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

Statistics and Probability Letters 83 (2013) 1901–1910

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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

article

abstract

info

Article history:
Received 19 November 2012
Received in revised form 16 April 2013
Accepted 21 April 2013
Available online 2 May 2013

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.

MSC:
60B11

60B12
60F15
60F25
60G42
Keywords:
p-uniformly smooth Banach spaces
Field of martingale differences
Strong law of large numbers

1. Introduction and preliminaries
Let q ≥ 1 and {Xn ; n ≥ 1} be a sequence of independent random variables. The Brunk–Prokhorov strong law of
large
(Brunk–Prokhorov SLLN) (see Brunk, 1948; Prokhorov, 1950) stated that if EXn = 0, for all n ≥ 1 and
∞ numbers
2q
q +1
E
|
X
|
/
n
< ∞, then
n
n =1
lim

n→∞

n

1

n k =1

Xk = 0

a.s.

Brunk–Prokhorov SLLN was extended to martingale differences e.g. in Fazekas and Klesov (2000) and Hu 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 and Noszaly 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 Section 1, 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 Section 2 we prove some useful lemmas and inequalities.
Section 3 contains 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: , (T.C. Son), (D.H. Thang).

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

1902

T.C. Son, D.H. Thang / Statistics and Probability Letters 83 (2013) 1901–1910

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 {Sn ; 1 ≤ n ≤ m}
E ∥Sm ∥p ≤ C

m


E ∥Sn − Sn−1 ∥p .

(1.1)

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 Lp (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 (see Woyczyn’ski, 1978) if E is of p-uniformly smooth,
then for all 1 ≤ q < ∞ there exists a finite constant C such that


q

E ∥Sm ∥ ≤ CE

 pq

m


∥Si − Si−1 ∥

p

.

(1.2)

i =1

Let d be a positive integer. For m = (m1 , . . . , md ), n = (n1 , . . . , nd ) ∈ Nd , denote m + n = (m1 + n1 , . . . , md + nd ), m − n =

(m1 − n1 , . . . , md − nd ), |n| = n1 .n2 . . . nd , ∥n∥ = min{n1 , . . . , nd }, 1 = (1, . . . , 1) ∈ Nd , di=1 (mi < ni ) means that there
is at least one of m1 < n1 , m2 < n2 , . . . , md < nd holds. We write m ≼ n (or n ≽ m) if mi ≤ ni , 1 ≤ i ≤ d; m ≺ n if m ≼ n
d
and m ̸= n; m ≪ n (or n ≫ m) if i=1 (mi < ni ).
Let (Ω , F , P ) be a probability space, E be a read separable Banach space, and B (E) be the σ -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 of F with respect to the partial order ≼ on Nd .
1. The field {Xn , Fn , 1 ≼ n ≼ N} is said to be an adapted field if Xn is Fn -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
E (E (X |Fm )|Fn ) = E (X |Fm∧n ) for all X ∈ L1

(1)

E (Xn |Fn ) = 0

(2)

and

for all 1 ≼ n ≼ N

(See Christofides and Serfling, 1990; Lagodowski, 2009)
3. The adapted field {Xn , Fn , 1 ≼ n ≼ N} is said to be a field of martingale differences if
E (Xn |Fn∗ ) = 0

for all 1 ≼ n ≼ N

where Fn = σ {Fl : ∨
∗

d
i =1

(3)

(li < ni )}, for 1 ≼ n ≼ N (see Son 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. Put Fn = σ (Xk , k ≼ n), then
E (Xn |Fn∗ ) = 0 and 1 ≼ n ≼ N. Therefore, {Xn , Fn , 1 ≼ n ≼ N} is a field of martingale differences.
Example 2. Let {Xn , Gn : n ≥ 1} is a sequences of martingale differences, set
Xn = Xn

