Mean Convergence Theorems for Double Arrays of Random Elements in Banach Spaces under Some Conditions of Uniform Integrability Le Van Dunga Abstract.. This convergence results are of the
Trang 1Mean Convergence Theorems for Double Arrays of Random Elements in Banach Spaces under Some
Conditions of Uniform Integrability
Le Van Dung(a)
Abstract In this paper, we establish some mean convergence theorems for the double sums Pu m
i=x m
Pv n
j=y nAmnijVij, for an array of random elements
{Vij; i, j ∈ Z}in a real separable Banach space and an array of random variables
{Amnij; xm 6 i 6 um, yn 6 j 6 vn, m > 1, n > 1}, where{xm, m ≥ 1},
{um, m ≥ 1},{yn, n ≥ 1}and{vn, n ≥ 1}are four sequences of integers.
I Introduction
Consider an array{Vij; i, j ∈ Z}of random elements defined on a probability space
(Ω, F , P)and an array{Amnij; xm 6 i 6 um, yn 6 j 6 vn, m > 1, n > 1}of random variables Let{xn, n ≥ 1},{un, n ≥ 1},{yn, n ≥ 1}and{vn, n ≥ 1}be four sequences
of integers such thatun−xn > 0for alln ≥ 1andun−xn → ∞asn → ∞,vn−yn> 0
for alln ≥ 1andvn− yn→ ∞asn → ∞ In this paper, mean convergence theorems will be established This convergence results are of the form
k
u m
X
i=x m
v n
X
j=y n
AmnijVijk−→ 0.Lp
Limit theorems for weighted sums (with or without random indies) for random variables (real-valued or Banach space-valued) are studied by many authors in [1], [4], [7] Recently, M Ordonez Cabrera and A Volodin obtained a mean convergence theorem for the weighted sumsPv n
j=u nAnjVj in L1 under some conditions of uniform integrability (see [2]) However, the same problems for double sums have not yet been studied
II Preliminaries
Theexpected valueormean of a random elementV, denoted byEV is defined to be thePettis integral provided it exists That is,V hasexpected value EV ∈ X if f (EV ) =
Ef (V )for everyf ∈ X∗, where X∗ denotes the (dual) space of all continuous linear functionals onX A sufficient condition forEV to exist is thatEkV k < ∞(see, e.g., Taylor [6], p.40)
1 NhËn bµi ngµy 15/8/2007 Söa ch÷a xong ngµy 12/12/2007.
Trang 2Consider an array of random variables {Anj; un 6 j 6 vn, n ≥ 1} and an array
{Vj, j ∈ Z} of random elements in a real separable Banach space X with a normal
k · k, defined on a probability space(Ω, F , P) Let{Fn, n ≥ 1}be a sequence of subσ -algebras ofF For eachn ≥ 1, denote byEFn(Y )the conditional expectation of random variableY toFn The array{Vj, j ∈ Z} is said to be{Anj}-conditionally uniformly
pth-order integrable relative to {Fn} if, for each > 0, there existsao = ao() > 0
such that
sup
n≥1
v n
X
j=u n
|Anj|pEFn(kVjkpI(kVjk > ao)) < a.s
The notion of conditionally uniformly integrable was introduced by M Ordonez Cabr-era and A Volodin [2] IfAnj = anj (nonrandom) a.s., for allun 6 j 6 vn, n ≥ 1, and
Fn = {∅, Ω}for all n ≥ 1, then {Anj}-conditionally uniformlypth-order integrable relative to{Fn} coincides{|anj|p}-uniform integrability for the array of random vari-ables{kVjkp, j ∈ Z}
Lemma 1 Let {kmn, m ≥ 1, n ≥ 1} be an array of positive integers such that
limm∨n→∞kmn = ∞, and let {Xij; i, j ∈ Z} be an array of random variables such that
sup
a>0
sup
n≥1
1
kmn
u m
X
i=x m
v n
X
i=y n
aP{|Xij| > a} 6 M < ∞, (2.1) and
lim
a→+∞sup
n≥1
1
kmn
u m
X
i=x m
v n
X
j=y n
aP{|Xij| > a} = 0 (2.2)
Then
1
kpmn
u m
X
i=x m
v n
X
i=y n
E(|Xij|pI(|Xij| 6 kmn)) → 0 as m ∨ n → ∞ (p > 1) (2.3)
Proof
1
kmnp
u m
X
i=x m
v n
X
j=y n
E(|Xij|pI(|Xij| 6 kmn)) =
= 1
kmnp
u m
X
i=x m
v n
X
j=y n
E(|Xij|pI(|Xij| 6 1)) + 1
kpmn
u m
X
i=x m
v n
X
j=y n
k mn
X
l=2
E(|Xij|pI(l − 1 < |Xij| 6 l))
=: Amn+ Bmn
Trang 3We first verify that
lim
m∨n→∞Amn= 0
Amn= 1
kpmn
u m
X
i=x m
v n
X
i=y n
E(|Xij|pI(|Xij| 6 1))
= 1
kpmn
u m
X
i=x m
v n
X
j=y n
∞
X
l=1
E(|Xij|pI( 1
l + 1 < |Xij| 6 1
l))
!
