## Renewal reward processes with heavy-tailed inter-renewal times and heavy-tailed rewards.(English)Zbl 0954.60071

This paper studies a superposition $$W^*$$ of renewal-reward processes having inter-renewal time and reward distributions with heavy tails of exponents $$\alpha$$ and $$\beta$$, respectively, where $$1< \alpha< 2$$, $$0<\beta<2$$. In the case $$\beta\leq\alpha$$, $$W^*$$ suitably normalized converges to Lévy stable motion with index $$\beta$$; see the authors [Ann. Sci. Math. Qué. 11, 95-110 (1987; Zbl 0646.60091)]. The present work re-iterates this result and establishes a limit for the case $$\beta>\alpha$$, which is also a self-similar, symmetric $$\beta$$-stable process with stationary increments; here, however, the increments are dependent.
Reviewer: J.Preater (Keele)

### MSC:

 60K05 Renewal theory 60K30 Applications of queueing theory (congestion, allocation, storage, traffic, etc.)

Zbl 0646.60091
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