Sharp Rates for Increments of Renewal Processes

Abstract

Let $\{N(t), t \geq 0\}$ be the renewal process associated to an i.i.d. sequence $X_1, X_2, \ldots$ of nonnegative interarrival times having finite moment generating function near the origin. In this article we give strong and weak limiting laws for the maximal and minimal increments $\sup_{0 \leq t \leq T - K}(N(t + K) - N(t))$ and $\inf_{0 \leq t \leq T - k}(N(t + K) - N(t))$, where $K = K_T$ is a function of $T$ such that $0 \leq K_T \leq T$.