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

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 alpha) and with rewards that have finite variance. We show here that if the rewards also have infinite variance (heavy tails with exponent beta) then the limit Z(beta) is a beta-stable self-similar process. If beta less than or equal to alpha, then Z(beta) is the Levy stable motion with independent increments; but if beta > alpha, then Z(beta) is a stable process with dependent increments and self-similarity parameter H = (beta - alpha + 1)/beta.