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.     

Banach空间上的随机加权和的收敛性

本文讨论了在P型和P凸Banach空间上的随机加权和un↑∑↑i=μnXni的r平均收敛性及依概率收敛性，并从给出了满足这些收敛性的充分与必要条件。在以往的文献中讨论的随机加权多为n↑∑↑i=1αniXi这种形式，而本文给出了在更一般情况下随机加权和的收敛性，并对以前的一些定理作了一些适当推广。     

A Note on the Rate of Complete Convergence for Weighted Sums of Arrays of Banach Space Valued Random Elements

A rate of complete convergence for weighted sums of arrays of rowwise independent Banach space valued random elements was obtained by Ahmed et al. [1 Ahmed , S.E. , Giuliano Antonini , R. , and Volodin , A. 2002 . On the rate of complete convergence for weighted sums of arrays of Banach space valued random elements with application to moving average processes . Statist. Probab. Lett. 58 : 185 – 194 . [Google Scholar]]. Recently, Sung and Volodin [2 Sung , S.H. , and Volodin , A.I. 2006. On the rate of complete convergence for weighted sums of arrays of random elements. J. Korean Math. Soc. 43:815–828.[Crossref], [Web of Science ®] , [Google Scholar]], Chen et al. [3 Chen , P. , Sung , S.H. , and Volodin , A.I. 2006 . Rate of complete convergence for arrays of Banach space valued random elements . Siberian Adv. Math. 16 : 1 – 14 . [Google Scholar]], and Kim and Ko [4 Kim , T.S. , and Ko , M.H. 2008 . On the complete convergence of moving average process with Banach space valued random elements . J. Theor. Probab. 21 : 431 – 436 . [Google Scholar]] solved an open question posed by Ahmed et al. In this article, we improve and complement the result of Ahmed et al. The method used in this article is simpler than those in Ahmed et al., Sung and Volodin, Chen et al., and Kim and Ko.     

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.     

On the Complete Convergence of Moving Average Process with Banach Space Valued Random Elements

Let {Y _i ;−∞<i<∞} be a doubly infinite sequence of independent random elements taking values in a separable real Banach space and stochastically dominated by a random variable X. Let {a _i ;−∞<i<∞} be an absolutely summable sequence of real numbers and set V _i =∑ _k=−∞^∞ a _i+k Y _i ,i≥1. In this paper, we derive that if and E|X| ^μ log  ^ρ |X|<0, for some μ (0<μ<2, μ≠1) and ρ>0 then for all ε>0. This work was partially supported by the Korean Research Foundation Grant funded by the Korean Government (KRF-2006-353-C00006, KRF-2006-251-C00026).     

A Note on Complete Convergence for Arrays of Rowwise Independent Banach Space Valued Random Elements

We obtain complete convergence results for arrays of rowwise independent Banach space valued random elements. Compared with similar results presented in the probabilistic literature our conditions are weaker.     

Convergence in the r－th Mean and Some Weak laws of Large Numbers for Random Weighted Sums of Random Elements in Banach Spaces

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Weak Laws with Random Indices for Arrays of Random Elements in Rademacher Type p Banach Spaces

For a sequence of constants {a _n,n1}, an array of rowwise independent and stochastically dominated random elements { V _nj, j1, n1} in a real separable Rademacher type p (1p2) Banach space, and a sequence of positive integer-valued random variables {T _n, n1}, a general weak law of large numbers of the form is established where {c _nj, j1, n1}, _n , b _n are suitable sequences. Some related results are also presented. No assumption is made concerning the existence of expected values or absolute moments of the {V _nj, j1, n1}. Illustrative examples include one wherein the strong law of large numbers fails.     

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.     

$B$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 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.     

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}.     

Convergence in the pth-mean and some Weak Laws of Large Numbers for weighted sums of random elements in separable normed linear spaces

Theorem. Let X_n, n ≥ 1, be a sequence of tight random elements taking values in a separable Banach space B such that |X_n|, n ≥ 1, is uniformly integrable. Let a_nk, n ≥ 1, k ≥ 1, be a double array of real numbers satisfying Σ_{k ≥ 1} |a_nk| ≤ Γ for every n ≥ 1 for some positive constant Γ. Then Σ_{k ≥ 1}a_nkX_k, n ≥ 1, converges to 0 in probability if and only if Σ_{k ≥ 1}a_nkf(X_k), n ≥ 1, converges to 0 in probability for every f in the dual space B¹.