Society for Industrial and Applied Mathematics: Theory of Probability & Its Applications: Table of Contents
Table of Contents for Theory of Probability & Its Applications. List of articles from both the latest and ahead of print issues.
https://epubs.siam.org/loi/tprbau?ai=s4&mi=3csssi&af=R
Society for Industrial and Applied Mathematics: Theory of Probability & Its Applications: Table of Contents
Society for Industrial and Applied Mathematics
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Theory of Probability & Its Applications
Theory of Probability & Its Applications
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https://epubs.siam.org/loi/tprbau?ai=s4&mi=3csssi&af=R
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Weakly Supercritical Branching Process in a Random Environment Dying at a Distant Moment
https://epubs.siam.org/doi/abs/10.1137/S0040585X97T991611?ai=s4&mi=3csssi&af=R
Theory of Probability &Its Applications, <a href="https://epubs.siam.org/toc/tprbau/68/4">Volume 68, Issue 4</a>, Page 537-558, February 2024. <br/> 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 $.
Theory of Probability & Its Applications, Volume 68, Issue 4, Page 537-558, February 2024. <br/> 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 $.
Weakly Supercritical Branching Process in a Random Environment Dying at a Distant Moment
10.1137/S0040585X97T991611
Theory of Probability & Its Applications
2024-02-07T09:31:03Z
© 2024, Society for Industrial and Applied Mathematics
V. I. Afanasyev
Weakly Supercritical Branching Process in a Random Environment Dying at a Distant Moment
68
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537
558
2024-02-06T08:00:00Z
2024-02-06T08:00:00Z
10.1137/S0040585X97T991611
https://epubs.siam.org/doi/abs/10.1137/S0040585X97T991611?ai=s4&mi=3csssi&af=R
© 2024, Society for Industrial and Applied Mathematics
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On Symmetrized Chi-Square Tests in Autoregression with Outliers in Data
https://epubs.siam.org/doi/abs/10.1137/S0040585X97T991623?ai=s4&mi=3csssi&af=R
Theory of Probability &Its Applications, <a href="https://epubs.siam.org/toc/tprbau/68/4">Volume 68, Issue 4</a>, Page 559-569, February 2024. <br/> 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 chi-square 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 $G(x)=A_{n,\Phi}(x):=(1-n^{-1/2})\Phi(x/\theta_0)+n^{-1/2}H(x)$, 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 $\gamma=0$.
Theory of Probability & Its Applications, Volume 68, Issue 4, Page 559-569, February 2024. <br/> 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 chi-square 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 $G(x)=A_{n,\Phi}(x):=(1-n^{-1/2})\Phi(x/\theta_0)+n^{-1/2}H(x)$, 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 $\gamma=0$.
On Symmetrized Chi-Square Tests in Autoregression with Outliers in Data
10.1137/S0040585X97T991623
Theory of Probability & Its Applications
2024-02-07T08:00:00Z
© 2024, Society for Industrial and Applied Mathematics
M. V. Boldin
On Symmetrized Chi-Square Tests in Autoregression with Outliers in Data
68
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559
569
2024-02-06T08:00:00Z
2024-02-06T08:00:00Z
10.1137/S0040585X97T991623
https://epubs.siam.org/doi/abs/10.1137/S0040585X97T991623?ai=s4&mi=3csssi&af=R
© 2024, Society for Industrial and Applied Mathematics
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Laplace Expansion for Bartlett--Nanda--Pillai Test Statistic and Its Error Bound
https://epubs.siam.org/doi/abs/10.1137/S0040585X97T991635?ai=s4&mi=3csssi&af=R
Theory of Probability &Its Applications, <a href="https://epubs.siam.org/toc/tprbau/68/4">Volume 68, Issue 4</a>, Page 570-581, February 2024. <br/> We construct asymptotic expansions for the distribution function of the Bartlett--Nanda--Pillai 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 matrix-valued functions.
Theory of Probability & Its Applications, Volume 68, Issue 4, Page 570-581, February 2024. <br/> We construct asymptotic expansions for the distribution function of the Bartlett--Nanda--Pillai 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 matrix-valued functions.
