On the strong law of large numbers and Lp -convergence for double arrays of random elements in p -uniformly smooth Banach spaces

Statistics and Probability Letters 79 (2009) 1891–1899

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On the strong law of large numbers and Lp -convergence for double arrays of random elements in p-uniformly smooth Banach spaces Nguyen Van Quang a,1 , Nguyen Van Huan b,∗ a

Department of Mathematics, Vinh University, Nghe An 42000, Vietnam

b

Department of Mathematics, Dong Thap University, Dong Thap 871000, Vietnam

article

abstract

info

Article history: Received 29 September 2008 Received in revised form 20 May 2009 Accepted 21 May 2009 Available online 30 May 2009

The aim of this paper is to establish some strong laws of large numbers and Lp -convergence for double arrays of random elements in p-uniformly smooth Banach spaces. We also provide a new characterization of p-uniformly smooth Banach spaces in terms of a strong law of large numbers for double arrays. © 2009 Elsevier B.V. All rights reserved.

MSC: 60B11 60B12 60F15 60F25 60G42

1. Introduction and preliminaries Smythe (1973) obtained the Kolmogorov strong law of large numbers for multidimensional arrays of random variables. The Marcinkiewicz–Zygmund law for double arrays was established by Gut (1978), Klesov (1995) and Hong and Volodin (1999) and was extended to double arrays of independent Banach space valued random elements by Mikosch and Norvaisa (1987) and Giang (1995) and Rosalsky and Thanh (2006) and other authors. Mean convergence theorems for double arrays of Banach space valued random elements have been studied by many authors. The reader may refer to Adler et al. (1997), Rosalsky and Sreehari (2001) and Ordóñez Cabrera and Volodin (2005). The aim of this paper is to establish some strong laws of large numbers and Lp -convergence for double arrays of random elements in p-uniformly smooth Banach spaces. We also provide a new characterization of p-uniformly smooth Banach spaces in terms of a strong law of large numbers for double arrays. Let us begin with some notations and definitions. Let (Ω , F , P) be a probability space, E be a real separable Banach space, and B (E) be the σ -algebra of all Borel sets in E. A real separable Banach space E is said to be p-uniformly smooth (1 6 p 6 2) if

ρ(τ ) = sup



kx + yk + kx − yk 2

− 1, ∀ x, y ∈ E; kxk = 1, kyk = τ



6 Cτ p

∗ Corresponding address: Department of Mathematics, Dong Thap University, 783 Pham Huu Lau, Cao Lanh city, Dong Thap province, Vietnam. Tel.: +84 989 291660 (Mobile), +84 676 291660 (Residential), +84 673 881624 (Office); fax: +84 673 881713. E-mail addresses: [email protected] (N.V. Quang), [email protected] (N.V. Huan). 1 Department of Mathematics, Vinh University, 182 Le Duan, Vinh city, Nghe An province, Vietnam. Tel.: +84 912 435734 (Mobile), + 84 383 847421 (Residential), + 84 383 855329 (Office); fax: +84 383 855269. 0167-7152/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2009.05.014

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for some constant C . Assouad (see Woyczyński (1978)) proved that a real separable Banach space E is p-uniformly smooth (1 6 p 6 2) if and only if for all q > 1, there exists a positive constant C such that for all E-valued martingales {Sn , Fn , n > 1}, we have q

EkSn k 6 C E

n X

!q/p kSi − Si−1 k

p

,

for all n ∈ N.

i=1

(The Marcinkiewicz–Zygmund inequality.) A real separable Banach space E is said to be martingale type p (1 6 p 6 2) if there exists a finite positive constant C such that for all martingales {Sn , Fn , n > 1} with values in E, we have sup EkSn kp 6 C n>1

∞ X

EkSn − Sn−1 kp .

