Persistence probabilities of MA(1) sequences with Laplace innovations and $q$-deformed zigzag numbers

TL;DR

This paper derives explicit formulas for the persistence probabilities of MA(1) sequences with Laplace innovations, using combinatorial and q-deformed functions, advancing understanding of stochastic process persistence.

Contribution

It introduces explicit computation methods for persistence probabilities of MA(1) processes with Laplace innovations using q-deformed combinatorial quantities.

Findings

Explicit formulas for persistence probabilities in terms of q-Pochhammer symbols.

Generating functions expressed via q-exponential and q-trigonometric functions.

Connections established between stochastic persistence and combinatorial q-analogues.

Abstract

We study the persistence probabilities of a moving average process of order one with innovations that follow a Laplace distribution. The persistence probabilities can be computed fully explicitly in terms of classical combinatorial quantities like certain $q$-Pochhammer symbols or $q$-deformed analogues of Euler's zigzag numbers, respectively. Similarly, the generating functions of the persistence probabilities can be written in terms of $q$-analogues of the exponential function or the $q$-sine/$q$-cosine functions, respectively.