A Complete Convergence Theorem for Row Sums from Arrays of Rowwise Independent Random Elements in Rademacher Type p Banach Spaces

We extend in several directions a complete convergence theorem for row sums from an array of rowwise independent random variables obtained by Sung, Volodin, and Hu [8 Sung , S.H. , Volodin , A.I. , and Hu , T.-C. ( 2005 ). More on complete convergence for arrays. Statist. Probab. Lett. 71:303–311.  [Google Scholar]] to an array of rowwise independent random elements taking values in a real separable Rademacher type p Banach space. An example is presented which illustrates that our result extends the Sung, Volodin, and Hu result even for the random variable case.     

Strong Laws of Large Numbers for Double Sums of Banach Space Valued Random Elements

For a double array {V_(m,n), m ≥ 1, n ≥ 1} of independent, mean 0 random elements in a real separable Rademacher type p(1 ≤ p ≤ 2) Banach space and an increasing double array {b_(m,n), m ≥1, n ≥ 1} of positive constants, the limit law ■ and in L_p as m∨n→∞ is shown to hold if ■ This strong law of large numbers provides a complete characterization of Rademacher type p Banach spaces. Results of this form are also established when 0 p ≤ 1 where no independence or mean 0 conditions are placed on the random elements and without any geometric conditions placed on the underlying Banach space.     

A weak law for normed weighted sums of random elements in rademacher type p banach spaces

For weighted sums Σ_{j = 1}ⁿa_jV_j of independent random elements {V_n, n ≥ 1} in real separable, Rademacher type p (1 ≤ p ≤ 2) Banach spaces, a general weak law of large numbers of the form (Σ_{j = 1}ⁿa_jV_j − v_n)/b_n →^p 0 is established, where {v_n, n ≥ 1} and b_n → ∞ are suitable sequences. It is assumed that {V_n, n ≥ 1} is stochastically dominated by a random element V, and the hypotheses involve both the behavior of the tail of the distribution of |V| and the growth behaviors of the constants {a_n, n ≥ 1} and {b_n, n ≥ 1}. No assumption is made concerning the existence of expected values or absolute moments of the {V_n, n >- 1}.     

Compact Laws of the Iterated Logarithm for B-Valued Random Variables with Two-Dimensional Indices

Let be a real separable Banach space and {X, X _{n, m}; (n, m) N ²} B-valued i.i.d. random variables. Set . In this paper, the compact law of the iterated logarithm, CLIL(D), for B-valued random variables with two-dimensional indices ranging over a subset D of N ² is studied. There is a gap between the moment conditions for CLIL(N ¹) and those for CLIL(N ²). The main result of this paper fills this gap by presenting necessary and sufficient conditions for the sequence to be almost surely conditionally compact in B, where, for 0, 1 r 2, N ^r (, ) = {(n, m) N ²; n m n exp{(log n) ^r–1 (n)}} and (·) is any positive, continuous, nondecreasing function such that (t)/(log log t) is eventually decreasing as t , for some > 0.     

$B$值独立随机元序列加权和的完全收敛性和强大数律

In this paper, we obtain theorems of complete convergence and strong laws of large numbers for weighted sums of sequences of independent random elements in a Banach space of type p （1 ≤ p ≤ 2）. The results improve and extend the corresponding results on real random variables obtained by [1] and [2].     

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

For a double array of independent random elements {Vmn,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 mi=1 nj=1Vij,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.     

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 {V _nk ,k1,n1} of rowwise independent random elements in a real separable Banach space with almost surely convergent row sums , we provide criteria for S _n –A _n to be stochastically bounded or for the weak law of large numbers to hold where {A _n ,n1} is a (nonrandom) sequence in .     

随机变量阵列加权和最大值的收敛性

本文研究了形如maxun≤j≤un｜∑ji=un aniXni｜的弱大数律和Lr收敛性,其中0＜r≤p,0＜p≤2,{ani,un≤i≤vn,n≥1}是实数阵列,{Xni,un≤i≤vn,n≥1}当0＜p＜1时是任意随机变量阵列,当1≤p≤2时是均值为零的行为NA的随机变量阵列.所得结果丰富和推广了许多已知的结果.     

B值鞅差型阵列的大数定律

主要研究了Ｂ值鞅差型阵列的大数定律,并讨论了它们与Ｂａｎａｃｈ空间几何持征的依赖关系。     

Banach空间中随机微分包含的弱解存在性定理

本文对Banach空间中的微分包含及随机微分包含引入了弱解的概念,并给出了它们的存在性定理。     

不同分布NA序列的一个弱大数定律

研究了不同分布NA序列加权和最大值的弱大数定律,推广了前人的结果.     

On Complete Convergence for Arrays of Random Elements and Variables

Abstract

We obtain complete convergence results for arrays of row-wise independent Banach space valued random elements. The main result deals with two cases that usually are considered separately: when no assumptions are made concerning the geometry of the underlying Banach space and when the Banach space is of Rademacher type p.