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Pré-Publication, Document De Travail Année : 2024

Persistence exponents via perturbation theory: MA(1)-processes

Résumé

For the moving average process $X_n=\rho \xi_{n-1}+\xi_n$, $n\in\mathbb{N}$, where $\rho\in\mathbb{R}$ and $(\xi_i)_{i\ge -1}$ is an i.i.d. sequence of normally distributed random variables, we study the persistence probabilities $\mathbb{P}(X_0\ge 0,\dots, X_N\ge 0)$, for $N\to\infty$. We exploit that the exponential decay rate $\lambda_\rho$ of that quantity, called the persistence exponent, is given by the leading eigenvalue of a concrete integral operator. This makes it possible to study the problem with purely functional analytic methods. In particular, using methods from perturbation theory, we show that the persistence exponent $\lambda_\rho$ can be expressed as a power series in $\rho$. Finally, we consider the persistence problem for the Slepian process, transform it into the moving average setup, and show that our perturbation results are applicable.
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Dates et versions

hal-04644436 , version 1 (11-07-2024)

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Frank Aurzada, Dieter Bothe, Pierre-Étienne Druet, Marvin Kettner, Christophe Profeta. Persistence exponents via perturbation theory: MA(1)-processes. 2024. ⟨hal-04644436⟩
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