6 1
kmnp
u m
X
i=x m
v n
X
j=y n
∞
X
l=1
1
lpP{ 1
l + 1 < |Xij| 6 1
l}
!
= 1
kpmn
u m
X
i=x m
v n
X
j=y n
∞
X
l=1
1
lp
P{|Xij| > 1
l + 1} − P{|Xij| > 1
l}
!
= 1
kpmn
u m
X
i=x m
v n
X
j=y n
∞
X
l=1
1
lp − 1 (l + 1)p
P{|Xij| > 1
l + 1}
!
= 1
kmnp−1
∞
X
l=1
1
lp − 1 (l + 1)p
(l + 1) 1
kmn
u m
X
i=x m
v n
X
j=y n
1
l + 1P{|Xij| > 1
l + 1}
!
6 M 1
kmnp−1
∞
X
l=1
1
lp − 1 (l + 1)p
(l + 1) (by (2.1))
= M 1
kmnp−1
(
∞
X
l=1
1
Next, we will show that
lim
m∨n→∞Bmn= 0
In deed, since
k mn
X
l=2
(lp− (l − 1)p)
kp−1mn(l − 1) =
1
kmnp−1
k mn
X
l=2
lp
(l − 1)l +
kmn
kmn− 1 −
2p−1
kp−1mn
6 2
kp−1mn
k mn
X
l=2
lp−2+ kmn
kmn− 1 6 4,
Trang 4By (2.2) we have
Bmn = 1
kmnp
u m
X
i=x m
v n
X
j=y n
k mn
X
l=2
E(|Xij|pI(l − 1 < |Xij| 6 l))
!
6 1
kpmn
u m
X
i=x m
v n
X
j=y n
k mn
X
l=2
lpP{l − 1 < |Xij| 6 l}
!
= 1
kmnp
u m
X
i=x m
v n
X
j=y n
k mn
X
l=2
lp[P{|Xij| > l − 1} − P{|Xij| > l}]
!
= 1
kmnp
u m
X
i=x m
v n
X
j=y n
k mn
X
l=2
[lp− (l − 1)p]P{|Xij| > l − 1}
!
=
k mn
X
l=2
"
(lp− (l − 1)p)
kmnp−1(l − 1)
1
kmn
u m
X
i=x m
v n
X
i=y n
(l − 1)P{|Xij| > l − 1}
!#
6 4 1
kmn
u m
X
i=x m
v n
X
j=y n
(l − 1)P{|Xij| > l − 1}
6 4 sup
m≥1,n≥1
1
kmn
u m
X
i=x m
v n
X
j=y n
(l − 1)P{|Xij| > l − 1}
So the conclusion (2.3) follows from (2.4) and (2.5)
Corollary 1 Let {amnij; xm 6 i 6 um, yn 6 j 6 vn, m ≥ 1, n ≥ 1} be an array of constants satisfying
u m
X
i=x m
v n
X
j=y n
|amnij| 6 M < ∞ and
sup
x m 6i6u m ,y n 6j6v n
|amnij| → 0 as m ∨ n → ∞
Suppose that the array of random variables{Xij; i, j ∈ Z}is{|amnij|}-uniformly inte-grable in the sense that
lim
a→+∞ sup
m≥1,n≥1
u m
X
j=x
u n
X
j=y
|amnij|E(|Xij|I(|Xij| > a)) = 0
Trang 5Setcmn = 1
sup
x m 6i6u m ,y n 6j6v n
|amnij|, then
u m
X
i=x m
v n
X
j=y n
|amnij|qE(|Xij|qI(|Xij| 6 cmn)) → 0 as m ∨ n → ∞ (q > 1)
Proof Applying Lemma 1 with kmn= [cmn] + 1 and Xij is replaced by amnijcmnXij
Let{Yn, n ≥ 1}be a symmetricBernoulli sequence, that is, {Yn, n ≥ 1}is a se-quence of independent and identically random variables withP{Y1 = 1} = P{Y1 =
−1} = 1/2 LetX∞= X × X × X × and define
C(X ) = {(v1, v2, ) ∈ X∞ :
∞
X
n=1
vnvn converges in probability}
