Small ball probabilities of Gaussian fields

Shao, Qi-Man; Wang, D.

doi:10.1007/bf01198847

Cited by 29 publications

(12 citation statements)

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“…In 1994, Talagrand [28] succeeded in proving the sharpness of (5) for d=2. Later on other authors adapted the methods of Bass and Talagrand to other processes (see, e.g., [23,30]). Surprisingly, the methods of [28] meet intrinsic difficulties when one tries to apply them for dimension d>2.…”

Section: Definitions and Resultsmentioning

confidence: 98%

Metric Entropy of Integration Operators and Small Ball Probabilities for the Brownian Sheet

Dunker

Linde

Kühn

et al. 1999

Journal of Approximation Theory

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Section: Definitions and Resultsmentioning

confidence: 98%

Metric Entropy of Integration Operators and Small Ball Probabilities for the Brownian Sheet

Dunker

Linde

Kühn

et al. 1999

Journal of Approximation Theory

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“…Persistence for (the absolute value of) the planar Gaussian analytic function was studied by Sodin and Tsirelson [18] and Nishry [15]. Also of interest are the results of Shao and Wang [16].…”

Section: Brief Review Of Past Resultsmentioning

confidence: 99%

Persistence probabilities in centered, stationary, Gaussian processes in discrete time

Krishna

Krishnapur

2016

Indian J Pure Appl Math

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THE PROBLEM AND OUR RESULTSLet X = (X m ) m∈Z d be a centered, stationary Gaussian process on Z d . This means that for any k ≥ 1 and any m 0 , . . . , m k ∈ Z d , the vector (X mj +m0 ) 1≤j≤k has a multivariate Gaussian distribution with zero mean and a covariance matrix that does not depend on m 0 . For basics on Gaussian processes, consult for example, the book by Adler [1].For a subset A ⊆ Z d , we define the persistence probability (also called gap probability or hole probability) of X in A asIn particular, one may be interested inThis paper is exclusively about getting bounds on the persistence probability under some additional conditions on the Gaussian process.A stationary Gaussian process on Z d is uniquely described by its covariance kernel Cov(X m , X n ) = Cov(X 0 , X m−n ). Further, there exists a unique finite Borel measure µ on T d that is symmetric about the origin (i.e., µ(I) = µ(−I) for any Borel set I ⊆ T d ) such that Cov(X 0 , X m ) =μ(m) whereμ(m) = T d e i m,t dµ(t) with the usual notation for the inner product m, t = m 1 t 1 + . . . + m d t d . The measure µ is called the spectral measure of the process X. Write dµ(t) = b(t)dλ(t) + dµ s (t) where µ s is singular to Lebesgue measure and b ∈ L 1 (T d , λ) is non-negative. In all the results of this paper, it will be assumed that b is not identically zero. In other words, the spectral measure is not singular. Lastly, for a subset A ⊆ Z d , we denote the covariance matrix of (X m ) m∈A by Σ A := (μ(j − k)) j,k∈A and the cardinality of A by |A|.These notations will be maintained throughout the paper without further mention. In addition, there will appear many constants denoted by C, c, γ etc. Unless otherwise mentioned, the constants depend on the given process X (or equivalently, on the spectral measure µ).We now state our results and then give an overview of past results in the literature in Section 2. Our first theorem has already been proved by N. Feldheim and O. Feldheim [8] and but we explain in Section 2 why we include it here nevertheless.

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“…Indeed, as can be surmised from Kuelbs and Li [16], this problem is often equivalent to open problems in functional analysis. Known results for well-behaved processes (especially, Brownian motion and related processes) can be found for example in the recent works [4], [7], [15]- [18], [21]- [23], [27]- [31].…”

Section: Introductionmentioning

confidence: 99%