O Journal of Inequalities and Applications

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RESEARCH

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On complete convergence for weighted sums of martingale-difference random fields

1i Hwa Ko*

Correspondence: songhack@wonkwang.ac.kr Division of Mathematics and InformationalStatistics, Wonkwang University, Jeonbuk, 570-749, Korea

Abstract

Let (ani,n e Z+, i < n} be an array of real numbers, and let (Xi, i e Z+} be the martingale differences with respect to (Gn,n e Z+} satisfying E(E(X\Gk)\Gm) = E(X\GkAm) a.s., where k am denotes componentwise minimum, (Gk, k e Z+} is a family of a-algebras such that Vk < n, Gk c Gn c G, and X is any integrable random variable defined on the initial probability space. The aim of this paper is to obtain some results concerning complete convergence of weighted sums

Ei<n aniXi.

MSC: 60F05; 60F15

Keywords: complete convergence; weighted sums; martingale difference; maximal moment inequality; a-algebra

1 Introduction

The concept of complete convergence for sums of independent and identically distributed random variables was introduced by Hsu and Robbins [1] as follows: A sequence of random variables {Xn} is said to be completely to a constant c if

J2pdxn - ci >e) < ^ foral1 e >°

ringer

This result has been generalized and extended to the random fields {Xn, n e Z+} of random variables. For example, Fazekas and Tomacs [2] and Czerebak-Mrozowicz et al. [3] for fields of pairwise independent random variables, and Gut and Stadtmuller [4] for random fields of i.i.d. random variables.

Let Z+ be the set of positive integers. For fixed d e Z+, set Z+ = {n = (n1, n2,..., nd): ni e Z+,i = 1,2,...,d} with coordinatewise partial order, <, i.e., for m = (m1,m2,...,md),n = (ni, n2,..., nd) e Z+, m < n if and only if mi < n, i = 1,2,..., d. For n = (m, n2,..., nd) e Zif, let |n| = ]"[¿=1 ni. For a field {an, n e Z+1} of real numbers, the limit superior is defined by infr>1 sup|n|>r an and is denoted by limsup|n^TO an.

Note that |n| —> ^ is equivalent to max{n1, n2,..., nd} ^ to, which is weaker than the condition min{n1, n2,..., nd} — to when d > 2.

Let {Xn, n e Z+} be a field of random variables, and let {an,k, n e Zf, k < n} be an array of real numbers. The weighted sum^k<n an,kXk can play an important role in various applied and theoretical problems, such as those of the least squares estimators (see Kafles

©2013 Ko; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the originalworkis properly cited.

and Bhaskara Rao [5]) and M-estimates (see Rao and Zhao [6]) in linear models, the non-parametric regression estimators (see Priestley and Chao [7]), etc. So, the study of the limiting behavior of the weighted sums is very important and significant (see Chen and Hao [8]).

Now, we consider the notion of martingale differences. Let {Gk, k e Z+} be a family of a-algebras such that

Gk c Gn c G, Vk < n,

and for any integrable random variable X defined on the initial probability space,

£(£(X|Gk)IGm) = £(X|GkAm) a.s., (1.1)

where k A m denotes the componentwise minimum.

An {Gk, k e Z+ }-adapted, integrable process {Yk, k e Z+} is called a martingale if and only if

E(YnIGm) = YmAn a.s. Let us observe that for martingale {(Yn, Gn), n e Z+}, the random variables

Xn = £ (-1)^1 a'Yn-a,

ae{0,1}d

where a = (ai, a2,..., ad) and n e Z+, are martingale differences with respect to {Gn, n e Zf} (see Kuczmaszewska and Lagodowski [9]).

For the results concerning complete convergence for martingale arrays obtained in the one-dimensional case, we refer to Lagodowski and Rychlik [10], Elton [11], Lesigne and Volny [12], Stoica [13] and Ghosal and Chandra [14]. Recently, complete convergence for martingale difference random fields was proved by Kuczmaszewska and Lagodowski [9].

The aim of this paper is to obtain some results concerning complete convergence of weighted sum^i<n an,iXi, where {an,i, n e Zf, i < n} is an array of real numbers, and {Xi, i e Zf} is the martingale differences with respect to {Gn, n e Z+} satisfying (1.1).

2 Results

The following moment maximal inequality provides us a useful tool to prove the main results of this section (see Kuczmaszewska and Lagodowski [9]).

Lemma 2.1 Let {(Yn, Gn), n e Z+} be a martingale, and let {(Xn, Gn), n e Z+} be the martingale differences corresponding to it. Let q >1. There exists a finite and positive constant C depending only on q and d such that

E(max|Yk|q) < CE(^XA" . (.1)

k<n k<n

Let us denote Gi* = a {Gj: j < i}. Now, we are ready to formulate the next result.

