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A class of fractional Ornstein–Uhlenbeck processes mixed with a Gamma distribution
Volume 10, Issue 1 (2023), pp. 37–57
Luigi Amedeo Bianchi ORCID icon link to view author Luigi Amedeo Bianchi details   Stefano Bonaccorsi ORCID icon link to view author Stefano Bonaccorsi details   Luciano Tubaro ORCID icon link to view author Luciano Tubaro details  

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https://doi.org/10.15559/22-VMSTA216
Pub. online: 21 November 2022      Type: Research Article      Open accessOpen Access

Received
14 October 2021
Revised
7 November 2022
Accepted
7 November 2022
Published
21 November 2022

Abstract

We consider a sequence of fractional Ornstein–Uhlenbeck processes, that are defined as solutions of a family of stochastic Volterra equations with a kernel given by the Riesz derivative kernel, and leading coefficients given by a sequence of independent Gamma random variables. We construct a new process by taking the empirical mean of this sequence. In our framework, the processes involved are not Markovian, hence the analysis of their asymptotic behaviour requires some ad hoc construction. In our main result, we prove the almost sure convergence in the space of trajectories of the empirical means to a given Gaussian process, which we characterize completely.

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Keywords
Fractional Ornstein–Uhlenbeck processes empirical means Gamma mixing stochastic Volterra equations generalized Wright function 60G22 60G17

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