Open Access
2016 Local limit theorems and renewal theory with no moments
Kenneth S. Alexander, Quentin Berger
Electron. J. Probab. 21: 1-18 (2016). DOI: 10.1214/16-EJP13

Abstract

We study i.i.d. sums τk of nonnegative variables with index 0: this means P(τ1=n)=φ(n)n1, with φ() slowly varying, so that E(τ1ϵ)= for all ϵ>0. We prove a local limit and local (upward) large deviation theorem, giving the asymptotics of P(τk=n) when n is at least the typical length of τk. A recent renewal theorem in [22] is an immediate consequence: P(nτ)P(τ1=n)/P(τ1>n)2 as n. If instead we only assume regular variation of P(nτ) and slow variation of Un:=k=0nP(kτ), we obtain a similar equivalence but with P(τ1=n) replaced by its average over a short interval. We give an application to the local asymptotics of the distribution of the first intersection of two independent renewals. We further derive downward moderate and large deviations estimates, that is, the asymptotics of P(τkn) when n is much smaller than the typical length of τk.

Citation

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Kenneth S. Alexander. Quentin Berger. "Local limit theorems and renewal theory with no moments." Electron. J. Probab. 21 1 - 18, 2016. https://doi.org/10.1214/16-EJP13

Information

Received: 31 March 2016; Accepted: 6 November 2016; Published: 2016
First available in Project Euclid: 26 November 2016

zbMATH: 1354.60107
MathSciNet: MR3580032
Digital Object Identifier: 10.1214/16-EJP13

Subjects:
Primary: 60F10 , 60G50 , 60K05

Keywords: i.i.d. sums , local large deviation , local limit theorem , Renewal theorem , reverse renewal theorems , slowly varying tail distribution

Vol.21 • 2016
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