DSpace at VNU: Rate of complete convergence for maximums of moving average sums of martingale difference fields in Banach spaces

Contents lists available atSciVerse ScienceDirect Statistics and Probability Letters journal homepage:www.elsevier.com/locate/stapro Rate of complete convergence for maximums of moving a

Contents lists available atSciVerse ScienceDirect Statistics and Probability Letters

journal homepage:www.elsevier.com/locate/stapro

Rate of complete convergence for maximums of moving average sums of martingale difference fields in Banach spaces

Ta Cong Sona,∗, Dang Hung Thanga, Le Van Dungb

aFaculty of Mathematics, Hanoi University of Science, 334 Nguyen Trai, Thanh Xuan, Hanoi, Viet Nam

bFaculty of Mathematics, Danang University of Education, 459 Ton Duc Thang, Lien Chieu, Danang, Viet Nam

a r t i c l e i n f o

Article history:

Received 3 April 2012

Received in revised form 14 June 2012

Accepted 16 June 2012

Available online 25 June 2012

The authors dedicate the paper to professor

Nguyen Duy Tien on the occasion of his

seventieth birthday.

MSC:

60B11

60B12

60F15

60G42

Keywords:

Complete convergence

Marcinkiewicz–Zygmund strong laws of

larger numbers

p-uniformly smooth Banach spaces

Martingale difference fields

a b s t r a c t

We obtain the rate of complete convergence for maximums of moving average sums

of martingale difference fields in p-uniformly smooth Banach spaces, and extend

Marcinkiewicz–Zygmund strong laws Our results extend the results of Gut and Stadtmüller(2009),Quang and Huan(2009),Dung and Tien(2010) and some other ones

© 2012 Elsevier B.V All rights reserved

1 Introduction

The concept of complete convergence for sums of independent and identically distributed random variables was introduced byHsu and Robbins(1947).Li et al.(1992) andChen et al.(2006) investigated the rate of complete convergence

for partial sums for moving average sequences of random variables taking values in Banach spaces of type p.

Many authors have investigated the Marcinkiewcz–Zygmund strong laws of large numbers for fields{Xn,n ∈ Nd}of random variables For example,Fazekas and Tómács(1998) proved that|n|−1/r Sn→0 a.s (for some 0<r<1) for fields

of pairwise independent random variables, andCzerebak-Mrozowicz et al.(2002) showed that|n|−1/r(Sn−ESn) →0 a.s (for some 1 < r < 2) for fields of pairwise independent random variables Recently,Gut and Stadtmüller(2009) have studied an asymmetric Marcinkiewicz–Zygmund law of large numbers for random fields of i.i.d random variables,Quang and Huan(2009) andDung and Tien(2010) studied Marcinkiewicz–Zygmund strong laws of large numbers for fields of

random variables taking values in p-uniformly smooth Banach spaces.

∗Corresponding author.

E-mail addresses:congson82@gmail.com (T.C Son), hungthang.dang53@gmail.com (D.H Thang), lvdunght@gmail.com (L.V Dung).

0167-7152/$ – see front matter © 2012 Elsevier B.V All rights reserved.

Let Zd be the integer d-dimensional lattice points, where d is a positive integer Consider a random field of martingale

differences{Xn,Fn;n∈Zd}defined on a probability space(Ω,F,P)taking values in a p-uniformly smooth Banach space

E(1≤p≤2)with norm∥ · ∥ In this paper, we study rate of complete convergence for maximum of moving average sums

of martingale difference fields taking values in p-uniformly smooth Banach spaces Namely, let{an;n∈Zd}be an absolutely

summable field of real numbers and set Tk= 

i∈Zd aiXi+k,k≽1 and Sn= n

k=1Tk for n≽1 and we try to find conditions

to ensure that



n≽1

|n|−1P{max

1≼k≼n

∥Sk∥ > ϵbn} for everyϵ >0,

where{bn,n≽1}is some field of positive constants As a consequence, we establish Marcinkiewicz–Zygmund strong law

of large numbers (SLLN)