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).
Let Fn = 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. Put Fn = 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 any ϵ > 0 there exists nϵ ∈ Nd such that for all n ≽ nϵ then ∥an − a∥ < ϵ.
2. We say that an → a strongly as n → ∞ if for any ϵ > 0 there exists nϵ ∈ Nd such that for all n ̸≼ nϵ then ∥an − a∥ < ϵ
(See Lagodowski, 2009).
Clearly, an → a strongly as n → ∞ then an → a as n → ∞, but the converse is not true. For example, let
a ̸= b, a(n1 ,1...,1) = b and an = a if otherwise, then an → a but an ̸→ strongly as n → ∞.
It is easy to see that in the case d = 1, the strong convergence and the convergence are equivalent.


T.C. Son, D.H. Thang / Statistics and Probability Letters 83 (2013) 1901–1910

1903

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


q

q/p

 




p
E
X  ≤ CE
∥Xk ∥
.
1≼k≼N k 
1≼k≼N

(2.1)

Proof. For the field of E-valued martingale differences {Xk , Fk ; 1 ≼ k ≼ N}, we define a sequence {Yk , Gk ; 1 ≤ k ≤ |N|} by
putting
Y1 = X(1,...,1,1) ;
Y2 = X(1,...,1,2) ; . . . ;
YNd = X(1,...,1,Nd ) ;
YNd +1 = X(1,...,2,1) ;
YNd +1 = X(1,...,2,2) ; . . . ;
Y|N| = XN ,
and for all 1 ≤ k ≤ |N|, let Gk = σ {Yi ; 1 ≤ i ≤ k}. Since
E (Yk |Gk−1 ) = E (E (Xi |Fi∗ )|Gk−1 ) = 0,
(where Yk = Xi ) so {Yk ; Gk ; 1 ≤ k ≤ |N|} is a sequence of E-valued martingale differences. From the inequality (1.2) we get
(2.1).
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 ≼

(s)

k ≼ N}, put Fn = σ {Fk : k = (k1 , . . . , kd ), 0 ≤ ki ≤ Ni (i ̸= s), and ks = ns } for all 1 ≤ s ≤ d and Sn =
k≼n Xk . Then we
have
E (∥S(n,s,ns +1) ∥q |Fn(s) ) ≥ ∥Sn ∥q .

(2.2)

where
S(n,s,α) = S(n1 ,...,ns−1 ,α,ns+1 ,...,nd ) .
(s)

⊂ F(∗i,s,ns +1) for all (i, s, 1) ≼ n = (n1 , . . . , ns , . . . , nd ), then




 (s)
(s)
∗
E (S(n,s,ns +1) − Sn |Fn ) = E
E (X(i,s,ns +1) |F(i,s,ns +1) )Fn
= 0,

Proof. We have S(n,s,ns +1) − Sn =



(i,s,1)≼n

X(i,s,ns +1) , and Fn

(i,s,1)≼n

so
E (∥S(n,s,ns +1) ∥q |Fn(s) ) ≥ ∥E (S(n,s,ns +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 all ε > 0,




sup ∥Sk ∥ > ε

lim P

∥n∥→∞

= 0.

(2.3)

k̸≼n

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


lim P

∥n∥→∞

sup ∥Sk ∥ > ε



= 0.

(2.4)

k≽ n

Proof. 1. Necessary. Suppose that Sn → 0 a.s. as |n| → ∞. For each ϵ > 0, let Aϵn = supk̸≼n ∥Sk ∥ > ϵ and Aϵ =
then P (Aϵ ) = 0, n = (n1 , n2 , . . . , nd ) = (n1 , n1 ), by the continuity from below theorem, so




ϵ

0 = P (A ) = P




ϵ

An

n ≽1



= lim P
n1 →∞




ϵ

An

n 1 ≽1

= · · · = lim P (Aϵn ),
∥n∥→∞

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


Ain

= sup ∥Sk ∥ ≥
k̸≼n

1
i


and A =


i≥1 n≽1

Ain .





