Inheritance of strong mixing and weak dependence under renewal sampling

TL;DR

This paper investigates how renewal sampling affects the dependence properties of continuous-time processes, showing that strong mixing or weak dependence is preserved under certain conditions, enabling the use of existing limit theorems.

Contribution

It provides general conditions and explicit formulas for the dependence coefficients of renewal sampled processes, extending the applicability of limit theorems.

Findings

Dependence properties are preserved under renewal sampling.

Explicit formulas for dependence coefficients are derived.

Results apply to a wide class of strongly mixing or weakly dependent processes.

Abstract

Let $X$ be a continuous-time strongly mixing or weakly dependent process and $T$ a renewal process independent of $X$ with inter-arrival times $\tau$. We show general conditions under which the sampled process $(X_{T_i},T_i-T_{i-1})^{\top}$ is strongly mixing or weakly dependent. Moreover, we explicitly compute the strong mixing or weak dependence coefficients of the renewal sampled process and show that exponential or power decay of the coefficients of $X$ is preserved (at least asymptotically). Our results imply that essentially all central limit theorems available in the literature for strongly mixing or weakly dependent processes can be applied when renewal sampled observations of the process $X$ are at disposal.