A phase transition in zero overcrowding and undercrowding probabilities for stationary gaussian processes
Naomi Feldheim,Ohad N. Feldheim +1 more
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In this paper , the authors studied the probability that a real stationary Gaussian process has at least ηT zeros in [0, T ] (overcrowding), or at most this number (undercrowded) and showed that if the spectral measure of the process is supported on ± [B, A ], overcrowding probability transitions from exponential decay to Gaussian decay at η = Aπ , while undercrowdness probability undergoes the reverse transition at δ = Bπ .Abstract:
. We study the probability that a real stationary Gaussian process has at least ηT zeros in [0 , T ] (overcrowding), or at most this number (undercrowding). We show that if the spectral measure of the process is supported on ± [ B, A ], overcrowding probability transitions from exponential decay to Gaussian decay at η = Aπ , while undercrowding probability undergoes the reverse transition at η = Bπ .read more
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Mathematical analysis of random noise
TL;DR: In this paper, the authors used the representations of the noise currents given in Section 2.8 to derive some statistical properties of I(t) and its zeros and maxima.
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Random Fields and Geometry
Robert J. Adler,Jonathan Taylor +1 more
TL;DR: Random Fields and Geometry as discussed by the authors is a comprehensive survey of the general theory of Gaussian random fields with a focus on geometric problems arising in the study of random fields, including continuity and boundedness, entropy and majorizing measures, Borell and Slepian inequalities.
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Lectures on entire functions
TL;DR: In this article, the authors considered the problem of the growth of an entire function and the distribution of its zeros, and they gave a lower bound on the maximum number of zeros of a function in the half-plane.
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The one-sided barrier problem for Gaussian noise
TL;DR: In this paper, the authors considered the probability that a stationary Gaussian process with mean zero and covariance function r(τ) be nonnegative throughout a given interval of duration T. Several strict upper and lower bounds for P were given, along with some comparison theorems that relate P's for different covariance functions.
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Level sets and extrema of random processes and fields
Jean-Marc Azaïs,Mario Wschebor +1 more
TL;DR: In this article, the authors present a generalization of the Rice series for Gaussian processes with continuous paths and show that it is invariant under orthogonal transformations and translations.