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Vol 229, No 6 (2018)

Article

Probabilistic Models of the Conservation and Balance Laws in Switching Regimes

Belopolskaya Y.I.

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.

Journal of Mathematical Sciences. 2018;229(6):601-625
pages 601-625 views

On Distributions of Integral Functionals of Diffusions Stopped at Inverse Range Time

Borodin A.N.

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.

Journal of Mathematical Sciences. 2018;229(6):626-631
pages 626-631 views

Distributions of Functionals of Switching Diffusions

Borodin A.N.

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.

Journal of Mathematical Sciences. 2018;229(6):632-650
pages 632-650 views

Asymptotic Efficiency of New Distribution-Free Tests of Symmetry for Generalized Skew Alternatives

Bookiya G.T., Nikitin Y.Y.

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.

Journal of Mathematical Sciences. 2018;229(6):651-663
pages 651-663 views

Correlation Functions of Real Zeros of Random Polynomials

Götze F., Kaliada D., Zaporozhets D.

Abstract

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

Journal of Mathematical Sciences. 2018;229(6):664-670
pages 664-670 views

On Probabilities of Moderate Deviations of Empirical Measures for Contiguous Distributions

Ermakov M.S.

Abstract

The large deviation principle for moderate deviation probabilities of empirical measures for contiguous distributions is proved.

Journal of Mathematical Sciences. 2018;229(6):671-677
pages 671-677 views

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

Zaikin A.A.

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.

Journal of Mathematical Sciences. 2018;229(6):678-697
pages 678-697 views

Arak’s Inequalities for the Generalized Arithmetic Progressions

Zaitsev A.Y.

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.

Journal of Mathematical Sciences. 2018;229(6):698-701
pages 698-701 views

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

Ibragimov I.A., Smorodina N.V., Faddeev M.M.

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.

Journal of Mathematical Sciences. 2018;229(6):702-713
pages 702-713 views

Estimation of the Tail of Probability Distribution Through its Characteristic Function

Karlová A., Klebanov L.B.

Abstract

A method for estimation of a probability distribution tail in terms of characteristic function is given.

Journal of Mathematical Sciences. 2018;229(6):714-718
pages 714-718 views

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

Korchevsky V.M.

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.

Journal of Mathematical Sciences. 2018;229(6):719-726
pages 719-726 views

On Random Partitions Induced by Random Maps

Krachun D., Yakubovich Y.

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.

Journal of Mathematical Sciences. 2018;229(6):727-740
pages 727-740 views

An Estimate of the Absolute Constant in the Inequality for the Uniform Distance Between the Distributions of Sequential Sums of Independent Random Variables

Maistrenko E.L.

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.

Journal of Mathematical Sciences. 2018;229(6):741-743
pages 741-743 views

A Probabilistic Representation of Solution to the Cauchy Problem for Evolution Equation with Differential Operator of Order Greater Than 2

Platonova M.V.

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

Journal of Mathematical Sciences. 2018;229(6):744-755
pages 744-755 views

Ranking and Selection of Populations on the Base of Sample Means

Revyakov M.

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.

Journal of Mathematical Sciences. 2018;229(6):756-766
pages 756-766 views

Small Deviation Probabilities of a Sum of Independent Positive Random Variables, the Common Distribution of Which Decreases at Zero Not Faster Than Exponential Function

Rozovsky L.V.

Abstract

We investigate small deviation probabilities of the cumulative sum of independent positive random variables, the common distribution of which decreases at zero not faster than exponential function.

Journal of Mathematical Sciences. 2018;229(6):767-771
pages 767-771 views

Adaptive Estimation of Function Observed in Gaussian Stationary Noise

Solev V.N.

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.

Journal of Mathematical Sciences. 2018;229(6):772-781
pages 772-781 views

On Integral of a Semi-Markov Diffusion Process

Harlamov B.P.

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)).

Journal of Mathematical Sciences. 2018;229(6):782-791
pages 782-791 views

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

Khartov A.A.

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.

Journal of Mathematical Sciences. 2018;229(6):792-802
pages 792-802 views