Persistence of Gaussian stationary processes: a spectral perspective

Feldheim, Naomi; Feldheim, Ohad N.; Nitzan, Shahaf

doi:10.48550/arxiv.1709.00204

Cited by 6 publications

(14 citation statements)

References 16 publications

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“…There are sufficient conditions in the literature for truly exponential decay (cf. [11,16,18,19]), but in our understanding none of them are both necessary and sufficient. Definition 3.1 For any r ∈ R and a stationary non negative correlation function A(.)…”

Section: General Toolsmentioning

confidence: 95%

Persistence Probabilities of Two-Sided (Integrated) Sums of Correlated Stationary Gaussian Sequences

Аурзада

Buck

2018

J Stat Phys

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With {ξ i } i≥0 being a centered stationary Gaussian sequence with non negative correlation function ρ(i) := E [ξ 0 ξ i ] and {σ(i)} i≥1 a sequence of positive reals, we study the asymptotics of the persistence probability of the weighted sum ℓ i=1 σ(i)ξ i , ℓ ≥ 1. For summable correlations ρ, we show that the persistence exponent is universal. On the contrary, for non summable ρ, even for polynomial weight functions σ(i) ∼ i p the persistence exponent depends on the rate of decay of the correlations (encoded by a parameter H) and on the polynomial rate p of σ. In this case, we show existence of the persistence exponent θ(H, p) and study its properties as a function of (p, H). During the course of our proofs, we develop several tools for dealing with exit problems for Gaussian processes with non negative correlations -e.g. a continuity result for persistence exponents and a necessary and sufficient criterion for the persistence exponent to be zero -that might be of independent interest.

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Section: General Toolsmentioning

confidence: 95%

Persistence Probabilities of Two-Sided (Integrated) Sums of Correlated Stationary Gaussian Sequences

Аурзада

Buck

2018

J Stat Phys

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“…In this case the density of the spectral measure is a continuous non-vanishing function is a neighbourhood of the origin. For such functions Theorem 1 of [14] states that log P(Z = 0) ≈ L.…”

Section: Sharp Bounds and Phase Transitionmentioning

confidence: 99%

Intermediate and small scale limiting theorems for random fields

Beliaev¹,

Maffucci²

2019

Preprint

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In this paper we study the nodal lines of random eigenfunctions of the Laplacian on the torus, the so called 'arithmetic waves'. To be more precise, we study the number of intersections of the nodal line with a straight interval in a given direction. We are interested in how this number depends on the length and direction of the interval and the distribution of spectral measure of the random wave. We analyse the second factorial moment in the short interval regime and the persistence probability in the long interval regime. We also study relations between the Cilleruelo and Cilleruelo-type fields. We give an explicit coupling between these fields which on mesoscopic scales preserves the structure of the nodal sets with probability close to one.

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“…In [17], the hole probability asymptotics have been worked out for general L. In the process, a surprising discovery is made to the effect that the form of the asymptotics (including its dependence on L) depend crucially on whether L is sub-critical (0 < L < 1), critical (L = 1) or super-critical (L > 1). Another interesting family of results involves gap probabilities (essentially, hole probabilities in 1D) for important families of 1D Gaussian processes, in particular connecting these asymptotics with simple properties of their spectral measures and so-called "persistence probabilities" ( [25], [26], [4], [21], [22]). In [57] and [58] the author obtains fine quantitative estiamates on various aspects of the hole probability and the hole event for the Ginibre ensemble and related determinantal processes associated with higher Landau levels.…”

Section: Hole Events and Hole Probabilitiesmentioning

confidence: 99%

Point Processes, Hole Events, and Large Deviations: Random Complex Zeros and Coulomb Gases

Ghosh

Nishry

2018

Constr Approx

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We consider particle systems (also known as point processes) on the line and in the plane, and are particularly interested in "hole" events, when there are no particles in a large disk (or some other domain). We survey the extensive work on hole probabilities and the related large deviation principles (LDP), which has been undertaken mostly in the last two decades. We mainly focus on the recent applications of LDP-inspired techniques to the study of hole probabilities, and the determination of the most likely configurations of particles that have large holes.As an application of this approach, we illustrate how one can confirm some of the predictions of Jancovici, Lebowitz, and Manificat for large fluctuation in the number of points for the (two-dimensional) β-Ginibre ensembles. We also discuss some possible directions for future investigations.

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Persistence of Gaussian stationary processes: a spectral perspective

Abstract: We study the persistence probability of a centered stationary Gaussian process on Z or R, that is, its probability to remain positive for a long time. We describe the delicate interplay between this probability and the behavior of the spectral measure of the process near zero and infinity.

Cited by 6 publications

References 16 publications

Persistence Probabilities of Two-Sided (Integrated) Sums of Correlated Stationary Gaussian Sequences

Persistence Probabilities of Two-Sided (Integrated) Sums of Correlated Stationary Gaussian Sequences

Intermediate and small scale limiting theorems for random fields

Point Processes, Hole Events, and Large Deviations: Random Complex Zeros and Coulomb Gases

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