Let1 6 p 6 2 Then the real separable Banach spaceX is said to be ofRademacher type pif there exists a constantC (0 < C < ∞)such that
Ek
∞
X
n=1
Ynvnkp
6 C
∞
X
n=1
kvnkp for all (v1, v2, v3, ) ∈ C(X ) (2.6)
Hoffmann-Jφrgensen and Pisier [6] proved for1 6 p 6 2that a real separable Banach space is ofRademacher type pif only if there exists a constant C (0 < C < ∞)such that
E
n
P
j=1
Vjk
p
6 C
n
X
j=1
for every finite collection{V1, V2, , Vn}of independent mean 0 random elements
If a real separable Banach space is of Rademacher type p for some1 < p 6 2, then
it is of Rademacher type r for all1 6 r < p Every real separable Banach space is of Rademacher type (at least) 1 while theLp-space andlp-space are of Rademacher type
2 ∧ pfor p ≥ 1 Every real separable Hilbert space and real finite dimensional Banach space is of Rademacher type 2
Fora, b ∈ R, min{a, b}andmax{a, b} will be denoted, respectively, bya ∧ band
a∨b Throughout this paper, the symbolCwill denote a generic constant(0 < C < ∞)
which is not necessarily the same one in each appearance
III Main results
Theorem 1 Let 1 6 r < p 6 2 Let {Vij; i, j ∈ Z}be an array of mean 0 random elements defined on a probability space(Ω, F , P)and taking values in a real separable
Trang 6Rademacher typepBanach spaceX,{Amnij; xm 6 i 6 um, yn 6 j 6 vn, m > 1, n > 1}
be an array of random variables satisfying
u m
X
i=x m
v n
X
j=y n
E|Amnij|r
and
sup
x m 6i6u m ,y n 6j6v n
E|Amnij|r→ 0 as m ∨ n → ∞ (3.2)
Let{Fmn; m ≥ 1, n ≥ 1} be an array of subσ-algebras of F such thatAmnij, xm 6
i 6 um, yn 6 j 6 vn are Fmn-measurable Suppose that {Vij; i, j ∈ Z} is{Amnij} -conditionally uniformlyrth-order integrable relative to{Fmn}in the sense that
lim
a→+∞ sup
m≥1,n≥1
u m
X
i=x m
v n
X
j=y n
|Amnij|rEFmn(kVijkrI(kVijk > a)) = 0 a.s (3.3)
Suppose for allm ≥ 1andn ≥ 1 the array{AmnijVij; xm 6 i 6 um, yn 6 j 6 vn}
is comprised of independent random elements, Amnij and Vij are independent for all
m ≥ 1, n ≥ 1, xm 6 i 6 um, yn6 j 6 vn Then
u m
X
i=x m
v n
X
j=y n
AmnijVij
L r
Proof Since (3.3) there exists ao > 0 such that
E
u m
X
j=x m
v n
X
j=y n
|Amnij|rEFmn(kVijkrI(kVijk > ao))
!