(1.1)

n =1

By the Marcinkiewicz–Zygmund inequality, we derive that a p-uniformly smooth Banach space is a martingale type p Banach space. Throughout this paper, the symbol C will denote a generic positive constant which is not necessarily the same one in each appearance. For a, b ∈ R, min{a, b} and max{a, b} will be denoted, respectively, by a ∧ b and a ∨ b. The logarithms are to the base 2, for a ∈ R, log(a ∨ 1) will be denoted by log+ a. The number of divisors of a positive integer k will be denoted by dk . Let N be the set of positive integers, {Xmn , m > 1, n > 1} be a double array of random elements in a real separable Banach space E with indices in N × N. Let {Fmn , m > 1, n > 1} be an arbitrary double array of sub-σ -algebras of F such that Xmn − is Fmn /B (E)-measurable for all m > 1, n > 1, and Fmn = σ (∪Fij : i < m or j < n), F11− = {∅, Ω }. A double array {Xmn , Fmn , m > 1, n > 1} is said to be an array of martingale differences if for all m > 1, n > 1, − ) = 0. E(Xmn |Fmn

(1.2)

Example 1. Let {Xmn , m > 1, n > 1} be a double array of independent mean 0 random elements. For all m > 1, n > 1, let − Fmn be the σ -algebra generated by Xmn , then E(Xmn |Fmn ) = EXmn = 0. Therefore, {Xmn , Fmn , m > 1, n > 1} is an array of martingale differences. Example 2. Let (Xn , Fn , n > 1) be an arbitrary martingale difference sequence such that (Xn , n > 1) is not a sequence of independent random elements. For n > 1, set Xmn = Xn

if m = 1

and

Xmn = 0

if m > 1,

Fmn = Fn if m = 1 and Fmn = {∅, Ω } if m > 1. We obtain the following Xmn ∈ Fmn

for all m > 1, n > 1,

− Fmn = Fn−1 if m = 1,

− Fmn = σ

∞ [

Fn



if m > 1.

n=1

It follows that {Xmn , Fmn , m > 1, n > 1} is an array of martingale differences, but it is not an array of independent mean 0 random elements. Thus, the set of all arrays of martingale differences is really larger than the set of all arrays of independent mean 0 random elements. Now we present some lemmas which will be needed in what follows. The key tool for proving Theorem 2.1 is the maximal inequality provided by the following lemma. Lemma 1.1. Let E be a real separable p-uniformly smooth Banach space (1 6 p 6 2). Then, there exists a positive constant C such that for all arrays of martingale differences {Xmn , Fmn , m > 1, n > 1} with values in E, we have

p k X l m X n

X

X

E max Xij 6 C EkXij kp , 16k6m

i=1 j=1 i=1 j=1

for all (m, n) ∈ N × N.

(1.3)

16l6n

Proof. We easily obtain the conclusion (1.3) in the case p = 1. Now we consider the case 1 < p 6 2. Set Skl = i =1 j =1 Xij , Yl = max16k6m kSkl k and let σl be the σ -algebra generated by the family of random elements {Xij , 1 6 i 6 m, 1 6 j 6 l}.

Pk Pl

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For each l, 1 < l 6 n, we have σl−1 ⊂ Fil− for all i > 1. Thus,

E(Skl |σl−1 ) = E(Sk,l−1 |σl−1 ) +

k X

E(Xil |σl−1 )

i=1

= Sk,l−1 +

k X

E(E(Xil |Fil− )|σl−1 ) = Sk,l−1 .

(1.4)

i=1

This means that for each k = 1, 2, . . . , m, {Skl , σl , 1 6 l 6 n} is a martingale. As in Scalora (1961), {kSkl k, σl , 1 6 l 6 n} is a nonnegative submartingale. It is easy to show that {Yl , σl ; 1 6 l 6 n} is a nonnegative submartingale. Applying Doob’s inequality (see Chow and Teicher (1997)), we obtain

E max kSkl kp = E max Yl 16k6m 16l6n

16l6n


p

6 C EYnp .

(1.5)

On the other hand, since E(Xij |Fij− ) = 0, we have that {Skn , Fk = Fk− +1,1 , 1 6 k 6 m} is a martingale. By (1.1), we get

EYnp

p n m

X

X

Xkj . = E max kSkn k 6 C E

16k6m j =1 k =1 p

Pl

For each k, 1 6 k 6 m, we again have that {

j =1

(1.6)

Xkj , Gl = Fk− ,l+1 , 1 6 l 6 n} is a martingale. Thus

p

p

n l n

X

X X



EkXkl kp . Xkj 6 C Xkj 6 E max E

j =1 16l6n l =1 j=1 Combining (1.5)–(1.7) yields the conclusion (1.3).

