scholarly journals On the structure of splitting fields of stationary Gaussian processes with finite multiple Markovian property

1974 ◽  
Vol 54 ◽  
pp. 191-213 ◽  
Author(s):  
Yasunori Okabe
Keyword(s):  
Gaussian Process ◽  
Gaussian Processes ◽  
Markovian Property ◽  
Stationary Gaussian Processes ◽  
Continuous Gaussian Process

Let X = (X(t); t ∈ R) be a real stationary mean continuous Gaussian process with expectation zero which is purely nondeterministic. In this paper we shall investigate the structure of splitting fields of X having finite multiple Markovian property using the results in [6]. We follow the notations and terminologies in [6].

Approximation to the exit probability of a continuous Gaussian process over a U-shaped boundary of increasing curvature

1995 ◽  
Vol 32 (2) ◽  
pp. 429-442
Author(s):  
A. N. Balabushkin
Keyword(s):  
Brownian Motion ◽  
Gaussian Process ◽  
Gaussian Processes ◽  
Simple Approximation ◽  
Minimum Point ◽  
Second Derivative ◽  
Numerical Examples ◽  
Exit Probability ◽  
Continuous Gaussian Process

A simple approximation to the probability of crossing a U-shaped boundary by a Brownian motion is given. The larger the second derivative of the curve at a minimum point, the higher the accuracy of the approximation. The result is also extended to a class of continuous Gaussian processes with definite properties. Numerical examples are given.

Approximation to the exit probability of a continuous Gaussian process over a U-shaped boundary of increasing curvature

1995 ◽  
Vol 32 (02) ◽  
pp. 429-442
Author(s):  
A. N. Balabushkin
Keyword(s):  
Brownian Motion ◽  
Gaussian Process ◽  
Gaussian Processes ◽  
Simple Approximation ◽  
Minimum Point ◽  
Second Derivative ◽  
Numerical Examples ◽  
Exit Probability ◽  
Continuous Gaussian Process

A simple approximation to the probability of crossing a U-shaped boundary by a Brownian motion is given. The larger the second derivative of the curve at a minimum point, the higher the accuracy of the approximation. The result is also extended to a class of continuous Gaussian processes with definite properties. Numerical examples are given.

Weak approximation for a class of Gaussian processes

2000 ◽  
Vol 37 (02) ◽  
pp. 400-407 ◽  
Author(s):  
Rosario Delgado ◽  
Maria Jolis
Keyword(s):  
Brownian Motion ◽  
Poisson Process ◽  
Gaussian Process ◽  
Gaussian Processes ◽  
Stochastic Integral ◽  
Hurst Parameter ◽  
Standard Wiener Process ◽  
Weak Approximation ◽  
The Law ◽  
Continuous Gaussian Process

We prove that, under rather general conditions, the law of a continuous Gaussian process represented by a stochastic integral of a deterministic kernel, with respect to a standard Wiener process, can be weakly approximated by the law of some processes constructed from a standard Poisson process. An example of a Gaussian process to which this result applies is the fractional Brownian motion with any Hurst parameter.

Weak approximation for a class of Gaussian processes

2000 ◽  
Vol 37 (2) ◽  
pp. 400-407 ◽  
Author(s):  
Rosario Delgado ◽  
Maria Jolis
Keyword(s):  
Brownian Motion ◽  
Poisson Process ◽  
Gaussian Process ◽  
Gaussian Processes ◽  
Stochastic Integral ◽  
Hurst Parameter ◽  
Standard Wiener Process ◽  
Weak Approximation ◽  
The Law ◽  
Continuous Gaussian Process

We prove that, under rather general conditions, the law of a continuous Gaussian process represented by a stochastic integral of a deterministic kernel, with respect to a standard Wiener process, can be weakly approximated by the law of some processes constructed from a standard Poisson process. An example of a Gaussian process to which this result applies is the fractional Brownian motion with any Hurst parameter.

Joint distribution of successive zero crossing distances for stationary Gaussian processes

1987 ◽  
Vol 24 (02) ◽  
pp. 378-385 ◽  
Author(s):  
Igor Rychlik
Keyword(s):  
Gaussian Process ◽  
Closed Form ◽  
Gaussian Processes ◽  
Joint Distribution ◽  
Closed Form Expression ◽  
Regression Curve ◽  
Form Expression ◽  
Zero Crossing ◽  
Stationary Gaussian Processes ◽  
Approximative Formula

As has been shown by de Maré, in a stationary Gaussian process the length of the successive zero-crossing intervals cannot be independent, except for the degenerate case of a pure cosine process. However, no closed-form expression of the distribution of these quantities is known at present. In this paper we present an accurate explicit approximative formula, derived by replacing the Slepian model process by its regression curve.

scholarly journals Optimalities for random functions Lee-Wiener’s network and non-canonical representation of stationary Gaussian processes

1998 ◽  
Vol 149 ◽  
pp. 9-17 ◽  
Author(s):  
Win Win Htay
Keyword(s):  
Brownian Motion ◽  
Gaussian Process ◽  
Gaussian Processes ◽  
Theoretical Approach ◽  
Canonical Representation ◽  
Prediction Theory ◽  
Random Functions ◽  
Canonical Representations ◽  
Stationary Gaussian Processes ◽  
Information Theoretical Approach

Abstract.Representation of a Gaussian process in terms of a Brownian motion is a powerful tool in the investigation of its structure. Among various representations is the canonical representation which is viewed as the best one from the viewpoint of the prediction theory. We have discovered some significance of non-canonical representations and discuss their optimality in an information theoretical approach.

Joint distribution of successive zero crossing distances for stationary Gaussian processes

1987 ◽  
Vol 24 (2) ◽  
pp. 378-385 ◽  
Author(s):  
Igor Rychlik
Keyword(s):  
Gaussian Process ◽  
Closed Form ◽  
Gaussian Processes ◽  
Joint Distribution ◽  
Closed Form Expression ◽  
Regression Curve ◽  
Form Expression ◽  
Zero Crossing ◽  
Stationary Gaussian Processes ◽  
Approximative Formula

As has been shown by de Maré, in a stationary Gaussian process the length of the successive zero-crossing intervals cannot be independent, except for the degenerate case of a pure cosine process. However, no closed-form expression of the distribution of these quantities is known at present. In this paper we present an accurate explicit approximative formula, derived by replacing the Slepian model process by its regression curve.

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.

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