R E S E A R C H
Open Access
On complete convergence for weighted sums
of martingale-difference random fields
Mi Hwa Ko
**Correspondence:
[email protected] Division of Mathematics and Informational Statistics, Wonkwang University, Jeonbuk, 570-749, Korea
Abstract
Let{an,i,n∈Z+d,i≤n}be an array of real numbers, and let{Xi,i∈Z+d}be the
martingale differences with respect to{Gn,n∈Z+d}satisfying
E(E(X|Gk)|Gm) =E(X|Gk∧m) a.s., wherek∧mdenotes componentwise minimum, {Gk,k∈Z+d}is a family of
σ
-algebras such that∀k≤n,Gk⊂Gn⊂G, andXis 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 sumsi≤nan,iXi. MSC: 60F05; 60F15
Keywords: complete convergence; weighted sums; martingale difference; maximal moment inequality;
σ
-algebra1 Introduction
The concept of complete convergence for sums of independent and identically distributed random variables was introduced by Hsu and Robbins [] as follows: A sequence of random variables{Xn}is said to be completely to a constantcif
∞
n=
P|Xn–c|>
<∞ for all> .
This result has been generalized and extended to the random fields{Xn,n∈Zd+}of random variables. For example, Fazekas and Tómács [] and Czerebak-Mrozowiczet al.[] for fields of pairwise independent random variables, and Gut and Stadtmüller [] for random fields of i.i.d. random variables.
LetZ+be the set of positive integers. For fixedd∈Z+, setZd+={n= (n,n, . . . ,nd) :ni∈
Z+,i= , , . . . ,d}with coordinatewise partial order, ≤,i.e., form= (m,m, . . . ,md),n=
(n,n, . . . ,nd)∈Zd+,m≤nif and only ifmi≤ni,i= , , . . . ,d. Forn= (n,n, . . . ,nd)∈Zd+, let|n|=di=ni. For a field{an,n∈Z+d}of real numbers, the limit superior is defined by infr≥sup|n|≥ranand is denoted bylim sup|n|→∞an.
Note that|n| → ∞is equivalent tomax{n,n, . . . ,nd} → ∞, which is weaker than the
conditionmin{n,n, . . . ,nd} → ∞whend≥.
Let{Xn,n∈Z+d}be a field of random variables, and let{an,k,n∈Zd+,k≤n}be an array of real numbers. The weighted sumsk≤nan,kXkcan play an important role in various applied and theoretical problems, such as those of the least squares estimators (see Kafles
and Bhaskara Rao []) andM-estimates (see Rao and Zhao []) in linear models, the non-parametric regression estimators (see Priestley and Chao []),etc. So, the study of the limiting behavior of the weighted sums is very important and significant (see Chen and Hao []).
Now, we consider the notion of martingale differences. Let{Gk,k∈Z+d}be a family of
σ-algebras such that
Gk⊂Gn⊂G, ∀k≤n,
and for any integrable random variableXdefined on the initial probability space,
EE(X|Gk)|Gm
=E(X|Gk∧m) a.s., (.)
wherek∧mdenotes the componentwise minimum.
An{Gk,k∈Zd+}-adapted, integrable process{Yk,k∈Zd+}is called a martingale if and only if
E(Yn|Gm) =Ym∧n a.s.
Let us observe that for martingale{(Yn,Gn),n∈Z+d}, the random variables
Xn=
a∈{,}d
(–)di=aiY
n–a,
wherea= (a,a, . . . ,ad) andn∈Zd+, are martingale differences with respect to{Gn,n∈Z+d} (see Kuczmaszewska and Lagodowski []).
For the results concerning complete convergence for martingale arrays obtained in the one-dimensional case, we refer to Lagodowski and Rychlik [], Elton [], Lesigne and Volny [], Stoica [] and Ghosal and Chandra []. Recently, complete convergence for martingale difference random fields was proved by Kuczmaszewska and Lagodowski [].
The aim of this paper is to obtain some results concerning complete convergence of weighted sums i≤nan,iXi, where{an,i,n∈Z+d,i≤n} is an array of real numbers, and {Xi,i∈Zd+}is the martingale differences with respect to{Gn,n∈Z+d}satisfying (.).
2 Results
The following moment maximal inequality provides us a useful tool to prove the main results of this section (see Kuczmaszewska and Lagodowski []).
Lemma . Let{(Yn,Gn),n∈Zd+}be a martingale,and let{(Xn,Gn),n∈Z+d}be the
mar-tingale differences corresponding to it.Let q> .There exists a finite and positive constant C depending only on q and d such that
E
max
k≤n|Yk|
q≤CE
k≤n
Xk q/
. (.)
