Almost periodic stationary processes

TL;DR

This paper characterizes almost periodicity in stochastic processes, providing conditions for various classes including infinitely divisible, Lévy-driven, and Ornstein-Uhlenbeck processes, and establishes a central limit theorem for m-dependent cases.

Contribution

It offers new necessary and sufficient conditions for almost periodic finite-dimensional distributions and characterizes almost periodicity for Lévy-based and Ornstein-Uhlenbeck processes.

Findings

Characterization of almost periodic finite-dimensional distributions.

Conditions for Lévy processes to be almost periodic.

Central limit theorem for m-dependent almost periodic processes.

Abstract

We derive a necessary and sufficient condition for stochastic processes to have almost periodic finite dimensional distributions; in particular, we obtain characterizations for infinitely divisible processes to be almost periodic in terms of their characteristic triplets. Furthermore, we derive conditions when the process $(X_t)_{t\in\R}$ defined by the stochastic integral $X_t:= \int_{\R^d} f(t,s) dL(s)$ is almost periodic stationary and also when it is almost periodic in probability, where $f(t,\cdot)\in L^1(\R^d,\R)\cap L^2(\R^d,\R)$ is deterministic and $L$ is a L\'evy basis. Moreover, we discuss almost periodic Ornstein-Uhlenbeck-type processes, and obtain a central limit theorem for $m$-dependent processes with almost periodic finite dimensional distributions.