The limit of a renewal reward process with heavy-tailed rewards is not a linear fractional stable motion. (English) Zbl 0963.60032

Pipiras, Vladas; Taqqu, Murad S.

Bernoulli 6, No. 4, 607-614 (2000).

There are considered a renewal reward process with both inter-renewal times and rewards that have heavy tails of exponents \(\alpha\) and \(\beta\), respectively, where \(1< \alpha< 2\), \(0<\beta<2\). It was proved by J. B. Levy and M. S. Taqqu [ibid. 6, No. 1, 23-44 (2000; Zbl 0954.60071)] that the suitably normalized renewal reward process converges to Lévy stable motion with index \(\beta\), possesses stationary increments and is self-similar in the case \(\beta>\alpha\). The limit process was identified through its finite-dimensional characteristic functions. The authors provide an integral representation for the process and show that it does not belong to the family of linear fractional stable motions.

Reviewer: Valentin Topchii (Omsk)

MSC:

60G18	Self-similar stochastic processes
60K05	Renewal theory

Keywords:

stable distributions; self-similar processes; renewal reward processes; stationary increments