Báo cáo hóa học: " Research Article Summability of Double Independent Random Variables potx

Patterson 1 and Ekrem Savas¸ 2 1 Department of Mathematics and Statistics, University of North Florida, 1 UNF Drive, Jacksonville, FL 32224, USA 2 Department of Mathematics, Istanbul com

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Volume 2008, Article ID 948195, 12 pages

doi:10.1155/2008/948195

Research Article

Summability of Double Independent

Random Variables

Richard F Patterson 1 and Ekrem Savas¸ 2

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

FL 32224, USA

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

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

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 X k,l be a factorable double sequence of independent, identically distributed random

variables with E|X k,l | < ∞ and EX k,l μ Let A a m,n,k,lbe a factorable double sequence to double sequence transformation defined as

Ax m,n ∞,∞

k,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 matrix A is RH-regular if and only

if

RH1: P-limm,n a m,n,k,l 0 for each k and l;

RH2: P-limm,n

k,l a m,n,k,l 1;

RH3: P-limm,n

k |a m,n,k,l | 0 for each l;

RH4: P-limm,n

l |a m,n,k,l | 0 for each k;

RH5:

k,l |a m,n,k,l| is P-convergent; and

RH6: there exist positive numbers A and B such that

k,l>B |a m,n,k,l | < A.

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Throughout this paper, we will denote ∞,∞

k,l 1,1 a m,n,k,l X k,l by Y m,n and examine Y m,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 that Ym,n ˘Y m ˘˘Y n P-converges to μ in probability is that max k,l |a m,n,k,l| maxk,l |a m,k a n,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 maxk,l |a m,n,k,l| maxk,l |a m,k a n,l | Om −γ1On −γ2, γ1, γ2 > 0,

then

E | ˘X|11/γ 1 < ∞, E | ˘˘X|11/γ 2 <∞ 1.2

implies that Y m,n → μ almost 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 2.1see 3 A double sequence x xk,l has Pringsheim limit L denoted by P-lim x L provided that given > 0 there exists N ∈ N such that |x k,l − L| < whenever

k, l > N We will describe such an x more briefly as “P-convergent.”

Definition 2.2 A double sequence x is called definite divergent, if for everyarbitrarily large

G > 0 there exist two natural numbers n1and n2such that|x n,k | > G for n ≥ n1, k ≥ n2.

Throughout this paper, we will also denote∞,∞

k,l 1,1 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 matrix A is RH-regular if and only if

RH1: P-lim m,n a m,n,k,l 0 for each k and l;

RH2: P-lim m,n

k,l a m,n,k,l 1;

RH3: P-lim m,n

k |a m,n,k,l | 0 for each l;

RH4: P-lim m,n

l |a m,n,k,l | 0 for each k;

RH5:

k,l |a m,n,k,l | is P-convergent; and

RH6: there exist positive numbers A and B such that

k,l>B |a m,n,k,l | < A.

Following Robison and Hamilton work, Patterson in6 presented the following two notions of subsequence of a double sequence

Definition 2.4 The double sequence y is a double subsequence of the sequence x provided

that there exist two increasing double index sequences{n j } and {k j } such that if z j x n ,k,

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then y is 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 sequencex provided that there exists a subsequence y of x that has Pringsheim limit

β: P-lim y β.

Using these definitions, Patterson presented a series of four-dimensional matrix characterizations of such sequence spaces Let{x k,l} be a double sequence of real numbers

and, for each n, let α n supn {x k,l : k, l ≥ n} Patterson 7 also extended the above notions with the presentation of the following The Pringsheim limit superior of x is defined as

follows:

1 if α ∞ for each n, then P-lim sup x : ∞;

2 if α < ∞ for some n, then P-lim sup x : inf n {α n }.

Similarly, let β n infn {x k,l : k, l ≥ n} Then the Pringsheim limit inferior of x is defined as

follows:

1 if β n −∞ for each n, then P-lim inf x : −∞;

2 if β n > −∞ for some n, then P-lim inf x : sup n {β n }.

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 that Y m,n ˘Y m ˘˘Y n P-converges to μ in probability

is that max k,l |a m,n,k,l| maxk,l |a m,k a n,l | converges to 0 in the Pringsheim sense.

