Functional limit theorems for sums of independent random variables with random coefficients
type: Item
instance of: scholarly article
title: Functional limit theorems for sums of independent random variables with random coefficients
zbMATH Open document ID: 0958.60006
DOI: 10.1007/BF02465816
author: N. Chuprunov
published in: Lithuanian Mathematical Journal
publication date: 2001-04-09T00:00:00Z
review text: Let \(\eta_n\), \(n\in N\), be i.i.d. random variables belonging to the domain of normal attraction of symmetric \(p\)-stable random variables \(\gamma_p\), \(0<p\leq 2\). Consider the sums \(S_n= \sum^{k(n)}_{i=1} \xi_{ni} \eta_i\), where the random coefficients \(\xi_{ni}\) are independent of \(\{\eta_n\}\). Conditions under which the convergence in distribution of \(S_n\) is equivalent to the convergence in distribution of \(T_n= \sum^{k(n)}_{i=1} |\xi_{ni} |^p\) are found. One of the results states that if \(\xi_{ni}\) are infinitesimal and \(\eta_n\) are symmetric for \(p\leq 1\) or centered for \(p>1\), then \(S_n\) converges in distribution to \(s\gamma_p\) for some positive \(s\) iff \(T_n\) converges in distributon to \(s^p\). The functional limit theorem for corresponding step functions is also obtained.
reviewed by: T. Inglot
Mathematics Subject Classification ID: 60E07
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 1465958
zbMATH Keywords: central limit theorem
zbMATH Keywords: random multipliers
zbMATH Keywords: stable law
zbMATH Keywords: functional limit theorem
zbMATH Keywords: infinitely divisible law
MaRDI profile type: Publication
cites work: Q3673748
cites work: Q5829358
cites work: Q3240992
cites work: Q3683263
cites work: Q5649715
cites work: Modern Theory of Summation of Random Variables
cites work: Q3866823
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description: scientific article; zbMATH DE number 1465958