Random Elements In Banach Spaces Research Articles

Almost sure summability of the maximal normed partial sums of m-dependent random elements in Banach spaces

Almost sure summability of the maximal normed partial sums of m-dependent random elements in Banach spaces

Optimal moment conditions for complete convergence for maximal normed weighted sums from arrays of rowwise independent random elements in Banach spaces

Optimal moment conditions for complete convergence for maximal normed weighted sums from arrays of rowwise independent random elements in Banach spaces

Strong Limit Theorems for Weighted Sums of Random Elements in Banach Spaces

We propose an approach to the strong law of large numbers for weighted sums of random elements, where the result of Jajte [4] will be extended to the Banach space setting. Some typical applications of the main results are given.

On the complete convergence of sequences of random elements in Banach spaces

For a sequence $${\{X_{n}, {n \geqslant 1}\}}$$ of independent random elements taking values in a Rademacher type p Banach space with the k-th partial sum $${S_{k} (k \geqslant 1)}$$ , we provide necessary and sufficient conditions for the convergence of $${\sum_{n=1}^{\infty} \frac{1}{n}\,\mathbb{P} ({\rm max}_{1 \leqslant{k}\leqslant{n}} \|S_{k}\| > \varepsilon{n}^{\alpha})}$$ and $${\sum_{n=1}^{\infty} \frac{{\rm log} n}{n} \, \mathbb{P} (\max_{1\leqslant{k}\leqslant{n}} \|S_{k}\| > \varepsilon{n}^{\alpha})}$$ for every $${\varepsilon > 0}$$ .

Necessary and sufficient conditions for complete convergence of double weighted sums of pairwise independent identically distributed random elements in Banach spaces

This paper provides necessary and sufficient conditions for complete convergence of double weighted sums of pairwise independent identically distributed random elements in Banach spaces. The main theorem extends Theorem 1.2 in [1] to the double weighted sum setting. The sharpness of the main result is illustrated by showing that the main theorem can fail if we replace the identical distribution condition by a slightly weaker condition, even when the random elements are independent and uniformly almost surely bounded.

On complete convergence in mean for double sums of independent random elements in Banach spaces

For a double array of random elements {T m,n , m ≥ 1, n ≥ 1} in a real separable Banach space X, we study the notion of T m,n converging completely to 0 in mean of order p where p is a positive constant. This notion is stronger than (i) T m,n converging completely to 0 and (ii) T m,n converging to 0 in mean of order p as max{m, n} →∞. When X is of Rademacher type p (1 ≤ p ≤ 2), for a double array of independent mean 0 random elements {V m,n , m ≥ 1, n ≥ 1} in X and a double array of constants {b m,n , m ≥ 1, n ≥ 1}, conditions are provided under which max1≤k≤m,1≤l≤n||Ʃi=1 kƩj=1 l V i,j||/b m,n converges completely to 0 in mean of order p. Moreover, these conditions are shown to provide an exact characterization of Rademacher type p (1 ≤ p ≤ 2) Banach spaces. Illustrative examples are provided.

On complete convergence in mean for double sums of independent random elements in Banach spaces

Conditions are provided under which a normed double sum of independent random elements in a real separable Rademacher type p Banach space converges completely to 0 in mean of order p. These conditions for the complete convergence in mean of order p are shown to provide an exact characterization of Rademacher type p Banach spaces. In case the Banach space is not of Rademacher type p, it is proved that the complete convergence in mean of order p of a normed double sum implies a strong law of large numbers.

On the laws of large numbers for double arrays of independent random elements in Banach spaces

For a double array of independent random elements {V mn ,m ≥ 1, n ≥ 1} in a real separable Banach space, conditions are provided under which the weak and strong laws of large numbers for the double sums Σ i=1 m Σ j=1 n V ij , m ≥ 1, n ≥ 1 are equivalent. Both the identically distributed and the nonidentically distributed cases are treated. In the main theorems, no assumptions are made concerning the geometry of the underlying Banach space. These theorems are applied to obtain Kolmogorov, Brunk-Chung, and Marcinkiewicz-Zygmund type strong laws of large numbers for double sums in Rademacher type p (1 ≤ p ≤ 2) Banach spaces.

Complete moment convergence for weighted sums of sequences of independent random elements in Banach spaces

In this paper, we obtain complete moment convergence results for weighted sums of sequences of independent random elements in a real separable Banach spaces without any geometric conditions imposed on the Banach space. Our results improve and extend some well known results from the literature. Furthermore, we obtain a complete moment convergence results for weighted sums of arrays of arbitrary random elements.

Almost sure convergence for double arrays of block-wiseM-dependent random elements in Banach spaces

For a double array of blockwise M-dependent random elements {Vmn : m 1, n 1} 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 , m 1, n 1. The paper treats two cases: (i) {Vmn : m 1, n 1} is block-wise M-dependent with EVmn 0, m, n 1, and (ii) {Vmn : m 1, n 1} is block-wise p-orthogonal. The conditions for case (i) are shown to provide exact characterizations 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.

Almost Sure and Complete Convergence of Randomly Weighted Sums of Independent Random Elements in Banach Spaces

This work develops almost sure and complete convergence of randomly weighted sums of independent random elements in a real separable Banach space. Sufficient conditions for almost sure convergence and complete convergence in the sense defined by Hsu and Robbins are provided. Examples showing that the conditions cannot be removed or weakened are given. It is also demonstrated that some of the known theorems in the literature are special cases of our results

Marcinkiewicz-Zygmund type law of large numbers for double arrays of random elements in Banach spaces

In this paper we establish Marcinkiewicz-Zygmund type laws of large numbers for double arrays of random elements in Banach spaces. Our results extend those of Hong and Volodin [6].

