Asymptotic efficiency of the arithmetic-mean estimator of the unknown mean of a homogeneous random field

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.

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

Journal of Applied Probability ◽

10.1017/s0021900200035737 ◽

1971 ◽

Vol 8 (03) ◽

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(x 1, ···, 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.

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The effects of zeros in the spectrum on the behavior of the variance for an arithmetic-mean estimate of a homogeneous random field

Journal of Mathematical Sciences ◽

10.1007/bf02111018 ◽

1994 ◽

Vol 70 (1) ◽

pp. 1600-1606

Author(s):

L. V. Shilo

Keyword(s):

Random Field ◽

Arithmetic Mean ◽

Homogeneous Random Field

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

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025799000333 ◽

1999 ◽

Vol 02 (04) ◽

pp. 569-615 ◽

Cited By ~ 8

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.

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