Intensity of Crossings of a Given Level by a Homogeneous Random Field

2017 ◽  
Vol 53 (5) ◽  
pp. 785-792
Author(s):  
D. V. Yevgrafov
Keyword(s):  
Random Field ◽  
Homogeneous Random Field

A note on the estimation of the mean of a homogeneous random field

1971 ◽  
Vol 8 (3) ◽  
pp. 626-629
Author(s):  
Michael Skalsky
Keyword(s):  
Finite Number ◽  
Random Field ◽  
Random Process ◽  
Mathematical Expectation ◽  
Covariance Function ◽  
Arithmetic Mean ◽  
Stationary Random Process ◽  
Homogeneous Random Field ◽  
The Mean

An important problem, arising in connection with the estimation of mathematical expectation of a homogeneous random field X(x1, ···, xn) in Rn by means of the arithmetic mean of observed values, is to determine the number of observations for which the variance of the estimate attains its minimum. Vilenkin [2] has shown, that in the case of a stationary random process X(x) such a finite number exists, provided that the covariance function satisfies certain conditions.

SELF-INTERSECTION LOCAL TIME FOR ${\mathcal S}' ({\mathbb R}^d)$-WIENER PROCESSES AND RELATED ORNSTEIN–UHLENBECK PROCESSES

1999 ◽  
Vol 02 (04) ◽  
pp. 569-615 ◽  
Author(s):  
TOMASZ BOJDECKI ◽  
LUIS G. GOROSTIZA
Keyword(s):  
Random Field ◽  
Local Time ◽  
Fourier Transforms ◽  
Stable Process ◽  
Local Times ◽  
Spherically Symmetric ◽  
Homogeneous Random Field ◽  
Intersection Local Time ◽  
Wiener Processes ◽  
First Results

Existence and continuity results are obtained for self-intersection local time of [Formula: see text]-valued Ornstein–Uhlenbeck processes [Formula: see text], where X0 is Gaussian, Wt is an [Formula: see text]-Wiener process (independent of X0), and T't is the adjoint of a semigroup Tt on [Formula: see text]. Two types of covariance kernels for X0 and for W are considered: square tempered kernels and homogeneous random field kernels. The case where Tt corresponds to the spherically symmetric α-stable process in ℝd, α∈(0,2], is treated in detail. The method consists in proving first results for self-intersection local times of the ingredient processes: Wt, T't X0 and [Formula: see text], from which the results for Xt are derived. As a by-product, a class of non-finite tempered measures on ℝd whose Fourier transforms are functions is identified. The tools are mostly analytical.

Simulation of stochastic waves in a non-homogeneous random field

1995 ◽  
Vol 14 (5) ◽  
pp. 387-396 ◽  
Author(s):  
Junji Kiyono ◽  
Kenzo Toki ◽  
Tadanobu Sato ◽  
Haruhiro Mizutani
Keyword(s):  
Random Field ◽  
Homogeneous Random Field ◽  
Stochastic Waves

Natural Frequencies and Mode Shapes of Deterministic and Stochastic Non-Homogeneous Rods

2006 ◽  
Vol 5-6 ◽  
pp. 207-216
Author(s):  
S. Nachum ◽  
E. Altus
Keyword(s):  
Grain Size ◽  
Perturbation Method ◽  
Random Field ◽  
Mode Shape ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Homogeneous Random Field ◽  
Geometrical Properties ◽  
Stochastic Case ◽  
Point Correlation

In this study the natural frequencies and mode shapes of the kth order of nonhomogeneous (deterministic and stochastic) rods are found. The solution is based on the functional perturbation method (FPM). The natural frequency and mode shape of the kth order is found analytically to any desired degree of accuracy. In the deterministic it is shown that the FPM accuracy range for the frequency ω and the mode shape is less then 1%. The stochastic case demonstrates the power of this method. The material and geometrical properties will be considered as statistically homogeneous random field with exponential two-point correlation. It is shown that the accuracy depends on the stochastic information used, the correlation distance (roughly the “grain size”), and whether we are interested in the properties of ω or ω2.

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