The Annals of Probability

The Hurst Index of Long-Range Dependent Renewal Processes

D. J. Daley

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.

Article information

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

Dates
First available in Project Euclid: 31 May 2002

https://projecteuclid.org/euclid.aop/1022874827

Digital Object Identifier
doi:10.1214/aop/1022874827

Mathematical Reviews number (MathSciNet)
MR1742900

Zentralblatt MATH identifier
0961.60083

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

Citation

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. https://projecteuclid.org/euclid.aop/1022874827

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