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

Joshua B. Levy and Murad S. Taqqu

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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 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 Zβ is a β-stable self-similar process. If β≤α, then Zβ is the Lévy stable motion with independent increments; but if β> α, then Zβ is a stable process with dependent increments and self-similarity parameter H = (β- α+ 1)/β.

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Bernoulli, Volume 6, Number 1 (2000), 23-44.

First available in Project Euclid: 22 April 2004

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computer networks infinite variance self-similar processes stable processes telecommunications


Levy, Joshua B.; Taqqu, Murad S. Renewal reward processes with heavy-tailed inter-renewal times and heavy-tailed rewards. Bernoulli 6 (2000), no. 1, 23--44.

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