Theorem . Let{an,i,n∈Z+d,i≤n}be an array of real numbers,and let{Xn,n∈Z+d}be
the martingale differences with respect to{Gn,n∈Z+d}satisfying(.).Forαp> ,p> and
α>
,we assume that
(i) n|n|αp–
i≤nP{|an,iXi|>|n|α}<∞,
(ii) n|n|α(p–q)–+q/i≤n|an,i|qE(|Xi|qI[|an,iXi| ≤ |n|α]) <∞forq≥,
(ii) n|n|α(p–q)–
i≤n|an,i|qE(|Xi|qI[|an,iXi| ≤ |n|α]) <∞for <q< and
(iii) n|n|αp–P{maxj≤n|
i≤jE(an,iXiI[|an,iXi| ≤ |n|α]|Gi∗)|>|n|α}<∞for all> .
Then we have
n
|n|αp–P
max
j≤n |Sj|>|n|
α
<∞ for all> , (.)
where Sn=
≤i≤nan,iXi.
Proof Let us notice that the seriesn|n|αp–is finite, then (.) always holds. Therefore, we consider only the case such thatn|n|αp–is divergent. LetXn,i=XiI[|an,iXi| ≤ |n|α],
Xn,i∗ =Xn,i–E(Xn,i|Gi∗) andS∗n,j=
i≤jan,iXn,i∗ . Then
n
|n|αp–P
max
j≤n |Sj|>|n|
α
≤
n
|n|αp–P|a
n,iXi|>|n|α
+
n
|n|αp–P
max
j≤n
i≤j
an,iXiI
|an,iXi| ≤ |n|α>|n|α
≤
n
|n|αp– i≤n
P|an,iXi|>|n|α
+
n
|n|αp–P
max
j≤n
i≤j
an,iXiI
|an,iXi| ≤ |n|α
–Ean,iXiI
|an,iXi| ≤ |n|α
|G∗
i>
|n|
α
+
n
|n|αp–P
max j≤n
i≤j
Ean,iXiI
|an,iXi| ≤ |n|α
|G∗
i>
|n|
α
=I+I+I.
Clearly,I<∞by (i), andI<∞by (iii). It remains to prove thatI<∞. Thus, the proof will be completed by proving that
n
|n|αp–P
max
j≤n
S∗n,j>|n|α
<∞.
To prove it, we first observe that{(S∗n,j,Gj),j≤n}is a martingale. In fact, ifi>j, thenGi∧j⊂
G∗
i and by (.), we have
Ean,iX∗n,i|Gj
=Ean,iXn,i–E
an,iXn,i|Gi∗
|Gi
=EEan,iXn,i–E
an,iXn,i|G∗i
|Gi
=Ean,iXn,i–E
an,iXn,i|Gi∗
|Gi∧j
= .
Then, by the Markov inequality and Lemma ., there exists some constantCsuch that
P
max
j≤n S∗
n,j>|n|
α≤CE(maxj≤n|S ∗
n,j|q) |n|αq
≤| C n|αqE
i≤n
an,iXn,i∗
q/ =I.
Caseq≥; we get
I≤
C
|n|αq|n| q/–
i≤n
Ean,iXn,i∗
q
≤C|n|q/––αq i≤n
E|an,iXi|qI
|an,iXi| ≤ |n|α
.
Note that the last estimation follows from the Jensen inequality. Thus, we have
n
|n|αp–Pmax j≤n
S∗n,j>|n|α
≤C
n
|n|αp––q(α–/) i≤n
E|an,iXi|qI
|an,iXi| ≤ |n|α
<∞
by assumption (ii). Case <q< ; we get
I≤
C
|n|αq
i≤n
Ean,iXn,i∗
q
≤C|n|–αq i≤n
E|an,iXi|qI
|an,iXi| ≤ |n|α
.
Therefore, for <q< , we obtain
n
|n|αp–P
max
j≤n
S∗n,j>|n|α
≤C
n
|n|α(p–q)– i≤n
E|an,iXi|qI
|an,iXi| ≤ |n|α
<∞
by assumption (ii). Thus,I<∞for allq> , and the proof of Theorem . is complete.
Corollary . Let{an,i,n∈Zd+,i≤n}be an array of real numbers.Let{Xn,n∈Zd+}be
martingale differences with respect to{Gn,n∈Z+d}satisfying(.),and EXn= forn∈Z+d.
Let p≥,α>andαp> .Assume that(i)and for some q> , (ii)or(ii)hold respectively.If
max j≤n
i≤j
Ean,iXiI
|an,iXi| ≤ |n|α
|G∗
i
=o|n|α, (.)
Proof It is easy to see that (.) implies (iii). We omit details that prove it.
The following corollary shows that assumption (iii) in Theorem . is natural, and in the case of independent random fields, it reduces to the known one.
Corollary . Let{an,i,n∈Zd+,i≤n}be an array of real numbers.Let{Xn,n∈Z+d}be a
field of independent random variables such that EXn= forn∈Zd+.Let p≥,α> and
αp> .Assume that(i)and for some q> , (ii)or(ii)hold respectively.If
|n|αmaxj≤n
i≤j
Ean,iXiI
|an,iXi| ≤ |n|α
→ as|n| → ∞, (.)
then(.)holds.