1

bn 1max≼k≼n

Throughout this paper, the symbol C will denote a generic constant (0<C < ∞) which is not necessarily the same one

in each appearance

2 Preliminaries and some useful lemmas

Throughout this paper, we consider E as a real separable Banach space For a E-valued random variable X and subσ -algebraGofF, the conditional expectation E(X|G)is defined analogously to that in the random variable case and enjoys similar properties (the reader may refer toScalora, 1961)

E is said to be p-uniformly smooth (1 ≤ p ≤ 2) if there exists a finite positive constant C such that for all E-valued

martingales{S n;n≥1}

sup

n≥1

E∥S n∥p≤C

∞



n=1

E∥S n−S n−1∥p.

Clearly every real separable Banach space is 1-uniformly smooth and the real line (the same as any Hilbert space) is

2-uniformly smooth If a real separable Banach space is p-2-uniformly smooth for some 1<p≤2 then it is r-uniformly smooth for all r∈ [1,p)

Let d be a positive integer For m = (m1, ,m d),n = (n1, ,n d) ∈ Zd, α = (α1, , αd) ∈ Nd denote

2 n = (2n1, ,2n d), |nα| = d

i=1nαi

i , 1 = (1, ,1) ∈ Ndand d

i=1(m i < n i)means that there is at least one of

m1 < n1,m2 < n2, ,m d < n dholds We write m ≼ n (or n ≽ m) if m i ≤ n i,1 ≤ i ≤ d; m ≺ n if m ≼ n and

m̸=n.

Let{Xn,n∈Z d}be a field of E-valued random variables and{Fn,n∈Z d}be a field of nondecreasing sub-σ-algebras of

F with respect to the partial order≼on Z d Assume that Xnis an adapted field with respect toFnin the sense that XnisFn measurable for all n∈Z d

The notation of martingale difference double array is introduced byQuang and Huan(2009) Naturally, the notation of martingale difference field is defined as follows

The field{Xn,Fn,n∈Z d}is said to be a martingale difference field if E(Xn|F∗

n) =0 for all n∈ Z d, whereF∗

n = σ {Fl :

∨d i=1(l i<n i)}

To prove the main result we need the following lemmas

Lemma 2.1 Let E be a real separable p-uniformly smooth Banach space for some 1≤p≤2, and{X k,1≼k≼n}be a field of

E-random variables We have

E max

1≼k≼n









1≼i≼k

(Xi−E(Xi|Fi∗))







p

1≼k≼n

E∥Xk∥p.

Proof The proof is completely similar to that of Lemma 1.1 inQuang and Huan(2009) by replacing S kl= k

i=1

l

j=1X ijby

Sn= 

1≼i≼k(Xi−E(Xi|Fi∗)) 

Remark If{Xn;n≽1}is a E-valued martingale difference field and d=2, we have Lemma 1.1 inQuang and Huan(2009), and note that{Xn−E(Xn|F∗

n);n≽1}is not a E-valued martingale difference field

Lemma 2.2 Let 1< p≤2, α1, , αd be positive constants, q be the number of integers s such thatαs= min{ α1, , αd}

and let X be a E-valued random variable Set r= α1, ,α .

(i) If r<p and E∥X∥r(log+∥X∥ )q−1

< ∞, then

(1) 

n

P(∥X∥ ≥ |nα| ) < ∞,

(2) 

n≽1

1

|nα|p

 |nα|p 0

P{∥X∥p≥t}dt < ∞.

(ii) If r>1 and E∥X∥r(log+∥X∥ )q−1

< ∞, then



n≽1

1

|nα|

|nα|

P{∥X∥ ≥t}dt < ∞.

(iii) If r =1 and E∥X∥ (log+∥X∥ )q

< ∞, then



n≽1

1

|nα|

|nα|

P{∥X∥ ≥t}dt < ∞.

