Stabilité de sommes pondérées de variables aléatoires vectorielles. (Stability of weighted sums of vector random variables)
type: Item
instance of: scholarly article
title: Stabilité de sommes pondérées de variables aléatoires vectorielles. (Stability of weighted sums of vector random variables)
zbMATH Open document ID: 0677.60005
DOI: 10.1016/0047-259X(89)90081-X
author: Jean-Christian Alt
published in: Journal of Multivariate Analysis
publication date: 1989-01-01T00:00:00Z
review text: This paper studies the stochastic convergence and the almost sure behaviour of weighted sums of random variables of the form \(S_ n=\sum^{k=k_ n}_{k=1}a_{nk}X_ k\) where \(a_{nk}\) are scalars and \(X_ k\) are independent random variables with values in a Banach space B. The following set of assumptions is used: \[ \sum^{k_ n}_{k=1}| a_{nk}|^{\alpha}\leq \Gamma <\infty,\quad \lim_{n\to \infty}\sup_{1\leq k\leq k_ n}| a_{nk}| =0, \] \[ \lim_{s\to \infty}\sup_{n}\sup_{t\geq s}P\{\| X_ n\| \geq t\}=0. \] Then if B is of stable type \(\alpha\) \((1<\alpha <2)\) and if the variables are mean zero, the sequence \(S_ n\) converges to 0 in probability. A similar conclusion is obtained under various variants of the above assumptions. The simpler case \(\alpha <1\) is also considered (then, independence is no longer essential). Moreover, the author gives a condition under which the stochastic convergence (i.e. convergence in probability) of the sequence \(S_ n\) to 0 implies its almost sure convergence to 0.
Mathematics Subject Classification ID: 60B12
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 46B03
zbMATH DE Number: 4109713
zbMATH Keywords: stochastic convergence
zbMATH Keywords: almost sure behaviour
zbMATH Keywords: weighted sums of random variables
reviewed by: Gilles Pisier
MaRDI profile type: Publication
cites work: Q3914124
cites work: Q3914125
cites work: Convergence of weighted averages of independent random variables
cites work: Characterizations of almost surely continuous p-stable random Fourier series and strongly stationary processes
cites work: Q3744958
cites work: Q5541672
cites work: Q5609799
cites work: Q4748883
cites work: A strong law for weighted averages of independent, identically distributed random variables with arbitrarily heavy tails
description: scientific article; zbMATH DE number 4109713