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Persistence and Ball Exponents for Gaussian Stationary Processes

Naomi Feldheim, Ohad Feldheim, Sumit Mukherjee

TL;DR

The paper addresses the persistence and ball exponents for Gaussian stationary processes by linking the existence and positivity of θρ^ℓ and ψρ^ℓ to the spectral measure near the origin. It develops a robust framework combining spectral methods with covariance-based arguments, including three key comparison lemmata (level continuity, smoothing, and measure continuity) that enable transferring results from smooth, compactly supported densities to general ρ in the class 𝓛. A central finding is that θρ^ℓ exists for ρ ∈ 𝓛 ∩ 𝓜, and θρ^ℓ > 0 when ρ_ac ≠ 0 if and only if ρ'(0) < ∞; analogous statements hold for ψρ^ℓ, with ball exponents being positive exactly when ρ_ac ≠ 0. The work also establishes continuity results of the exponents in the level ℓ and in the spectral measure under suitable metrics, and proves convergence of exponents under dense sampling when ρ is compactly supported, complemented by a non-existence example and a non-convergence example under sampling to highlight sharpness of assumptions. Overall, the results provide a precise, spectrally guided understanding of persistence and small-deviation phenomena for continuous- and discrete-time Gaussian processes, with implications for statistical physics and signal processing.

Abstract

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.

Persistence and Ball Exponents for Gaussian Stationary Processes

TL;DR

The paper addresses the persistence and ball exponents for Gaussian stationary processes by linking the existence and positivity of θρ^ℓ and ψρ^ℓ to the spectral measure near the origin. It develops a robust framework combining spectral methods with covariance-based arguments, including three key comparison lemmata (level continuity, smoothing, and measure continuity) that enable transferring results from smooth, compactly supported densities to general ρ in the class 𝓛. A central finding is that θρ^ℓ exists for ρ ∈ 𝓛 ∩ 𝓜, and θρ^ℓ > 0 when ρ_ac ≠ 0 if and only if ρ'(0) < ∞; analogous statements hold for ψρ^ℓ, with ball exponents being positive exactly when ρ_ac ≠ 0. The work also establishes continuity results of the exponents in the level ℓ and in the spectral measure under suitable metrics, and proves convergence of exponents under dense sampling when ρ is compactly supported, complemented by a non-existence example and a non-convergence example under sampling to highlight sharpness of assumptions. Overall, the results provide a precise, spectrally guided understanding of persistence and small-deviation phenomena for continuous- and discrete-time Gaussian processes, with implications for statistical physics and signal processing.

Abstract

Consider a real Gaussian stationary process , indexed on either or and admitting a spectral measure . We study , the persistence exponent of . We show that, if has a positive density at the origin, then the persistence exponent exists; moreover, if has an absolutely continuous component, then if and only if this spectral density at the origin is finite. We further establish continuity of in , in (under a suitable metric) and, if is compactly supported, also in dense sampling. Analogous continuity properties are shown for , the ball exponent of , and it is shown to be positive if and only if has an absolutely continuous component.
Paper Structure (43 sections, 39 theorems, 263 equations)

This paper contains 43 sections, 39 theorems, 263 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Background
  4. Gaussian stationary processes and persistence
  5. Related processes and events
  6. Comparison lemmata for persistence probabilities
  7. Examples
  8. Outline of the paper
  9. Preliminaries
  10. Gaussian measures on Euclidean spaces
  11. Operations on GSPs
  12. Decompositions of GSPs
  13. Classical Gaussian tools
  14. Supremum
  15. Bounds on ball and persistence exponents
  16. ...and 28 more sections

Key Result

Theorem 1

Let $\rho\in \mathcal{L}\cap \mathcal{M}$. Then, for all $\ell\in\mathbb R$ a persistence exponent $\theta_\rho^\ell\in[0,\infty)$ exists. Moreover, under the further assumption $\rho_{ac}\ne 0$, we have $\theta_\rho^\ell>0$ if and only if $\rho'(0)<\infty$.

Theorems & Definitions (77)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 2
  • Theorem 3
  • Remark 4
  • Theorem 4
  • Remark 5
  • Theorem 5
  • ...and 67 more