Proof (ii) and (i.2) are proved byDung and Tien(2010) in Lemma 2.2, and (i.1) is implied from (i.2) by

1

|nα|p

 |nα|p

0

P{∥X∥p≥t}dt≥P(∥X∥ ≥ |nα| ).

Now, the proof is similar to that of Lemma 2.2 inDung and Tien(2010), and we get



n≽1

1

|nα|

|nα|

P{∥X∥ ≥t}dt ≤C

∞



j=1

jP{j≤ ∥X∥ <j+1} (log j)q−1

j



k=1

1

k

+C

∞



j=1

P{j≤ ∥X∥ <j+1}j(log j)q−1

∞



j=1

P{j≤ ∥X∥ <j+1}j(log j)q< ∞

which implies that (iii) holds 

Lemma 2.3 Let{bn,n ≽ 1}be a field of positive constants such that bn ≤ bmfor all n ≼ m and supn≽1b2n+1

b2n < ∞ Let

{Xn,n≽1}be a field of E-valued random variables, and set Sn= 

1≼k≼nXk Then



n≽1

1

|n|P{max

1≼k≼n

if and only if



n≽1

P



max

1≼k≼2 n∥Sk∥ > ϵb2 n



Moreover,(2.1)implies that the SLLN

1

bn 1max≼k≼n

∥Sk∥ →0 a.s as|n| → ∞

holds.

Proof The proof is similar to that of Theorem 3.1 inDung and Tien(2010) and so is omitted 

Let{bn;n≽1}be an array of positive numbers We define

N(x) =Card{n:bn≤x} ,

and suppose that N(x) < ∞, ∀x>0

Now we define two other functions L(x)and R p(x)which are little different than those ofSu and Tong(2004):

L(x) =  x

0

N(t)logd+−1N(t)

x

N(t)logd+−1N(t)

for x>0 and p>0 We have following lemma

Lemma 2.4 Let{bn;n≽1}be a field of positive numbers satisfying for each n≽1,bn ≤bmfor all n≼m and bn→ ∞as

|n| → ∞ Let X be a E-valued random variable.

(i) If E∥X∥L(∥X∥ ) < ∞then



n≽1

1

bn

bn

(ii) If E∥X∥p R p(∥X∥ ) < ∞for some p>0 then



n≽1

and



n≽1

1

b pn

 bn

0

Proof First, we prove (i) Suppose that E∥X∥L(∥X∥ ) < ∞, we have by Lemma 3 ofStadtmüller and Thalmaier(2009) that

1≤n1 , ,n d≤j

1 ∼ C j(log j)d−1 (d−1)! as j→ ∞ .

Denote1g(k) =g(k) −g(k−1)and note that N(x)is non-decreasing Letting s=bnt, we have



1≼n≼k

1

bn

bn

P(∥X∥ >s)ds= 

1≼n≼k

1

P ∥X∥

t >bn



dt=

1



1≼n≼k

P



∥X∥

t >bn



dt

≤

1



1≼n≼k

P



N ∥X∥ t



≥N(bn)



dt≤

1



1≼n≼k

P



N ∥X∥ t



≥ |n|



dt

≤

1

∞



k=1

1g(k)P



N ∥X∥ t



≥k



dt

=

1



j=1

P



j≤N ∥X∥

t



<j+1



k=1

1g(k)



dt

1



j=1

j log d−1(j)P



j≤N ∥X∥

t



<j+1

 

dt

1

EN ∥X∥ t



logd+−1N ∥X∥

t



dt

0

1

Nx t



logd+−1Nx

t



dt



dP(∥X∥ ≤x)

0

x



 x

0

N(y)logd+−1N(y)



dP(∥X∥ ≤x) =CE∥X∥L(∥X∥ ) < ∞,

for all k∈Nd, and then we obtain(2.3)

To prove (ii), we have



n≽1

P(∥X∥ >bn) ≤ 

n≽1

P(N(∥X∥ ) ≥ |n| ) =

∞



k=1

1g(k)P(N(∥X∥ ) ≥k)

∞



k=1

k log d−1(k)P(k≤N(∥X∥ ) <k+1)

≤CE(N(∥X∥ )logd−1N(∥X∥ )) =C·p·E



∥X∥p

 +∞

∥X∥

N(∥X∥ )logd+−1N(∥X∥ )



≤CE∥X∥p R p(∥X∥ ) < ∞.

