MEAN CONVERGENCE THEOREM FOR MULTIDIMENSIONAL ARRAYS OF RANDOM ELEMENTS IN BANACH SPACES

where (n1,n2, . . . ,nd)= n∈ Z+. Recently, Thanh [11] proved (1.1) under condition of uniform integrability of {|Xn|p, n∈ Z+}. Mean convergence theorems for sums of random elements Banach-valued are studied by many authors. The reader may refer to Wei and Taylor [12], Adler et al. [2], Rosalsky and Sreehari [9], or more recently, Rosalsky et al. [10], Cabrera and Volodin [3]. However, we are unaware of any literature of investigation on the mean convergence for multidimensional arrays of random elements in Banach spaces. Consider a d-dimensional array {Vn, n ∈ Z+} of independent random elements defined on a probability space (Ω, ,P) and taking values in a real separable Banach space


Introduction
where d is a positive integer, denote the positive integer d-dimensional lattice points.The notation m ≺ n, where m = (m 1 ,m 2 ,...,m d ) and n = (n 1 ,n 2 ,...,n d ) ∈ Z d + , means that m i n i , 1 i d, |n| is used for d i=1 n i .Gut [5] proved that if {X, X n ,n ∈ Z d + } is a d-dimensional array of i.i.d.random variables with E|X| p < ∞ (0 < p < 2) and EX = 0 if 1 p < 2, then j≺n X j |n| 1/p −→ 0 in L p as min where (n 1 ,n 2 ,...,n d ) = n ∈ Z d + .Recently, Thanh [11] proved (1.1) under condition of uniform integrability of Mean convergence theorems for sums of random elements Banach-valued are studied by many authors.The reader may refer to Wei and Taylor [12], Adler et al. [2], Rosalsky and Sreehari [9], or more recently, Rosalsky et al. [10], Cabrera and Volodin [3].However, we are unaware of any literature of investigation on the mean convergence for multidimensional arrays of random elements in Banach spaces.
Consider a d-dimensional array {V n , n ∈ Z d + } of independent random elements defined on a probability space (Ω,Ᏺ,P) and taking values in a real separable Banach space ᐄ with norm • .In the current work, we establish the convergence in mean of order p (1 p < 2) of the sums j≺n V j /|n|  [11].While the proof of Theorem 2.1 and the proof of the main result in Thanh [11] are similar, we will show in Theorem 2.2 that the implication in Theorem 2.1 indeed completely characterizes stable-type p Banach spaces.
Let 0 < p 2 and let {θ n , n 1} be independent and identically distributed stable random variables each with characteristic function φ(t) = exp{−|t p |}.The real separable Banach space ᐄ is said to be of stable-type p if ∞ n=1 θ n v n converges a.s.whenever Equivalent characterizations of a Banach space being of stable-type p, properties of stable-type p Banach spaces, as well as various relationships between the conditions Rademacher-type p, and stable-type p may be found by Woyczy ński in [13], by Marcus and Woyczy ński in [7], and by Pisier in [8], see also the discussion by Adler et al. in [1].We now mention explicitly some characterizations of this concept.The first theorem was obtained by Mandrekar and Zinn [6] and by Marcus and Woyczy ński [7].
Theorem 1.1.Let 1 p < 2 and let ᐄ be a real separable Banach space.Then the following statements are equivalent.
(i) ᐄ is of stable-type p.
(ii) For every symmetric random elements V , the condition where {V j , j 1} are independent copies of V .
where { n , n 1} is a Rademacher sequence.
The symbol C denotes throughout a generic constant (0 < C < ∞) which is not necessarily the same one in each appearance.

Main results
We can now present the main results.Theorem 2.1 is a stable-type p Banach space version of the main result of Thanh [11].
Le Van (2.2) Proof.For arbitrary > 0, there exists M > 0 such that Since ᐄ is of stable-type p and p < 2, it is of Rademacher-type q for some p < q < 2. Thus (2.5)While the proof of Theorem 2.1 and the proof of the main result in Thanh [11]

imply (2.2).
Proof.The implication ((i)⇒(ii)) is precisely Theorem 2.1, whereas the implication ((ii)⇒(iii)) is immediate.It remains to verify the implication ((iii)⇒(i)).For reasons of clarity, we collect some of the steps in the following lemmas.The first lemma is a slight modification of de Acosta [4, Theorem 3.1] which holds for sequences of independent identically distributed random elements.The proof of the following modification can be obtained from de Acosta [4, Theorem 3.1] line by line, and so will be omitted.
if and only if Lemma 2.4.Let 1 p < 2 and let ᐄ be a real separable Banach space.Suppose that for every sequence {V ,W k , k 1} of independent mean-zero random elements in ᐄ, the conditions (2.14) is an array of independent mean-zero random elements, and (2.17)By Lemma 2.4, ᐄ is of stable-type p.
Remark 2.5.In Theorem 2.1, if 0 < p < 1, then the independence hypothesis and the hypothesis that the {V n , n ∈ Z d + } have mean-zero are not needed for the theorem to hold.Indeed, for arbitrary > 0, define V n and V n , n ∈ Z d + as in the proof of Theorem 2.

( 2 .
18) 1/p , n ∈ Z d + , under the condition that { V n p , n ∈ Z d + } is uniformly integrable.The main results of this paper are Theorems 2.1 and 2.2.Theorem 2.1 is a stable-type p Banach space version of the main result of Thanh

Thanh 3
Theorem 2.1.Let {V n , n ∈ Z d + } be a d-dimensional array of independent mean-zero random elements in a real separable stable-type p (1 p < 2) Banach space ᐄ.If are similar, we now show in Theorem 2.2 that the implication ((2.1)⇒(2.2)) in Theorem 2.1 indeed completely characterizes stable-type p Banach spaces.Theorem 2.2.Let 1 p < 2 and let ᐄ be a real separable Banach space.Then the following statements are equivalent.(i) ᐄ is of stable-type p. (ii) For every d-dimensional array {V n , n ∈ Z d + } of independent mean-zero random elements in ᐄ, the condition (2.1) implies (2.2). (iii) For every d-dimensional array {V ,V n , n ∈ Z d + } of independent mean-zero random elements in ᐄ, the conditions Proof ofLemma 2.4.Let {ε k , k 1} be a Rademacher sequence and let {x k , k 1} be a sequence of elements in ᐄ such that ᐄ is of stable-type p.The proof of Lemma 2.4 is completed.We now prove the implication ((iii)⇒(i)).If d = 1, then the conclusion follows directly from Lemma 2.4.So, we can assume that d 2. Let {V ,W k , k 1} be a sequence of independent mean-zero random elements in