Proof Since{Xn,n∈Zd+}is a field of independent random variables, we have
|n|αmaxj≤n
i≤j
Ean,iXiI
|an,iXi| ≤ |n|α
|G∗
i
=
|n|αmaxj≤n
i≤j
Ean,iXiI
|an,iXi| ≤ |n|α
.
Now, it is easy to see that (.) implies (iii) of Theorem .. Thus, by Theorem ., result
(.) follows.
Remark Theorem . and Corollary . are extensions of Theorem . and Corollary . in Kuczmaszewska and Lagodowski [] to the weighted sums case, respectively.
Corollary . Let{an,i,n∈Z+d,≤i≤n}be an array of real numbers.Let{Xn,n∈Z+d}
be the martingale differences with respect to{Gn,n∈Z+d}satisfying(.)and EXn= .Let
p≥,α> andαp> and E|Xn|+λn<∞forλnwith <λn< forn∈Z+d.If
n
|n|αp–|n|–α(+λn) i≤n
|an,i|+λnE|Xi|+λn<∞, (.)
max ≤j≤n
i≤j
E|an,iXi|I
|an,iXi| ≤ |n|α
|G∗
i
=o|n|α, (.)
then(.)holds.
Proof If the seriesn|n|αp–<∞, then (.) always holds. Hence, we only consider the casen|n|αp–=∞. It follows from (.) that
|n|–α(+λn)
i≤n
|an,i|+λnE|Xi|+λn< .
By (.) and the Markov inequality,
n
|n|αp–P|an,iXi|>|n|α
≤
n
|n|αp–|n|–α(+λn)
i≤n
|an,i|+λnE|Xi|+λn<∞, (.)
As the proof of Corollary ., (.) implies (iii) of Theorem .. It remains to show that Theorem .(ii) or (ii)is satisfied. For <q< , take +λn<q. Then we have
n
|n|α(p–q)– i≤n
|an,i|qE
|Xi|qI
|an,iXi| ≤ |n|α
≤
n
|n|αp–|n|–α(+λn)|n|–αq+α(+λn)|n|αq–α(+λn) i≤n
|an,i|+λnE|Xi|+λn
=
n
|n|αp–|n|–α(+λn)
i≤n
|an,i|+λnE|Xi|+λn<∞ by (.),
which satisfies Theorem .(ii). Hence, the proof is complete.
Corollary . Let{an,i,n∈Z+d,≤i≤n}be an array of real numbers,and let{Xn,n∈
Zd
+}be the martingale differences with respect to{Gn,n∈Zd+}satisfying(.),EXn= and
E|Xn|p<∞for <p< .Letα>,αp> and <p< .If
≤i≤n
|an,i|pE|Xi|p=O
|n|δ for <δ< , (.)
and Theorem.(iii)hold,then(.)holds.
Proof By (.) and the Markov inequality,
n
|n|αp– i≤n
P|an,iXi|>|n|α
≤
n
|n|αp– i≤n
|an,i|pE|Xi|p |n|αp
≤C
n
|n|–+δ<∞. (.)
By takingq<p, we have
n
|n|α(p–q)– i≤n
|an,i|qE
|Xi|qI
|an,iXi| ≤|n|α
≤
n
|n|– i≤n
|an,i|pE|Xi|p
≤C
n
|n|–+δ<∞. (.)
Hence, by (.) and (.), conditions (i) and (ii)in Theorem . are satisfied, respectively. To complete the proof, it is enough to note that byEXn= forn∈Z+dand by (.), we get forj≤n
|n|–α i≤j
|an,i|E|Xi|I
|an,iXi| ≤|n|α
→ as|n| → ∞. (.)
Corollary . Let{Xn,n∈Z+d}be the martingale differences with respect to{Gn,n∈Z+d}
satisfying(.),let EXn= and E|Xn|p<∞for <p< and be stochastically dominated
by a random variable X,i.e.,there is a constant D such that P(|Xn|>x)≤DP(|X|>x)for
all x≥andn∈Zd
+.Let{an,i,n∈Z+d,i≤n}be an array of real numbers satisfying
i≤n
|an,i|p=O
|n|δ for <δ< . (.)
If Theorem.(iii)holds,then(.)holds.
Proof From (.), (.) follows. Hence, by Corollary ., we obtain (.).
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 Kimet al.[]).
LetYk=
i≥ak+iXi, where{ai,i∈Zd+}is a field of real numbers with
i≥|ai|<∞, and{Xi,i∈Zd+}is a field of the martingale difference random variables.
Definean,i=
≤k≤nai+k. Then we have
≤k≤n
Yk=
≤k≤n
i≥
ai+kXi=
i≥
≤k≤n
ai+kXi=
i≥
an,iXi.
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