Finally, we easily prove(3.5)by using the method of the proof similar to that of(2.3) 

The field of E-valued random variables{Xn,n∈Nd}is said to be stochastically dominated by the E-valued random variable

X if, for some 0<C< ∞,

1

|n|



k≺n

P{∥Xk∥ ≥x} ≤CP{∥X∥ ≥x}

for all n∈Nd and x>0

3 Main results

Let{Xn,Fn;n ∈ Zd}be a field of E-valued martingale differences Throughout the paper,{an;n ∈ Zd}is always an

absolutely summable field of real numbers such that Tk= 

i∈ Zd aiXi+kconverges a.s and put

1≼k≼n

Tk.

The following theorem characterizes the p-uniformly smooth Banach spaces.

Theorem 3.1 Let 1≤p≤2, and let E be a separable Banach space, then the following two statements are equivalent (i) E is p-uniformly smooth.

(ii) For every E-valued martingale difference field{Xn,Fn;n ∈ Zd}, for every absolutely summable field of real numbers

{an;n∈Zd}such that Tk = 

i∈ Zd aiXi+kconverges a.s for all k≽1 and for every field of positive constants{bn,n≽1}

such that bn≤bmfor all n≼m and

1< inf

1≼n≺m

b2 m

b2 n

≤sup

n≽1

b2 n+1

If



n∈Zd

E∥Xn∥p· ϕ(n) < ∞,

whereϕ(n) = k≽1

an−k

b pk , then



n≽1

1

|n|P{max

k≼n

for everyϵ >0.

In addition, the SLLN

1

bn 1max≼k≼n

holds.

Proof We note that Sn= 

1≼k≼nTk= 

i∈Zdai

i+1≼j≼i+nXj ByLemma 2.3, in order to prove[ (i) ⇒ (ii)]we show that



n≽1

P{ max

1≼k≼2 n∥Sk∥ > ϵb2 n} < ∞ for everyϵ >0.

Applying the Markov inequality, the Holder inequality andLemma 2.1we have that



n≽1

P{ max

1≼k≼2 n∥Sk∥ > ϵb2 n} ≤ 

n≽1

1

ϵp b p2 n

E( max

1≼k≺2 n∥Sk∥p) ≤ 

n≽1

1

ϵp b p2 n

E





i∈ Zd

|ai| max

1≼k≼2 n









i+1≼j≼i+k

Xj







p

n≽1

1

ϵp b p2 n





i∈Zd

|ai|

p−1



i∈Zd

|ai|E max

1≼k≺2 n









i+1≼j≼i+k

Xj







p

n≽1

1

ϵp b p2 n





i∈ Zd

|ai| 

i+1≼j≼i+2 n

E∥Xj∥p

 (byLemma 2.1)

n≽1

1

ϵp b p2 n





d

|ai| 

i+1≼j≼i+2 n

E∥Xj∥p



k≽1



d

|ai|E∥Xi+k∥p 

n:k≼2 n

1

b p2 n

i∈Zd

|ai| 

k≼1

E∥Xi+k∥p

b pk (by(3.1))

j∈Zd

E∥Xj∥p

i≼j−1

|ai|

b pj−i =C



j∈Zd

E∥X∥p· ϕ(j) < ∞.