Theorem 2.2 Let {an, i, n e Zf, i < n} be an array of real numbers, and let {Xn, n e Z+} be the martingale differences with respect to {Gn, n e Z+} satisfying (1.1). For ap > 1, p >1 and

a >2, we assume that

(i) En |n|ap-2Ei<n-P{|an,iXi| > |n|a} <

(ii) En |n|a<p-q)-3+q/2 Ei<n |an,i|qE(|Xi|q/[|an,iXi| < |n|a ]) < to for q > 2,

(ii)' En |n|a(p-q)-2 Ei<n |an,i|qE(|Xi|qI[|an,iXi| < |n|a ]) < to for 1 < q <2 and

(iii) En |n|ap-2P{maxj<n | Ei<j E(aniXiI[|aniXi| < |n|a ] | G*) | > e|n|a} < to for all e > 0.

Then we have

£>r-2p{max |5j| > e |n|aJ < to for all e >0,

where Sn = E1<i<n an,iXi.

Proof Let us notice that the series En |n|ap 2 is finite, then (2.2) always holds. Therefore, we consider only the case such that En |n|ap-2 is divergent. Let Xn,i = XiI[|an,iXi| < |n|a], Xn,i = Xn,i - E(Xn,i G) and Sn,j = Ei<j an,iXn4. Then

y|n|^p-2P max |Sj| > e|n|

<YJ|n|aP-2P{|an,iXi| > |n|a}

+ J2 |nr-2pj max J2an,iXil[|an,iXi| < |n|a]

n j < n

> e|n|a

<J2 Kp-2EP{|an,iXi| > |n|a} + £ |nr-2p{max J>n,iXiI[|an,iXi| < |n|a]

n i< n n j n i< j

> - |n|a 2

-E(an,iXi![|an,iXi| < |n|a]|G*))

+ ^ |nr-2p{max J2E(an,iXil[|an,iXi| < |n|a]|G*

n j < n

= I1 +12 +13.

> e |n|a 2

Clearly, Ii < to by (i), and I3 < to by (iii). It remains to prove that I2 < to. Thus, the proof will be completed by proving that

£>r-2p{max|sn,j| > e|n|aj < to.

To prove it, we first observe that {(SW,j, Gj), j < n} is a martingale. In fact, if i > j, then GiAj c G* and by (1.1), we have

E(an,iX^|Gj) = E(an,iXn,i -E(an4Xn4|Gi*)|Gi)

= E(E(an,iXn,i - E(an,iXn,i|Gi*) |Gi) |Gj)

= £(an,iXn4 - £(an,iXn,ilG*)GiAj) = 0.

Then, by the Markov inequality and Lemma 2.1, there exists some constant C such that

rii 1 £(maxj<n ISn^)

P max|^n . | > c|n|a < C—-J< n,)

I j<n 1 njl 'J " iniaq

r / \ q/2

< — EiYanXnA = I4.

- |n|aq n,i n,i^ 4

Case q > 2; we get

I4 |n|q/2-^£|aniXni|q

4- |n|aq1 1 I n,i n,il

< qn|q/2-1-aq^E(|an,iXi|qI[|an,iXi| < |n|a]).

Note that the last estimation follows from the Jensen inequality. Thus, we have £>r-2p{max^n,;! > e|n|aj

< cJ2 mr-3-q(a-1/2)£ e^^I [^nX < |n|a ]) < to

by assumption (ii). Case 1 < q < 2; we get

I4 < VE|an X Jq 1 n,i n,i|

< C|nГqJ2E(|an,iXiIqI[|an,iXi| < |n|a]).

Therefore, for 1 < q <2, we obtain £>r-2P{max|5n,j| > c|n|a

< C£ |n|a(p-q)-2 £E(|an,iXi|qI[|an,iXi| < |n|a]) < to

by assumption (ii)'. Thus, I2 < to for all q >1, and the proof of Theorem 2.2 is complete.

Corollary 2.3 Let {an, i, n e Z+, i < n} be an array of real numbers. Let {Xn, n e Z+} be martingale differences with respect to {Gn, n e Z+f} satisfying (1.1), and EXn = Ofor n e Zd. Letp > 1, a >2 and ap > 1. Assume that (i) and for some q > 1, (ii) or (ii)' hold respectively. If

maxEE(an,iXil[|an,iXi| < |n|a]|G*) = o(|n|a), (2.3)

j<n i<j

then (2.2) holds.

Proof It is easy to see that (2.3) implies (iii). We omit details that prove it. □

The following corollary shows that assumption (iii) in Theorem 2.2 is natural, and in the case of independent random fields, it reduces to the known one.

Corollary 2.4 Let {an, i, n e Z+, i < n} be an array of real numbers. Let {X^ n e Z+} be a

field of independent random variables such that EXn = Ofor n e Z{. Let p > 1, a > 1 and

ap > 1. Assume that (i) and for some q >1, (ii) or (ii)' hold respectively. If

— maxVE(aniXiI\\aniXi\ <\n\a 1) ^ O as \n\^TO, (2.4)

in!" j<n ^ ' L J/

then (2.2) holds.

Proof Since {Xn, n e Z+} is a field of independent random variables, we have

-i- max VE(an,iXiI[\aniXi| < \n\a]\£*) = -L max VE(an,iXJ[\aniXi\ < \n\a]).

|n|a j£„ v L J ' |n|a j<n v L J/

i<j i<j

Now, it is easy to see that (2.4) implies (iii) of Theorem 2.2. Thus, by Theorem 2.2, result (2.2) follows. □

Remark Theorem 2.2 and Corollary 2.4 are extensions of Theorem 4.1 and Corollary 4.1 in Kuczmaszewska and Lagodowski [9] to the weighted sums case, respectively.

