DSpace at VNU: Almost sure convergence for double arrays of block-wise M-dependent random elements in Banach spaces

DSpace at VNU: Almost sure convergence for double arrays of block-wise M-dependent random elements in Banach spaces tài...

Almost sure convergence for double arrays

of block-wise M-dependent random elements

in Banach spaces

Nguyen Van Quang, Le Van Thanh and Nguyen Duy Tien

Dedicated to Professor Nicholas Vakhania on the occasion of his 80th birthday

Abstract For a double array of blockwise M-dependent random elements¹VmnW m  1;

n  1º taking values in a real separable Rademacher type p (1  p  2) Banach

space, we provide conditions to obtain the almost sure convergence for double sums

Pm

i D1

Pn

j D1Vij, m  1; n  1 The paper treats two cases: (i) ¹Vmn W m  1; n  1º

is block-wise M-dependent with EVmnD 0, m; n  1, and (ii) ¹VmnW m  1; n  1º is

block-wise p-orthogonal The conditions for case (i) are shown to provide exact

charac-terizations of Rademacher type p and stable type p Banach spaces Examples are given

showing that the conditions cannot be removed or weakened It is also demonstrated that

some of the well-known theorems in the literature are special cases of our results

Keywords Blockwise M-dependent random elements, strong law of large numbers,

double array of random elements, Rademacher type p Banach space, stable type p

Banach space

2010 Mathematics Subject Classification 60F15, 60B11, 60B12

1 Introduction

Móricz [15] introduced the concept of block-wise m-dependence for a sequence of

random variables and extended the classical strong law of large numbers (SLLN)

of Kolmogorov (see, e.g., Chow and Teicher [6, p 124]) to the block-wise

m-dependent case Móricz’s result [15] was extended by Gaposhkin [8] Based on a

lemma of Chobanyan, Levental and Mandrekar [3], Rosalsky and Thanh [23] gave

a simple proof of strong laws for sequences of block-wise m-dependent random

elements in Banach spaces (see also [4, 7] for more details about this approach)

The first and second author were supported in part by the National Foundation for Science and

Tech-nology Development, Vietnam (NAFOSTED), no 101.02.32.09 The third author was supported by

the National Foundation for Science Technology Development, Vietnam (NAFOSTED), no

10103-2010.6.

The SLLN for double arrays of block-wise independent random variables was

also studied by Quang and Thanh [19] Recently, Móricz, Stadtmüller and

Thal-maier [16] introduced the concept of block-wise M-dependence for a double array

of random variables and established a double array version of the Kolmogorov

SLLN for double arrays of random variables which are block-wise M-dependent

with respect to the blocks¹ Œ2k; 2kC1/ Œ2l; 2lC1/ W k  0; l  0º The results

of Móricz, Stadtmüller and Thalmaier [16] were generalized by Stadtmüller and

Thanh [27]

In the present paper, we study this problem for double arrays of block-wise

M-dependent random elements in Banach spaces Moreover, the conditions for

the strong law of large numbers are shown to provide exact characterizations

of Rademacher type p and stable type p Banach spaces The “asymmetric”

Marcinkiewicz–Zygmund type SLLN for double arrays are also considered Some

results in the literature, such as those in Gut [9], Gut and Stadtmüller [10], Móricz,

Stadtmüller and Thalmaier [16], Móricz, Su and Taylor [17], Quang and Thanh

[19], and Rosalsky and Thanh [21, 22] are improved and extended

The following notation will be used throughout this paper For x  0, let Œx

denote the greatest integer less than or equal to x For a; b 2 R, min¹a; bº and

max¹a; bº will be denoted, respectively, by a^b and a_b We use log to denote the

logarithm to the base 2 The symbol C denotes a generic constant (0 < C < 1)

which is not necessarily the same one in each appearance

The paper is organized as follows Technical definitions, notation, and the

lem-mas used in the proofs of the main results are presented in Section 2 The main

results are stated and proved in Sections 3 and 4 In Section 5, some examples are

presented to illustrate the sharpness of the main results

2 Preliminaries

Some definitions, notation, and preliminary results will be presented prior to

es-tablishing the main results LetX be a real separable Banach space with normk
k