On Cantrell–Rosalsky’s strong laws of large numbers

Starting from a result of almost sure (a.s.) convergence for nonnegative random variables, a simplified proof of a strong law of large numbers (SLLN) established in ([Cantrell, A., Rosalsky, A., 2003. Some strong law of large numbers for Banach space valued summands irrespective of their joint distributions. Stochastic Anal. Appl. 21, 79–95], Theorem 1) for random elements in a real separable Banach space is presented, and some other results of a.s. convergence related to SLLNs in ([Cantrell, A., Rosalsky, A., 2004. A strong law for compactly uniformly integrable sequences of independent random elements in Banach spaces. Bull. Inst. Math. Acad. Sinica 32, 15–33], Th. 3.1) and ([Cantrell, A., Rosalsky, A., 2003. Some strong law of large numbers for Banach space valued summands irrespective of their joint distributions. Stochastic Anal. Appl. 21, 79–95], Theorem 2) are derived. No conditions of independence or on the joint distribution of random elements are required. Likewise, no geometric condition on the Banach space where random elements take values is imposed. Some applications to weighted (for an array of constants) sums of random elements and to the case of random sets are also considered.

Integrals of random fuzzy sets

This paper tries to give a systematic investigation of integration of random fuzzy sets. Besides the widely used Aumann-integral adaptions of Pettis- and Bochner-integration for random elements in Banach spaces are introduced. The mutual relationships of these competing concepts will be explored comprehensively, completing and improving former results from literature. As a by product dominated convergence theorems, strong laws of large numbers and central limit theorems for random fuzzy sets can be derived. They are based on weaker assumptions than previous versions from literature.

On Complete Convergence in Mean of Normed Sums of Independent Random Elements in Banach Spaces

For a sequence of random elements {T n , n ≥ 1} in a real separable Banach space 𝒳, we study the notion of T n converging completely to 0 in mean of order p where p is a positive constant. This notion is stronger than (i) T n converging completely to 0 and (ii) T n converging to 0 in mean of order p. When 𝒳 is of Rademacher type p (1 ≤ p ≤ 2), for a sequence of independent mean 0 random elements {V n , n ≥ 1} in 𝒳 and a sequence of constants b n → ∞, conditions are provided under which the normed sum converges completely to 0 in mean of order p. Moreover, these conditions for converging completely to 0 in mean of order p are shown to provide an exact characterization of Rademacher type p Banach spaces. Illustrative examples are provided.

On the Weak Limiting Behavior of Almost Surely Convergent Row Sums from Infinite Arrays of Rowwise Independent Random Elements in Banach Spaces

For an array {Vnk,k≥1,n≥1} of rowwise independent random elements in a real separable Banach space \(X\) with almost surely convergent row sums \(S_n = \sum {_{k = 1}^\infty {\text{ }}V_{nk} ,n \geqslant 1} \), we provide criteria for Sn−An to be stochastically bounded or for the weak law of large numbers \(S_n - A_n \xrightarrow{P}0\) to hold where {An,n≥1} is a (nonrandom) sequence in \(X\).

Strong Limit Theorems of Cesaro Type Means for Arrays of OrthogonalRandom Elements with Multi-dimensional Indices in Banach Spaces

It will investigate the strong limit behavior of the Cesaro type average of arrays of James-type orthogonal random elements in Banach spaces. Particularly, it will show that the d-dimensional, Banach spaces version conditions is sufficient to yield

On conditional compactly uniform pth-order integrability of random elements in Banach spaces

The notion of conditional compactly uniform pth-order integrability of an array of random elements in a separable Banach space concerning an array of random variables and relative to a sequence of σ-algebras is introduced and characterized. We state a conditional law for randomly weighted sums of random elements in a Banach space with the bounded approximation property, and we prove that, under the introduced condition, the problem can be reduced to a similar problem for random elements in a finite-dimensional space.

An application of the Ryll-Nardzewski–Woyczyński theorem to a uniform weak law for tail series of weighted sums of random elements in Banach spaces

For a sequence of Banach space valued random elements { V n , n⩾1} (which are not necessarily independent) with the series ∑ n=1 ∞ V n converging unconditionally in probability and an infinite array a={a ni, i⩾n, n⩾1} of constants, conditions are given under which (i) for all n⩾1, the sequence of weighted sums ∑ i=n m a niV i converges in probability to a random element T n ( a) as m→∞, and (ii) T n(a) → P 0 uniformly in a as n→∞ where a is in a suitably restricted class of infinite arrays. The key tool used in the proof is a theorem of Ryll-Nardzewski and Woyczyński (1975, Proc. Amer. Math. Soc. 53, 96–98).

On convergence of series of independent random elements in banach spaces

The rate of convergence for an almost certainly convergent series independent random elements in a real separable Banach space is studied in this paper. More specifically, when S n converges almost certainly to a random element S, the tail series is a well defined sequence of random elements with almost certainly. The main result establishes for a sequence of positive constants with b j ⩽ Const. b n whenever the equivalence between the tail series weak law of large numbers and the limit law thereby extending a result of Nam and Rosalsky [20] to a Banach space setting while also simplifying the argument used in the earlier result. The quasimonotonicity proviso on cannot be dispensed with