Strong Laws for Small Increments of Renewal Processes

Abstract

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.