A functional limit theorem for Erdős and Rényi's law of large numbers
type: Item
instance of: scholarly article
title: A functional limit theorem for Erdős and Rényi's law of large numbers
zbMATH Open document ID: 0794.60018
DOI: 10.1007/BF01311345
author: Gabriela R. Sanchis
publication date: 1994-08-28T00:00:00Z
review text: We use the theory of large deviations on function spaces to extend Erdős and Rényi's law of large numbers. In particular, we show that with probability 1, the double-indexed set of paths \(\{W_{N,n}\}\) defined by \[ W_{N,n} (t,\omega)={S_{n+[h(N)t]} (\omega)-S_ n (\omega) \over h(N)}+{h(N)t-[h(N)t] \over h(N)} X_{n+[h(N)t]+1} (\omega), \] where \(S_ n={1 \over n} \sum^ n_ 1X_ i\), \(\{X_ i:i \geq 1\}\) is an i.i.d. sequence of random variables, and \(h(N)=[c \log N]\) is relatively compact; the limit set is given by the set \([x:I^*(x) \leq 1/c]\) where \(I^*(x)=\int^ 1_ 0 I(x'(t)) dt\) and \(I\) is Cramér's rate function.
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 14E20
Mathematics Subject Classification ID: 46E25
zbMATH DE Number: 569019
zbMATH Keywords: large deviations
zbMATH Keywords: Erdős and Rényi's law of large numbers
zbMATH Keywords: limit set
zbMATH Keywords: Cramér's rate function
MaRDI profile type: Publication
cites work: Q3794956
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full work available at URL: https://doi.org/10.1007/bf01311345
OpenAlex ID: W4248390526
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 569019