### Book

## Сборник материалов V-й Международной конференции по стохастическим методам: The 5th International Conference on Stochastic Methods (ICSM5). 23-27 November 2020, Russia, Moscow.

The materials of the 5th International conference on stochastic methods are presented including the following directions: probability and statistics (analytic modelling, asymptotic methods and limit theorems, stochastic analysis, Markov processes and martingales, actuarial and financial mathematics, et al.); applications of stochastic methods (queueing theory and stochastic networks, reliability theory and risk analysis, probability in indistry, economics and other areas, computer science and computer networks, machine learning and data analysis, etc.).

A new notion of ``Markov up'' processes is discussed. A toy model of this notion is proposed and discussed; recurrence and ergodic properties are studied.

We consider high-dimension low-sample-size data taken from the standard multivariate normal distribution under assumption that dimension is a random variable. The second order Chebyshev–Edgeworth expansions for distributions of an angle between two sample observations and corresponding sample correlation coefficient are constructed with error bounds. Depending on the type of normalization, we get three different limit distributions: Normal, Student’s t-, or Laplace distributions. The paper continues studies of the authors on approximation of statistics for random size samples.

Two--sided bounds are constructed for a probability density function of a weighted sum of chi- square variables. Both cases of central and non-central chi-square variables are considered. The upper and lower bounds have the same dependence on the parameters of the sum and differ only in absolute constants. The estimates obtained will be useful, in particular, when comparing two Gaussian random elements in a Hilbert space and in multidimensional central limit theorems, including the infinite-dimensional case.

We consider the typical behavior of the weighted sums of independent identically distributed random vectors in k-dimensional space. It is shown that in this case the rate of convergence in the multivariate central limit theorem is of order O(1/n) up to logarithmic factor. This extends the one-dimensional Klartag and Sodin result.

We prove that a joint distribution of a locally integrable increasing process X◦ and its compensator A◦ at a terminal moment of time can be realized as a joint terminal distribution of another locally integrable increasing process X* and its compensator A*, A* being continuous.

Developing a new approach to the problem of optimal control in the open dynamical model of a three-sector economy

We study the problem of testing composite hypotheses versus composite alternatives when there is a slight deviation between the model and the real distribution. The used approach, which we called sub-optimal testing, implies an extension of the initial model and a modification of a sequential statistically significant test for the new model. The sub-optimal test is proposed and a non-asymptotic border for the loss function is obtained. Also we investigate correlation between the sub-optimal test and the sequential probability ratio test for the initial model.

Background: Connectivity networks, which reflect multiple interactions between genes and proteins, possess not only a descriptive but also a predictive value, as new connections can be extrapolated and tested by means of computational analysis. Integration of different types of connectivity data (such as co-expression and genetic interactions) in one network has proven to benefit ‘guilt by association’ analysis. However predictive values of connectives of different types, that had their specific functional meaning and topological characteristics were not obvious, and have been addressed in this analysis. Methods: eQTL data for 3 experimental C.elegans age groups were retrieved from WormQTL. WormNet has been used to obtain pair-wise gene interactions. The Shortest Path Function (SPF) has been adopted for statistical validation of the co-expressed gene clusters and for computational prediction of their potential gene expression regulators from a network context. A new SPF-based algorithm has been applied to genetic interactions sub-networks adjacent to the clusters of co-expressed genes for ranking the most likely gene expression regulators causal to eQTLs. Results: We have demonstrated that known co-expression and genetic interactions between C. elegans genes can be complementary in predicting gene expression regulators. Several algorithms were compared in respect to their predictive potential in different network connectivity contexts. We found that genes associated with eQTLs are highly clustered in a C. elegans co-expression sub-network, and their adjacent genetic interactions provide the optimal functional connectivity environment for application of the new SPF-based algorithm. It was successfully tested in the reverse-prediction analysis on groups of genes with known regulators and applied to co-expressed genes and experimentally observed expression quantitative trait loci (eQTLs). Conclusions: This analysis demonstrates differences in topology and connectivity of co-expression and genetic interactions sub-networks in WormNet. The modularity of less continuous genetic interaction network does not correspond to modularity of the dense network comprised by gene co-expression interactions. However the genetic interaction network can be used much more efficiently with the SPF method in prediction of potential regulators of gene expression. The developed method can be used for validation of functional significance of suggested eQTLs and a discovery of new regulatory modules.

The complexity of today’s statistical data calls for modern mathematical tools. Many fields of science make use of mathematical statistics and require continuous updating on statistical technologies. Practice makes perfect, since mastering the tools makes them applicable. Our book of exercises and solutions offers a wide range of applications and numerical solutions based on R. In modern mathematical statistics, the purpose is to provide statistics students with a number of basic exercises and also an understanding of how the theory can be applied to real-world problems. The application aspect is also quite important, as most previous exercise books are mostly on theoretical derivations. Also we add some problems from topics often encountered in recent research papers. The book was written for statistics students with one or two years of coursework in mathematical statistics and probability, professors who hold courses in mathematical statistics, and researchers in other fields who would like to do some exercises on math statistics.

The functionals related to the quality of the system control are obtained

in the analytic form. The statement that the optimal strategy of controlling

the system is a deterministic strategy is proved. Analytic form representation

for the function the absolute extremum of which is determined as the optimal

control strategy is obtained also.

The purpose of this paper is the presentation of the ideas and concepts that

form the basis of the concept of mathematical model control some processes

occurring in the Russian market of cereals. The estimated model must have a

stochastic nature, i.e. constitute some random process. Indeed, in a free market

there are objectively random factors that cannot be described by deterministic.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.