
Theory of Probability and its Applications
Journal Prestige (SJR): 0.411 Number of Followers: 2 Hybrid journal (It can contain Open Access articles) ISSN (Print) 0040585X  ISSN (Online) 10957219 Published by Society for Industrial and Applied Mathematics [17 journals] 
 Joint Distributions of Generalized Integrable Increasing Processes and
Their Generalized Compensators
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Authors: D. A. Borzykh
Pages: 1  24
Abstract: Theory of Probability & Its Applications, Volume 69, Issue 1, Page 124, May 2024.
Let $\Lambda$ be the set of all boundary joint laws $\operatorname{Law} ([X_a, A_a], [X_b, A_b])$ at times $t=a$ and $t=b$ of integrable increasing processes $(X_t)_{t \in [a, b]}$ and their compensators $(A_t)_{t \in [a, b]}$, which start at the initial time from an arbitrary integrable initial condition $[X_a, A_a]$. We show that $\Lambda$ is convex and closed relative to the $\psi$weak topology with linearly growing gauge function $\psi$. We obtain necessary and sufficient conditions for a probability measure $\lambda$ on $\mathcal{B}(\mathbf{R}^2 \times \mathbf{R}^2)$ to lie in the class of measures $\Lambda$. The main result of the paper provides, for two measures $\mu_a$ and $\mu_b$ on $\mathcal{B}(\mathbf{R}^2)$, necessary and sufficient conditions for the set $\Lambda$ to contain a measure $\lambda$ for which $\mu_a$ and $\mu_b$ are marginal distributions.
Citation: Theory of Probability & Its Applications
PubDate: 20240502T07:00:00Z
DOI: 10.1137/S0040585X97T991714
Issue No: Vol. 69, No. 1 (2024)

 Stability Properties of Feature Selection Measures

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Authors: A. V. Bulinski
Pages: 25  34
Abstract: Theory of Probability & Its Applications, Volume 69, Issue 1, Page 2534, May 2024.
In this paper, we prove that the monotonicity property of the stability measure for the feature (factor) selection introduced by Nogueira, Sechidis, and Brown [J. Mach. Learn. Res., 18 (2018), pp. 154] may not hold. Another monotonicity property takes place. We also show the cases in which it is possible to compare by certain parameters the matrices describing the operation of algorithms for identifying relevant features.
Citation: Theory of Probability & Its Applications
PubDate: 20240502T07:00:00Z
DOI: 10.1137/S0040585X97T991726
Issue No: Vol. 69, No. 1 (2024)

 Universal Nonparametric KernelType Estimators for the Mean and Covariance
Functions of a Stochastic Process
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Authors: Yu. Yu. Linke, I. S. Borisov
Pages: 35  58
Abstract: Theory of Probability & Its Applications, Volume 69, Issue 1, Page 3558, May 2024.
Let $f_1(t), \dots, f_n(t)$ be independent copies of some a.s. continuous stochastic process $f(t)$, $t\in[0,1]$, which are observed with noise. We consider the problem of nonparametric estimation of the mean function $\mu(t) = \mathbf{E}f(t)$ and of the covariance function $\psi(t,s)=\operatorname{Cov}\{f(t),f(s)\}$ if the noise values of each of the copies $f_i(t)$, $i=1,\dots,n$, are observed in some collection of generally random (in general) time points (regressors). Under wide assumptions on the time points, we construct uniformly consistent kernel estimators for the mean and covariance functions both in the case of sparse data (where the number of observations for each copy of the stochastic process is uniformly bounded) and in the case of dense data (where the number of observations at each of $n$ series is increasing as $n\to\infty$). In contrast to the previous studies, our kernel estimators are universal with respect to the structure of time points, which can be either fixed rather than necessarily regular, or random rather than necessarily formed of independent or weakly dependent random variables.
Citation: Theory of Probability & Its Applications
PubDate: 20240502T07:00:00Z
DOI: 10.1137/S0040585X97T991738
Issue No: Vol. 69, No. 1 (2024)