A random element inX will be denoted by V or by Vmn, etc

The expected value or mean of a random element V , denoted by EV , is

de-fined to be the Pettis integral provided it exists That is, V has an expected value

EV 2 X if f EV / D E.f V // for every f 2 XwhereXdenotes the (dual)

space of all continuous linear functionals on X (see, e.g., Vakhania, Tarieladze

and Chobanyan [30, p 113]) If EkV k < 1, then (see, e.g., Taylor [28, p 40]) V

has an expected value But the expected value can exist when EkV k D 1, see,

e.g., Taylor [28, p 41]

A double array of random elements¹Vmn W m  1; n  1º is said to be

stochas-tically dominated by a random element V if for some constant C <1

P¹kVmnk > tº  CP ¹kV k > tº; t 0; m  1; n  1: (2.1)

For a double array¹Vmn W m  1; n  1º of identically distributed random

ele-ments, this condition will automatically hold with V D V11and C D 1 It follows

from Lemma 5.2.2 of [28, p 123] that stochastic dominance can be accomplished

by the random elements ¹Vmn W m  1; n  1º having a bounded absolute rth

moment (r > 0) Specifically, if

X

m1;n1

EkVmnkr <1;

then there exists a random element V such that EkV kp <1 for all 0 < p < r

and (2.1) holds with C D 1 (The condition r > 1 of [28, Lemma 5.2.2] is not

needed, as was pointed out by Adler, Rosalsky and Taylor [2])

Let¹YnW n  1º be a symmetric Bernoulli sequence; that is, ¹YnW n  1º be a

sequence of independent and identically distributed (i i d.) random variables with

P¹Y1D 1º D P ¹Y1D 1º D 1=2 Let X1D X  X  X 


and define

Let 1  p  2 Then X is said to be of Rademacher type p if there exists a

constant 0 < C <1 such that

Hoffmann–Jørgensen and Pisier [11] proved for 1  p  2 that a real separable

Banach space is of Rademacher type p if and only if there exists a constant 0 <

for every finite collection¹V1; : : : ; Vnº of independent mean 0 random elements

If a real separable Banach space is of Rademacher type p for some 1 < p 

2, then it is of Rademacher type q for all 1  q < p Every real separable

Banach space is of Rademacher type (at least) 1, while theLp- and `p-spaces are

of Rademacher type 2^ p for p  1 Every real separable Hilbert space and real

separable finite-dimensional Banach space is of Rademacher type 2 In particular,

the real line R is of Rademacher type 2

Let 0 < p  2 and let ¹n W n  1º be independent and identically

dis-tributed stable random variables, each with E exp.i t  / D exp¹ jtjpº The

sepa-rable Banach spaceX is said to be of stable type p ifP1

nD1nvnconverges a.s

whenever vn 2 X, n  1 withP1

nD1kvnkp <1

Equivalent characterizations of a Banach space being of stable type p,

proper-ties of stable type p Banach spaces, as well as various relationships between the

conditions Rademacher type p and stable type p may be found in [1,14,18,25,31]

Some of these properties and relationships will now be summarized

(i) Every separable Banach spaceX is of stable type p for all 0 < p < 1

(ii) For q  2, the Lq-spaces and lq-spaces are of stable type 2, while for 1

q < 2, the Lq-spaces and lq-spaces are of stable type p for all 0 < p < q

but are not of stable type q

(iii) Every separable Hilbert space and separable finite dimensional Banach space

is of stable type 2

(iv) For 1 p < 2, X is of stable type p if and only if X is of Rademacher type

p1for some p12 pI 2

(v) For pD 2, X is of stable type 2 if and only if X is of Rademacher type 2

The concept of block-wise M-dependence was introduced by Móricz,

Stadt-müller and Thalmaier [16] and by StadtStadt-müller and Thanh [27] as follows Let M

be a nonnegative integer A finite collection of random elements¹V11; : : : ; Vmnº

