An estimate for the tails of the distribution of the supremum for a class of stationary multiparameter Gaussian processes

1981 ◽  
Vol 18 (2) ◽  
pp. 536-541 ◽  
Author(s):  
E. M. Cabaña ◽  
M. Wschebor
Keyword(s):  
Gaussian Process ◽  
Gaussian Processes ◽  
Stationary Gaussian Process ◽  
Parameter Domain ◽  
Standard Of Comparison

Using Slepian processes as a standard of comparison, estimates are given for the probability that a centered multiparameter stationary Gaussian process reaches a constant barrier u on a subset of the parameter domain.

An estimate for the tails of the distribution of the supremum for a class of stationary multiparameter Gaussian processes

1981 ◽  
Vol 18 (02) ◽  
pp. 536-541 ◽  
Author(s):  
E. M. Cabaña ◽  
M. Wschebor
Keyword(s):  
Gaussian Process ◽  
Gaussian Processes ◽  
Stationary Gaussian Process ◽  
Parameter Domain ◽  
Standard Of Comparison

Using Slepian processes as a standard of comparison, estimates are given for the probability that a centered multiparameter stationary Gaussian process reaches a constant barrier u on a subset of the parameter domain.

Occupation times of stationary gaussian processes

1970 ◽  
Vol 7 (03) ◽  
pp. 721-733
Author(s):  
Simeon M. Berman
Keyword(s):  
Gaussian Process ◽  
Gaussian Processes ◽  
Covariance Function ◽  
Stationary Gaussian Process ◽  
Occupation Times ◽  
Unit Variance ◽  
Stationary Gaussian Processes ◽  
Image Position

Let X(t), t ≧ 0, be a stationary Gaussian process with zero mean, unit variance and continuous covariance function r(t). Suppose that, for some ε > 0 so that there is a version of the process whose sample functions are continuous [1].

scholarly journals Exponential Concentration for Zeroes of Stationary Gaussian Processes

2018 ◽  
Vol 2020 (23) ◽  
pp. 9769-9796
Author(s):  
Riddhipratim Basu ◽  
Amir Dembo ◽  
Naomi Feldheim ◽  
Ofer Zeitouni
Keyword(s):  
Gaussian Process ◽  
Gaussian Processes ◽  
Compact Support ◽  
Spectral Measure ◽  
Mean Value ◽  
Stationary Gaussian Process ◽  
Exponential Moments ◽  
Additional Regularity ◽  
Stationary Gaussian Processes ◽  
Exponentially Small

Abstract We show that for any centered stationary Gaussian process of absolutely integrable covariance, whose spectral measure has compact support, or finite exponential moments (and some additional regularity), the number of zeroes of the process in $[0,T]$ is within $\eta T$ of its mean value, up to an exponentially small in $T$ probability.

A class of almost nowhere differentiable stationary Gaussian processes which are somewhere differentiable

1973 ◽  
Vol 10 (3) ◽  
pp. 682-684 ◽  
Author(s):  
P. L. Davies
Keyword(s):  
Gaussian Process ◽  
Gaussian Processes ◽  
Covariance Function ◽  
Stationary Gaussian Process ◽  
Counter Example ◽  
Sample Paths ◽  
Stationary Gaussian Processes ◽  
Almost All ◽  
Set Of Points ◽  
The Continuum

A stationary Gaussian process is exhibited with the following property: the covariance function of the process is not differentiable at the origin and yet almost all the sample paths of the process are differentiable in a set of points of the power of the continuum. The process provides a counter example to a statement of Slepian.

scholarly journals On the Continuity of Stationary Gaussian Processes

1969 ◽  
Vol 34 ◽  
pp. 89-104 ◽  
Author(s):  
Makiko Nisio
Keyword(s):  
Fourier Series ◽  
Gaussian Process ◽  
Gaussian Processes ◽  
Covariance Function ◽  
Stationary Gaussian Process ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Stationary Gaussian Processes ◽  
Necessary And Sufficient ◽  
Special Case

Let us consider a stochastically continuous, separable and measurable stationary Gaussian process X = {X(t), − ∞ < t < ∞} with mean zero and with the covariance function p(t) = EX(t + s)X(s). The conditions for continuity of paths have been studied by many authors from various viewpoints. For example, Dudley [3] studied from the viewpoint of ε-entropy and Kahane [5] showed the necessary and sufficient condition in some special case, using the rather neat method of Fourier series.

