## The Annals of Probability

- Ann. Probab.
- Volume 20, Number 3 (1992), 1229-1247.

### On Generalized Renewal Measures and Certain First Passage Times

#### Abstract

Let $X_1,X_2,\ldots$ be i.i.d. random variables with common mean $\mu \geq 0$ and associated random walk $S_0 = 0, S_n = X_1 + \cdots + X_n, n \geq 1$. For a regularly varying function $\phi(t) = t^\alpha L(t), \alpha > -1$ with slowly varying $L(t)$, we consider the generalized renewal function $U_\phi(t) = \sum_{n \geq 0} \phi(n)P(S_n \leq t),\quad t \in \mathbb{R},$ by relating it to the family $\tau = \tau(t) = \inf\{n \geq 1: S_n > t\} t \geq 0$. One of the major results is that $U_\phi(t) < \infty$ for all $t \in \mathbb{R}, \operatorname{iff} \phi(t)^{-1}U_\phi(t) \sim 1/(\alpha + 1)\mu^{\alpha + 1}$ as $t \rightarrow \infty, \operatorname{iff} E(X^-_1)^2\phi(X^-_1) < \infty$, provided $\phi$ is ultimately increasing $(\Rightarrow \alpha \geq 0)$. A related result is proved for $U_\phi(t + h) - U_\phi(t)$ and $U^+_\phi(t) = \sum_{n \geq 0}\phi(n)P(M_n \leq t)$, where $M_n = \max_{0 \leq j \leq n} S_j$. Our results form extensions of earlier ones by Heyde, Kalma, Gut and others, who either considered more specific functions $\phi$ or used stronger moment assumptions. The proofs are based on a regeneration technique from renewal theory and two martingale inequalities by Burkholder, Davis and Gundy.

#### Article information

**Source**

Ann. Probab., Volume 20, Number 3 (1992), 1229-1247.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176989690

**Digital Object Identifier**

doi:10.1214/aop/1176989690

**Mathematical Reviews number (MathSciNet)**

MR1175261

**Zentralblatt MATH identifier**

0759.60088

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

Secondary: 60K05: Renewal theory

**Keywords**

Random walk generalized renewal measure first passage time ladder height regularly varying function martingale

#### Citation

Alsmeyer, Gerold. On Generalized Renewal Measures and Certain First Passage Times. Ann. Probab. 20 (1992), no. 3, 1229--1247. doi:10.1214/aop/1176989690. https://projecteuclid.org/euclid.aop/1176989690