DSpace at VNU: The Brunk–Prokhorov strong law of large numbers for fields of martingale differences taking values in a Banach space

Contents lists available atSciVerse ScienceDirectStatistics and Probability Letters journal homepage:www.elsevier.com/locate/stapro The Brunk–Prokhorov strong law of large numbers for fi

Contents lists available atSciVerse ScienceDirect

Statistics and Probability Letters

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

The Brunk–Prokhorov strong law of large numbers for fields

of martingale differences taking values in a Banach space

Ta Cong Son∗, Dang Hung Thang

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

a r t i c l e i n f o

Article history:

Received 19 November 2012

Received in revised form 16 April 2013

Accepted 21 April 2013

Available online 2 May 2013

MSC:

60B11

60B12

60F15

60F25

60G42

Keywords:

p-uniformly smooth Banach spaces

Field of martingale differences

Strong law of large numbers

a b s t r a c t

In this paper, we define a new type of fields of martingale differences taking values in Banach spaces and establish the Brunk–Prokhorov strong laws of large numbers and the convergence rate in the strong laws of large numbers for such fields

© 2013 Elsevier B.V All rights reserved

1 Introduction and preliminaries

Let q ≥ 1 and{X n;n ≥ 1} be a sequence of independent random variables The Brunk–Prokhorov strong law of large numbers (Brunk–Prokhorov SLLN) (seeBrunk,1948;Prokhorov,1950) stated that if EX n = 0, for all n ≥ 1 and

∞

n= 1E|X n|2q/n q+1< ∞,then

lim

n→∞

1

n

n



k= 1

X k=0 a.s

Brunk–Prokhorov SLLN was extended to martingale differences e.g inFazekas and Klesov(2000) andHu et al.(2008) For the field of random variables with multidimensional index,Lagodowski(2009) established the Brunk–Prokhorov SLLN for fields of independent E-valued random variables andNoszaly and Tomacs(2000) proved the Brunk–Prokhorov SLLN for fields of real-valued martingale differences

In this paper, we introduce a new type of fields of E-valued martingale differences and establish the Brunk–Prokhorov SLLN for such fields In Section1, a new type of fields of E-valued martingale differences is defined, illustrated by some non-trivial examples and compared with the usual definition In Section2we prove some useful lemmas and inequalities Section3contains the main results including the Brunk–Prokhorov SLLN for such fields of E-valued martingale differences

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

in each appearance

∗Corresponding author Tel.: +84 1989318669.

E-mail addresses:congson82@gmail.com , congson82@hus.edu.com.vn (T.C Son), hungthang.dang53@gmail.com (D.H Thang).

0167-7152/$ – see front matter © 2013 Elsevier B.V All rights reserved.

Let E be a real separable Banach space.(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;1≤n≤m}

E∥S m∥p≤C

m



n= 1

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

2-uniformly smooth and the space L p(1≤p≤2)is of p-uniformly smooth If a real separable Banach space of p-uniformly

smooth(1<p≤2)then it is of r-uniformly smooth for all r∈ [1,p)

Using classical methods from martingale theory, it was shown that (seeWoyczyn’ski, 1978) if E is of p-uniformly smooth,

then for all 1≤q< ∞there exists a finite constant C such that

E∥S m∥q≤CE

 m



i= 1

∥S i−S i− 1∥p

q

Let d be a positive integer For m= (m1, ,m d),n= (n1, ,n d) ∈Nd, denote m+n= (m1+n1, ,m d+n d),m−n= (m1−n1, ,m d−n d), |n| =n1.n2 .n d, ∥n∥ =min{n1, ,n d} ,1= (1, ,1) ∈Nd, 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; m≪n (or n≫m) ifd

i= 1(m i<n i).

Let(Ω,F,P)be a probability space, E be a read separable Banach space, andB(E)be theσ-algebra of all Borel sets in E

Definition 1 Let{Xn,1≼n≼N}be a field of E-valued random variables and{Fn,1≼n≼N}be a field of nondecreasing sub-σ-algebras ofF with respect to the partial order≼on Nd

1 The field{Xn,Fn,1≼n≼N}is said to be an adapted field if XnisFn -measurable for all 1≼n≼N.

2 The adapted field{Xn,Fn,1≼n≼N}is said to be a field of martingale differences in the usual sense if

and

(SeeChristofides and Serfling, 1990;Lagodowski,2009)

