Limit theorems for methods of summation of independent random variables. I
type: Item
instance of: scholarly article
title: Limit theorems for methods of summation of independent random variables. I
zbMATH Open document ID: 0637.60007
DOI: 10.1007/BF00972025
author: R. Norvajsha
author: Thomas Mikosh
published in: Lithuanian Mathematical Journal
publication date: 1987-01-01T00:00:00Z
review text: Let \(Z(t)=\sum^{\infty}_{i=1}c_ i(t)X_ i\), \(t\in T\), be a class of series formed by independent random variables \(X_ i\), with values in a linear measurable space, and real numbers \(c_ i(t)\) such that: 1. for all \(i\geq 1\), \(\lim_{t\to t_ 0}c_ i(t)=0\); 2. for some \(k_ 0\geq 1\) \[ \limsup_{t\to t_ 0}\sum_{k\geq k_ 0}var\{c_ i(t)\}_{i\in I_ k}\equiv C(Z)<\infty, \] where \[ I_ k=(n_{k- 1},n_ k],\quad b_ k=| \inf \{| c_ i(t_ k)|: t\in I_ k\}|^{-1},\quad and \] \[ var\{c_ i(t)\}_{i\in I_ k}=\sum^{n_ k-1}_{n=n_{k-1}+1}| c_ i(t)-c_{i+1}(t)| +| c_{n_ k}(t)| \text{ for some } (n_ k)\subset N\quad and\quad (t_ k)\subset T. \] The methods of summation of sequences of Banach-valued random elements are considered. The equivalence of analogues of the law of large numbers for a family \(\{Z(t), t\in T\}\) is proved. One of the results gives conditions under which from the estimate \(\limsup_{t\to t_ 0}\| Z(t)\| \leq M\) follows that \[ \sum_{n\in N}P(\| \sum_{i\in I_ k}(X_ i-X'_ i)\| \geq \alpha b_ k)<\infty \text{ for } \alpha >4M, \] \(X'_ i\) a random element independent of \(X_ i\) and having the same distribution. Moreover, if the series \[ \sum_{k\in N}P(\| \sum_{i\in I_ k}X_ i\| >\alpha b_ k)<\infty \text{ for each }\alpha >\Lambda, \] and \[ \limsup_{k\to \infty}\max_{i\in I_ k}P(\| S_{n_ k}- S_ i\| >Lb_ k)\leq 1,\quad then: \limsup_{t\to t_ 0}\| Z(t)\| \leq (\Lambda +L)C(Z)\quad a.s. \]
reviewed by: Dominik Szynal
Mathematics Subject Classification ID: 60B12
zbMATH DE Number: 4038883
zbMATH Keywords: methods of summation of sequences of Banach-valued random elements
MaRDI profile type: Publication
cites work: Limit theorems for moving averages of independent random vectors
cites work: Q4171336
cites work: Strong Laws for Ruled Sums
cites work: Summability methods and almost sure convergence
cites work: Q4049853
cites work: Limiting behavior of weighted sums of independent random variables
cites work: On Random Fourier Series
cites work: Riemann R 1 -Summability of Independent, Identically Distributed Random Variables
cites work: Probabilities of Moderate Deviations in a Banach Space
cites work: The central limit theorem for summability methods of I.I.D. random variables
cites work: The Discounted Central Limit Theorem and its Berry-Esseen Analogue
cites work: Approximation for Abel sums of independent, identically distributed random variables
cites work: A central limit theorem for generalized discounting
cites work: Summability Methods for Independent, Identically Distributed Random Variables
cites work: Limit theorems for delayed sums
cites work: Q4056771
cites work: Q3339855
cites work: The bounded law of the iterated logarithm for the weighted empirical distribution process in the non-i.i.d. case
cites work: On the Cesaro means of orthogonal sequences of random variables
cites work: Q3774276
cites work: A limit theorem for discounted sums
cites work: On error–estimates in the central limit theorem for generalized linear discounting
cites work: Q4776687
cites work: Zum Gesetz der großen Zahlen für allgemeinere Mittel
cites work: Stochastic Abelian and Tauberian theorems
cites work: Q3689920
cites work: Some Strassen-type laws of the iterated logarithm for multiparameter stochastic processes with independent increments
cites work: Q3952903
cites work: Der zentrale grenzwertsatz und wahrseheinliehkeiten grober abweichungen f¨r summen
full work available at URL: https://doi.org/10.1007/bf00972025
OpenAlex ID: W4245604905
description: scientific article; zbMATH DE number 4038883