Persistence probabilities for MA(1) sequences with uniform innovations

TL;DR

This paper analyzes the persistence probabilities of MA(1) processes with uniform innovations, identifying different behavioral regions and deriving explicit generating functions and combinatorial expressions for these probabilities.

Contribution

It provides a comprehensive characterization of persistence probabilities for MA(1) with uniform innovations, including explicit formulas and combinatorial representations across different parameter regions.

Findings

Persistence probabilities vary qualitatively across regions.

Explicit generating functions are derived for all regions.

In some regions, probabilities are expressed via combinatorial quantities.

Abstract

We study the persistence probabilities of a moving average process of order one with uniform innovations. We identify a number of regions, characterized by the location of the uniform distribution and the coupling parameter of the process, where the persistence probabilities have qualitatively different behaviour. We obtain the generating functions of the persistence probabilities explicitly in all possible regions. In some of the regions, the persistence probabilities can be expressed explicitly in terms of various combinatorial quantities.