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We provide fundamental results of the structural theory of computably separable models and consider applications of this theory to the theory of effective linear orders and theoretical informatics., CONTENTS 1. Preliminaries 746 2. Computable Separability Criterion 749 3. Uniformity 753 4. Computably Separable Linear Orders 754 5. Specifications 762 References 766 1. Preliminaries Base notions used in this [...]

In this paper, we construct an analog of local time for an arbitrary Levy process with finite second moment. When our process is a Wiener process, this object coincides with the local time. Bibliography: 4 titles., Let [xi](t), t [greater than or equal to] 0, be a measurable, real-valued random process. By the change of measure theorem, for any measurable function f, the identity [Please download [...]

A Poisson process [X.sub.[member of]](t) with intensity density function [[epsilon].sup.-1][lambda](t) is observed on interval [a, b]. The problem is to estimate the function [lambda] (t). It is assumed that the unknown function [lambda] (t) belongs to a given class of functions analytic in a given region G [contains] [a, b] and is bounded there by a given constant M. The parameter [epsilon] is supposed to be known and the problem is considered as [epsilon] [right arrow] 0. Bibliography: 7 titles., 1. PROBLEM AND RESULTS Below we consider the following nonparametric estimation problem. A Poisson process [X.sub.[epsilon]](t) with intensity density [[epsilon].sup.-1][lambda](t) is observed on an interval [a, b], - [infinity] < [...]

We consider some problems concerning a probabilistic interpretation of the Cauchy problem solution for the equation [partial derivative]u/[partial derivative]t = 1/2 (S[nabla], [nabla]V)u, where S is a symmetric complex matrix such that Re S [greater than or equal to] 0. Bibliography: 6 titles., UDC 519.2 1. Introduction In the papers [1, 2], the authors obtained results on probabilistic representation and probabilistic approximation of the solution of the Cauchy problem u(0, x) = [phi](x) [...]

The main problem of the paper looks as follows. A functional parameter [theta] [member of] [THETA] [subset] [L.sub.2](- -[infinity], [infinity]) is observed in Gaussian noise. The problem is to estimate the value F([theta]) of a given function F. A construction of asymptotically efficient estimates for F([theta]) is suggested under the condition that [THETA] admits approximations by subspaces [H.sub.T] [subset] [L.sub.2] with reproducing kernels [K.sub.T](t, s), [K.sub.T](t, t) [less than or equal to] T. Bibliography: 10 titles., 1. Introduction. Main result 1.1. Consider the following general problem of statistical estimation. We have a statistical experiment {X, [??], {[P.sub.[theta]], [theta] [member of] [THETA]}} generated by an observation [X.sub.[epsilon]] [...]

A new method of probabilistic approximation of solution to the Cauchy problem for the unperturbed Schrodinger 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. Bibliography: 9 titles, UDC 519.2 1. INTRODUCTION In papers [4, 5], the authors proved a variant of a limit theorem, in which the asymptotic distribution is absolutely continuous with the (nonprobabilistic) density [mathematical [...]

Mikhail Stepanovich Nikulin was an active member of the POMI Laboratory of Statistical Methods for many years and a long-term editor of the Statistical Series of Proceedings of Scientific Seminars [...]

Let X be a random variable taking positive integer values and let P{X = k} = [theta](k). We consider the problem of estimation of the parameter [theta] = ([theta](1), [theta](2), ...) on the base of a sample [X.sub.1],[X.sub.2], ..., [X.sub.n], where the observations [X.sub.j] are independent copies of X. Bibliography: 5 titles., UDC 519 1. Statement of the problem Let X be a random variable with positive integer values and let P{X = k} = [theta](k), k = 1, 2, .... One [...]

A Poisson random field with intensity density function [lambda](x)/[epsilon] is observed in a bounded region G [subset or equal to] [R.sup.d]. It is supposed that the unknown function [lambda] belongs to a known class of entire functions. The parameter [epsilon] is supposed to be known. The problem is to estimate the value [lambda] (x) at the points x [not member of]G. An asymptotic setup of the problem as [epsilon] 0 is considered. Bibliography: 13 titles., UDC 519.2 1. PROBLEM STATEMENT AND MAIN RESULTS Let [[PI].sub.[epsilon]] be a Poisson random field in [R.sup.d] with intensity density [lambda](x)/[epsilon] with respect to the Lebesgue measure, where [epsilon] > [...]

5 Dependence on the Boundary Data In this section, we study the continuous dependence of the solution to the initial boundary value problem (3.1) on the boundary data [psi](x, t). [...]

The paper is devoted to some problems associated with a probabilistic representation and probabilistic approximation of the Cauchy problem solution for the family of equations [partial derivative]u/[partial derivative]t = [[sigma].sup.2]/2 [DELTA]u with a complex parameter [sigma] such that Re [[sigma].sup.2] [greater than or equal to] 0. The above family includes as a particular case both the heat equation (Im [sigma] = 0) and the Schrodinger equation (Re [[sigma].sup.2] =0). Bibliography: 10 titles., UDC 519.2 1. INTRODUCTION Consider the equation partial derivative]u/[partial derivative]t = [[sigma].sup.2]/2 [DELTA]u, (1) where x [member of] [R.sup.d], t [greater than or equal to] 0, and a is a [...]

In 1962, N. N. Chentsov suggested the following method of estimation of a functional parameter [theta] belonging to a Hilbert space H. He suggested to project [theta] to finite-dimensional subspaces of H and consider as estimates of [theta] estimates of these projections. In this paper, we suggest to consider the projections on all reproducing kernel subspaces of H. Bibliography: 15 titles., UDC 519.2 1. Projection estimates first appeared in 1962 in the N. N. Chentsov's paper [1] as a method of construction of estimates of an infinite-dimensional (for example, functional) parameter. [...]

Yurii Izmailovich Ingster died on August 8, 2012. He was a remarkable scientist, one of the most active participants of the St.Petersburg Seminar on Probability and Statistics, and a permanent [...]

In our paper, we construct an analog of probabilistic representation of the solution of the Cauchy problem, for the equation ∂u/∂t + [[σ.sup.2]/2] [[[partial derivative].sup.2]u/∂[x.sup.2]] + f(x)u = 0, where σ is a complex number. Bibliography: 7 titles., 1. INTRODUCTION In this paper, we construct an analog of probabilistic representation of solutions of the Cauchy problem for some evolution equations. Let w(t), t ∈ [0, T], be the [...]

In 1990, I. Ibragimov and R. Khasminskii have established some general minimax lower bounds of risk, functions for probability density estimates. Here analogous results are proved for some other nonparametric estimation problems. Bibliography: 6 titles., 1. Introduction In the paper [1], the authors considered the following problem of nonparametrie estimation. One observes a sample [X.sub.1], ..., [X.sub.n] whose elements take values in a measure space [...]

To the eightieth anniversary of professor A. A. Ivanov, a leading researcher in POMI, doctor in physics and mathematics, who works in LOMI since 1950. Aleksandr Aleksandrovich Ivanov was born [...]