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

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 is an element of [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 --> infinity and then M --> infinity, Taqqu and Levy have shown that the properly normalized total reward process W*(T., M) converges to the stable Levy motion, but, if M --> infinity followed by T --> infinity, 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) T-->infinity as T --> infinity. 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 --> infinity. We also show that when the tails of the rewards are heavier than the tails of the renewals, the limit is stable Levy motion as M = M(T) --> infinity, irrespective of the function M(T).