n ≽1

Aϵn ,


1904

T.C. Son, D.H. Thang / Statistics and Probability Letters 83 (2013) 1901–1910

by the continuity from below theorem, then






P (A) = P

= lim lim P (Ain ) = lim 0 = 0,

Ain

i→∞ ∥n∥→∞

i≥1 n≽1

i→∞

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 ≤ bm for all n ≼ m. Then for all ε > 0, q ≥ 1 we have

εP

∥Sn ∥q


max

≥ϵ

q

bn

1≼n ≼N








1

1≼n≼N−1

bn

≤ min

q

1≤s≤d

−



1



E ∥Sn ∥q

N−1≺n≼N

bn

q

E ∥Sn ∥ +

q

b(n,s,ns +1)



q

where b(n,s,α) = b(n1 ,...,ns−1 ,α,ns+1 ,...,nd ) .
Proof. Let ‘‘<’’ be the lexicographic order on Nd , i.e., n = (n1 , . . . , nd ) < m = (m1 , . . . , md ) if and only
m1 or
 if either n1 <

∥S ∥q
there exists 1 ≤ k < d such that ni = mi for all 1 < i ≤ k and nk < mk . Define the sets A, An by, A = max1≼n≼N nq ≥ ϵ ,
bn


An =
then

∥Sk ∥q
q
bk



1≼n≼N

< ϵ if k < n, and

An = A, An



∥Sn ∥q



≥ϵ ,

q

bn

(1)

Am = ∅ for all i ̸= j and Ak ∈ Fn

for all k ≼ n. By Lemma 2.2 so

for all k ≼ n.






E ∥S(n,1,n1 +1) ∥q IAk ≥ E ∥Sn ∥q IAk

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






ϵ P (A) = ϵ P

=ϵ

An

n ≼N

 

E

∥S(k,n′ ) ∥q
q

b(k,n′ )

n′ ≼N′ k≤N1



≤







1

n ′ ≼N ′

k≤N1

b(k,n′ )

q


≤

P (An ) =

n ≼N


=





1

1≼n≼N

q
bn

−



E (ϵ IAn ) ≤

n≼N

I


i≤k

−

A(i,n′ )

−E

q

q

bn

n ≼N

∥S(k,n′ ) ∥q
q

b(k,n′ )

I

i≤k−1

E ∥S(k,n′ ) ∥ + E



A(i,n′ )

∥S(N1 ,n′ ) ∥q
q

b(k,n′ )



E ∥S n ∥q

N−1≺n≼N

bn

E ∥S n ∥q +


IA n




q

b(k+1,n′ )

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

E



1

1

  ∥S n ∥q

q


I


k≤N1

A(k,n′ )

.

Similarly, it can be shown that:


ϵ P (A) ≤



1

1≼n≼N−1

bn

q

−



1
q

b(n,s,ns +1)



E ∥S n ∥q

N−1≺n≼N

bn

E ∥Sn ∥q +

q

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 ≤ bm for 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

max

∥n∥→∞ n−1≺k≼n

E ∥S k ∥q
q

bk

= 0.

(2.5)

Then for all ε > 0, q ≥ 1, we have



ε P sup
n ≽1

∥Sn ∥q
q

bn

≥ϵ




≤ min

1≤s≤d

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





1

n ≽1

bn

q

−

1
q

b(n,s,ns +1)




E ∥Sn ∥

q


T.C. Son, D.H. Thang / Statistics and Probability Letters 83 (2013) 1901–1910

1905

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

= 0.

q

bk

By the continuity from below theorem and Theorem 2.4, we have



ε P sup

∥Sn ∥q

≥ε

q

bn

n ≽1



= lim ε P

∥S n ∥q


sup

n→∞






1

1≼n≼Nn −1

bn

≤ lim min

q

n→∞ 1≤s≤d

= min



 1

Remark. If lim∥n∥→∞

E ∥Sn ∥q
q

bn

−

q

1≤s≤d

≥ε

q

bn

1≼n≼Nn

bn

n ≽1




1

−

q



1

E ∥S n ∥ +

q

b(n,s,ns +1)

E ∥S n ∥q

Nn −1≺n≼Nn

bn



q


E ∥S n ∥

q

b(n,s,ns +1)



q

.