Corollary 2.5 Let {an, i, n e Z+ 1 < i < n} be an array of real numbers. Let {Xn, n e Zd+} be the martingale differences with respect to G, n e Z+} satisfying (1.1) and EXn = O. Let p > 1, a >i and ap >1 and E|Xn|1+An < to for X„ with O < X„ < 1for n e Zf. If

J2\nr-2\nГ(1+^J2 |an,i|1+AnE|Xi|1+An < to, (2.5)

maxxZ! K amX^\aniXi\ < \n\a] \g*) = o( \n\a), (2.6)

" " i<j

then (2.2) holds.

Proof If the series En \n\ap-2 < to, then (2.2) always holds. Hence, we only consider the case En \n\ap-2 = to. It follows from (2.5) that

|-a(i+An) V^ \a

\n\-a(i+An^ \an,i\1+A"E\Xi\1+An < 1.

By (2.5) and the Markov inequality, V |nrp-2P(|an,iXi| > \n\a)

< J2 \n\ap-2\n\-a(1+A-^ \a„,i\1+AnE\Xi\1+An < to, (2.7)

which satisfies (i) of Theorem 2.2.

As the proof of Corollary 2.3, (2.6) implies (iii) of Theorem 2.2. It remains to show that Theorem 2.2(ii) or (ii)' is satisfied. For 1 < q <2, take 1 + Xn < q. Then we have

|n|a(p-q)-2 ^ |an.|q£(|Xi|q/[\aniXi\< K])

< |n|aP-2 |n|-«(l+^n) |n|-«?+«(l+An) |n|«?-«(l+An^ |flni|1+Àn£|Xi|1+Àn n i< n

= |nr-2|nr(1+^£ |fln,i|1+AnEXi^ < TO by (2.5),

which satisfies Theorem 2.2(ii)'. Hence, the proof is complete. □

Corollary 2.6 Let {an, i, n e Z+, 1 < i < n} be an array of real numbers, and let {Xn, n e Z+} be the martingale differences with respect to {Gn, n e Zf} satisfying (1.1), EXn = 0 and E|Xn|p < to for 1 <p <2. Let a >2, ap >1 and 1 <p < 2. If

J2 |an,i|pE|Xi|p = O(|n|S) for 0 < S < 1, (2.8)

and Theorem 2.2(iii) hold, then (2.2) holds. Proof By (2.8) and the Markovinequality, £ |n|ap-2 £P(|an,iXi| > |n|a) |an,i|pE|Xi|p

<Emr-2E-

< CJ2 |n|-2+5 < TO. (.9)

By taking q < p,we have

£ |n|a(p-q)-2 £ |an,i|qE(|Xi|q/[|an,iXi| < e|n|a]) n i< n

< £ |n|-2 E |an,i|pE|Xi|p

n i< n

< CJ2 |n|-2+5 < TO. (.10)

Hence, by (2.9) and (2.10), conditions (i) and (ii)' in Theorem 2.2 are satisfied, respectively.

To complete the proof, it is enough to note that by EXn = 0 for n e Z+ and by (2.8), we get for j < n

|n|-a J2 |an,i|E|Xi|/[|An,iXi| < e|n|a] ^ 0 as |n| ^ to. (2.11)

Hence, the proof is complete. □

Corollary 2.7 Let {X„, n e Z?} be the martingale differences with respect to {G„, n e Z?} satisfying (1.1), letEXn = O andE\Xnn\p < to for 1 <p <2 and be stochastically dominated by a random variable X, i.e., there is a constant D such thatP(\Xn\ > x) < DP(\X\ > x) for allx > O and n e Z+. Let {an, i, n e Z+, i < n} be an array of real numbers satisfying

J2\a*i\p = 0(\n\5) for O <S <1. (2.12)

If Theorem 2.2(iii) holds, then (2.2) holds.

Proof From (2.12), (2.8) follows. Hence, by Corollary 2.6, we obtain (2.2). □

Remark Linear random fields are of great importance in time series analysis. They arise in a wide variety of context. Applications to economics, engineering, and physical science are extremely broad (see Kim etal. [15]).

Let Yk = Ei>1 ak+iXi, where {ai, i e Z?} is a field of real numbers with Ei > 1\ai \ < to, and {Xi, i e Z+} is a field of the martingale difference random variables. Define an,i = E 1<k<n ai+k. Then we have

^ Yk = ^ ^ai+kXi = ^ ^ ai+kXi = ^ a^Xi.

1<k<n 1<k<n i>1 i>1 1<k<n i>1

Hence, we can use the above results to investigate the complete convergence for linear random fields.

Competing interests

The author declares that she has no competing interests.

Received: 21 June 2013 Accepted: 23 September 2013 Published: 07 Nov 2013

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10.1186/1029-242X-2013-473

Cite this article as: Ko: On complete convergence for weighted sums of martingale-difference random fields. Journal of Inequalities and Applications 2013, 2013:473