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

Frank Aurzada; Kilian Raschel

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arXiv · 2512.14152

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

Frank Aurzada·Kilian Raschel

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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.

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Frank Aurzada, Kilian Raschel. 2025-12-16. Persistence probabilities of MA(1) sequences with Laplace innovations and $q$-deformed zigzag numbers. https://arxiv.org/abs/2512.14152

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