## Electronic Journal of Probability

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

### Local limit theorems and renewal theory with no moments

Kenneth S. Alexander and Quentin Berger

#### 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, Number (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

**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