Mixing and moments properties of a non-stationary copula-based Markov process

We provide conditions under which a non-stationary copula-based Markov process is geometric β-mixing and geometric ρ-mixing. Our results generalize some results of Beare who considers the stationary case. As a particular case we introduce a stochastic process, that we call convolution-based Markov process, whose construction is obtained by using the C-convolution operator which allows the increments to be dependent. Within this subclass of processes we characterize a modified version of the standard random walk where copulas and marginal distributions involved are in the same elliptical family. We study mixing and moments properties to identify the differences compared to the standard case.