< 1, m ≥ 1, n ≥ 1
Thus
EkAmnijVijI(kVijk > ao)k < 1 for all xm 6 i 6 um, yn6 j 6 vn, m ≥ 1, n ≥ 1
(3.5) For all m ≥ 1, n ≥ 1, xm 6 i 6 um, yn6 j 6 vn, (by (3.1) and (3.5) we have
EkAmnijVijk = EkAmnijVijI(kVijk 6 ao)k + EkAmnijVijI(kVijk > ao)k
6 aoE|Amnij| + EkAmnijVijI(kVijk > ao)k < ∞
implying that E(AmnijVij) exists Set cmn = 1
sup
x m 6i6u m ,y n 6j6v n
E|Amnij|r,
Vmnij0 = VijI(kVijk 6 cmn), Vmnij00 = VijI(kVijk > cmn),
Trang 7b0mnij = EVmnij0 , b00mnij = EVmnij00 Observe that for each i and j, xm 6 i 6 um, yn 6 j 6 vn, then Vij = (Vmnij0 − b0
mnij) + (Vmnij00 − b00
mnij) And since Amnij and Vij are independent for each m, n, i, j we have E(Amnij(Vij0 − b0mnij)) = E(Amnij(Vmnij00 − b00mnij)) = 0
Hence
E
u m
X
i=x m
v n
X
j=y n
AmnijVij
r
= E
u m
X
i=x m
v n
X
j=y n
Amnij(Vmnij0 − b0mnij) +
u m
X
i=x m
v n
X
j=y n
Amnij(Vmnij00 − b00mnij)
r
6 CE
u m
X
i=x m
v n
X
j=y n
Amnij(Vmnij0 − b0mnij)
r
+ CE
u m
X
i=x m
v n
X
j=y n
Amnij(Vmnij00 − b00mnij)
r
(by cr-inequality)
6 C E
u m
X
i=x m
v n
X
j=y n
Amnij(Vmnij0 − b0mnij)
p!r/p
+ CE
u m
X
i=x m
v n
X
j=y n
Amnij(Vmnij00 − b00mnij)
r
6 C
u m
X
i=x m
v n
X
j=y n
EkAmnij(Vmnij0 − b0mnij)kp
!r/p
+ C
u m
X
i=x m
v n
X
j=y n
EkAmnij(Vmnij00 − b00mnij)kr
6 C
u m
X
i=x m
v n
X
j=y n
E|Amnij|pE(kVijkpI(kVijk 6 cmn))
!r/p
+ C
u m
X
i=x m
v n
X
j=y n
E(kAmnijVijkrI(kVijk > cmn))
Now, by (3.3), for arbitrary > 0 there exists ao > 0 such that for all a ≥ ao We have
m≥1,n≥1
u m
X
j=x m
v n
X
j=y n
|Amnij|rEFmn(kVijkrI(kVijk > a))
!
< (3.6) This implies
sup
m≥1,n≥1
u m
X
i=x
v n
X
j=y
E|Amnij|rE(kVijkrI(kVijk > a)) < ∀a ≥ ao (3.7)
Trang 8Note that (3.6) means {kVijkr
; i, j ∈ Z} is {E|Amnij|r}-uniformly integrable, and then
by Corollary 1 with q = p/r, Xij = kVijkr and amnij = E|Amnij|r we get
u m
X
i=x m
v n
X
j=y n
|Amnij|pE(kVijkpI(kVijk 6 cmn)) → 0 as m ∨ n → ∞
On the other hand (3.6) also implies
u m
X
i=x m
v n
X
j=y n
E(kAmnijVijkrI(kVijk > cmn) → 0 as m ∨ n → ∞
Thus
E
u m
X
i=x m
v n
X
j=y n
AmnijVij
r
→ 0 as m ∨ n → ∞
The proof is completed
Theorem 2 Let 0 < r < 1 Let{Vij; i, j ∈ Z} be an array of random elements in a real separable Banach space,{Amnij; xm 6 i 6 um, yn 6 j 6 vn, m > 1, n > 1}be an array of random variables satisfying
u m
X
i=x m
v n
X
j=y n
(E|Amnij|)r 6 M < ∞ (3.8) and
sup
x m 6i6u m ,y n 6j6v n
E|Amnij| → 0 as m ∨ n → ∞ (3.9)
Let {Fmn; m ≥ 1, n ≥ 1} be an array of sub σ-algebras of F, and suppose that
Amnij, xm 6 i 6 um, yn 6 j 6 vn are Fmn-measurable Suppose that {Vij; i, j ∈ Z}
is{Amnij}-conditionally uniformlyrth-order integrable relative to {Fmn}in the sense that(3.3)holds
Then
u m
X
i=x m
v n
X
j=y n
AmnijVij −→ 0 as m ∨ n → ∞.Lr (3.10)
Proof By (3.3), for arbitrary > 0 there exists a > 0 such that
m≥1,n≥1
u m
X
j=x m
v n
X
j=y n
|Amnij|rEFmn(kVijkrI(kVijk > a))
!
<
2, this implies
u m
X
i=x
v n
X
j=y
E(kAmnijVijkrI(kVijk > a)) <
2, m ≥ 1, n ≥ 1.
