The limit of a renewal reward process with heavy-tailed rewards is not a linear fractional stable motion

Abstract

Levy and Taqqu (2000) considered a renewal reward process with both inter-renewal times and rewards that have heavy tails with exponents α and β, respectively. When 1<α<β< 2 and the renewal reward process is suitably normalized, the authors found that it converges to a symmetric β-stable process Z_β(t), t∈[0,1] which possesses stationary increments and is self-similar. They identified the limit process through its finite-dimensional characteristic functions. We provide an integral representation for the process and show that it does not belong to the family of linear fractional stable motions.