Renewal reward processes with heavy-tailed inter-renewal times and heavy-tailed rewards

Abstract

It is well known that fractional Brownian motion can be obtained as the limit of a superposition of renewal reward processes with inter-renewal times that have infinite variance (heavy tails with exponent α) and with rewards that have finite variance. We show here that if the rewards also have infinite variance (heavy tails with exponent β) then the limit Z_β is a β-stable self-similar process. If β≤α, then Z_β is the Lévy stable motion with independent increments; but if β> α, then Z_β is a stable process with dependent increments and self-similarity parameter H = (β- α+ 1)/β.