Random Field and Stochastic Process

2021 ◽  
pp. 227-274
Keyword(s):  
Stochastic Process ◽  
Random Field

Innovations for Random Fields

1998 ◽  
Vol 01 (04) ◽  
pp. 499-509 ◽  
Author(s):  
Takeyuki Hida ◽  
Si Si
Keyword(s):  
Stochastic Process ◽  
Random Field ◽  
White Noise ◽  
Random Fields ◽  
Noise Analysis ◽  
Variational Calculus ◽  
Modern Theory ◽  
White Noise Analysis ◽  
Future Directions ◽  
Probabilistic Structure

There is a famous formula called Lévy's stochastic infinitesimal equation for a stochastic process X(t) expressed in the form [Formula: see text] We propose a generalization of this equation for a random field X(C) indexed by a contour C. Assume that the X(C) is homogeneous in a white noise x, say of degree n, we can then appeal to the classical theory of variational calculus and to the modern theory of white noise analysis in order to discuss the innovation for the X(C) and hence its probabilistic structure. Some of future directions are also mentioned.

A VARIATIONAL FORMULA FOR SOME RANDOM FIELDS: AN ANALOGUE OF ITO'S FORMULA

1999 ◽  
Vol 02 (02) ◽  
pp. 305-313
Author(s):  
SI SI
Keyword(s):  
Stochastic Process ◽  
Random Field ◽  
White Noise ◽  
Random Fields ◽  
Gaussian Random Field ◽  
Nonlinear Function ◽  
Variational Formula ◽  
Canonical Representation ◽  
Two Dimensional ◽  
Dimensional Parameter

We shall first establish a canonical representation of a Gaussian random field X(C) indexed by a smooth contour C in terms of two-dimensional parameter white noise. Then, we take a nonlinear function F(X(C)) of the X(C) and obtain its variation when C deforms slightly. The variational formula is analogous to the Ito formula for a stochastic process X(t), but somewhat simpler.

Excursions above a fixed level by n-dimensional random fields

1976 ◽  
Vol 13 (2) ◽  
pp. 276-289 ◽  
Author(s):  
Robert J. Adler
Keyword(s):  
Stochastic Process ◽  
Random Field ◽  
Random Fields ◽  
Mean Value ◽  
Regularity Conditions ◽  
Gaussian Field ◽  
Differential Topology ◽  
One Dimensional ◽  
Excursion Set ◽  
The Mean

For an n-dimensional random field X(t) we define the excursion set A of X(t) by A = [t ∊ S: X(t) ≧ u] for real u and compact S ⊂ Rn. We obtain a generalisation of the number of upcrossings of a one-dimensional stochastic process to random fields via a characteristic of the set A related to the Euler characteristic of differential topology. When X(t) is a homogeneous Gaussian field satisfying certain regularity conditions we obtain an explicit formula for the mean value of this characteristic.

scholarly journals ON FUNCTIONAL CENTRAL LIMIT THEOREMS FOR LINEAR RANDOM FIELDS WITH DEPENDENT INNOVATIONS

2008 ◽  
Vol 49 (4) ◽  
pp. 533-541 ◽  
Author(s):  
MI-HWA KO ◽  
HYUN-CHULL KIM ◽  
TAE-SUNG KIM
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
Random Field ◽  
Central Limit ◽  
Random Fields ◽  
Limit Theorems ◽  
Sufficient Conditions ◽  
Standard Brownian Motion ◽  
Functional Central Limit Theorems ◽  
Functional Central Limit

AbstractFor a linear random field (linear p-parameter stochastic process) generated by a dependent random field with zero mean and finite qth moments (q>2p), we give sufficient conditions that the linear random field converges weakly to a multiparameter standard Brownian motion if the corresponding dependent random field does so.