(1.7)



Remark 1. In the case 0 < p 6 1, inequality (1.3) of Lemma 1.1 holds for all double arrays of random elements in a real separable Banach space. Remark 2. In the case E = R, a similar lemma was proved by Hong and Volodin (1999). By putting Fmn = σ (Xij : i < m or i = m and j < n), σl = σ (Xij : 1 6 i 6 m, 1 6 j 6 l) and the hypothesis

E(Xmn |Fmn ) = 0.

(1.8)

Hong and Volodin (1999) proved that (1.4) follows from (1.8). We think that, under assumption (1.8), it is difficult to guarantee that ?

E(Xil |σl−1 ) = E E(Xil |Fml )|σl−1 = 0,




1 6 i 6 m, 1 6 l 6 n.

Lemma 1.2. Let {Xmn , m > 1, n > 1} be a double array of (real-valued) random variables which are stochastically dominated by a random variable X .
(i) If E |X |(log+ |X |)2 < ∞, then


∞ X ∞ X E |Xmn | I (|Xmn | > mn) m=1 n=1

mn

< ∞.

(ii) If E(|X |q log+ |X |) < ∞, for some q > 0, then 1
∞ X ∞ X E |Xmn |r I (|Xmn | > (mn) q ) < ∞ for all 0 < r < q; r (mn) q m=1 n=1 1
∞ X ∞ X E |Xmn |p I (|Xmn | 6 (mn) q ) < ∞ for all p > q. p (mn) q m=1 n=1 Proof. (i) Let F be the distribution of X . By using the fact that i X dk k=1

k

= O(log2 i),

(1.9)

(1.10)

(1.11)

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we obtain


∞ X ∞ X E |Xmn | I (|Xmn | > mn) mn

m=1 n=1

=C

Z ∞ X dk k=1

=C

k

xdF (x) = C k

6C

∞ Z X i=1

∞ ∞ Z X dk X

k i=k

k =1

Z ∞ X i X dk k

Z ∞ X ∞ X 1 m=1 n=1

∞

i=1 k=1

6C

∞ X

i +1

xdF (x) 6 C i

mn

∞

xdF (x) mn

i+1

xdF (x) i

log2 i

i+1

Z

xdF (x) i

i=1

i+1

x log2 xdF (x) 6 C E |X |(log+ |X |)2 < ∞.




i


r = O i1− q log i , 0 < r < q, we can obtain (1.10) by the same method. p
P∞ d 1− Next, by using the fact that k=i pk = O i q log i , p > q, we have (ii) Noting that

Pi

dk

k=1

r kq

kq

1

∞ X ∞ X E |Xmn |p I (|Xmn | 6 (mn) q )




p

(mn) q Z (mn) 1q

m=1 n=1

6

∞ X ∞ X

1 p

m=1 n=1

6C

∞ X dk p

k=1

6C

k q i=1

∞ k X dk X k=1

6C

(mn) q k Z X

k

p q

i=1 k=i

6C

6C

1

iq

1

(i−1) q

p

p

P{|X |q > i − 1} i q − (i − 1) q

k

p q

p

p

P{|X |q > i − 1} i q − (i − 1) q p

log i P{|X |q > i − 1}




p

p

−1

iq

∞ X

∞ X

∞ X




i q − (i − 1) q

i=1

log i P{|X |q > i − 1} = C

i =1

i=1

6C

pxp−1 P{|X | > x}dx

i=1

∞ X ∞ X dk

∞ X

pxp−1 P{|Xmn | > x}dx

0

log i

∞ X

P{k − 1 6 |X |q < k}

k=i

k log k P{k − 1 6 |X |q < k} 6 C E(|X |q log+ |X |) < ∞.

k=1

The proof is completed.



