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Mixtures in non stable Levy processes

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  • Nicola Cufaro Petroni

Abstract

We analyze the Levy processes produced by means of two interconnected classes of non stable, infinitely divisible distribution: the Variance Gamma and the Student laws. While the Variance Gamma family is closed under convolution, the Student one is not: this makes its time evolution more complicated. We prove that -- at least for one particular type of Student processes suggested by recent empirical results, and for integral times -- the distribution of the process is a mixture of other types of Student distributions, randomized by means of a new probability distribution. The mixture is such that along the time the asymptotic behavior of the probability density functions always coincide with that of the generating Student law. We put forward the conjecture that this can be a general feature of the Student processes. We finally analyze the Ornstein--Uhlenbeck process driven by our Levy noises and show a few simulation of it.

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  • Nicola Cufaro Petroni, 2007. "Mixtures in non stable Levy processes," Papers math/0702058, arXiv.org.
    • Handle: RePEc:arx:papers:math/0702058
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    Cited by:

    1. Till Massing, 2018. "Simulation of Student–Lévy processes using series representations," Computational Statistics, Springer, vol. 33(4), pages 1649-1685, December.
    2. Grothe, Oliver & Schmidt, Rafael, 2010. "Scaling of Lévy–Student processes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(7), pages 1455-1463.
    3. Berg, C. & Vignat, C., 2010. "On the density of the sum of two independent Student t-random vectors," Statistics & Probability Letters, Elsevier, vol. 80(13-14), pages 1043-1055, July.

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