A class of almost nowhere differentiable stationary Gaussian processes which are somewhere differentiable

1973 ◽  
Vol 10 (03) ◽  
pp. 682-684
Author(s):  
P. L. Davies
Keyword(s):  
Gaussian Process ◽  
Gaussian Processes ◽  
Covariance Function ◽  
Stationary Gaussian Process ◽  
Counter Example ◽  
Sample Paths ◽  
Stationary Gaussian Processes ◽  
Almost All ◽  
Set Of Points ◽  
The Continuum

A stationary Gaussian process is exhibited with the following property: the covariance function of the process is not differentiable at the origin and yet almost all the sample paths of the process are differentiable in a set of points of the power of the continuum. The process provides a counter example to a statement of Slepian.

Occupation times of stationary gaussian processes

1970 ◽  
Vol 7 (3) ◽  
pp. 721-733 ◽  
Author(s):  
Simeon M. Berman
Keyword(s):  
Gaussian Process ◽  
Gaussian Processes ◽  
Covariance Function ◽  
Stationary Gaussian Process ◽  
Occupation Times ◽  
Unit Variance ◽  
Stationary Gaussian Processes ◽  
Image Position

Let X(t), t ≧ 0, be a stationary Gaussian process with zero mean, unit variance and continuous covariance function r(t). Suppose that, for some ε > 0 so that there is a version of the process whose sample functions are continuous [1].

The excursions of a stationary Gaussian process outside a large two-dimensional region

2001 ◽  
Vol 33 (1) ◽  
pp. 141-159
Author(s):  
Robert Illsley
Keyword(s):  
Gaussian Process ◽  
Density Function ◽  
Covariance Matrix ◽  
Asymptotic Distribution ◽  
Gaussian Processes ◽  
Asymptotic Distributions ◽  
Piecewise Smooth ◽  
Stationary Gaussian Process ◽  
Two Dimensional ◽  
Marginal Density

Let X(t) be a continuous two-dimensional stationary Gaussian process with mean zero, having a marginal density function p[x] and covariance matrix R(t). Let Δ = {∂L; L > 0} be a family of piecewise smooth boundaries of similar two-dimensional star-shaped regions ΓL. We show that, under two conditions on R(t), the asymptotic distribution of the duration of an excursion of X(t) outside ΓL, for large L, depends on the position of the maximum of p[x] on ∂L and on whether R′(0) is zero or not, should the maximum occur at a vertex. We obtain the asymptotic distributions of the duration of an excursion for each of the three cases that arise. We also generalise some results of Breitung (1994) on the asymptotic crossing rates of vector Gaussian processes.

Minimal distributions in a stochastic partial ordering and bounds for Gaussian processes

1983 ◽  
Vol 20 (03) ◽  
pp. 529-536
Author(s):  
W. J. R. Eplett
Keyword(s):  
Gaussian Process ◽  
Gaussian Processes ◽  
Partial Ordering ◽  
Boundary Crossing ◽  
Conditional Probabilities ◽  
Life Distribution ◽  
Life Distributions ◽  
Bounded Below ◽  
Crossing Probabilities

A natural requirement to impose upon the life distribution of a component is that after inspection at some randomly chosen time to check whether it is still functioning, its life distribution from the time of checking should be bounded below by some specified distribution which may be defined by external considerations. Furthermore, the life distribution should ideally be minimal in the partial ordering obtained from the conditional probabilities. We prove that these specifications provide an apparently new characterization of the DFRA class of life distributions with a corresponding result for IFRA distributions. These results may be transferred, using Slepian's lemma, to obtain bounds for the boundary crossing probabilities of a stationary Gaussian process.

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