= 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 ∥Sn ∥ ≥ ε
q

q/p







≤

1≼n≼N



q

E ∥Sn ∥ ≤ CE

N−1≺n≼N

∥X n ∥

p

.

1≼n≼N

Proof. By Theorem 2.4 with bn = 1 for all n ≽ 1 and Lemma 2.1, we have

εP



max ∥Sn ∥q ≥ ε





≤

1≼n≼N

E ∥Sn ∥q

N−1≺n≼N

q/p




≤C

E

N−1≺n≼N



∥X k ∥

p

1 ≼ k≼ n

q/p




d

≤ 2 ·C ·E

p

∥X n ∥

.

1≼n ≼N

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 ≤ bm for 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)

ck .

(3.1)

<∞

(3.2)

1≼k≼n

Then, the condition


min

1≤s≤d




where ψs (n) =
Sn
bn

cn .ψs (n)

n∈Nd




k≽ n

1
q
bk

−

q

1

b(k,s,n +1)
s



ξ (n), implies

→ 0 strongly a.s. as |n| → ∞.

Proof. By Lemma 2.3, we have to show that


lim P

∥N∥→∞

sup
n̸≼N

∥Sn ∥
bn


≥ϵ

= 0 for any ϵ > 0,


1906

T.C. Son, D.H. Thang / Statistics and Probability Letters 83 (2013) 1901–1910

where N = (N1 , . . . , Nk , . . . , Nd ). Put N(k) = (1, . . . , Nk , . . . , 1). By Corollary 2.5 and the condition (3.1) we have



∥Sn ∥

ϵ P sup
q

bn

n̸≼N





≥ϵ

=ϵ P
q

d




sup

≤ϵ

q

d



P

sup


≤ d min

1≤s≤d

∥Sn ∥
bn

n≽N (k)

k=1



1

n̸≼N

bn

q


≤ C · min

1≤s≤d

≥ϵ

bn

n≽N (k)

k=1



∥S n ∥

1

n̸≼N

q
bn

≥ϵ


1

−





E ∥S n ∥q

q

b(n,s,ns +1)

−



1

ξ (n)

q
b(n,s,ns +1)



ck .

1≼k≼n

By (3.2), we have


min

1≤s≤d



1

n ≽1

q
bn

−

= min

1≤s≤d

ck

k≽ 1

ξ (n )

q
b(n,s,ns +1)






1



1

n ≽k

bn

ck

1 ≼ k≼ n

−

q




1


ξ (n) = min

q

b(n,s,ns +1)

1≤s≤d




cn · ψs (n)

< ∞.

n∈Nd

Hence


lim C · min

1≤s≤d

∥N∥→∞



1

n̸≼N

q
bn

−



1

ξ (n)

q
b(n,s,ns +1)



ck = 0

1≼k≼n

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 ≤ bm for all n ≼ m. If
1.
E ∥Sk ∥pq
lim inf max
= 0.
(3.3)
pq
∥n∥→∞ n−1≺k≼n
bk
2.


min

1≤s≤d



E ∥Xn ∥pq .ϕs (n)

< ∞,

(3.4)

n∈Nd

where ϕs (n) =




k≽ n

1
pq
bk

−

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



|k|q−1 ;

then
Sn
bn

→ 0 strongly a.s. as |n| → ∞

(3.5)

Proof ((i) ⇒ (ii)). By Lemma 2.1 and Cr -inequality, we have

q


E ∥Sn ∥

pq

≤ CE



∥X k ∥

p

≤ C |n|q−1

k≼ n



E ∥Xk ∥pq

k≼n

Applying Theorem 3.1 with ξ (n) = |n| , cn = E ∥Xn ∥pq , we have (ii).
Now we prove [(ii) ⇒ (i)]. Let {Yn1 , Gn1 ; n1 ≥ 1} be an arbitrary sequence of E-valued martingale differences such that
q −1

∞

E ∥Yn1 ∥p
p

n 1 =1

n1

< ∞.