Proof First, note that

lim

˘T→∞ ˘TP| ˘X| ≥ ˘T 0, lim

because E| ˘X| < ∞ and E| ˘˘X| < ∞ Let T ˘T ˘˘T, X m,n,k,l ˘X m,k X˘˘n,l , a m,n,k,l X k,l a m,k X˘

k a n,l X˘˘

l,

and Z m,n ˘Z m Z˘˘n k,l X m,n,k,l For suﬃciently large m and n and since maxk,l |a m,n,k,l| is

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a P-null sequence, it follows from3.1 that

P Z m,n / Y m,n ≤

k,l

P X m,n,k,l / a m,n,k,l X k,l

k,l

P

| ˘X| ≥ |a1

m,k|;| ˘˘X| ≥

1

|a n,l|

≤ k,l

|a m,n,k,l|

≤ M,

3.2

where M is define by RH6of regularity conditions Therefore, it suﬃces to show that

P-lim

Observe that

E Z m,n − μ

k,l

a m,n,k,l

| ˘x|<1/|a m,k|x d ˘F˘

| ˘˘x|<1/|a n,l|˘˘x d ˘˘F − μ

 μ

k,l

a m,n,k,l− 1 , 3.4 which is a P-null sequence Since

1

˘T ˘˘T

| ˘x|< ˘T

| ˘˘x|< ˘˘T x˘2˘˘x2d ˘F d ˘˘F 1

˘T ˘˘T {− ˘T

2

P | ˘X| ≥ ˘T · − ˘˘T2P | ˘˘X| ≥ ˘˘T}

1

˘T ˘˘T

2

˘T

0

˘

xP | ˘X| ≥ ˘xd ˘x · 2

˘˘T

0

˘˘xP | ˘˘X| ≥ ˘˘xd ˘˘x

is a P-null sequence with respect to T, we have

k,l

Var X m,n,k,l≤

k,l

|a m,n,k,l|2

| ˘x|<1/|a m,k|x˘2d ˘F

| ˘˘x|<1/|a n,l|˘˘x2d ˘˘F ≤

k,l

|a m,n,k,l | ≤ M 3.6

for m and n suﬃciently large, where F ˘F ˘˘F and x ˘x ˘˘x It is also clear that Ek,l x m,n,k,l2is finite Thus,

k,l Var X m,n,k,l Var

k,l

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,

let U k,l X k,l − μ and consider the transformation T m,n k,l a m,n,k,l U k,l Our goal become

showing that T m,n P-converges in probability to 0 Which imply that T m,nP-converges in law

to 0 Let us consider the characteristic function of T m,n,that is,

E e uT m,n Ee u

k,l a m,n,k,l U k,l EΠ k,l e ua m,n,k,l U k,l Πk,l E e ua m,n,k,l U k,l : Πk,l g ua m,n,k,l .

3.8

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Observe that

P-lim

Because

|Πk,l g ua m,n,k,l | ≤ |gua m,n,k,l| ≤ 1 3.10 for allm, n we have that

P-lim

for allk, l Clearly, there exists u0such that|gua m,n,k,l | < 1 for 0 < |u| < u0 Let u u0/2M

then there exists a double subsequencea m,n,k m ,l n such that

|ua m,n,k m ,l n | ≤ Mu u0

Thus P-limm,n ua m,n,k m ,l n 0 Therefore, clearly we can choose k m , l n such that

|a m,n,k m ,l n| max

Theorem 3.2 If E| ˘X|11/γ1 < ∞, E| ˘˘X|11/γ2 < ∞, and max k,l |a m,n,k,l| maxk |a m,k| · maxl |a n,l | ≤ ˘Bm −γ1˘˘Bn −γ2, then for every > 0

m,n

P |a m,n,k,l X k,l | ≥ for some k, l < ∞, 3.14

that is,

m,n

P |a m,k X˘

k | ≥ ; |a n,l X˘˘

l | ≥ for some k, l < ∞. 3.15

Proof Let

N m,n x N m,n ˘x ˘˘x

{k,l:1/|a m,k |≤ ˘x; 1/|a n,l |≤ ˘˘x}

|a m,n,k,l |. 3.16

Note x ˘x ˘˘x, and observe that N m,n x 0 for ˘x < m γ1, ˘˘ x < n γ2, and ∞0 d N m,n x

k,l |a m,n,k,l | ≤ M If

G x P|X| ≥ x P| ˘X| ≥ ˘xP| ˘˘X| ≥ ˘˘x G ˘xG ˘˘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 |a m,n,k,l x k,l| ≥ 1

k,l

G

1

|a m,n,k,l|

k,l

1

|a m,n,k,l|G

1

|a m,n,k,l|

∞

0

xG xdN m,n x

N m,n TTGT|∞

0|∞

0 −

∞

0

N m,n xdxGx

lim

T→∞N m,n TTGT −

∞

0

N m,n xdxGx

≤ M

∞

m γ1

∞

n γ2 |dxGx|

M

∞

m γ1

∞

n γ2 |d ˘xG ˘xd ˘˘xG ˘˘x|.