We have(3.2), then(3.3)is implied by (3.2)and Lemma 2.3 Now we prove[ (ii) ⇒ (i)] Assume that (ii) holds Let

{Y n1,Gn1;n1≥1}be an arbitrary sequence of martingale difference taking values in E such that

∞



n1 =1

E∥Y n1∥p

n p1 < ∞.

For n= (n1, ,n d) ∈Z d, set

Xn=Y n1 if n1≥1, n2= · · · =n d=1 otherwise Xn=0 andFn= σ{Xi;i≼n}

Then{X n,Fn;n ∈ Z d}is the martingale difference field taking values in E Let bn = |n| ,ai = 0 if i ̸= 0 and a0 = 1 so

Sn= n1

i=1Y iand



n≽1

E∥Xn∥pϕ(n) =

∞



n1 =1

E∥Y n1∥p

n p1 < ∞.

By (ii),

1

n1, ,n d

n1



i1 =1

X i1 →0 a.s as|n| → ∞

Taking n2= · · · =n d=1 and letting n1→ ∞we obtain

1

n1

n1



j=1

W j→0 a.s as n1→ ∞

Then by Theorem 2.2 ofHoffmann-Jørgensen and Pisier(1976), E is p-uniformly smooth. 

The next theorem provides an another sufficient condition for(3.2)to hold in terms of functions L(x)and R p(x)

Theorem 3.2 Let{Xn,Fn;n ∈ Zd}be a martingale difference field taking values in a separable real p-uniformly smooth E Let {bn,n ≽ 1}be a field of positive constants such that bn ≤ bmfor all n ≼ m, and bn → +∞as|n| → +∞ Set

N(x) =card{n;bn≤x}for all x>0.

If {Xn;n∈Z d}is stochastically dominated by a E-valued random variable X such that

E∥X∥p R p(∥X∥ ) < ∞, E∥X∥L(∥X∥ ) < ∞,

then



n≽1

1

|n|P{max

k≼n

In addition, if supn≽1b2n+1

b2n < ∞, then the SLLN

1

bn 1max≼k≼n

holds.

Proof For 1≼i≼n we set

Yni=XiI{∥ Xi∥ >bn}−E(XiI{∥ Xi∥ >bn}|Fi∗), Zni=XiI{∥ Xi∥≤bn}−E(XiI{∥ Xi∥≤bn}|Fi∗).

It is clear that Xn=Yni+Zni for all 1≼i≼n.

For an arbitraryϵ >0, by using the Chebyshev inequality,Lemmas 2.1and2.4we get

n≽1

1

|n|P



max

1≼k≼n









i∈ Zd

i+1≼j≼i+k

Yni







≥ ϵbn



n≽1

1

|n|bnϵ



i∈ Zd

|ai|E max

1≼k≼n









i+1≼j≼i+k

Xj







n≽1

1

bnϵ



d

|ai|E∥X∥I{∥ X∥≥bn}≤C

n≽1

1

bn

bn

P(∥X∥ >s)ds+C

n≽1

P(∥X∥ >bn) < ∞.

Again using the Chebyshev inequality, the Holder inequality,Lemmas 2.1and2.4we get

n≽1

1

|n|P



max

1≼k≼n









i∈Zd

i+1≼j≼i+k

Zni







≥ ϵbn



n≽1

1

|n|b pnϵp





i∈Zd

|ai|

p−1



i∈Zd

|ai|E max

1≼k≼n









i+1≼j≼i+k

Zni







p

n≽1

1

|n|b pnϵp





i∈ Zd

|ai|

p−1



i∈ Zd

|ai|





i+1≼j≼i+n

E∥Xn∥p I{∥ X i∥≤bn}



n≽1

1

b pnE∥X∥

p I{∥ X∥≤bn}≤C

n≽1

1

b pn

bn

s p−1P(∥X∥ >s)ds< ∞.

