Estimates of the rate of convergence of sums of independent random variables in a banach space. II

1982 ◽  
Vol 22 (4) ◽  
pp. 344-353 ◽  
Author(s):  
V. Bentkus ◽  
A. Račkauskas
Keyword(s):  
Banach Space ◽  
Rate Of Convergence ◽  
Random Variables ◽  
Independent Random Variables

scholarly journals Some results on the span of families of Banach valued independent, random variables

1991 ◽  
Vol 14 (2) ◽  
pp. 381-384
Author(s):  
Rohan Hemasinha
Keyword(s):  
Banach Space ◽  
Probability Space ◽  
Linear Span ◽  
Random Variables ◽  
Isomorphic Copy ◽  
Independent Random Variables ◽  
Closed Linear Span

LetEbe a Banach space, and let(Ω,ℱ,P)be a probability space. IfL1(Ω)contains an isomorphic copy ofL1[0,1]then inLEP(Ω)(1≤P<∞), the closed linear span of every sequence of independent,Evalued mean zero random variables has infinite codimension. IfEis reflexive orB-convex and1<P<∞then the closed(in LEP(Ω))linear span of any family of independent,Evalued, mean zero random variables is super-reflexive.

scholarly journals Complete convergence for weighted sums of pairwise independent random variables

2017 ◽  
Vol 15 (1) ◽  
pp. 467-476
Author(s):  
Li Ge ◽  
Sanyang Liu ◽  
Yu Miao
Keyword(s):  
Rate Of Convergence ◽  
Complete Convergence ◽  
Moving Average ◽  
Random Variables ◽  
Independent Random Variables ◽  
Weighted Sums ◽  
Pairwise Independent Random Variables ◽  
Moving Average Processes

Abstract In the present paper, we have established the complete convergence for weighted sums of pairwise independent random variables, from which the rate of convergence of moving average processes is deduced.

scholarly journals Convergence of weighted sums of independent random variables and an extension to Banach space-valued random variables

1979 ◽  
Vol 2 (2) ◽  
pp. 309-323
Author(s):  
W. J. Padgett ◽  
R. L. Taylor
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Random Variables ◽  
Almost Sure Convergence ◽  
Independent Random Variables ◽  
Weighted Sums ◽  
Real Numbers ◽  
Martingale Theory ◽  
Random Elements ◽  
Schauder Bases

Let{Xk}be independent random variables withEXk=0for allkand let{ank:n≥1, k≥1}be an array of real numbers. In this paper the almost sure convergence ofSn=∑k=1nankXk,n=1,2,…, to a constant is studied under various conditions on the weights{ank}and on the random variables{Xk}using martingale theory. In addition, the results are extended to weighted sums of random elements in Banach spaces which have Schauder bases. This extension provides a convergence theorem that applies to stochastic processes which may be considered as random elements in function spaces.

A Central Limit Theorem for Cumulative Processes

1994 ◽  
Vol 26 (01) ◽  
pp. 104-121 ◽  
Author(s):  
Allen L. Roginsky
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Rate Of Convergence ◽  
Central Limit ◽  
Remainder Term ◽  
Limit Theorems ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Independent Random Variables ◽  
Cumulative Processes

A central limit theorem for cumulative processes was first derived by Smith (1955). No remainder term was given. We use a different approach to obtain such a term here. The rate of convergence is the same as that in the central limit theorems for sequences of independent random variables.

The convergence of a sum of independent random variables

1963 ◽  
Vol 59 (2) ◽  
pp. 411-416
Author(s):  
G. De Barra ◽  
N. B. Slater
Keyword(s):  
Distribution Function ◽  
Euclidean Space ◽  
Rate Of Convergence ◽  
Radial Distribution Function ◽  
Random Variables ◽  
Covariance Matrices ◽  
Independent Random Variables ◽  
Second Moments ◽  
Image Position ◽  
Real Euclidean Space

Let Xν, ν= l, 2, …, n be n independent random variables in k-dimensional (real) Euclidean space Rk, which have, for each ν, finite fourth moments β4ii = l,…, k. In the case when the Xν are identically distributed, have zero means, and unit covariance matrices, Esseen(1) has discussed the rate of convergence of the distribution of the sumsIf denotes the projection of on the ith coordinate axis, Esseen proves that ifand ψ(a) denotes the corresponding normal (radial) distribution function of the same first and second moments as μn(a), thenwhere and C is a constant depending only on k. (C, without a subscript, will denote everywhere a constant depending only on k.)

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