is said to be M-dependent if either m _ n  M C 1 or m _ n > M C 1

and the random elements¹V11; : : : ; Vijº are independent of the random elements

¹Vkl; : : : ; Vmnº whenever k i /_ l j / > M A double array of random

vari-ables¹Vmn W m  1; n  1º is said to be M-dependent if for each m  1; n  1,

the random elements¹V11; : : : ; Vmnº are M-dependent

The notion of p-orthogonality of random elements was introduced by

Howell and Taylor [12], and by Móricz, Su and Taylor [17] A finite collection

of random elements¹V11; : : : ; Vmnº is said to be p-orthogonal (1  p < 1) if

EkVijkp<1 for all 1  i  m, 1  j  n and

for all choices of 1 i  k  m, 1  j  l  n for all constants ¹a11; : : : ; aklº,

and for all permutations 1and 2of the integers¹1; : : : ; kº and ¹1; : : : ; lº,

re-spectively An array of random elements ¹Vmn W m  1; n  1º is said to be

p-orthogonal(1 p < 1) if ¹V11; : : : ; Vmnº is p-orthogonal for all m ^ n  2.

We refer to Howell and Taylor [12] and Móricz, Su and Taylor [17] for a detailed

discussion of p-orthogonality Móricz, Su and Taylor [17] also established the

Rademacher–Menshov type SLLN for double arrays of p-orthogonal random

el-ements Chobanyan and Mandrekar [5] and Quang and Thanh [20] studied the

SLLN problem for p-orthogonal random elements under rearrangements

Let ¹!.k/ W k  1º and ¹.k/ W k  1º be strictly increasing sequences of

positive integers with !.1/D .1/ D 1 and set

kl D!.k/; !.k C 1/
 .l/; .l C 1/
:

We say that an array ¹Vij W i  1; j  1º of random elements is block-wise

M-dependent (resp., block-wise p-orthogonal (1  p < 1)) with respect to

the blocks ¹kl W k  1; l  1º, if for each k and l, the random variables

¹Vij W i; j / 2 klº are M-dependent (resp., p-orthogonal) Thus the random

ele-ments with indices in each block are M-dependent (resp., p-orthogonal) but there

are no M-dependence (resp., p-orthogonality) requirements between the random

elements with indices in different blocks; even repetitions are permitted

For¹!.k/ W k  1º, ¹.k/ W k  1º and ¹kl W k  1; l  1º as above, and for

m 0; n  0; k  1; l  1, we introduce the following notation:

In [19], the following relations are verified.

(i) If !kD k D 2k 1, k 1, then

cmnD 1; m 0; n  0'.m; n/D 1; .m; n/D 1; m 1; n  1: (2.3)(ii) If !kD k D bqk 1c for all large k where q > 1, then

cmnD O.1/ and '.m; n/ D O.1/; .m; n/ D O.1/: (2.4)

(iii) If !kD b2k˛c, kD b2kˇc for all large k where 0 < ˛; ˇ < 1, then

cmnD O.m.1 ˛/=˛n.1 ˇ /=ˇ/;

'.m; n/D O Log m/.1 ˛/=˛.Log n/.1 ˇ /=ˇ/:

(2.5)

(iv) If !kD bk˛c, kD bkˇc, k  1 where ˛ > 1; ˇ > 1, then

cmnD O.2m=˛2m=ˇ/ and '.m; n/D O.m1=˛n1=ˇ/: (2.6)

(v) If ˇk D kD k, k  1, then

cmnD 2mCn; m; n 0 and '.m; n/ D O.mn/: (2.7)The following lemma establishes the maximal inequality for double sums of

independent random variables which is due to Rosalsky and Thanh [21]

Lemma 2.1 Let¹Vij W 1  i  m; 1  j  nº be a double array of independent

mean0 random elements in a real separable Rademacher type p (1  p  2)

Banach space Then

E max

1km 1ln

j D1Vij,C is a constant independent of m and n

Using the method of Móricz, Stadtmüller and Thalmaier [16] and Lemma 2.1,

we can establish the maximal inequality for double sums of M-dependent random

elements In the proof of Lemma 2.2, we sometimes denote Vij by V iI j / for

convenience

Lemma 2.2 Let¹Vij W 1  i  m; 1  j  nº be a double array of

M-depen-dent mean0 random elements in a real separable Rademacher type p (1 p  2)

Banach space Then

E max

1km 1ln

j D1Vij,C is a constant independent of m and n

Proof If m_ n  M C 1, Lemma 2.2 is trivial So, we only need to consider

m_ n  M C 1 In the case p D 1, note that for all m  1 and n  1

The next lemma is a Rademacher–Menshov maximal inequality for double

ar-rays of p-orthogonal random elements in Banach spaces For a proof see Móricz,

Su and Taylor [17]

