Заседание семинара "Методы Монте-Карло в вычислительной математике и математической физике" 01.10.2019 в 10-30. "Моделирование случайного поля с заданным распределением одномерного интеграла", Е.Г. Каблукова, Г.А. Михайлов, В.А. Огородников, С.М. Пригарин

The problem of constructing a numerically realizable model of a
three-dimensional homogeneous random field in a layer $ 0<z<H$   with given
one-dimensional  distribution and correlation function of  the integral over
coordinate $z$  is solved. The gamma distribution with shape parameter $\nu$
and scale  parameter $\theta$ is used in the work.  An aggregate of $n$
independent elementary horizontal layers of thickness $h=H/n$ vertically
shifted by a random value uniformly distributed in the interval $(0, h)$ is
considered as a  basic model. For each elementary random field, the
normalized correlation function of the corresponding integral over $z$
coincides with the given one,  gamma distribution with parameters depending
on the number of horizontal layers  is used as a one-dimensional
distribution. It is proved that for the constructed model, the normalized
correlation function of the integral over $z$ coincides with the given
normalized “horizontal” correlation function, and the parameters of the
one-dimensional distribution converge to given  values asymptotically with
$n \to +\infty$, but the corresponding mathematical expectation and variance
coincide with given values exactly.  To extend the class of possible models,
additional randomization of the basic model is considered. In the conclusion
the results of computations for the realistic version  of the problem are
presented.