A class of fractional Ornstein-Uhlenbeck processes mixed with a Gamma distribution

TL;DR

This paper studies a sequence of non-Markovian fractional Ornstein-Uhlenbeck processes with Gamma-distributed coefficients, showing that their empirical mean converges almost surely to a fully characterized Gaussian process.

Contribution

It introduces a novel class of fractional Ornstein-Uhlenbeck processes with Gamma mixing, analyzing their asymptotic behavior and establishing convergence to a Gaussian process.

Findings

Empirical means converge almost surely to a Gaussian process.

The limiting Gaussian process is explicitly characterized.

The processes are non-Markovian, requiring specialized analysis.

Abstract

We consider a sequence of fractional Ornstein-Uhlenbeck processes, that are defined as solutions of a family of stochastic Volterra equations with 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.