TWO-DIMENSIONAL MARKOV FIELD AND STATISTICS OF INTEGRAL QUADRATIC FUNCTIONAL BASED ON 2D-FIELD

Abstract

Random normal processes and random normal fields possessing the Markov property have widespread applications in various fields, such as communication theory and statistical radiophysics. The object of this paper is to study the statistics of integral quadratic functionals based on Markov processes and fields. The one-dimensional Markovian process of Ornstein-Uhlenbeck is examined. The solid function of the division of the integral quadratic functional based on the one-dimensional normal Ornstein-Uhlenbeck process is considered. An explicit expression for the rigid function and the division of the integral quadratic functional has been extracted. The two-dimensional Markovian 2D-field and the based for the new integral quadratic functional are examined. An analysis of the rigid function of the division of the integral quadratic functional based on a two-dimensional Markovian 2D-field has been carried out. An explicit expression for this integral quadratic functional has been found. We list the general properties of the found solution. From the theory of such functionals, based on solutions of stochastic differential equations, it follows: all zeros of the generating function are simple; the density function in the fluctuation region decays faster than any polynomial and is identically zero at h = 0; the density function in the peripheral region has an exponential asymptotic behavior with decrement n; the density function has one maximum and two inflection points; the formula for the autoconvolution of the density function also has the form density function. The above allows us to speak about the Laguerre property of the found probability distribution density. A formalized solution for a stationary normal Markovian rich-atomic field is presented. A further step in this direction is to include a signal component with deterministic properties in the observation functional J[H].We note that complete information about the probability distribution density of the random variable J[H] under consideration makes it possible to successfully resolve evaluation and decision-making problems based on statistical criteria.