T.C. Son, D.H. Thang / Statistics and Probability Letters 83 (2013) 1901–1910

1907

For n = (n1 , . . . , nd ) ∈ Nd . Set
if n1 ≥ 1, n2 = · · · = nd = 1 otherwise Xn = 0

Xn = Yn1

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 =

ϕ1 ((n1 , 1, . . . , 1)) =





k 1 ≥n


1≤s≤d

E ∥Xn ∥p ϕs (n)

≤

n ≽1

E ∥Sn ∥p

Moreover,

−

1

(k1 +1)p

=

1
p
n1

p
bn



E ∥Xn ∥p ϕ1 (n) =

1

n1


n1 . . . nd

i 1 =1

E ∥Sn1 ∥p
p

n1

1
p
p,
n2 ...nd

Xi1 → 0

∞


E ∥Yn1 ∥p

i=1

Yi ,

< ∞.

p

n1 =1

n≽1

=

 n1

and




min

1
p
k1



n1
E ∥S ∥p

so lim inf∥n∥→∞ maxn−1≺k≼n |kk|p

= 0. By (ii),

a.s. as |n| → ∞.

Taking n2 = · · · = nd = 1 and letting n1 → ∞, we obtain
n1
1 

n1 j = 1

Yj → 0

strongly a.s. as n1 → ∞.

Then by Theorem 2.2 of Hoffmann-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

 E ∥Xn ∥pq
<∞
1≤s≤d
n |n|pq−q

n ≽1 s
min

(3.6)

then
Sn

lim

|n|→0

|n|

= 0 strongly a.s.

Proof. When bn = |n|, we have


 1

1
1
|k|q−1 =
−
pq−q+1
pq
pq
|
k

|
|(
k
,
s
,
n
+
1
)|
k
|
k
|
s
s
k≽ n
k≽ n


d



1
1
1
=
≤C
,

pq−q+2
pq−q+1
pq−q
n
|
n
s |
ks ≥ns ks
i=1,i̸=s k ≥n ks

ϕs (n) =

i

i

then (3.4) is satisfied. Next, we have

 E ∥Sn ∥pq
n ≽1

ns ∥n∥pq

≤


n ≽1

=



1



ns ∥n∥pq−q+1 k≼n
E ∥Xn ∥pq



E ∥Xk ∥pq
1
pq−q+1

k |k|
k≽ n s
 E ∥Xn ∥pq
≤C
< ∞,
n ∥n∥pq−q
n ≽1 s
n ≽1

then (3.3) is satisfied, By Theorem 3.2, we have
lim

|n|→0

Sn

|n|

= 0 strongly a.s.

(by (3.6))


1908

T.C. Son, D.H. Thang / Statistics and Probability Letters 83 (2013) 1901–1910

Applying Theorem 3.3 for d = 1 and Theorem 3.1 in Son et al. (2012) for bn = |n|, ai = 0 if i ̸= 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 {Xn , Fn ; n ≥ 1} be a sequence of E-valued
martingale differences such that

 E ∥Xn ∥pq

< ∞,

npq−q+1

n≥1

then
Sn

lim

n→0

= 0 a.s.

n

Since the Hilbert space is of 2-uniformly smooth, then we have
Corollary 3.5. Let q ≥ 1, E be a Hilbert space, {Xn , Fn ; n ≥ 1} be a sequence of E-valued martingale differences such that

 E ∥Xn ∥2q
n2q−1

n ≥1

< ∞,

then
Sn

lim

n→0

= 0 a.s.

n

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



E ∥Xn ∥q .φ(n) < ∞.

