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October 1999 The Hurst Index of Long-Range Dependent Renewal Processes
D. J. Daley
Ann. Probab. 27(4): 2035-2041 (October 1999). DOI: 10.1214/aop/1022874827

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.

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D. J. Daley. "The Hurst Index of Long-Range Dependent Renewal Processes." Ann. Probab. 27 (4) 2035 - 2041, October 1999. https://doi.org/10.1214/aop/1022874827

Information

Published: October 1999
First available in Project Euclid: 31 May 2002

zbMATH: 0961.60083
MathSciNet: MR1742900
Digital Object Identifier: 10.1214/aop/1022874827

Subjects:
Primary: 60K05
Secondary: 60G55

Rights: Copyright © 1999 Institute of Mathematical Statistics

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Vol.27 • No. 4 • October 1999
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