Wave-length and amplitude in Gaussian noise

1972 ◽  
Vol 4 (1) ◽  
pp. 81-108 ◽  
Author(s):  
Georg Lindgren
Keyword(s):  
Gaussian Process ◽  
Density Function ◽  
Gaussian Noise ◽  
Covariance Function ◽  
Wave Length ◽  
Local Maximum ◽  
Stationary Gaussian Process ◽  
Numerical Examples ◽  
Vertical Distance

We give moment approximations to the density function of the wavelength, i. e., the time between “a randomly chosen” local maximum with height u and the following minimum in a stationary Gaussian process with a given covariance function. For certain processes we give similar approximations to the distribution of the amplitude, i. e., the vertical distance between the maximum and the minimum. Numerical examples and diagrams illustrate the results.

Wave-length and amplitude in Gaussian noise

1972 ◽  
Vol 4 (01) ◽  
pp. 81-108 ◽  
Author(s):  
Georg Lindgren
Keyword(s):  
Gaussian Process ◽  
Density Function ◽  
Gaussian Noise ◽  
Covariance Function ◽  
Wave Length ◽  
Local Maximum ◽  
Stationary Gaussian Process ◽  
Numerical Examples ◽  
Vertical Distance

We give moment approximations to the density function of the wavelength, i. e., the time between “a randomly chosen” local maximum with height u and the following minimum in a stationary Gaussian process with a given covariance function. For certain processes we give similar approximations to the distribution of the amplitude, i. e., the vertical distance between the maximum and the minimum. Numerical examples and diagrams illustrate the results.

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].

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.

A Stochastic Model for the Inheritance of the Cancer Proneness Phenotype

1987 ◽  
Vol 26 (03) ◽  
pp. 117-123
Author(s):  
P. Tautu ◽  
G. Wagner
Keyword(s):  
Stochastic Model ◽  
Gaussian Process ◽  
Covariance Function ◽  
Neoplastic Disease ◽  
Disease Phenotype ◽  
Probabilistic Representation ◽  
Temporal Behaviour ◽  
Continuous Trait ◽  
Continuous Parameter ◽  
Level Zero

SummaryA continuous parameter, stationary Gaussian process is introduced as a first approach to the probabilistic representation of the phenotype inheritance process. With some specific assumptions about the components of the covariance function, it may describe the temporal behaviour of the “cancer-proneness phenotype” (CPF) as a quantitative continuous trait. Upcrossing a fixed level (“threshold”) u and reaching level zero are the extremes of the Gaussian process considered; it is assumed that they might be interpreted as the transformation of CPF into a “neoplastic disease phenotype” or as the non-proneness to cancer, respectively.

scholarly journals On Asymptotic Efficiency of the M2M4 Signal-to-Noise Estimator for Deterministic Complex Sinusoids

Sensors ◽  
2021 ◽  
Vol 21 (15) ◽  
pp. 4950
Author(s):  
Gianmarco Romano
Keyword(s):  
Covariance Matrix ◽  
Gaussian Noise ◽  
Asymptotic Efficiency ◽  
Noise Power ◽  
Finite Sample ◽  
Signal To Noise ◽  
Numerical Examples ◽  
Unknown Phase ◽  
The Moment ◽  
Asymptotically Efficient

The moment-based M2M4 signal-to-noise (SNR) estimator was proposed for a complex sinusoidal signal with a deterministic but unknown phase corrupted by additive Gaussian noise by Sekhar and Sreenivas. The authors studied its performances only through numerical examples and concluded that the proposed estimator is asymptotically efficient and exhibits finite sample super-efficiency for some combinations of signal and noise power. In this paper, we derive the analytical asymptotic performances of the proposed M2M4 SNR estimator, and we show that, contrary to what it has been concluded by Sekhar and Sreenivas, the proposed estimator is neither (asymptotically) efficient nor super-efficient. We also show that when dealing with deterministic signals, the covariance matrix needed to derive asymptotic performances must be explicitly derived as its known general form for random signals cannot be extended to deterministic signals. Numerical examples are provided whose results confirm the analytical findings.

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