Hence, the conclusion(3.4)holds from

P(max

1≼i≼n

∥Si∥ ≥2ϵbn) ≤P



max

1≼k≼n









i∈Zd

i+1≼j≼i+k

Yni







≥ ϵbn



+P



max

1≼k≼n









i∈Zd

i+1≼j≼i+k

Zni







≥ ϵbn



< ∞.

The conclusion(3.5)follows from(3.4)andLemma 2.3 

Finally, in the case bn= |nα|, we obtain some new sufficient conditions for(3.2)to hold.

Theorem 3.3 Let{Xn,Fn;n∈Zd}be a martingale difference field taking values in a separable real p-uniformly smooth E with

1<p≤2 Letα1, , αd be positive constants satisfying 1/p<min{ α1, , αd} <1, let q be the number of integers s such

thatαs=min{ α1, , αd} If {Xn;n∈Zd}is stochastically dominated by a E-random variable X such that

E(∥X∥rlogq−1∥X∥ ) < ∞ with r = 1

min{ α1, , αd} ,

then



n≽1

1

|n|P{max

1≼k≺n

and the SLLN

1

holds.

Proof For 1≼i≼n, we set

Yni=XiI{∥ Xi∥ >|nα|}−E(XiI{∥ Xi∥ >|nα|}|Fi∗), Zni=XiI{∥ Xi∥≤|nα|}−E(XiI{∥ Xi∥≤|nα|}|Fi∗).

It is clear that Xn=Yni+Zni for all 1≼i≼n We have that

P(max

i≼n

∥Si∥ ≥2ϵ|nα| ) ≤P



max

k≼n









i∈ Zd

i+1≼j≼i+k

Yni







≥ ϵ|nα|



+P



max

k≼n









i∈ Zd

i+1≼j≼i+k

Zni







≥ ϵ|nα|



Then, in order to prove(3.4)it is enough to prove that

n≽1

1

|n|P



max

1≼k≼n









i∈ Zd

i+1≼j≼i+k

Yni







≥ ϵ|nα|



< ∞,

n≽1

1

|n|P



max

1≼k≼n









i∈Zd

i+1≼j≼i+k

Zni







≥ ϵ|nα|



< ∞.

For A by the same argument as in the proof ofTheorem 3.2and byLemma 2.2, we have

A≤CP{∥X∥ ≥ |nα|} +C 1

|nα|

|nα|

P{∥X∥ ≥t}dt< ∞.

For B by the same argument as in the proof ofTheorem 3.2and byLemma 2.2, we have

n≽1

1

|nα|p

 |nα|p

0

P{∥X∥p≥t}dt< ∞.

The proof is completed 

Remark Note that when ai=0 if i̸=0 and a0=1 so Sn= 

1≼i≼nXnthenTheorem 3.3is an extension of Theorem 3.3

inDung and Tien(2010) and Theorem 2.1 ofGut and Stadtmüller(2009)

Theorem 3.4 Let{Xn,Fn;n∈Zd}be a martingale difference field in a separable real p-uniformly smooth E with 1<p≤2.

Letα1, , αd be positive constants satisfying min{ α1, , αd} = 1, let q be the number of integers s such that αs = 1 =

min{ α1, , αd} If {Xn;n ∈ Zd}is stochastically dominated by a E-random variable X such that E(∥X∥logq∥X∥ ) < ∞ Then(3.6)and the SLLN(3.7)hold.

Proof The proof is similar to that ofTheorem 3.3and using (i) and (iii) ofLemma 2.2 

Remark FromTheorems 3.3and 3.5 with ai= 0 if i̸= 0 and a0 =1,d=2 andα1 = α2 =1we obtain Theorem 2.4 in

Quang and Huan(2009)

Acknowledgements

The authors would like to express their gratitude to the referee for his/her detailed comments and valuable suggestions which helped them to improve the manuscript The research of the third author (grant no 10103-2012.07) and the second author has been partially supported by Vietnam’s National Foundation for Science and Technology Development (NAFOSTED)

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