A strong law for a set-indexed partial sum process with applications to exchangeable and stationary sequences
type: Item
instance of: scholarly article
title: A strong law for a set-indexed partial sum process with applications to exchangeable and stationary sequences
zbMATH Open document ID: 0634.60026
DOI: 10.1016/0304-4149(87)90181-5
author: Andrew Rosalsky
published in: Stochastic Processes and their Applications
publication date: 1987-01-01T00:00:00Z
review text: Let \(\{Y_ n\), \(n\geq 1\}\) be a sequence of identically distributed random variables, not necessarily independent nor with finite mean, and let \(\{J_ n\), \(n\geq 1\}\) be an increasing sequence of finite sets of positive integers. Set \(S_ n=\sum_{j\in J_ n}Y_ j\), and let \(\{\) b(n), \(n\geq 1\}\) be any sequence of constants such that \[ n^{- 1}b(n)\to \infty \quad and\quad \sum_{n\geq 1}P[| Y_ 1| >b(n)]<\infty. \] Then, provided an ergodic theorem holds for the partial sums of \(g(| Y_ j|)\), for any \(g\in L_ 1\), \(S_ n/b(| J_ n|)\to 0\) a.s. The result is applied to obtain laws of large numbers for symmetric functions of exchangeable sequences, for stationary sequences and for sequences of pairwise independent random variables.
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60G09
Mathematics Subject Classification ID: 60G10
zbMATH DE Number: 4032704
zbMATH Keywords: set-indexed partial sum process
zbMATH Keywords: strong law of large numbers
zbMATH Keywords: pointwise ergodic theorem
zbMATH Keywords: stationary sequences
zbMATH Keywords: sequences of pairwise independent random variables
reviewed by: A. D. Barbour
MaRDI profile type: Publication
full work available at URL: 0304-4149(87)90181-5
OpenAlex ID: W2014685602
cites work: Q4197800
cites work: Q4183968
cites work: An elementary proof of the strong law of large numbers
cites work: A Limit Theoerm for Random Variables with Infinite Moments
cites work: A Short Proof of a Known Limit Theorem for Sum of Independent Random Variables with Infinite Expectations
cites work: Upper and lower functions for martingales and mixing processes
cites work: Sums of Stationary Sequences Cannot Grow Slower Than Linearly
cites work: Uses of exchangeability
cites work: On the law of large numbers for stationary sequences
cites work: On a Theorem of Feller
cites work: Q4057867
cites work: Convergence Properties of $S_n$ Under Moment Restrictions
cites work: Q4084474
cites work: A zero-one law for stationary sequences
description: scientific article; zbMATH DE number 4032704