Almost periodic stationary 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 $(X_t{)}_{t\in \mathbb{R}}$ defined by the stochastic integral $X_t:={\int }_{{\mathbb{R}}^d}f(t,s)dL(s)$ is almost periodic stationary and also when it is almost periodic in probability, where $f(t,\cdot )\in L^1({\mathbb{R}}^d,\mathbb{R})\cap L^2({\mathbb{R}}^d,\mathbb{R})$ is deterministic and $L$ is a Lévy 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.