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.
Citation
Gerold Alsmeyer. "On Generalized Renewal Measures and Certain First Passage Times." Ann. Probab. 20 (3) 1229 - 1247, July, 1992. https://doi.org/10.1214/aop/1176989690
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