Lemma 2.3 Let¹Vij W 1  i  m; 1  j  nº be a double array of random

elements in a real separable Rademacher typep (1  p  2) Banach space If

j D1Vij,C is a constant independent of m and n

The following lemma establishes the strong law of large numbers for double

arrays of arbitrary random elements It is based on Thanh [29, Theorem 3.1] and

the remark thereafter concerning the case 0 < p 1

Lemma 2.4 Let¹Vij W i  1; j  1º be a double array of random elements and

Lemma 2.5 below can be found in [24] concerning the characterization of stable

type p Banach spaces It may also be compared with [31, Theorem V.9.1]

Lemma 2.5 Let1  p < 2 and let X be a real separable Banach space For

every sequence¹Wk W k  1º of independent and symmetric random elements in

X which are stochastically dominated by a random element V with EkV kp <1

suppose that

Pn kD1Wk

n1=p ! 0 a.s

ThenX is of stable type p

The final lemma is a double sum analogue of the Toeplitz lemma (see, e.g.,

Loève [13, p 250]) and is due to Stadtmüller and Thanh [27]

Lemma 2.6 Let¹amnij W 1  i  m; 1  j  n; m  1; n  1º be an array of

positive constants such that

3 Strong laws for the non-identically distributed case

We are now in the position to present the first main result of the paper The

fol-lowing theorem extends the main results of Rosalsky and Thanh [21] to the

block-wise M-dependent case It demonstrates that the geometric condition of a Banach

space being of Rademacher type p and the limit theorem (3.3)) (3.4) holding

completely characterize each other

Theorem 3.1 LetX be a real separable Banach space and let 1 p  2 Then

the following two statements are equivalent

(i) X is of Rademacher type p

(ii) For every double array of mean 0 random elements ¹VmnW m  1; n  1º

which is block-wise M-dependent with respect to some array of blocks

¹kl W k  1; l  1º and every nondecreasing array of positive constants

¹anW n  1º and ¹bn W n  1º satisfying

Proof We first prove the implication (i)) (ii) Set

It follows from (3.3) thatP1

It follows from (3.2) that

The conclusion (3.4) follows from (3.5), (3.6), (3.7) and Lemma 2.6

To verify the implication (ii)) (i), assume that (ii) holds Let ¹WnW n  1º be

a sequence of independent mean 0 random elements inX such that

Then¹Vmn W m  1; n  1º is a double array of block-wise 0-dependent random

elements with respect to the blocks¹Œ2k; 2kC1/ Œ2l; 2lC1/W k  0; l  0º in X

Set an D bn D n, n  1 Then (3.1) and (3.2) hold and, recalling (2.3), we see

that (3.3) and (3.8) coincide Thus, by (ii),

lim

m_n!1

1mn

We now present an alternative version of Theorem 3.1 When M D 0, am m,

bn  n and X is the real line, the implication (i) ) (ii) of Theorem 3.2 was

obtained by Quang and Thanh [19]

Theorem 3.2 LetX be a real separable Banach space and let 1 p  2 Then

the following two statements are equivalent

(i) X is of Rademacher type p

(ii) For every double array of mean 0 random elements ¹VmnW m  1; n  1º

which is block-wise M-dependent with respect to some array of blocks

¹kl W k  1; l  1º and every nondecreasing array of positive constants

¹anW n  1º and ¹bn W n  1º satisfying (3.1) and (3.2), the condition

Proof We verify the implication (i)) (ii) Define Tkl.mn/, k; l /2 Im, m  0,

n 0 as in the proof of Theorem 3.1 and set

It follows from (3.9) thatP1

i D1

P1

j D1E.ij/p <1 The rest of the argument issimilar to that at the end of the proof of Theorem 3.1

The next theorem, when specialized to¹Vmn W m  1; n  1º being

p-orthogo-nal, am  m˛ and bn  nˇ where ˛ > 0, ˇ > 0, reduces to a result of Rosalsky

and Thanh [22], and Móricz, Su and Taylor [17] by taking

klD2k 1; 2k
 2l 1; 2l
; k 1; l  1and recalling (2.3)