 Energy Saving Approximation of Wiener Process under Unilateral Constraints

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Authors: M. A. Lifshits, S. E. Nikitin
Pages: 59  70
Abstract: Theory of Probability & Its Applications, Volume 69, Issue 1, Page 5970, May 2024.
We consider an energy saving approximation of a Wiener process under unilateral constraints. We show that, almost surely, on large time intervals the minimal energy necessary for the approximation logarithmically depends on the interval length. We also construct an adaptive approximation strategy optimal in a class of diffusion strategies and providing the logarithmic order of energy consumption.
Citation: Theory of Probability & Its Applications
PubDate: 20240502T07:00:00Z
DOI: 10.1137/S0040585X97T99174X
Issue No: Vol. 69, No. 1 (2024)

 Markov Branching Random Walks on $\mathbf{Z}_+$: An Approach Using
Orthogonal Polynomials. I
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Authors: A. V. Lyulintsev
Pages: 71  87
Abstract: Theory of Probability & Its Applications, Volume 69, Issue 1, Page 7187, May 2024.
We consider a continuoustime homogeneous Markov process on the state space $\mathbf{Z}_+=\{0,1,2,\dots\}$. The process is interpreted as the motion of a particle. A particle may transit only to neighboring points $\mathbf{Z}_+$, i.e., for each single motion of the particle, its coordinate changes by 1. The process is equipped with a branching mechanism. Branching sources may be located at each point of $\mathbf{Z}_+$. At a moment of branching, new particles appear at the branching point and then evolve independently of each other (and of the other particles) by the same rules as the initial particle. To such a branching Markov process there corresponds a Jacobi matrix. In terms of orthogonal polynomials corresponding to this matrix, we obtain formulas for the mean number of particles at an arbitrary fixed point of $\mathbf{Z}_+$ at time $t>0$. The results obtained are applied to some concrete models, an exact value for the mean number of particles in terms of special functions is given, and an asymptotic formula for this quantity for large time is found.
Citation: Theory of Probability & Its Applications
PubDate: 20240502T07:00:00Z
DOI: 10.1137/S0040585X97T991751
Issue No: Vol. 69, No. 1 (2024)

 On a Periodic Branching Random Walk on $\mathbf{Z}^{{d}}$ with an Infinite
Variance of Jumps
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Authors: K. S. Ryadovkin
Pages: 88  98
Abstract: Theory of Probability & Its Applications, Volume 69, Issue 1, Page 8898, May 2024.
We consider periodic branching random walks with periodic branching sources. It is assumed that the transition intensities of the random walk satisfy some symmetry conditions and obey a condition which ensures infinite variance of jumps. In this case, we obtain the leading term for the asymptotics of the mean population size of particles at an arbitrary point of the lattice for large time.
Citation: Theory of Probability & Its Applications
PubDate: 20240502T07:00:00Z
DOI: 10.1137/S0040585X97T991763
Issue No: Vol. 69, No. 1 (2024)

 Conditional Functional Limit Theorem for a Random Recurrence Sequence
Conditioned on a Large Deviation Event
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Authors: A. V. Shklyaev
Pages: 99  116
Abstract: Theory of Probability & Its Applications, Volume 69, Issue 1, Page 99116, May 2024.
Let $\{Z_n,\, n\ge 0\}$ be a branching process in an independent and identically distributed (i.i.d.) random environment and $\{S_n,\, n\,{\ge}\, 1\}$ be the associated random walk with steps $\xi_i$. Under the Cramér condition on $\xi_1$ and moment assumptions on a number of descendants of one particle, we know the asymptotics of the large deviation probabilities $\mathbf{P}(\ln Z_n> x)$, where $x/n> \mu^*$. Here, $\mu^*$ is a parameter depending on the process type. We study the asymptotic behavior of the process trajectory under the condition of a large deviation event. In particular, we obtain a conditional functional limit theorem for the trajectory of $(Z_{[nt]},\, t\in [0,1])$ given $\ln Z_n>x$. This result is obtained in a more general model of linear recurrence sequence $Y_{n+1}=A_n Y_n + B_n$, $n\ge 0$, where $\{A_i\}$ is a sequence of i.i.d. random variables, $Y_0$, $B_i$, $i\ge 0$, are possibly dependent and have different distributions, and we need only some moment assumptions on them.
Citation: Theory of Probability & Its Applications
PubDate: 20240502T07:00:00Z
DOI: 10.1137/S0040585X97T991775
Issue No: Vol. 69, No. 1 (2024)