3 The adapted field{Xn,Fn,1≼n≼N}is said to be a field of martingale differences if

whereF∗

n = σ{Fl: ∨d

i= 1(l i<n i)},for 1≼n≼N (seeSon et al., 2012)

Remark For a field of martingale differences the condition(1)about{Fn,1≼n≼N}is not required but the condition(3) seems to be stronger than the condition(2)

Example 1 Let{Xn,1 ≼n ≼N}be a field of independent random variables with mean 0 PutFn = σ (Xk,k≼n), then

E(Xn|F∗

n) =0 and 1≼n≼N Therefore,{Xn,Fn,1≼n≼N}is a field of martingale differences

Example 2 Let{X n,Gn:n≥1}is a sequences of martingale differences, set

Xn=X n if n= (n,n, ,n) and Xn=0 if n ̸= (n,n, ,n);

Gn=Gn if n= (n,n, ,n) and Gn= {∅ ,Ω} if n ̸= (n,n, ,n).

LetFn = σ {Gk,k ≼ n}for all n ≽ 1,then{Xn,Fn : n ≽ 1}is a field of martingale differences, but it is not a field of independent random variables

Example 3 Let{Xn,1 ≼n ≼N}be a field of independent random variables with mean 0 PutFn = σ(Xk,k ≼ n)and

Yn = 

k≼n Xk, if EYn < ∞for all n≼N, then E(Yn|Fn∗) = 0 and 1≼n≼N Therefore,{Xn,Fn,1≼n≼N}is a field of martingale differences, but it is not a field of independent random variables At the same time, it is not a field of martingale differences in the usual sense since the condition(1)does not hold for{Fn,1≼n≼N}

Definition 2 Let{an,n∈Nd}be a field of elements in E

1 We say that an→a as n→ ∞if for anyϵ >0 there exists nϵ∈Ndsuch that for all n≽nϵthen∥an−a∥ < ϵ.

2 We say that an→a strongly as n→ ∞if for anyϵ >0 there exists nϵ∈Ndsuch that for all n̸≼nϵthen∥an−a∥ < ϵ

Clearly, an → a strongly as n → ∞then an → a as n → ∞, but the converse is not true For example, let

a̸=b,a(n1 , 1 , 1 )=b and an=a if otherwise, then an→a but an̸→strongly as n→ ∞

It is easy to see that in the case d=1, the strong convergence and the convergence are equivalent

2 Some useful lemmas and inequalities

Lemma 2.1 Let 1≤p≤2,q≥1 and E be a real separable p-uniformly smooth Banach space Then there exists positive C such that for each field of E-valued martingale differences{Xk,Fk;1≼k≼N}, we have

E









1≼k≼N

Xk







q

≤CE





1≼k≼N

∥Xk∥p

q/p

Proof For the field of E-valued martingale differences{Xk,Fk;1≼k≼N}, we define a sequence{Y k,Gk;1≤k≤ |N|}by putting

Y1=X( 1 , , 1 , 1 ); Y2=X( 1 , , 1 , 2 ); ; Y N d =X( 1 , , 1 ,N d);

Y N d+ 1=X( 1 , , 2 , 1 ); Y N d+ 1=X( 1 , , 2 , 2 ); ; Y|N|=XN,

and for all 1≤k≤ |N|, letGk= σ{Y i;1≤i≤k} Since

E(Y k|Gk− 1) =E(E(Xi|Fi∗)|Gk− 1) =0,

(where Y k=Xi) so{Y k;Gk;1≤k≤ |N|}is a sequence of E-valued martingale differences From the inequality(1.2)we get

Lemma 2.2 Let q≥1 and E be a real separable Banach space For each field of E-valued martingale differences{Xk,Fk;1≼

k≼N}, putFn(s)= σ{Fk:k= (k1, ,k d),0≤k i≤N i(i̸=s), and k s=n s}for all 1≤s≤d and Sn= 

k≼nXk Then we have

where

S(n,s,α)=S(n1 , ,n s− 1 ,α,n s+ 1 , ,n d).

Proof We have S(n,s,n s+ 1 )−Sn= 

(i,s, 1 )≼nX(i,s,n s+ 1 ), andFn(s)⊂F(∗i,s,n s+1)for all(i,s,1) ≼n= (n1, ,n s, ,n d), then

E(S(n,s,n s+ 1 )−Sn|Fn(s)) =E





(i,s, 1 )≼n

E(X(i,s,n s+ 1 )|F(∗i,s,n s+1))  Fn(s)



=0,

so

E(∥S(n,s,n s+ 1 )∥q|Fn(s)) ≥ ∥E(S(n,s,n s+ 1 )|Fn(s))∥q= ∥Sn∥q. 