3.18

Our goal now is to get an estimate for ∞m γ1 ∞n γ2 |dxGx| To this end observe that, for z < y

yG y − 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 − G ˘y · ˘˘y − ˘˘zG ˘˘z ˘˘yG ˘˘z − G ˘˘Y. 3.20 The last inequality grant us the following:

∞

m γ1

∞

n γ2 |d ˘xG ˘xd ˘˘xG ˘˘x|

∞,∞

i,j m,n

i1 γ1

i γ1

j1 γ2

j γ2 |d ˘xG ˘xd ˘˘xG ˘˘x|

≤ ∞,∞

i,j m,n

{i 1 γ1− i γ1Gi γ1 · j 1 γ2− j γ2Gj γ2}

∞,∞

i,j m,n

{i 1 γ1Gi γ1 − Gi 1 γ1· j 1 γ2Gj γ2 − Gj 1 γ2}.

3.21

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∞

m γ1

∞

n γ2 |d ˘xG ˘xd ˘˘xG ˘˘x|

≤ 2∞,∞

i,j m,n

{i 1 γ1Gi γ1 − Gi 1 γ1· j 1 γ2Gj γ2 − Gj 1 γ2}.

∞

m,n

P |a m,n,k,l X k,l | ≥ for some k, l

≤∞

m,n

∞

k,l

P |a m,n,k,l X k,l | ≥ 

≤ 2M ∞,∞ m,n 1,1

∞,∞

i,j m,n

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

2M ∞,∞ i,j 1,1

{i 1 γ1Gi γ1 − Gi 1 γ1· j 1 γ2Gj γ2 − Gj 1 γ2}

≤ 21γ 121γ2M

| ˘x|11/γ1| ˘˘x|11/γ2d ˘F ˘xd ˘˘F ˘˘x

< ∞.

3.22

Theorem 3.3 Let x and F be define as in Theorem 3.2 If E | ˘X|11/γ 1 < ∞, E| ˘˘X|11/γ 2 < ∞, and

m,n

P |a m,n,k,l X k,l | ≥ m α1n α2 for at least two pairs k, l < ∞, 3.23

that is,

m,n

P |a m,k X˘

k | ≥ m α1;|a n,l X˘˘

l | ≥ n α2 for at least two pairs k, l < ∞. 3.24

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

m

P |a m,k X˘

k | ≥ m α1 ≤ |a m,k|11/γ 1E | ˘x|11/γ1m α111/γ1 ,

n

P |a n,l X˘˘

l | ≥ n α2 ≤ |a n,l|11/γ 2E | ˘˘x|11/γ 2n α211/γ2 .

3.25

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m,n

P |a m,k X˘

k | ≥ m α1;|a n,l X˘˘

l | ≥ n α2 for at least two pairsk, l

i / k, j / l

P |a m,i X˘

i | ≥ m α1;|a m,k X˘

k | ≥ m α1;|a n,j X˘˘

j | ≥ n α2;|a n,l X˘˘

l | ≥ n α2

≤ E| ˘x|11/γ 12m 2α111/γ1

i / k

|a m,i|11/γ 1|a m,k|11/γ 1

· E| ˘˘x|11/γ 22n 2α211/γ2

j / l

|a n,j|11/γ 2|a n,l|11/γ 2

≤ E| ˘x|11/γ 12· E| ˘˘x|11/γ 22˘B 2/γ1˘˘B 2/γ2

M4m2−1α111/γ1 n2−1α211/γ2 ,

3.26

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

γ2/2 γ2 1

Theorem 3.4 Let x and F be define as in Theorem 3.2 If μ 0, E| ˘X|11/γ 1 < ∞, E| ˘˘X|11/γ 2 < ∞,

m,n

P

k,l

|a m,n,k,l X k,l | ≥

where

k,l

a m,n,k,l X k,l

{k:|a m,k X k |<m −α1 l: |a n,l X l |<n −α2}

a m,n,k,l X k,l , 3.28

α1< γ1, and α2< γ2.

Proof Let

X m,n,k,l:

⎧

⎪

X m,k; if|a m,k X k | < m −α1,

X n,l; if|a n,l X l | < n −α2,

0; otherwise,

3.29

and β m,n,k,l EX m,n,k,l If a m,n,k,l 0, then β m,n,k,l μ 0 and if a m,n,k,l / 0, then

|β m,n,k,l|

μ −| ˘x|≥m −α1 |a

m,k| −1

| ˘˘x|≥m −α2 |a n,l| −1x dF

≤

| ˘x|≥m −α1 ˘B−1m γ1

| ˘˘x|≥n −α2 ˘˘B−1n γ2 |x|dF.

