Persistence exponents via perturbation theory: MA(1)-processes
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866907973386764288 |
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| author | Aurzada, Frank Bothe, Dieter Druet, Pierre-Étienne Kettner, Marvin Profeta, Christophe |
| author_facet | Aurzada, Frank Bothe, Dieter Druet, Pierre-Étienne Kettner, Marvin Profeta, Christophe |
| contents | For the moving average process $X_n=ρξ_{n-1}+ξ_n$, $n\in\mathbb{N}$, where $ρ\in\mathbb{R}$ and $(ξ_i)_{i\ge -1}$ 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 $λ_ρ$ 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 $λ_ρ$ can be expressed as a power series in $ρ$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_06870 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Persistence exponents via perturbation theory: MA(1)-processes Aurzada, Frank Bothe, Dieter Druet, Pierre-Étienne Kettner, Marvin Profeta, Christophe Probability Functional Analysis For the moving average process $X_n=ρξ_{n-1}+ξ_n$, $n\in\mathbb{N}$, where $ρ\in\mathbb{R}$ and $(ξ_i)_{i\ge -1}$ 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 $λ_ρ$ 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 $λ_ρ$ can be expressed as a power series in $ρ$. 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. |
| title | Persistence exponents via perturbation theory: MA(1)-processes |
| topic | Probability Functional Analysis |
| url | https://arxiv.org/abs/2407.06870 |