2. Main results With the preliminaries accounted for, the first main result may be established. Theorem 2.1 is a double sum analogue of a strong law of large numbers of Hoffmann-Jørgensen and Pisier (1976), Woyczyński (1978), concerning partial sums from a martingale difference sequence in a p-uniformly smooth Banach space (1 6 p 6 2). Theorem 2.1 also provides a new characterization of p-uniformly smooth Banach spaces in terms of a strong law of large numbers for double arrays. Theorem 2.1. Let E be a real separable Banach space and 1 6 p 6 2. Then the following two statements are equivalent: (i) The Banach space E is p-uniformly smooth. (ii) For every array of martingale differences {Xmn , Fmn , m > 1, n > 1} and every choice of constants α > 0β > 0, the condition ∞ X ∞ X EkXmn kp m=1 n=1

m α p nβ p

<∞

(2.1)

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implies



P

l

k P

max Xij

16k6m, 16l6n i=1 j=1

→0

m α nβ Proof. (i) ⇒ (ii): Set Skl =

a.s. and in Lp as m ∨ n → ∞.

Pk Pl i=1

j =1

(2.2)

Xij . By using Lemma 1.1, we get

p 

P

n 2k P 2l P

m P

max Xij 

EkXij kp  ∞ ∞ ∞ X ∞ ∞ ∞ X X  16m62k , 16n62l i=1 j=1  X X X i=1 j=1 EkXij kp  6C E 6 C < ∞ (by (2.1)).   α k β l α k β l p 2 2 (2 2 ) (iα jβ )p   k=1 l=1 i =1 j =1 k=1 l=1 Applying Markov’s inequality, we obtain



P n

m P max Xij

16m62k , 16n62l i=1 j=1 2α k 2β l

→ 0 a.s. and in Lp as k ∨ l −→ ∞.

Next, for m > 1 and n > 1, let k > 1 and l > 1 be such that 2k−1 6 m < 2k and 2l−1 6 n < 2l . Then



P s

r P Xij max

16r 6m, 16s6n i=1 j=1 mα nβ

= 2α+β

6



P s

r P Xij max

16r 62k , 16s62l i=1 j=1 (k−1)α 2(l−1)β

2

P r P s

max Xij

16r 62k , 16s62l i=1 j=1

2kα 2lβ → 0 a.s. and in Lp as m ∨ n −→ ∞

(by (2.3))

thereby establishing (2.2). (ii) ⇒ (i): Assume that (ii) holds. Let (Xn , Fn , n > 1) be an arbitrary martingale difference sequence such that ∞ X EkXn kp n =1

np

< ∞.

For n > 1, set Xmn = Xn

if m = 1

and

Xmn = 0

if m > 1,

Fmn = Fn if m = 1 and Fmn = {∅, Ω } if m > 1. We obtain the following Xmn ∈ Fmn

for all m > 1, n > 1,

− Fmn = Fn−1 if m = 1,

− Fmn = σ

∞ [

Fn



if m > 1.

n =1

Then {Xmn , m > 1, n > 1} is an array of martingale differences with ∞ X ∞ X EkXmn kp m=1 n=1

m p np

=

∞ X EkXn kp n =1

np

< ∞.

By (ii), m P n P i=1 j=1

mn

Xjj

→ 0 a.s. as m ∨ n → ∞.

(2.3)

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This implies, by taking m = 1 and letting n → ∞, that n P

lim

Xj

j =1

n

n→∞

→ 0 a.s.

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



Remark 3. In the case 0 < p 6 1, the implication ((2.1) ⇒ (2.2)) of Theorem 2.1 holds for all double arrays of random elements in a real separable Banach space. The next corollary follows immediately from Theorem 2.1. In the special case where α = β = 1 and E = R (p = 2), from Corollary 2.2 we get the two-dimensional version of Kolmogorov’s theorem which was proved by Smythe (1973). Note here that Smythe’s proof proceeds as in the one-dimensional case in that it is based on the ‘‘convergence of random series, Kronecker lemma approach’’ whereas our proof is based on the ‘‘method of subsequences’’. Corollary 2.2. Let {Xmn , m > 1, n > 1} be a double array of independent mean 0 random elements in a real separable p-uniformly smooth Banach space E (1 6 p 6 2). If ∞ X ∞ X EkXmn kp m=1 n=1

m α p nβ p

<∞

then



P l

k P max Xij

16k6m, 16l6n i=1 j=1 mα nβ m P n P

→ 0 a.s. and in Lp as m ∨ n → ∞,

Xij

i =1 j =1

mα nβ

→ 0 a.s. and in Lp as m ∨ n → ∞.

