


Vol 229, No 6 (2018)
- Year: 2018
- Articles: 19
- URL: https://journal-vniispk.ru/1072-3374/issue/view/14893
Article
Probabilistic Models of the Conservation and Balance Laws in Switching Regimes
Abstract
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.



On Distributions of Integral Functionals of Diffusions Stopped at Inverse Range Time
Abstract
Methods for computing the distributions of integral functionals of diffusions stopped at inverse range time are developed. The moment, which is the minimum of the inverse range time and exponentially distributed stopping time independent of the diffusion, is also considered. An interesting example of the applications of these methods is presented.



Distributions of Functionals of Switching Diffusions
Abstract
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.



Asymptotic Efficiency of New Distribution-Free Tests of Symmetry for Generalized Skew Alternatives
Abstract
The Bahadur efficiency of new nonparametric tests of symmetry recently proposed by Nikitin and Ahsanullah is calculated. In contrast to this result, where only location alternatives were discussed, in the present paper generalized skew alternatives are of interest. It is shown that the new tests are highly efficient for a large class of skew alternatives. The problem of most favorable alternatives is also studied.









Asymptotic Expansion of Posterior Distribution of Parameter Centered by a \( \sqrt{n} \)-Consistent Estimate
Abstract
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.



Arak’s Inequalities for the Generalized Arithmetic Progressions
Abstract
In 1980s, Arak has obtained powerful inequalities for the concentration functions of sums of independent random variables. Using these results, he has solved an old problem stated by Kolmogorov. In this paper, one of Arak’s results is modified to include generalized arithmetic progressions in the statement.



On a Limit Theorem Related to Probabilistic Representation of Solution to the Cauchy Problem for the Schrödinger Equation
Abstract
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.






On the Rate of Convergence in the Strong Law of Large Numbers for Nonnegative Random Variables
Abstract
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.



On Random Partitions Induced by Random Maps
Abstract
The partition lattice of the set [n] with respect to refinement is studied. Any map ϕ: [n] → [n], is associated with a partition of [n] by taking preimages of the elements. Assume that t partitions p1, p2, . . . , pt are chosen independently according to the uniform measure on the set of mappings [n] → [n]. It is shown that the probability for the coarsest refinement of all the partitions pi to be the finest partition {{1}, . . . , {n}} tends to 1 for any t ≥ 3 and to e−1/2 for t = 2. It is also proved that the probability for the finest coarsening of the partitions pi to be the one-block partition tends to 1 as t(n) − log n→∞ and tends to 0 as t(n) − log n→−∞. The size of the maximal block of the finest coarsening of all the pi for a fixed t is also studied.



An Estimate of the Absolute Constant in the Inequality for the Uniform Distance Between the Distributions of Sequential Sums of Independent Random Variables
Abstract
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 Probabilistic Representation of Solution to the Cauchy Problem for Evolution Equation with Differential Operator of Order Greater Than 2
Abstract
Let m be a positive integer. A probabilistic representation of a solution to the Cauchy problem for the high order heat type equation \( \frac{\partial u}{\partial t}={c}_m\frac{\partial^mu}{\partial^mx} \) is constructed. Bibliography: 11 titles



Ranking and Selection of Populations on the Base of Sample Means
Abstract
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.






Adaptive Estimation of Function Observed in Gaussian Stationary Noise
Abstract
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.



On Integral of a Semi-Markov Diffusion Process
Abstract
Let (X(t)) (t ≥ 0) be a semi-Markov diffusion process. The process (J(T )) (T ≥ 0) equal to the integral of (X(t)) on interval [0, T ) is studied. The relation between one-dimensional differential equation of the second order of elliptical type and asymptotics of a solution to Dirichlet problem on an interval with length tending to zero is established. This relation is used to derive a differential equation for the Laplace transform of the semi-Markov generating function of the process (J(t)).



Characteristic Functions and Compactness of Distributions of Sums of Independent Random Variables
Abstract
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.