Lemma 2.3 Let{Sn;n≽1}be a field of E-valued random variables Then,

1 Sn→0 strongly a.s as|n| → ∞if only if for allε >0,

lim

∥n∥→∞P



sup

k̸≼n

∥Sk∥ > ε



2 Sn→0 a.s as|n| → ∞if only if for allε >0,

lim

∥n∥→∞P



sup

k≽n

∥Sk∥ > ε



Proof 1 Necessary Suppose that Sn→0 a.s as|n| → ∞ For eachϵ >0, let Aϵ

supk̸≼n∥Sk∥ > ϵand Aϵ = 

n≽1Aϵ

n,

then P(Aϵ) =0,n= (n1,n2, ,n d) = (n1,n1),by the continuity from below theorem, so

0=P(Aϵ) =P





n≽1

Aϵ

n



= lim

n1 →∞P





n 1≽1

Aϵ

n



= · · · = lim

∥n∥→∞P(Aϵ

n),

we have(2.3)

Sufficient Suppose(2.3)holds, for all n∈Nd and i∈N, we let

A in=



sup

k̸≼n

∥Sk∥ ≥ 1

i



and A=  A in.

by the continuity from below theorem, then

P(A) =P





i≥ 1



n≽1

A in



= lim

i→∞ lim

∥n∥→∞P(A in) = lim

i→∞0=0,

i.e Sn→0 a.s as|n| → ∞

2 The prove of(2)is similar to the one of(1) 

Theorem 2.4 Let E be a Banach space,{Xn,Fn;1≼n≼N}be a field of E-valued martingale differences and{bn,1≼n≼N}

be a field of positive constants such that bn≤bmfor all n≼m Then for allε >0,q≥1 we have

εP



max

1≼n≼N

∥Sn∥q

b qn ≥ ϵ



≤ min

1 ≤s≤d





1≼n≼N−1



1

b qn−

1

b q(n,s,n s+ 1 )



E∥Sn∥q+ 

N−1≺n≼N

E∥Sn∥q

b qn



where b(n,s,α)=b(n1 , ,n s− 1 ,α,n s+ 1 , ,n d).

Proof Let ‘‘<’’ be the lexicographic order on Nd, i.e., n= (n1, ,n d) <m= (m1, ,m d)if and only if either n1<m1or there exists 1≤k<d such that n i=m ifor all 1<i≤k and n k<m k.Define the sets A,Anby, A=



max1≼n≼N

∥Sn∥q

b qn

≥ ϵ ,

An=  ∥Sk∥q

b qk < ϵif k<n,and ∥Sn∥

q

b qn ≥ ϵ

 ,

then

1≼n≼NAn=A,An

Am= ∅for all i̸=j and Ak ∈Fn(1)for all k≼n ByLemma 2.2so

E∥S(n,1,n

1 + 1 )∥q I Ak ≥E∥Sn∥q I Ak for all k≼n.

Set n= (n1,n2, ,n d) = (n1,n′)where n′= (n2, ,n d), we have

ϵP(A) = ϵP





n≼N

An



= ϵ 

n≼N

P(An) = 

n≼N

E(ϵI An) ≤ 

n≼N

E ∥Sn∥

q

b qn I An



n′≼N′



k≤N1



E∥S(k,n′)∥q

b q(k,n′)

I

i≤k A( i,n′ )−E∥S(k,n′)∥q

b q(k,n′)

I 

i≤k− 1A( i,n′ )



n′≼N′





k≤N1



1

b q(k,n′)

b q(k+ 1 ,n′)



E∥S(k,n′)∥q+E∥S(N1 ,n′)∥q

b q(k,n′)

I

k≤N1 A( k,n′ )



1≼n≼N



1

b qn −

1

b q(n, 1 ,n1 + 1 )



E∥Sn∥q+ 

N−1≺n≼N

E∥Sn∥q

b qn .

Similarly, it can be shown that:

ϵP(A) ≤ 

1≼n≼N−1



1

b qn −

1

b q(n,s,n s+ 1 )



E∥Sn∥q+ 

N−1≺n≼N

E∥Sn∥q

b qn

for all 2≤s≤d Then the proof is completed. 

