Vol 229, No 6 (2018)

A probabilistic representation is constructed for classical solution to the Cauchy problem for system of semilinear parabolic equations such that the second order terms with different coefficients enter in diagonal way, while the lower order terms enter in nondiagonal way. The systems of this kind arise as mathematical models of the parabolic conservation and balance laws and as mathematical models of dynamical systems with switching regimes.

The paper deals with methods of computation of distributions of functionals of switching diffusions. The switching between two collections of diffusion coefficients occurs at the Poisson time moments which are independent of the initial diffusions. One can also consider more general diffusions when the choice is made from three or more collections of diffusion coefficients.

An explicit formula for the correlation functions of real zeros of a random polynomial with arbitrary independent continuously distributed coefficients is derived.

The paper studies asymptotic behavior of posterior distribution of a real parameter centered by a \( \sqrt{n} \)-consistent estimate. The uniform analog of the Bernstein–von Mises theorem is proved. This result is extended to asymptotic expansion of the posterior distribution in powers of n^−1/2. This expansion is generalized as the expansion of expectations of functions with polynomial majorant with respect to posterior distribution.

A new method of probabilistic approximation of solution to the Cauchy problem for the unperturbed Schrödinger equation by expectations of functionals of a random walk is suggested. In contrast to earlier papers of the authors, the existence of exponential moment for each step of the random walk is not assumed.

The rate of convergence in the strong law of large numbers for sequences of nonnegative random variables is studied without the independence assumption. Conditions for which an analog of the Baum–Katz theorem holds are obtained.

Some of known inequalities for the uniform distance between distributions of sequential sums of independent identically distributed random variables are considered. In the case where the distribution F has 0 as the q-quantile, an upper bound for the absolute constant in the inequality is given.

A number of directions is indicated in which for statistical problems of decision making related to ordering the parameters of distributions, it is expedient to lean on comparison of sample means. It is assumed that the corresponding parametric family has no nontrivial sufficient statistics. The key role is played by establishing conditions under which the reliability of inferences increases monotonically with increasing the sampling size. Examples of applications are given.

An adaptive estimation of unknown pseudoperiodic function observing in stationary noise with unknown spectral density from a given class is proposed. The accuracy of the proposed estimation is compared with the minimax risk and a lower bound for the minimax risk is established.

The sequences of distributions of centered sums of independent random variables are considered within the framework of the series scheme, without assuming the classical conditions for uniform asymptotic smallness and uniform limit constancy. Necessary and sufficient conditions are obtained for relative and stochastic compactness of such sequences in terms of the characteristic functions of summable random variables and with using their τ-centers.