On the non-commutative fractional Wishart process |
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Authors: | Juan Carlos Pardo José-Luis Pérez Victor Pérez-Abreu |
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Institution: | 1. Department of Probability and Statistics, Centro de Investigación en Matemáticas, Apartado Postal 402, Guanajuato GTO 36000, Mexico;2. Department of Probability and Statistics, IIMAS-UNAM, Mexico City, Mexico |
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Abstract: | We investigate the process of eigenvalues of a fractional Wishart process defined by , where B is the matrix fractional Brownian motion recently studied in 18]. Using stochastic calculus with respect to the Young integral we show that, with probability one, the eigenvalues do not collide at any time. When the matrix process B has entries given by independent fractional Brownian motions with Hurst parameter , we derive a stochastic differential equation in the Malliavin calculus sense for the eigenvalues of the corresponding fractional Wishart process. Finally, a functional limit theorem for the empirical measure-valued process of eigenvalues of a fractional Wishart process is obtained. The limit is characterized and referred to as the non-commutative fractional Wishart process, which constitutes the family of fractional dilations of the free Poisson distribution. |
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Keywords: | Fractional Wishart matrix process Measure valued process Young integral Fractional calculus |
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