The Annals of Probability

On Generalized Renewal Measures and Certain First Passage Times

Gerold Alsmeyer

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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

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

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]
Secondary: 60K05: Renewal theory

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


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

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