Electronic Journal of Probability

Local limit theorems and renewal theory with no moments

Kenneth S. Alexander and Quentin Berger

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We study i.i.d. sums $\tau _k$ of nonnegative variables with index $0$: this means ${\mathbf P}(\tau _1=n) = {\varphi }(n) n^{-1}$, with ${\varphi }(\cdot )$ slowly varying, so that ${\mathbf E}(\tau _1^{\epsilon })=\infty $ for all ${\epsilon }>0$. We prove a local limit and local (upward) large deviation theorem, giving the asymptotics of ${\mathbf P}(\tau _k=n)$ when $n$ is at least the typical length of $\tau _k$. A recent renewal theorem in [22] is an immediate consequence: ${\mathbf P}(n\in \tau ) \sim{\mathbf P} (\tau _1=n)/{\mathbf P}(\tau _1 > n)^2$ as $n\to \infty $. If instead we only assume regular variation of ${\mathbf P}(n\in \tau )$ and slow variation of $U_n:= \sum _{k=0}^n {\mathbf P}(k\in \tau )$, we obtain a similar equivalence but with ${\mathbf P}(\tau _1=n)$ 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 ${\mathbf P}(\tau _k \leq n)$ when $n$ is much smaller than the typical length of $\tau _k$.

Article information

Electron. J. Probab., Volume 21 (2016), paper no. 66, 18 pp.

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

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Zentralblatt MATH identifier

Primary: 60K05: Renewal theory 60G50: Sums of independent random variables; random walks 60F10: Large deviations

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

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Alexander, Kenneth S.; Berger, Quentin. Local limit theorems and renewal theory with no moments. Electron. J. Probab. 21 (2016), paper no. 66, 18 pp. doi:10.1214/16-EJP13. https://projecteuclid.org/euclid.ejp/1480129233

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