Electronic Journal of Probability

Local limit theorems and renewal theory with no moments

Kenneth S. Alexander and Quentin Berger

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1480129233

Digital Object Identifier
doi:10.1214/16-EJP13

Mathematical Reviews number (MathSciNet)
MR3580032

Zentralblatt MATH identifier
1354.60107

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

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

Rights
Creative Commons Attribution 4.0 International License.

Citation

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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References

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