Excursions above a fixed level by n-dimensional random fields

1976 ◽  
Vol 13 (02) ◽  
pp. 276-289 ◽  
Author(s):  
Robert J. Adler
Keyword(s):  
Stochastic Process ◽  
Random Field ◽  
Random Fields ◽  
Mean Value ◽  
Regularity Conditions ◽  
Gaussian Field ◽  
Differential Topology ◽  
One Dimensional ◽  
Excursion Set ◽  
The Mean

For an n-dimensional random field X(t) we define the excursion set A of X(t) by A = [t ∊ S: X(t) ≧ u] for real u and compact S ⊂ Rn. We obtain a generalisation of the number of upcrossings of a one-dimensional stochastic process to random fields via a characteristic of the set A related to the Euler characteristic of differential topology. When X(t) is a homogeneous Gaussian field satisfying certain regularity conditions we obtain an explicit formula for the mean value of this characteristic.

scholarly journals Balls left empty by a critical branching Wiener process

1996 ◽  
Vol 9 (4) ◽  
pp. 531-549
Author(s):  
Pál Révész
Keyword(s):  
Stochastic Process ◽  
Random Field ◽  
Wiener Process

At time t=0 we have a Poisson random field on ℝd. Each particle executes a critical branching Wiener process starting from its position at time t=0. Let RT be the radius of the largest ball around the origin of ℝd which does not contain any particle at time T. Our goal is to characterize the properties of the stochastic process {RT,T≥0}.This article is dedicated to the memory of Professor Roland L. Dobrushin.

Asymptotic distribution of a discrete transform of an arbitrarily sampled homogeneous random field

1975 ◽  
Vol 7 (01) ◽  
pp. 140-153
Author(s):  
David A. Swick
Keyword(s):  
Stochastic Process ◽  
Asymptotic Normality ◽  
Random Field ◽  
Real Data ◽  
Statistical Testing ◽  
One Dimension ◽  
Homogeneous Random Field ◽  
Uniform Sampling ◽  
Non Uniform Sampling ◽  
Discrete Transform

Multidimensional sampling of real data, e.g., in space and time, often requires observations at non-uniformly spaced intervals. A discrete transform of a multidimensional stationary stochastic process transforms a multivariate problem into an asymptotically univariate one if the spacing is uniform in at least one dimension. For both uniform and non-uniform sampling and a model of ‘signal’ imbedded in a ‘noise’ process, asymptotic normality and independence justifies statistical testing in each cell of the transformed domain of the hypothesis ‘noise alone’ versus the alternate ‘signal plus noise’.

Asymptotic distribution of a discrete transform of an arbitrarily sampled homogeneous random field

1975 ◽  
Vol 7 (1) ◽  
pp. 140-153
Author(s):  
David A. Swick
Keyword(s):  
Stochastic Process ◽  
Asymptotic Normality ◽  
Random Field ◽  
Real Data ◽  
Statistical Testing ◽  
One Dimension ◽  
Homogeneous Random Field ◽  
Uniform Sampling ◽  
Non Uniform Sampling ◽  
Discrete Transform

Multidimensional sampling of real data, e.g., in space and time, often requires observations at non-uniformly spaced intervals. A discrete transform of a multidimensional stationary stochastic process transforms a multivariate problem into an asymptotically univariate one if the spacing is uniform in at least one dimension. For both uniform and non-uniform sampling and a model of ‘signal’ imbedded in a ‘noise’ process, asymptotic normality and independence justifies statistical testing in each cell of the transformed domain of the hypothesis ‘noise alone’ versus the alternate ‘signal plus noise’.

Probability Distribution for Mobilized Shear Strengths of Saturated Undrained Clays Modeled by 2-D Stationary Gaussian Random Field - A 1-D Stochastic Process View

2014 ◽  
Vol 30 (3) ◽  
pp. 229-239 ◽  
Author(s):  
J. Ching ◽  
C.-J. Lin
Keyword(s):  
Stochastic Process ◽  
Shear Strength ◽  
Probability Distribution ◽  
Random Field ◽  
Gaussian Random Field ◽  
Extreme Value ◽  
Distribution Model ◽  
One Dimensional ◽  
Process View ◽  
Analytical Expressions

ABSTRACTThis paper shows that the mobilized shear strength of a two-dimensional (2-D) spatially variable saturated undrained clay is closely related to the extreme value of a one-dimensional (1-D) continuous stationary stochastic process. This 1-D stochastic process is the integration of the 2-D spatially variable shear strength along potential slip curves. Based on this finding, a probability distribution model for the mobilized shear strength of the 2-D clay is developed based on a probability distribution model for the extreme value of the 1-D stochastic process. The latter (the model for the 1-D extreme value) has analytical expressions. With the proposed probability distribution model, the mobilized shear strength of a 2-D clay can be simulated without the costly random field finite element analyses.

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