3.30

Therefore, P-limm,n β m,n,k,l 0 uniformly in k, l and P-lim m,n

k,l a m,n,k,l β m,n,k,l 0 Let

Z m,n,k,l Z m,k Z n,l X m,n,k,l − β m,n,k,l , 3.31

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so that EZ m,n,k,l 0, E|Z m,k|11/γ 1 < c1, and E |Z n,l|11/γ 2 < c2 for some c1 and c2 Also

|a m,k Z m,k | ≤ 2m −α1and|a n,l Z n,l | ≤ 2n −α2 Observe that

k,l

a m,n,k,l X k,l

k,l

a m,n,k,l X m,n,k,l

k,l

a m,n,k,l Z m,n,k,l

k,l

a m,n,k,l β m,n,k,l 3.32

Note for suﬃciently large m and n

k,l

a m,n,k,l X k,l

≥

⊂

k,l

a m,n,k,l Z m,n,k,l

≥

2

Thus it is suﬃcient to show that

m,n

P

k,l

|a m,n,k,l Z m,n,k,l|

≥

Let η1and η2be the least integers greater than 1/γ1and 1/γ2, respectively Our goal now is to produce an estimate for

E

k

a m,k Z m,k

2η1

l

a n,l Z n,l

2η2

Observe that

E

k

a m,k Z m,k

2η1

l

a n,l Z n,l

2η2

3.36

is equal to

k1,k2, ,k 2p ; l1,l2, ,l 2q

E

2p

i1

2q

j1

a m,n,k i ,l j Z m,n,k i ,l j 3.37

It happens to be the case that Ek a m,k Z m,k2η1l a n,l Z n,l2η2 is zero if k i , l i / k j , l j for i / j because the Z m,n,k,l ’s are independent and EZ m,n,k,l 0 Let us now consider the general term Thus

p1of the ks φ1, , p θ1of the ks φ θ1,

q1 of the ks ϕ1, , q θ2 of the ks ϕ θ2,

r1 of the ls κ1, , r τ1 of the ls κ τ1,

s1 of the ls ω1, , s τ2 of the ls ω τ2,

3.38

where 2≤ p i ≤ 1 1/γ1, q j > 1 1/γ1, 2≤ r λ ≤ 1 1/γ2, s χ > 1 1/γ2,

θ1

i1

p iθ2

j1

q j 2η1,

τ1

λ1

r iτ2

χ1

s χ 2η2.

3.39

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Now let us consider the following expectation:

E

θ

1

i1

a m,φ i Z m,φ ip i·θ2

j1

a m,ϕ j Z m,ϕ jq j·τ1

λ1

a n,κ λ Z n,κ λr λ

τ2

χ1

a n,ω χ Z n,ω χs χ

≤ 1 c1θ11 c2τ1·τ2

χ1

|a m,φ i|p i

τ1

λ1

|a n,κ λ|r λ

· E

θ

2

j1

a m,ϕ j Z m,ϕ jq j·τ2

χ1

a n,ω χ Z n,ω χs χ

≤ 1 c1θ11 c2τ1·θ1

i1

|a m,φ i|p i

τ1

λ1

|a n,κ λ|r λ

·θ2

j1

|a m,ϕ j|11/γ12m −α1q j −1−1/γ1·τ2

χ1

|a n,ω χ|11/γ 22n −α2s χ −1−1/γ2

≤ 1 c1θ11 c2τ1·θ1

i1

|a m,φ i ||a m,φ i|p i−1

·τ1

λ1

|a n,κ λ ||a n,κ λ|r λ−1·θ2

j1

|a m,ϕ j|11/γ 12m −α1q j −1−1/γ1

·τ2

χ1

|a n,ω χ|11/γ22n −α2s χ −1−1/γ2

≤ 1 c1θ11 c2τ1·θ1

i1

|a m,φ i|τ1

λ1

|a n,κ λ|θ2

j1

|a m,ϕ j|τ2

χ1

|a n,ω χ|

· ˘Bm −γ1θ1 i1p i −1θ2/γ12m −α1θ2 j1q j −1−1/γ1

· ˘˘Bn −γ2τ1 λ1r λ −1τ2/γ22n −α2τ2 χ1s χ −1−1/γ2 ,

3.40

where c1and c2are upper bound for E|Z m,k | and E|Z n,l|, respectively Now let us examine the negative exponents, that is,

γ1

θ1

i1

p i − 1 θ2 α1

θ2

j1

q j− 1 − 1

γ1

,

γ2

τ1

λ1

r λ − 1 τ2 α2

τ2

χ1

s χ− 1 − 1

γ2

.

3.41

Observe that, if θ2and τ2are 1 or large, then

θ2 α1

θ2

j1

q j− 1 − 1

γ1

≥ 1 α1

η1− 1

γ1

,

τ2 α2

τ2

χ1

s χ− 1 − 1

γ2

≥ 1 α2

η2− 1

γ2

,

3.42

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