The Hurst index of long-range dependent renewal processes. (English) Zbl 0961.60083

Daley, D. J.

Ann. Probab. 27, No. 4, 2035-2041 (1999).

Let \(N(t)= \#\{n: X(0)+\cdots+ X(n)\leq t\}\) be a stationary renewal process where \(X(i)\geq 0\), \(i\geq 0\), \(X(k)\) has d.f. \(F\), \(k\geq 1\), and \(X(0)\) has density \(\lambda(1-F(x))\), \(x\geq 0\), with \(\lambda^{-1}= EX(1)< \infty\). Put \(V(t)= \text{Var }N(t)\) and assume \(EX^2(1)= \infty\). Then (and only then) the renewal process is long-range dependent, i.e. \(\limsup V(t)/t= \infty\), \(t\to\infty\). It then has Hurst index \(\alpha\) in \([{1\over 2},1]\), i.e. \(\limsup t^{-2a}V(t)\) \(<\infty\) or \(=\infty\) according as \(a> \alpha\) or \(a<\alpha\). The paper connects the Hurst index to the moment index \(\kappa= \sup\{c> 0: EX(1)< \infty\}\), viz. by \(\alpha={1\over 2}(3- \kappa)\) when \(1< \kappa< 2\). Proofs start from the observation that \(V(t)\) has the same asymptotic behaviour as \(2\lambda \int^t_0 (U(y)-\lambda y) dy\) where \(U\) is the nonstationary renewal function. Then asymptotic estimates for \(U(t)-\lambda t\) in terms of \(\kappa\) are derived. An example is constructed to show that these estimates cannot be refined without extra conditions.

Reviewer: A.J.Stam (Winsum)

MSC:

60K05	Renewal theory
60G55	Point processes (e.g., Poisson, Cox, Hawkes processes)

Keywords:

Hurst index; moment index; asymptotic behaviour; nonstationary renewal function