 Limit Behavior of Order Statistics on Cycle Lengths of Random
$A$Permutations
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Authors: A. L. Yakymiv
Pages: 117  126
Abstract: Theory of Probability & Its Applications, Volume 69, Issue 1, Page 117126, May 2024.
We consider a random permutation $\tau_n$ uniformly distributed on the set of all permutations of degree $n$ whose cycle lengths lie in a fixed set $A$ (the socalled $A$permutations). Let $\zeta_n$ be the total number of cycles, and let $\eta_n(1)\leq\eta_n(2)\leq\dots\leq\eta_n(\zeta_n)$ be the ordered sample of cycle lengths of the permutation $\tau_n$. We consider a class of sets $A$ with positive density in the set of natural numbers. We study the asymptotic behavior of $\eta_n(m)$ with numbers $m$ in the lefthand and middle parts of this series for a class of sets of positive asymptotic density. A limit theorem for the rightmost terms of this series was proved by the author of this note earlier. The study of limit properties of the sequence $\eta_n(m)$ dates back to the paper by Shepp and Lloyd [Trans. Amer. Math. Soc., 121 (1966), pp. 340357] who considered the case $A=\mathbf N$.
Citation: Theory of Probability & Its Applications
PubDate: 20240502T07:00:00Z
DOI: 10.1137/S0040585X97T991787
Issue No: Vol. 69, No. 1 (2024)

 Utility Maximization of the Exponential Lévy Switching Models

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Authors: Y. Dong, L. Vostrikova
Pages: 127  149
Abstract: Theory of Probability & Its Applications, Volume 69, Issue 1, Page 127149, May 2024.
This article is devoted to maximization of HARA (hyperbolic absolute risk aversion) utilities of the exponential Lévy switching processes on a finite time interval via the dual method. The description of all $f$divergence minimal martingale measures and the expression of their RadonNikodým densities involving the Hellinger and KulbackLeibler processes are given. The optimal strategies in progressively enlarged filtration for the maximization of HARA utilities as well as the values of the corresponding maximal expected utilities are derived. As an example, the Brownian switching model is presented with financial interpretations of the results via the value process.
Citation: Theory of Probability & Its Applications
PubDate: 20240502T07:00:00Z
DOI: 10.1137/S0040585X97T991799
Issue No: Vol. 69, No. 1 (2024)

 Hellinger Distance Estimation for Nonregular Spectra

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Authors: M. Taniguchi, Y. Xue
Pages: 150  160
Abstract: Theory of Probability & Its Applications, Volume 69, Issue 1, Page 150160, May 2024.
For Gaussian stationary processes, a time series Hellinger distance $T(f,g)$ for spectra $f$ and $g$ is derived. Evaluating $T(f_\theta,f_{\theta+h})$ of the form $O(h^\alpha)$, we give $1/\alpha$consistent asymptotics of the maximum likelihood estimator of $\theta$ for nonregular spectra. For regular spectra, we introduce the minimum Hellinger distance estimator $\widehat{\theta}=\operatorname{arg}\min_\theta T(f_\theta,\widehat{g}_n)$, where $\widehat{g}_n$ is a nonparametric spectral density estimator. We show that $\widehat\theta$ is asymptotically efficient and more robust than the Whittle estimator. Brief numerical studies are provided.
Citation: Theory of Probability & Its Applications
PubDate: 20240502T07:00:00Z
DOI: 10.1137/S0040585X97T991805
Issue No: Vol. 69, No. 1 (2024)