Corollary 2.5 Let{bn,n ≽ 1}be a field of positive constants such that bn ≤ bmfor all n ≼ m,E be a Banach space and

{Xn,Fn;n≽1}be a field of E-valued martingale differences such that

lim inf

∥n∥→∞ max

n−1≺k≼n

E∥Sk∥q

Then for allε >0,q≥1, we have

εP



sup

n≽1

∥Sn∥q

b qn ≥ ϵ



≤ min

1 ≤s≤d





n≽1



1

b qn−

1

b q(n,s,n s+ 1 )



E∥Sn∥q



where b(n,s,α)=b(n1 , ,n s− 1 ,α,n s+ 1 , ,n d).

Proof By the condition (2.5), there exists field Nn ∈ Nd such that Nn ≺ Nn+1 for all n ∈ N, ∥Nn∥ → ∞ and limn→∞maxNn− 1 ≺k≼Nn E∥Sk∥

q

b qk =0.

By the continuity from below theorem andTheorem 2.4, we have

εP



sup

n≽1

∥Sn∥q

b qn ≥ ε



= lim

n→∞εP



sup

1≼n≼Nn

∥Sn∥q

b qn ≥ ε



≤ lim

n→∞ min

1 ≤s≤d





1≼n≼Nn−1



1

b qn−

1

b q(n,s,n s+ 1 )



E∥Sn∥q+ 

Nn− 1 ≺n≼Nn

E∥Sn∥q

b qn



= min

1 ≤s≤d





n≽1



1

b qn −

1

b q(n,s,n s+ 1 )



E∥Sn∥q

 

Remark If lim∥n∥→∞E∥Sn∥q

b qn

=0 then(2.5)holds

Corollary 2.6 Let 1≤p≤2,q≥1 and E be a real separable p-uniformly smooth Banach space Then there exists positive C such that for all field of E-martingale difference{Xk;1≼k≼N}, we have

εP



max

1≼n≼N

∥Sn∥q≥ ε



N−1≺n≼N

E∥Sn∥q≤CE





1≼n≼N

∥Xn∥p

q/p

.

Proof ByTheorem 2.4with bn=1 for all n≽1 andLemma 2.1, we have

εP



max

1≼n≼N

∥Sn∥q≥ ε



N−1≺n≼N

E∥Sn∥q

N−1≺n≼N

E





1≼k≼n

∥Xk∥p

q/p

≤2d·C·E





1≼n≼N

∥Xn∥p

q/p

. 

3 Main results

Theorem 3.1 Let q ≥ 1,E be a Banach space,{bn,n ≽ 1} and{cn,n ≽ 1} be a field positive constants such that

bn ≤ bmfor all n ≼ m, ξ(n),n ≽ 1 be a positive function,{Xn,Fn;n ≽ 1}be a field of E-valued martingale differences satisfying(2.5)and the following condition

E∥Sn∥q≤C.ξ(n) 

1≼k≼n

Then, the condition

min

1 ≤s≤d





n∈ Nd

cn.ψs(n)



whereψs(n) = k≽n

 1

b qk − 1

b q(k,s,ns+ 1 )

 ξ(n), implies

Sn

bn →0 strongly a.s as|n| → ∞

Proof ByLemma 2.3, we have to show that

lim

∥N∥→∞

P



sup∥Sn∥



=0 for anyϵ >0,

where N= (N1, ,N k, ,N d) Put N(k)= (1, ,N k, ,1) ByCorollary 2.5and the condition(3.1)we have

ϵq P



sup

n̸≼N

∥Sn∥

bn

≥ ϵ



= ϵq P

 d



k= 1



sup

n≽N( k)

∥Sn∥

bn

≥ ϵ



≤ ϵq

d



k= 1

P



sup

n≽N( k)

∥Sn∥

bn

≥ ϵ



≤ d min

1 ≤s≤d



n̸≼N



1

b qn−

1

b q(n,s,n s+ 1 )



E∥Sn∥q

≤ C· min

1 ≤s≤d



n̸≼N



1

b qn −

1

b q(n,s,n s+ 1 )

 ξ(n) 

1≼k≼n

ck.

By(3.2), we have

min

1 ≤s≤d



n≽1



1

b qn −

1

b q(n,s,n s+ 1 )

 ξ(n) 

1≼k≼n

ck

= min

1 ≤s≤d



k≽1

ck

n≽k



1

b qn−

1

b q(n,s,n s+ 1 )

 ξ(n) = min

1 ≤s≤d





n∈ Nd

cn· ψs(n)



< ∞.

