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Stratified Monte Carlo quadrature for continuous random fields
We consider the problem of numerical approximation of integrals of random
fields over a unit hypercube. We use a stratified Monte Carlo quadrature and
measure the approximation performance by the mean squared error. The quadrature
is defined by a finite number of stratified randomly chosen observations with
the partition (or strata) generated by a rectangular grid (or design). We study
the class of locally stationary random fields whose local behavior is like a
fractional Brownian field in the mean square sense and find the asymptotic
approximation accuracy for a sequence of designs for large number of the
observations. For the H\"{o}lder class of random functions, we provide an upper
bound for the approximation error. Additionally, for a certain class of
isotropic random functions with an isolated singularity at the origin, we
construct a sequence of designs eliminating the effect of the singularity
point.Comment: 17 pages, 6 figures, typos and references corrected, results
unchange