The Hurst Index of Long-Range Dependent Renewal Processes

Abstract

A stationary renewal process $N(\cdot)$ for which the lifetime distribution has its $k$th moment finite or infinite according as $k$ is less than or greater than $\kappa$ for some $1 \lt \kappa \lt 2$, is long-range dependent and has Hurst index $\alpha=1/2(3-\kappa)$ (this is the critical index $\alpha$ for which $\lim\sup_{t \to \infty} t^{-2a}$ var $N(0,t]$ is finite or infinite according as $\alpha$ is greater than or less than $\alpha$. This identification is accomplished by delineating the growth rate properties of the difference between the renewal function and its linear asymptote, thereby extending work of Täcklind.