# Theoretical and applied problems of Physics and Mathematics

# Complex-Valued Functions with Nondegenerate Groups of Regular Points

Complex-valued functions with nondegenerate groups of regular points are studied in the paper. A class of functions , which takes value on the complex plain and for which the limit exists, and is nonzero and finite for some from subset of positive real numbers is considered. It was received that this subset is multiplicative group, and it is called the group of regular points. Functions with nondegenerate groups of regular points generalize the class of RV functions. The corresponding limit functions are defined for complex-valued functions with nondegenerate groups of regular points.

# Consistency of Least Squares Estimator of Linear Regression Parameter in Case of Discrete Tme and Long-Range or Weak Dependent Regressors

Linear regression model with discrete time, long-range/weak dependent random noise and time dependent regressors, which are observed with long L range/weak dependent errors, is considered. Parameter estimation of such models is one of the important problems of statistics of random processes. Least squares estimator is chosen for the estimation. The aim of the work is to prove consistency of least squares estimator of such regression model.

# The Limit Theorems for Extreme Residuals in Nonlinear Regression Model with Gaussian Stationary Noise

In this paper non-linear regression model with Gaussian stationary random noise and continuous time is considered. The behavior of normalized in some way maximum residuals and maximum of residuals absolute values in which its the least squares estimator is substituted instead of unknown parameter of regression function. The convergence of distribution of these normalized maximum to double exponent law is proved which follows from the assumption of random noise normality.

# Moments Asymptotic Expansion of the Least Squares Estimator of the Vector-Parameter of Nonlinear Regression with Correlated Observations

A nonlinear regression model with continuous time and mean square continuous separable measurable Gaussian stationary random noise with zero mean and integrable covariance function is considered. Parameter estimation in the models of such kind is an important problem of statistics of random processes. In this paper, the first terms of asymptotic expansions of the bias vector and covariance matrix of the least square estimator of nonlinear regression function vector parameter are obtained.

# Asymptotic Uniqueness of the Non-Linear Regression Model Parameter with Least Squares Estimator

In the paper the nonlinear regression model with continuous time and random noise, which is a local functional of strongly dependent stationary Gaussian random process, is considered. Sufficient conditions of asymptotic uniqueness of the least squares estimator of regression function parameters are obtained. This result is applied to the least squares estimator of amplitude and angular frequencies of harmonic oscillations sum observed on the background of given random noise.

# Study of Distributive Law in Extended Interval Space

A study of the law of distributivity in the extended interval space is suggested. The research for interval in the center–radius form was conducted. The classification of the intervals is proposed. A set of intervals is represented as a union of three subsets which have defined by the relations of values the centers and the radii. We proved the lemma about the conditions under which the sum of the two intervals will own to same subset of the intervals you want to add. The conditions in which the sum of the two intervals belongs to the same subset as intervals, which are added.

# Continuous Solutions of a Class of Difference-Functional Equations

The main object of research in this article is a study of the continuous solutions set structure of difference-functional equations

# Direct Spectral Problems for the Block Jacobi Type Bounded Symmetric Matrices Related to the two dimensional real moment problem

The generalization of the classical moment problem and the spectral theory of self-adjoint Jacobi block matrix are well-known in one-dimensional case and it generalized on the two-dimensional case. Finite and infinite moment problem is solved using Yu.M. Berezansky generalized eigenfunction expansion method for respectively finite and infinite family of commuting self-adjoint operators. In the classical case one orthogonalize a family of polynomials , with respect to a measure on the real axis and shift operator on takes the form of ordinary Jacobi matrix.

# Operators of Stochastic Differentiation on Spaces of Regular Test and Generalized Functions in the Lévy White Noise Analysis

The operators of stochastic differentiation, which are closely related with stochastic integrals and with the Hida stochastic derivative, play an important role in the classical white noise analysis. In particular, one can use these operators in order to study properties of solutions of normally ordered stochastic equations, and properties of the extended Skorohod stochastic integral. So, it is natural to introduce and to study analogs of the mentioned operators in the Lévy white noise analysis.

# Estimates for Moments of Extreme Values of the Random Process with Superadditive Moment Function

This paper considers the stochastic process with superadditive moment function. The aim is to generalize the results of R. Serfling, which he received for a sequence of random variables with superadditive moment function. We have obtained the estimation for moments of supremum of a random process with the appropriate bounds for moments of this random process. We make no assumptions about the structure of the dependence of increments of a random process, but only the estimation for moments of random process.