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

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)
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