Slow, fast and arbitrary growth conditions for renewal-reward processes when both the renewals and the rewards are heavy-tailed

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∈[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→∞ and then M→∞, Taqqu and Levy have shown that the properly normalized total reward process W * (T⋅,M) converges to the stable Lévy motion, but, if M→∞ followed by T→∞, 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)→∞ as T→∞. We provide conditions on the growth of M for the total reward process W * (T⋅,M(T)) to converge to any of the limits stated above, as T→∞. 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)→∞, irrespective of the function M(T).

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