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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 in®nite variance (heavy tails with exponent á) and with rewards that have ®nite variance. We show here that if the rewards also have in®nite variance (heavy tails with exponent â) then the limit Z â is a â-stable self-similar process. If â < á, then Z â is the Le Âvy stable motion with independent increments; but if â . á, then Z â is a stable process with dependent increments and self-similarity parameter H (â À á 1)aâ.

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