(3.7)

n≽1

where φ(n) =

|k|p−α−1



pq

k≽ n

bk

. Then

1. If {bn ; n ≽ 1} satisfying bn ≤ bm for all n ≼ m, then


P

sup
k≽ n

∥S k ∥
bk

>ϵ




=o



1

as |n| → ∞ for every ϵ > 0.

|n|1−α

2. If {bn ; n ≽ 1} satisfying bn ≤ bm for all n ≪ m then


P

sup
k̸≼n

∥S k ∥
bk


>ϵ


=o



1

as |n| → ∞ for every ϵ > 0.

|n|1−α

Proof. 1. Firstly, we show that


 1 
P max ∥Sk ∥ > ϵ bn < ∞.
k≼ n
|n|α
n ≽1

(3.8)

Using Corollary 2.6 and Cr -inequality, we have


 1 
 |n|−α

P
max
∥
S
∥
>
ϵ
b
≤
C
k
n
pq E
k≼ n
|n|α
bn
n ≽1
n ≽1
≤C




n∈Nd

Then, we have (3.8).

∥X k ∥

bn

E ∥Xk ∥pq

p

1≼k≼n

E ∥Xk ∥pq

pq

k≽ 1

=



 |n|−α+p−1 
n ≽1

=C

q



1≼k≼n

 |n|−α+p−1
pq

n ≽k

bn

E ∥Xn ∥q .φ(n) < ∞

(by (3.7)).


T.C. Son, D.H. Thang / Statistics and Probability Letters 83 (2013) 1901–1910

1909

Next, with ϵ > 0, set An = {k, 2n−1 ≺ k ≼ 2n }.
If bn ≤ bm for all n ≼ m, we see



n−α P



1
sup b−
k ∥Sk ∥ > ϵ


=

k≽ n

n ≽1

i −1
 2

n−α P



1
sup b−
k ∥Sk ∥ > ϵ
k≽ n

i≽1 n=2i−1

≤C

i −1
 2


−i α

|2



≤C



sup bk ∥Sk ∥ > ϵ

|P

k≽2i−1



1
|2i(1−α) |P sup max b−
∥
S
∥
>
ϵ
k
k
u≽i k∈Au

i ≽1

|2

i(1−α)

|

i ≽1

≤C

 
P

1
b−
max ∥Sk ∥ > ϵ
2u−1

P




n≽1



k≼2u

u(1−α)

|2





k≼2u

 

|2i(1−α) |

i≼u

|P b2u maxu ∥Sk ∥ >
−1

k≼2

u ≽1

≤C

b2u−1 max ∥Sk ∥ > ϵ
−1

u ≽i

u ≽1

≤C


−1

i≽1 n=2i−1

≤C





1
|n|−α P b−
n max ∥Sk ∥ >
k≼ n

ϵ
M

ϵ



M



< ∞ (by (3.8)).

1
d
d

Since {P supk≽n b−
k ∥Sk ∥ > ϵ , n ∈ N } are non-increasing in n = (n1 , . . . , nd ) for order relationship ≼ in N , it follows
that




P



1
sup b−
k ∥S k ∥ > ϵ




=o

k≽n



1

|n|1−α

as |n| → ∞ for all ϵ > 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

 E ∥Xn ∥pq
<∞
|n|pq−q+1
d

n∈N

then



∥S k ∥
P sup
>ϵ
k̸≼n ∥k∥




=o

1



|n|1−α

as |n| → ∞ for every ϵ > 0

and


P

sup
k≽ n

∥S k ∥
>ϵ
|k|




=o

1

|n|1−α



as |n| → ∞ for every ϵ > 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).


1910

T.C. Son, D.H. Thang / Statistics and Probability Letters 83 (2013) 1901–1910

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