Hence

lim

∥N∥→∞C· min

1 ≤s≤d



n̸≼N



1

b qn −

1

b q(n,s,n s+ 1 )

 ξ(n) 

1≼k≼n

ck=0 and we are done 

The following theorem gives a characterization of p-uniformly smooth Banach spaces.

Theorem 3.2 Let 1≤p≤2 and E be a separable Banach space Then the following statements are equivalent:

(i) E is of p-uniformly smooth.

(ii) Let{Xn,Fn;n∈Nd}be a field of E-valued martingale differences, q≥1,d≥2, {bn,n≽1}be a field positive constants such that bn≤bmfor all n≼m If

1

lim inf

∥n∥→∞ max

n−1≺k≼n

E∥Sk∥pq

2

min

1 ≤s≤d





n∈ Nd

E∥Xn∥pq.ϕs(n)



whereϕs(n) = k≽n

 1

b pqk − 1

b pq(k,s,ns+ 1 )



|k|q− 1; then

Sn

Proof ((i)⇒(ii)) ByLemma 2.1and C r-inequality, we have

E∥Sn∥pq≤CE





k≼n

∥Xk∥p

q

≤C|n|q−1

k≼n

E∥Xk∥pq

ApplyingTheorem 3.1withξ(n) = |n|q− 1,cn=E∥Xn∥pq, we have (ii)

Now we prove [(ii)⇒(i)] Let{Y n1,Gn1;n1≥1}be an arbitrary sequence of E-valued martingale differences such that

∞

E∥Y n1∥p

n p1 < ∞.

For n= (n1, ,n d) ∈Nd Set

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

Then{Xn,Fn;n ∈ Nd} is the field of E-valued martingale differences Let bn = |n| ,q = 1, so Sn = n1

i= 1Y i,

ϕ1((n1,1, ,1)) = k1 ≥n

 1

k p − (k1 +11 )p



= 1

n p and

min

1 ≤s≤d





n≽1

E∥Xn∥pϕs(n)



n≽1

E∥Xn∥pϕ1(n) =

∞



n1 = 1

E∥Y n1∥p

n p1 < ∞.

Moreover,E∥Sn∥p

b pn

= E∥S n1∥

p

n p

1

n p n p, so lim inf∥n∥→∞maxn−1≺k≼nE∥Sk∥

p

|k|p =0 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

Y j→0 strongly a.s as n1→ ∞

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

Remark It should be noted that for d=1 [(i)→(ii)] but [(ii)̸→(i)]

Theorem 3.3 Let 1≤p≤2,q>1,E be a separable real p-uniformly smooth Banach space and{Xn,Fn;n∈Nd}be a field of E-valued martingale differences If

min

1 ≤s≤d



n≽1

E∥Xn∥pq

then

lim

|n|→ 0

Sn

|n| =0 strongly a.s.

Proof When bn= |n|, we have

ϕs(n) = 

k≽n



1

|k|pq− 1

| (k,s,n s+1)|pq



|k|q−1= 

k≽n

1

k s|k|pq−q+1

=





k s≥n s

1

k pq s−q+2



i= 1 ,i̸=s



k i≥n i

1

k pq s−q+1

n s|n|pq−q,

then(3.4)is satisfied Next, we have



n≽1

E∥Sn∥pq

n s∥n∥pq ≤ 

n≽1

1

n s∥n∥pq−q+ 1



k≼n

E∥Xk∥pq

n≽1

E∥Xn∥pq

k≽n

1

k s|k|pq−q+ 1

≤C

n≽1

E∥Xn∥pq

n s∥n∥pq−q < ∞, (by(3.6))

then(3.3)is satisfied, ByTheorem 3.2, we have

lim Sn

|n| =0 strongly a.s 

ApplyingTheorem 3.3for d=1 andTheorem 3.1inSon et al.(2012) for bn= |n| ,ai=0 if i̸=0 and a0=1,d=1, we obtain the following

Corollary 3.4 Let 1≤p≤2,q≥1,E be a p-uniformly smooth Banach space and{X n,Fn;n≥1}be a sequence of E-valued martingale differences such that



n≥ 1

E∥X n∥pq

n pq−q+ 1 < ∞,

then

lim

n→ 0

S n

n =0 a.s.

Since the Hilbert space is of 2-uniformly smooth, then we have

Corollary 3.5 Let q≥1,E be a Hilbert space,{X n,Fn;n≥1}be a sequence of E-valued martingale differences such that



n≥ 1

E∥X n∥2q

n 2q− 1 < ∞,

then

lim

n→ 0

S n

n =0 a.s.