Trang 9On the other hand, since
E
u m
X
j=x m
v n
X
j=y n
AmnijVijI(kVijk 6 a)
r!1/r
6 E
u m
X
j=x m
v n
X
j=y n
AmnijVijI(kVijk 6 a)
6
u m
X
j=x m
v n
X
j=y n
E(kAmnijVijkI(kVijk 6 a)) 6 a
u m
X
j=x m
v n
X
j=y n
E|Amnij|
6 a
u m
X
j=x m
v n
X
j=y n
(E|Amnij|)r
!
sup
x m 6i6u m ,y n 6j6v n
(E|Amnij|)1−r
x m 6i6u m ,y n 6j6v n
(E|Amnij|)1−r → 0 as m ∨ n → ∞, there exists mo, no such that for all (m ∨ n) ≥ (mo∨ no),
E
u m
X
i=x m
v n
X
j=y n
AmnijVijI(kVijk 6 a)
r
6
Hence,
E
u m
X
i=x m
v n
X
j=y n
AmnijVij
r
= E
u m
X
i=x m
v n
X
j=y n
AmnijVijI(kVijk 6 a) +
u m
X
i=x m
v n
X
j=y n
AmnijVijI(kVijk > a)
r
6 E
u m
X
i=x m
v n
X
j=y n
AmnijVijI(kVijk 6 a)
r
+ E
u m
X
i=x m
v n
X
j=y n
AmnijVijI(kVijk > a)
r
6 E
u m
X
i=x m
v n
X
j=y n
AmnijVijI(kVijk 6 a)
r
+
u m
X
i=x m
v n
X
j=y n
E(kAmnijVijkrI(kVijk > a)) < for all (m ∨ n) ≥ (mo∨ no),
which completes the proof
Remark A basic hypothesis in all of the previous theorems in this section is condi-tion "{Amnij; xm 6 i 6 um, yn 6 j 6 vn, m > 1, n > 1}areFmn-measurable for each
m ≥ 1,n ≥ 1" A particular case of great interest, in which this condition is satisfied,
is whenFmn = σ(Amnij, xm 6 i 6 um, yn 6 j 6 vn), i.e., whenFmnis the σ-algebra generated by{Amnij; xm 6 i 6 um, yn 6 j 6 vn, m > 1, n > 1}, for eachm ≥ 1and
Trang 10n ≥ 1.
references
[1] A Adler, A Rosalsky and R I Taylor, A weak law for normed weighted sum of ran-dom elements in Rademacher typepBanach space, Journal of Multivariate Analysis,
37 (2), 1991, 259-268
[2] M O Cabrera, A I Volodin, Convergence of randomly weighted sums of Banach-space-valued random elements under some conditions of uniform integrability, Journal
of Mathematical Sciences, 138 (1), 2006, 5450-5459
[3] J Hoffmann-Jφrgensen and G Pisier, The law of larger numbers and the central limit theorem in Banach space, Ann Probab., 4, 1976, 587-599
[4] A Rosalsky, M Sreehari, and A I Volodin, Mean convergence theorem with or with-out random indies for randomly weighted sums of random elements in Rademacher typepBanach space, Stochastic Analysis and Applications ,21 (5) (2003), 1169-1187 [5] A Rosalsky, M Sreehari and A I Volodin, A weak law with random indies for randomly weighted sums of rowwise independent random elements in Rademacher typepBanach space, Calcutta Staist Asoc Bull.,52, 2002, 85-98
[6] R L Taylor, Stochastic Convergence of Weighted Sums of Random Elements in Linear Space, Lecture Notes in Mathematic; Springer-Verlag: Berlin, 672, 1978 [7] L V Thanh, Mean convergence theorems and weak laws of large numbers for double arrays of random variables, Journal of Applied Mathemmatics and Stochastic Analy-sis, 2006, 1-15
Tóm tắt
Một số định lý hội tụ trung bình đối với mảng hai chỉ số các phần
tử ngẫu nhiên trong không gian Banach với điều kiện khả tích đều
Trong bài báo này, chúng tôi thiết lập một số định lý hội tụ trung bình cho tổng hai chỉ sốPu m
i=x m
Pv n
j=y nAmnijVij, đối với mảng các phần tử ngẫu nhiên{Vij; i, j ∈ Z}nhận giá trị trên không gian Banach thực và mảng các đại lượng ngẫu nhiên{Amnij; xm 6
i 6 um, yn 6 j 6 vn}, trong đó{xm, m ≥ 1},{um, m ≥ 1},{yn, n ≥ 1}và{vn, n ≥ 1}
là bốn dãy số nguyên
(a) 13thprobability class, Department of postgraduate, Vinh University.