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October, 1991 Strong Laws for Small Increments of Renewal Processes
Josef Steinebach
Ann. Probab. 19(4): 1768-1776 (October, 1991). DOI: 10.1214/aop/1176990234


Let $\{N(t), t \geq 0\}$ be the (generalized) renewal process associated with an i.i.d. sequence $X_1,X_2,\ldots$ of random variables having finite moment generating function on some left-sided neighborhood of the origin. Some strong limiting results are proved for the maximal increments $\sup_{0\leq t\leq T-K} (N(t + K) - N(t))$, where $K = K_T$ is a function of $T$ such that $K_T \uparrow \infty$, but $K_T/\log T \downarrow 0$ as $T \rightarrow \infty$. These provide analogs to a recent extension due to Mason (1989) of the Erdos-Renyi strong law of large numbers for partial sums.


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Josef Steinebach. "Strong Laws for Small Increments of Renewal Processes." Ann. Probab. 19 (4) 1768 - 1776, October, 1991.


Published: October, 1991
First available in Project Euclid: 19 April 2007

zbMATH: 0738.60022
MathSciNet: MR1127726
Digital Object Identifier: 10.1214/aop/1176990234

Primary: 60F15
Secondary: 60F10 , 60K05

Keywords: Erdos-Renyi law , Increments of renewal processes , large deviations , Strong law of large numbers

Rights: Copyright © 1991 Institute of Mathematical Statistics

Vol.19 • No. 4 • October, 1991
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