Persistence and Ball Exponents for Gaussian Stationary Processes
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866917975032856576 |
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| author | Feldheim, Naomi Feldheim, Ohad Mukherjee, Sumit |
| author_facet | Feldheim, Naomi Feldheim, Ohad Mukherjee, Sumit |
| contents | Consider a real Gaussian stationary process $f_ρ$, indexed on either $\mathbb{R}$ or $\mathbb{Z}$ and admitting a spectral measure $ρ$. We study $θ_ρ^\ell=-\lim\limits_{T\to\infty}\frac{1}{T} \log\mathbb{P}\left(\inf_{t\in[0,T]}f_ρ(t)>\ell\right)$, the persistence exponent of $f_ρ$. We show that, if $ρ$ has a positive density at the origin, then the persistence exponent exists; moreover, if $ρ$ has an absolutely continuous component, then $θ_ρ^\ell>0$ if and only if this spectral density at the origin is finite. We further establish continuity of $θ_ρ^\ell$ in $\ell$, in $ρ$ (under a suitable metric) and, if $ρ$ is compactly supported, also in dense sampling. Analogous continuity properties are shown for $ψ_ρ^\ell=-\lim\limits_{T\to\infty}\frac{1}{T} \log\mathbb{P}\left(\inf_{t\in[0,T]}|f_ρ(t)|\le \ell\right)$, the ball exponent of $f_ρ$, and it is shown to be positive if and only if $ρ$ has an absolutely continuous component. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2112_04820 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Persistence and Ball Exponents for Gaussian Stationary Processes Feldheim, Naomi Feldheim, Ohad Mukherjee, Sumit Probability Classical Analysis and ODEs Functional Analysis 60G15, 60G10, 42A38 Consider a real Gaussian stationary process $f_ρ$, indexed on either $\mathbb{R}$ or $\mathbb{Z}$ and admitting a spectral measure $ρ$. We study $θ_ρ^\ell=-\lim\limits_{T\to\infty}\frac{1}{T} \log\mathbb{P}\left(\inf_{t\in[0,T]}f_ρ(t)>\ell\right)$, the persistence exponent of $f_ρ$. We show that, if $ρ$ has a positive density at the origin, then the persistence exponent exists; moreover, if $ρ$ has an absolutely continuous component, then $θ_ρ^\ell>0$ if and only if this spectral density at the origin is finite. We further establish continuity of $θ_ρ^\ell$ in $\ell$, in $ρ$ (under a suitable metric) and, if $ρ$ is compactly supported, also in dense sampling. Analogous continuity properties are shown for $ψ_ρ^\ell=-\lim\limits_{T\to\infty}\frac{1}{T} \log\mathbb{P}\left(\inf_{t\in[0,T]}|f_ρ(t)|\le \ell\right)$, the ball exponent of $f_ρ$, and it is shown to be positive if and only if $ρ$ has an absolutely continuous component. |
| title | Persistence and Ball Exponents for Gaussian Stationary Processes |
| topic | Probability Classical Analysis and ODEs Functional Analysis 60G15, 60G10, 42A38 |
| url | https://arxiv.org/abs/2112.04820 |