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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 \(M(T)\).

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