Open Access
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


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.


Download Citation

D. J. Daley. "The Hurst Index of Long-Range Dependent Renewal Processes." Ann. Probab. 27 (4) 2035 - 2041, October 1999.


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

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

Primary: 60K05
Secondary: 60G55

Keywords: Hurst index , long-range dependence , moment index , regular variation , renewal function asymptotics , Renewal process

Rights: Copyright © 1999 Institute of Mathematical Statistics

Vol.27 • No. 4 • October 1999
Back to Top