The Annals of Probability

The Hurst Index of Long-Range Dependent Renewal Processes

D. J. Daley

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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.

Article information

Ann. Probab., Volume 27, Number 4 (1999), 2035-2041.

First available in Project Euclid: 31 May 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K05: Renewal theory
Secondary: 60G55: Point processes

Hurst index long-range dependence renewal process renewal function asymptotics regular variation moment index


Daley, D. J. The Hurst Index of Long-Range Dependent Renewal Processes. Ann. Probab. 27 (1999), no. 4, 2035--2041. doi:10.1214/aop/1022874827.

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