 News of Scientific Life  Information on the General Seminar of the
Department of Probability, Faculty of Mathematics and Mechanics, Lomonosov
Moscow State University, Moscow, Russia, Spring Term 2023
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Authors: A.N. Shiryaev, E.B. Yarkovaya, V.A. Kutsenko
Pages: 161  165
Abstract: Theory of Probability & Its Applications, Volume 69, Issue 1, Page 161165, May 2024.
This paper presents summaries of talks given during the 2023 spring semester of the General Seminar of the Department of Probability, Moscow State University. The seminar was held under the direction of A. N. Kolmogorov and B. V. Gnedenko. Current information about the seminar is available at http://new.math.msu.su/department/probab/seminar.html.
Citation: Theory of Probability & Its Applications
PubDate: 20240502T07:00:00Z
DOI: 10.1137/S0040585X97T991817
Issue No: Vol. 69, No. 1 (2024)

 In Memory of V. N. Tutubalin (10.15.19366.18.2023)

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Authors: S.A. Molchanov, D.D. Sokolov, E.B. Yarovaya
Pages: 166  167
Abstract: Theory of Probability & Its Applications, Volume 69, Issue 1, Page 166167, May 2024.
A remembrance of Professor Valerii Nikolaevich Tutubalin, who passed away on June 18, 2023. He held a position in the Department of Probability Theory at Moscow State University since 1965 and was known as a leading authority on probability theory and its applications.
Citation: Theory of Probability & Its Applications
PubDate: 20240502T07:00:00Z
DOI: 10.1137/S0040585X97T991829
Issue No: Vol. 69, No. 1 (2024)

 Weakly Supercritical Branching Process in a Random Environment Dying at a
Distant Moment
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Authors: V. I. Afanasyev
Pages: 537  558
Abstract: Theory of Probability & Its Applications, Volume 68, Issue 4, Page 537558, February 2024.
A functional limit theorem is proved for a weakly supercritical branching process in a random environment under the condition that the process becomes extinct after time $n\to \infty $.
Citation: Theory of Probability & Its Applications
PubDate: 20240207T09:31:03Z
DOI: 10.1137/S0040585X97T991611
Issue No: Vol. 68, No. 4 (2024)

 On Symmetrized ChiSquare Tests in Autoregression with Outliers in Data

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Authors: M. V. Boldin
Pages: 559  569
Abstract: Theory of Probability & Its Applications, Volume 68, Issue 4, Page 559569, February 2024.
A linear stationary model $\mathrm{AR}(p)$ with unknown expectation, coefficients, and the distribution function of innovations $G(x)$ is considered. Autoregression observations contain gross errors (outliers, contaminations). The distribution of contaminations $\Pi$ is unknown, their intensity is $\gamma n^{1/2}$ with unknown $\gamma$, and $n$ is the number of observations. The main problem here (among others) is to test the hypothesis on the normality of innovations $\boldsymbol H_{\Phi}\colon G (x)\in \{\Phi(x/\theta),\,\theta>0\}$, where $\Phi(x)$ is the distribution function of the normal law $\boldsymbol N(0,1)$. In this setting, the previously constructed tests for autoregression with zero expectation do not apply. As an alternative, we propose special symmetrized chisquare type tests. Under the hypothesis and $\gamma=0$, their asymptotic distribution is free. We study the asymptotic power under local alternatives in the form of the mixture $G(x)=A_{n,\Phi}(x):=(1n^{1/2})\Phi(x/\theta_0)+n^{1/2}H(x)$, where $H(x)$ is a distribution function, and $\theta_0^2$ is the unknown variance of the innovations under $\boldsymbol H_{\Phi}$. The asymptotic qualitative robustness of the tests is established in terms of equicontinuity of the family of limit powers (as functions of $\gamma$, $\Pi,$ and $H(x)$) relative to $\gamma$ at the point $\gamma=0$.
Citation: Theory of Probability & Its Applications
PubDate: 20240207T08:00:00Z
DOI: 10.1137/S0040585X97T991623
Issue No: Vol. 68, No. 4 (2024)