Corollary 2.3. Let α ∧β > 0 and {Xmn , Fmn , m > 1, n > 1} be an array of martingale differences in a real separable p-uniformly smooth Banach space E (1 6 p 6 2). If

EkXmn kp 6 C

mα p−1 nβ p−1 log(m + 1) log(n + 1)

(2.4)


1+ε

for some constants α > 0, β > 0, ε > 0, C < ∞, and all m, n > 1, then



P

l

k P

max Xij

16k6m, 16l6n i=1 j=1 mα nβ

→ 0 a.s. and in Lp as m ∨ n → ∞.

(2.5)

Proof. Note that by (2.4) ∞ X ∞ X EkXmn kp m=1 n=1

m α p nβ p

6C

∞ X ∞ X m=1 n=1

1


1+ε
1+ε < ∞ m log(m + 1) n log(n + 1)

and the conclusion (2.5) follows from Theorem 2.1.



A double array of random elements {Xmn , m > 1, n > 1} is said to be stochastically dominated by a random element X if there exists a constant C (0 < C < ∞) such that

P{kXmn k > t } 6 C P{kX k > t },

t > 0, m > 1, n > 1.

This condition is, of course, automatic with X = X11 and C = 1 if {Xmn , m > 1, n > 1} is a double array of identically distributed random elements. Remark 4. We note that if {Xmn , Fmn , m > 1, n > 1} is an array of martingale differences in a real separable p-uniformly smooth Banach space E (1 6 p 6 2) such that sup EkXmn kp < ∞,

m>1,n>1

(2.6)

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then (2.4) holds for all ε > 0 and some C < ∞ whenever α ∧ β > 1/p, and Corollary 2.3 then yields (2.5). The condition (2.6) is, of course, automatic if {Xmn , m > 1, n > 1} is stochastically dominated by a random element X with

EkX kp < ∞. When α = β = 1/q > 1/p, Theorem 2.4 shows that the condition EkX kp < ∞ can be replaced by the weaker condition. Theorem 2.4. Let {Xmn , Fmn , m > 1, n > 1} be an array of martingale differences in a real separable p-uniformly smooth Banach space E (1 < p 6 2). Suppose that {Xmn , m > 1, n > 1} is stochastically dominated by a random element X . (i) If E(kX kq log+ kX k) < ∞ for some q ∈ (1, p), then m P n P

Xij

i=1 j=1

→ 0 a.s. as m ∨ n → ∞.

(mn)1/q

(2.7)


(ii) If E kX k(log+ kX k)2 < ∞, then m P n P

Xij

i=1 j=1

mn

→ 0 a.s. as m ∨ n → ∞.

(2.8)

Proof. (i) For m > 1 and n > 1, set 1



0 Xmn = Xmn I kXmn k 6 (mn) q



1



,

00 Xmn = Xmn I kXmn k > (mn) q



.

Thus 0 0 − − E Xmn − E(Xmn |Fmn )|Fmn = 0,

00 00 − − E Xmn − E(Xmn |Fmn )|Fmn = 0,







0 0 − 00 00 − Xmn = Xmn − E(Xmn |Fmn ) + Xmn − E(Xmn |Fmn ),

and 0 0 − p 0 0 − EkXmn − E(Xmn |Fmn )k 6 E kXmn k + E(kXmn k|Fmn )


p

0 0 − p 6 2p−1 EkXmn kp + E(E(kXmn k|Fmn ))




0 0 − 6 2p−1 EkXmn kp + E(E(kXmn kp |Fmn ))




0 = 2p EkXmn kp .

(2.9)

By combining (2.9) and (1.11), we get ∞ X ∞ 0 0 − p X EkXmn − E(Xmn |Fmn )k p

(mn) q

m=1 n=1

6 2p

∞ X ∞ 0 X EkXmn kp p

m=1 n=1

(mn) q

< ∞.

