Persistence exponents via perturbation theory: MA(1)-processes
| Autor: | Aurzada, Frank, Bothe, Dieter, Druet, Pierre-Étienne, Kettner, Marvin, Profeta, Christophe |
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| Rok vydání: | 2024 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 27 pages |
| Databáze: | arXiv |
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