Том 51, № 3 (2018)

The author continues his previous works on preparation to develop generalized axiomatics of the probability theory. The approach is based on the study of set systems of a more general form than the traditional set algebras and their Boolean versions. They are referred to as Dynkin algebras. The author introduces the spectrum of a separable Dynkin algebra and an appropriate Grothendieck topology on this spectrum. Separable Dynkin algebras constitute a natural class of abstract Dynkin algebras, previously distinguished by the author. For these algebras, one can define partial Boolean operations with appropriate properties. The previous work found a structural result: each separable Dynkin algebra is the union of its maximal Boolean subalgebras. In the present note, leaning upon this result, the spectrum of a separable Dynkin algebra is defined and an appropriate Grothendieck topology on this spectrum is introduced. The corresponding constructions somewhat resemble the constructions of a simple spectrum of a commutative ring and the Zariski topology on it. This analogy is not complete: the Zariski topology makes the spectrum of a commutative ring an ordinary topological space, while the Grothendieck topology, which, generally speaking, is not a topology in the usual sense, turns the spectrum of a Dynkin algebra into a more abstract object (site or situs, according to Grothendieck). This suffices for the purposes of the work.

This is the second paper in a series of reviews devoted to the scientific achievements of the Leningrad and St. Petersburg school of probability and mathematical statistics from 1947 to 2017. This paper is devoted to the works on limit theorems for dependent variables (in particular, Markov chains, sequences with mixing properties, and sequences admitting a martingale approximation) and to various aspects of the theory of random processes. We pay particular attention to Gaussian processes, including isoperimetric inequalities, estimates of the probabilities of small deviations in various norms, and the functional law of the iterated logarithm. We present a brief review and bibliography of the works on approximation of random fields with a parameter of growing dimension and probabilistic models of systems of sticky inelastic particles (including laws of large numbers and estimates for the probabilities of large deviations).

A convex function defined on an open convex set of a finite-dimensional space is known to be continuous at every point of this set. In fact, a convex function has a strengthened continuity property. The notion of strong continuity is introduced in this study to show that a convex function has this property. The proof is based on only the definition of convexity and Jensen’s inequality. The definition of strong continuity involves a constant (the constant of strong continuity). An unimprovable value of this constant is given in the case of convex functions. The constant of strong continuity depends, in particular, on the form of a norm introduced in the space of arguments of a convex function. The polyhedral norm is of particular interest. It is straightforward to calculate the constant of strong continuity when it is used. This requires a finite number of values of the convex function.

In our recent paper [4], algorithms for generating normal record values were developed. The developed algorithms were faster and more efficient than currently existing algorithms for generating normal record values. Algorithm 2.2, presented in this paper, is the most efficient algorithm among the algorithms studied in [4]. It allows generating “long” sequences of record values (up to two billion record values). In the present paper, two algorithms for generating normal maxima are proposed, one of which is based on algorithm 2.2. It also allows the generation of maxima taken from “large” samples. An algorithm for generating record times in a general continuous case is also proposed in the present paper.

In the Kalman—Bucy filter problem, the observed process consists of the sum of a signal and a noise. The filtration begins at the same moment as the observation process and it is necessary to estimate the signal. As a rule, this problem is studied for the scalar and vector Markovian processes. In this paper, the scalar linear problem is considered for the system in which the signal and noise are not Markovian processes. The signal and noise are independent stationary autoregressive processes with orders of magnitude higher than 1. The recurrent equations for the filter process, its error, and its conditional cross correlations are derived. These recurrent equations use previously found estimates and some last observed data. The optimal definition of the initial data is proposed. The algebraic equations for the limit values of the filter error (the variance) and cross correlations are found. The roots of these equations make possible the conclusions concerning the criterion of the filter process convergence. Some examples in which the filter process converges and does not converge are given. The Monte Carlo method is used to control the theoretical formulas for the filter and its error.

The classical Kapitsa problem of the inverted flexible pendulum is generalized. We consider a thin homogeneous vertical rod with a free top end and pivoted or rigid attached lower end under the weight of the pendulum’s action and vertical harmonic vibrations of the support. In both cases of attachment, we have stability conditions for the vertical rod position. We take the influence of axial and bending rod vibrations and describe the bending vibrations using the Bernoulli–Euler beam model. The solution is built as a Fourier expansion by eigenfunctions of auxiliary boundary-value problems. As a result, the problem is reduced to the set of ordinary differential equations with periodic coefficients and a small parameter. The asymptotic method of two-scale expansions is used for its solution and to determine the critical level of vibration. The influence of longitudinal waves in the rod essentially decreases the critical load. The single-mode approximation has an acceptable accuracy. With pivoting support at the lower end of the rod, we find the explicit approximate solution. For the rigid attachment, we conduct numerical analysis of the critical level of vibrations depending on the problem parameters.

In astronomical problems, we must estimate the proximity of celestial body orbits. This can serve as a criterion for common origin (usually of a parent body fragmentation). Several submetrics were proposed for this in the latter half of the 20th century. We call the submetric a function defined for each pair of Keplerian orbits, and, satisfying the first two axioms of metric space, but not making obligatory the third, triangle axiom. During the last decade, for each of the proposed submetrics, one can indicate an open set of orbital pairs that this key axiom violates. Recently, new metrics were constructed satisfying all axioms of mertric space, as well as metrics induced by them, widespread in celestial mechanics factor-spaces of the space of nonrectilinear Keplerian orbits. In the present paper, we extended the examination of considered submetrics and metrics propertie; calculated corresponding subdistances and distances between planetary orbits in the Solar System; calculated distances between all pairs of orbits of numbered asteroids (in the space of orbits as well as in its three subspaces); and calculated distances between the orbit of the Chelyabinsk body and orbits of all numbered asteroids.