(2.10)

< ∞.

(2.11)

Similarly, for all r ∈ [1, q) ∞ X ∞ 00 00 − r X EkXmn − E(Xmn |Fmn )k

(mn)

m=1 n=1

r q

6 2r

∞ X ∞ 00 r X EkXmn k r

m=1 n=1

(mn) q

By (2.10) and (2.11) and Theorem 2.1, we get m P n P

(Xij0 − E(Xij0 |Fij− ))

i=1 j=1

(mn)1/q m P n P

→ 0 a.s. as m ∨ n → ∞,

(2.12)

→ 0 a.s. as m ∨ n → ∞.

(2.13)

(Xij00 − E(Xij00 |Fij− ))

i=1 j=1

(mn)1/q

Combining (2.12) and (2.13) yields (2.7).

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(ii) By the same method we obtain ∞ X ∞ ∞ X ∞ 0 0 − p 0 X X EkXmn − E(Xmn |Fmn )k EkXmn kp 6 2p , (mn)p (mn)p m=1 n=1 m=1 n=1 ∞ X ∞ 00 00 − X EkXmn − E(Xmn |Fmn )k

mn

m=1 n=1

62

∞ X ∞ 00 X EkXmn k

mn

m=1 n=1

(2.14)

.

(2.15)

By (2.14), (2.15), (1.9) and (1.11) and Theorem 2.1, we get m P n P

(Xij0 − E(Xij0 |Fij− ))

i =1 j =1

mn m P n P

→ 0 a.s. as m ∨ n → ∞,

(2.16)

→ 0 a.s. as m ∨ n → ∞.

(2.17)

(Xij00 − E(Xij00 |Fij− ))

i =1 j =1

mn

Combining (2.16) and (2.17) yields (2.8). The proof is completed.



The example below shows that the hypothesis in Theorem 2.4 that E is a real separable p-uniformly smooth Banach space

(1 < p 6 2) cannot be replaced by the hypothesis that E is a real separable p-uniformly smooth Banach space (1 6 p 6 2). This example concerns P∞ the real separable Banach space `1 consisting of absolutely summable real sequences x = {xk , k > 1} with norm kxk = k=1 |xk |. It is well known that the Banach space `1 is 1-uniformly smooth. The element having 1 in its kth position and 0 elsewhere will be denoted by x(k) , k > 1. Let ϕ : N × N → N be a one-to-one and onto map. Let {Xmn , m > 1, n > 1} be a double array of independent random elements in `1 by requiring the {Xmn , m > 1, n > 1} to be independent with P{Xmn = x(ϕ(mn)) } = P{Xmn = −x(ϕ(mn)) } =

1 2

,

m > 1, n > 1.

Then {Xmn , m > 1, n > 1} is a double array of independent, identically distributed mean 0 random elements, and so {Xmn , Fmn , m > 1, n > 1} is an array of martingale differences in `1 (Fmn is the σ -algebra generated by Xmn ). On the other hand,
it is easy to show that {Xmn , m > 1, n > 1} is stochastically dominated by X11 and E kX11 k(log+ kX11 k)2 < ∞, but



P n

m P Xij

i=1 j=1 mn

= 1 for all m > 1, n > 1,

and so (2.8) fails. The next corollary follows immediately from Theorem 2.4 and is an extension of a result of Thanh (2005). Corollary 2.5. Let {Xmn , m > 1, n > 1} be a double array of independent mean 0 random elements in a real separable p-uniformly smooth Banach space E (1 < p 6 2). Suppose that {Xmn , m > 1, n > 1} is stochastically dominated by a random element X . (i) If E(kX kq log+ kX k) < ∞ for some q ∈ (1, p), then m P n P

Xij

i =1 j =1

(mn)1/q

→ 0 a.s. as m ∨ n → ∞.


(ii) If E kX k(log+ kX k)2 < ∞, then m P n P i =1 j =1

mn

Xij

→ 0 a.s. as m ∨ n → ∞.

Acknowledgments The authors are grateful to an anonymous referee and the Co-Editor-in-Chief Professor Hira Koul for their numerous suggestions which led to improvements in the paper.

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