The rate of the convergence of strong laws of large numbers is given in the following theorem

Theorem 3.6 Let 1 ≤p≤2,q≥1, α < 1,E be a real separable p-uniformly smooth Banach space,{bn,n≽1}be a field positive constants and{Xn,Fn;n≽1}be a field of E-valued martingale differences satisfying the condition



n≽1

whereφ(n) = k≽n

|k|p−α− 1

b pqk Then

1 If{bn;n≽1}satisfying bn≤bmfor all n≼m, then

P



sup

k≽n

∥Sk∥

bk > ϵ



=o



1

|n|1 − α



as|n| → ∞for everyϵ >0.

2 If{bn;n≽1}satisfying bn≤bmfor all n≪m then

P



sup

k̸≼n

∥Sk∥

bk > ϵ



=o



1

|n|1 − α



as|n| → ∞for everyϵ >0.

Proof 1 Firstly, we show that



n≽1

1

|n|αP



max

k≼n

∥Sk∥ > ϵbn



UsingCorollary 2.6and C r-inequality, we have



n≽1

1

|n|αP



max

k≼n

∥Sk∥ > ϵbn



≤C

n≽1

|n|−α

b pqn E





1≼k≼n

∥Xk∥p

q

≤C

n≽1

|n|− α+p− 1

b pqn



1≼k≼n

E∥Xk∥pq

=C

k≽1

E∥Xk∥pq

n≽k

|n|−α+p− 1

b pqn

n∈ Nd

E∥Xn∥q.φ(n) < ∞ (by(3.7)).

Then, we have(3.8)

Next, withϵ >0, set An= {k,2n−1≺k≼2n}.

If bn≤bm for all n≼m, we see



n≽1

n−αPsup

k≽n

b−k1∥Sk∥ > ϵ



i≽1

2 i−1



n=2 i−1

n−αPsup

k≽n

b−k1∥Sk∥ > ϵ



≤C

i≽1

2 i−1



n=2 i−1

|2−iα|P



sup

k≽ 2i− 1

b−k1∥Sk∥ > ϵ



≤C

i≽1

|2i( 1 − α)|Psup

u≽i

max

k∈Au

b−k1∥Sk∥ > ϵ



≤C

i≽1

|2i( 1 − α)| 

u≽i

P



b−2 u1−1max

k≼2 u∥Sk∥ > ϵ



≤C

u≽ 1

P



b−1

2 u−1max

k≼2 u∥Sk∥ > ϵ





i≼u

|2i( 1 − α)|

≤C

u≽ 1

|2u( 1 − α)|Pb− 1

2 u max

k≼2 u∥Sk∥ > ϵ

M



≤C

n≽ 1

|n|−αPb− 1

k≼n

∥Sk∥ > ϵ

M



< ∞ (by(3.8)).

Since{P

supk≽n b−k1∥Sk∥ > ϵ ,n∈Nd}are non-increasing in n= (n1, ,n d)for order relationship≼in Nd, it follows that

P



sup

k≽n

b−k1∥Sk∥ > ϵ



=o



1

|n|1 − α



as|n| → ∞for allϵ >0.

2 the prove is the same as in the case (1) 

For bn= ∥n∥ ,n≽1 then bn≤bm for all n≪m For bn= |n| ,n≽1 then bn≤bm for all n≼m Hence we obtain

Corollary 3.7 Let 1≤p≤2,q≥1, α <1 and E be a real separable p-uniformly smooth Banach space,{Xn,Fn;n≽1}be a field of E-valued martingale differences.

If



n∈ Nd

E∥Xn∥pq

|n|pq−q+ 1 < ∞

then

P



sup

k̸≼n

∥Sk∥

∥k∥ > ϵ



=o



1

|n|1 − α



as|n| → ∞for everyϵ >0

and

P



sup

k≽n

∥Sk∥

|k| > ϵ



=o



1

|n|1 − α



as|n| → ∞for everyϵ >0.

Acknowledgments

The authors would like to express their gratitude to the referee for his/her helpful comments and valuable suggestions which have significantly improved this paper

The authors express their sincere thanks to the Advanced Math Program of Ministry of Education and Training, Viet Nam for sponsoring their working visit to University of Washington and to the Department of Mathematics, University of Washington for the hospitality

This research was supported in part by Vietnam’s National Foundation for Science and Technology Development (NAFOSTED)

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