 Laplace Expansion for BartlettNandaPillai Test Statistic and Its Error
Bound
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Authors: H. Wakaki, V. V. Ulyanov
Pages: 570  581
Abstract: Theory of Probability & Its Applications, Volume 68, Issue 4, Page 570581, February 2024.
We construct asymptotic expansions for the distribution function of the BartlettNandaPillai statistic under the condition that the null linear hypothesis is valid in a multivariate linear model. Computable estimates of the accuracy of approximation are obtained via the Laplace approximation method, which is generalized to integrals for matrixvalued functions.
Citation: Theory of Probability & Its Applications
PubDate: 20240207T08:00:00Z
DOI: 10.1137/S0040585X97T991635
Issue No: Vol. 68, No. 4 (2024)

 Kolmogorov's Last Discovery' (Kolmogorov and Algorithmic Statistics)

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Authors: A. L. Semenov, A. Kh. Shen, N. K. Vereshchagin
Pages: 582  606
Abstract: Theory of Probability & Its Applications, Volume 68, Issue 4, Page 582606, February 2024.
The definition of descriptional complexity of finite objects suggested by Kolmogorov and other authors in the mid1960s is now well known. In addition, Kolmogorov pointed out some approaches to a more finegrained classification of finite objects, such as the resourcebounded complexity (1965), structure function (1974), and the notion of $(\alpha,\beta)$stochasticity (1981). Later it turned out that these approaches are essentially equivalent in that they define the same curve in different coordinates. In this survey, we try to follow the development of these ideas of Kolmogorov as well as similar ideas suggested independently by other authors.
Citation: Theory of Probability & Its Applications
PubDate: 20240207T08:00:00Z
DOI: 10.1137/S0040585X97T991647
Issue No: Vol. 68, No. 4 (2024)

 On a Diffusion Approximation of a Prediction Game

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Authors: M. V. Zhitlukhin
Pages: 607  621
Abstract: Theory of Probability & Its Applications, Volume 68, Issue 4, Page 607621, February 2024.
This paper is concerned with a dynamic gametheoretic model, where the players place bets on outcomes of random events or random vectors. Our purpose here is to construct a diffusion approximation of the model in the case where all players follow nearly optimal strategies. This approximation is further used to study the limit dynamics of the model.
Citation: Theory of Probability & Its Applications
PubDate: 20240207T08:00:00Z
DOI: 10.1137/S0040585X97T991659
Issue No: Vol. 68, No. 4 (2024)

 On Complete Convergence of Moments in Exact Asymptotics under Normal
Approximation
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Authors: L. V. Rozovsky
Pages: 622  629
Abstract: Theory of Probability & Its Applications, Volume 68, Issue 4, Page 622629, February 2024.
For the sums of the form $\overline I_s(\varepsilon) = \sum_{n\geqslant 1} n^{sr/2}\mathbf{E} S_n ^r\,\mathbf I[ S_n \geqslant \varepsilon\,n^\gamma]$, where $S_n = X_1 +\dots + X_n$, $X_n$, $n\geqslant 1$, is a sequence of independent and identically distributed random variables (r.v.'s) $s+1 \geqslant 0$, $r\geqslant 0$, $\gamma>1/2$, and $\varepsilon>0$, new results on their behavior are provided. As an example, we obtain the following generalization of Heyde's result [J. Appl. Probab., 12 (1975), pp. 173175]: for any $r\geqslant 0$, $\lim_{\varepsilon\searrow 0}\varepsilon^{2}\sum_{n\geqslant 1} n^{r/2} \mathbf{E} S_n ^r\,\mathbf I[ S_n \geqslant \varepsilon\, n] =\mathbf{E} \xi ^{r+2}$ if and only if $\mathbf{E} X=0$ and $\mathbf{E} X^2=1$, and also $\mathbf{E} X ^{2+r/2}
Citation: Theory of Probability & Its Applications
PubDate: 20240207T08:00:00Z
DOI: 10.1137/S0040585X97T991660
Issue No: Vol. 68, No. 4 (2024)

