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Slow, fast and arbitrary growth conditions for renewal-reward processes when both the renewals and the rewards are heavy-tailed. (English) Zbl 1043.60040
Authors’ abstract: Consider $$M$$ independent and identically distributed renewal-reward processes with heavy-tailed renewals and rewards that have either finite variance or heavy tails. Let $$W^*(Ty, M)$$, $$y \in[0, 1]$$, denote the total reward process computed as the sum of all rewards in $$M$$ renewal-reward processes over the time interval $$[0, T]$$. If $$T\to\infty$$ and then $$M\to\infty$$, J. B. Levy and M. S. Taqqu [Bernoulli 6, No. 1, 23–44 (2000; Zbl 0954.60071)] have shown that the properly normalized total reward process $$W^*(T\cdot , M)$$ converges to the stable Lévy motion, but, if $$M\to\infty$$ followed by $$T\to\infty$$, the limit depends on whether the tails of the rewards are lighter or heavier than those of renewals. If they are lighter, then the limit is a self-similar process with stationary and dependent increments. If the rewards have finite variance, this self-similar process is fractional Brownian motion, and if they are heavy-tailed rewards, it is a stable non-Gaussian process with infinite variance. We consider asymmetric rewards and investigate what happens when $$M$$ and $$T$$ go to infinity jointly, that is, when $$M$$ is a function of $$T$$ and $$M = M(T) \to\infty$$ as $$T\to\infty$$. We provide conditions on the growth of $$M$$ for the total reward process $$W^*(T\cdot , M)$$ to converge to any of the limits stated above, as $$T\to\infty$$. We also show that when the tails of the rewards are heavier than the tails of the renewals, the limit is stable Lévy motion as $$M = M(T) \to\infty$$, irrespective of the function $$M(T)$$.

##### MSC:
 60G52 Stable stochastic processes 60K05 Renewal theory 60G18 Self-similar stochastic processes
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