|n^−1−α²^η²^−1/γ²^,

3.44

whereK1dependent onc1,γ1, ˘B; andc2,γ2,B, respectively.˘˘

Therefore,

E

k

a_m,kZ_m,k 2η1

≤K₃m^−1−α¹^η²^−1/γ¹^,

E

l

a_n,lZ_n,l 2η2

≤K₄n^−1−α²^η²^−1/γ²

3.45

for someK₃ andK₄ which independent onc₁,γ₁, ˘B,Mandc₂,γ₂,B,˘˘ M,respectively. With both independent ofm, n. Now the result follows from Markov’s inequality.

∞andE|X|˘˘ ^11/γ¹ <∞implies thatYm,n→μalmost sure P-convergence.

Proof. Observe that

k,l

a_m,n,k,lX_k,l

k,l

a_m,n,k,lXk,l−μ μ

k,l

a_m,n,k,l. 3.46

Note the last term P-converge toμbecause of the regularity ofA. We will only consider the case whenμ0. By the Borel-Cantelli lemma, it is suﬃcient to prove that for >0

m,n

P

k,l

a_m,n,k,lx_k,l ≥

≤ ∞. 3.47

At this point, the proof follows a path identical to Pruitt’s proof using the above theorems and as such, the rest is omitted.

Acknowledgment

This research was completed with the support of The Scientific and Technological Research Council of Turkey while the first author was a visiting scholar at Istanbul Commerce University, Istanbul, Turkey.

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