 One Limit Theorem for Branching Random Walks

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Authors: N. V. Smorodina, E. B. Yarovaya
Pages: 630  642
Abstract: Theory of Probability & Its Applications, Volume 68, Issue 4, Page 630642, February 2024.
The foundations of the general theory of Markov random processes were laid by A.N. Kolmogorov. Such processes include, in particular, branching random walks on lattices $\mathbf{Z}^d$, $d \in \mathbf{N}$. In the present paper, we consider a branching random walk where particles may die or produce descendants at any point of the lattice. Motion of each particle on $\mathbf{Z}^d$ is described by a symmetric homogeneous irreducible random walk. It is assumed that the branching rate of particles at $x \in \mathbf{Z}^d$ tends to zero as $\ x\ \to \infty$, and that an additional condition on the parameters of the branching random walk, which gives that the mean population size of particles at each point $\mathbf{Z}^d$ grows exponentially in time, is met. In this case, the walk generation operator in the righthand side of the equation for the mean population size of particles undergoes a perturbation due to possible generation of particles at points $\mathbf{Z}^d$. Equations of this kind with perturbation of the diffusion operator in $\mathbf{R}^2$, which were considered by Kolmogorov, Petrovsky, and Piskunov in 1937, continue being studied using the theory of branching random walks on discrete structures. Under the above assumptions, we prove a limit theorem on meansquare convergence of the normalized number of particles at an arbitrary fixed point of the lattice as $t\to\infty$.
Citation: Theory of Probability & Its Applications
PubDate: 20240207T08:00:00Z
DOI: 10.1137/S0040585X97T991672
Issue No: Vol. 68, No. 4 (2024)

 On Forward and Backward Kolmogorov Equations for Pure Jump Markov
Processes and Their Generalizations
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Authors: E. A. Feinberg, A. N. Shiryaev
Pages: 643  656
Abstract: Theory of Probability & Its Applications, Volume 68, Issue 4, Page 643656, February 2024.
In the present paper, we first give a survey of the forward and backward Kolmogorov equations for pure jump Markov processes with finite and countable state spaces, and then describe relevant results for the case of Markov processes with values in standard Borel spaces based on results of W. Feller and the authors of the present paper.
Citation: Theory of Probability & Its Applications
PubDate: 20240207T08:00:00Z
DOI: 10.1137/S0040585X97T991684
Issue No: Vol. 68, No. 4 (2024)

 On Sufficient Conditions in the MarchenkoPastur Theorem

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Authors: P. A. Yaskov
Pages: 657  673
Abstract: Theory of Probability & Its Applications, Volume 68, Issue 4, Page 657673, February 2024.
We find general sufficient conditions in the MarchenkoPastur theorem for highdimensional sample covariance matrices associated with random vectors, for which the weak concentration property of quadratic forms may not hold in general. The results are obtained by means of a new martingale method, which may be useful in other problems of random matrix theory.
Citation: Theory of Probability & Its Applications
PubDate: 20240207T08:00:00Z
DOI: 10.1137/S0040585X97T991696
Issue No: Vol. 68, No. 4 (2024)

 Abstracts of Talks Given at the 8th International Conference on Stochastic
Methods
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Authors: A. N. Shiryaev, I. V. Pavlov, P. A. Yaskov, T. A. Volosatova
Pages: 674  711
Abstract: Theory of Probability & Its Applications, Volume 68, Issue 4, Page 674711, February 2024.
This paper presents abstracts of talks given at the 8th International Conference on Stochastic Methods (ICSM8), held June 18, 2023 at Divnomorskoe (near the town of Gelendzhik) at the Raduga sports and fitness center of the Don State Technical University. This year's conference was dedicated to the 120th birthday of Andrei Nikolaevich Kolmogorov and was chaired by A. N. Shiryaev. Participants included leading scientists from Russia, Portugal, and Tadjikistan.
Citation: Theory of Probability & Its Applications
PubDate: 20240207T08:00:00Z
DOI: 10.1137/S0040585X97T991702
Issue No: Vol. 68, No. 4 (2024)