Laplace Expansion for Bartlett--Nanda--Pillai Test Statistic and Its Error Bound
10.1137/S0040585X97T991635
Theory of Probability & Its Applications
2024-02-07T08:00:00Z
© 2024, Society for Industrial and Applied Mathematics
H. Wakaki
V. V. Ulyanov
Laplace Expansion for Bartlett--Nanda--Pillai Test Statistic and Its Error Bound
68
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570
581
2024-02-06T08:00:00Z
2024-02-06T08:00:00Z
10.1137/S0040585X97T991635
https://epubs.siam.org/doi/abs/10.1137/S0040585X97T991635?ai=s4&mi=3csssi&af=R
© 2024, Society for Industrial and Applied Mathematics
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Kolmogorov's Last Discovery? (Kolmogorov and Algorithmic Statistics)
https://epubs.siam.org/doi/abs/10.1137/S0040585X97T991647?ai=s4&mi=3csssi&af=R
Theory of Probability &Its Applications, <a href="https://epubs.siam.org/toc/tprbau/68/4">Volume 68, Issue 4</a>, Page 582-606, February 2024. <br/> The definition of descriptional complexity of finite objects suggested by Kolmogorov and other authors in the mid-1960s is now well known. In addition, Kolmogorov pointed out some approaches to a more fine-grained classification of finite objects, such as the resource-bounded 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.
Theory of Probability & Its Applications, Volume 68, Issue 4, Page 582-606, February 2024. <br/> The definition of descriptional complexity of finite objects suggested by Kolmogorov and other authors in the mid-1960s is now well known. In addition, Kolmogorov pointed out some approaches to a more fine-grained classification of finite objects, such as the resource-bounded 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.
Kolmogorov's Last Discovery? (Kolmogorov and Algorithmic Statistics)
10.1137/S0040585X97T991647
Theory of Probability & Its Applications
2024-02-07T08:00:00Z
© 2024, Society for Industrial and Applied Mathematics
A. L. Semenov
A. Kh. Shen
N. K. Vereshchagin
Kolmogorov's Last Discovery? (Kolmogorov and Algorithmic Statistics)
68
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582
606
2024-02-06T08:00:00Z
2024-02-06T08:00:00Z
10.1137/S0040585X97T991647
https://epubs.siam.org/doi/abs/10.1137/S0040585X97T991647?ai=s4&mi=3csssi&af=R
© 2024, Society for Industrial and Applied Mathematics
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On a Diffusion Approximation of a Prediction Game
https://epubs.siam.org/doi/abs/10.1137/S0040585X97T991659?ai=s4&mi=3csssi&af=R
Theory of Probability &Its Applications, <a href="https://epubs.siam.org/toc/tprbau/68/4">Volume 68, Issue 4</a>, Page 607-621, February 2024. <br/> This paper is concerned with a dynamic game-theoretic 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.
Theory of Probability & Its Applications, Volume 68, Issue 4, Page 607-621, February 2024. <br/> This paper is concerned with a dynamic game-theoretic 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.
On a Diffusion Approximation of a Prediction Game
10.1137/S0040585X97T991659
Theory of Probability & Its Applications
2024-02-07T08:00:00Z
© 2024, Society for Industrial and Applied Mathematics
M. V. Zhitlukhin
On a Diffusion Approximation of a Prediction Game
68
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607
621
2024-02-06T08:00:00Z
2024-02-06T08:00:00Z
10.1137/S0040585X97T991659
https://epubs.siam.org/doi/abs/10.1137/S0040585X97T991659?ai=s4&mi=3csssi&af=R
© 2024, Society for Industrial and Applied Mathematics
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On Complete Convergence of Moments in Exact Asymptotics under Normal Approximation
https://epubs.siam.org/doi/abs/10.1137/S0040585X97T991660?ai=s4&mi=3csssi&af=R
Theory of Probability &Its Applications, <a href="https://epubs.siam.org/toc/tprbau/68/4">Volume 68, Issue 4</a>, Page 622-629, February 2024. <br/> For the sums of the form $\overline I_s(\varepsilon) = \sum_{n\geqslant 1} n^{s-r/2}\mathbf{E}|S_n|^r\,\mathbf I[|S_n|\geqslant \varepsilon\,n^\gamma]$, 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. 173--175]: 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 $\mathbf{E}|X|^{2+r/2}<\infty$ if $r < 4$, $\mathbf{E}|X|^r<\infty$ if $r>4$, and $\mathbf{E} X^4 \ln{(1+|X|)}<\infty$ if $r=4$. Here, $\xi$ is a standard Gaussian r.v.
Theory of Probability & Its Applications, Volume 68, Issue 4, Page 622-629, February 2024. <br/> For the sums of the form $\overline I_s(\varepsilon) = \sum_{n\geqslant 1} n^{s-r/2}\mathbf{E}|S_n|^r\,\mathbf I[|S_n|\geqslant \varepsilon\,n^\gamma]$, 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. 173--175]: 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 $\mathbf{E}|X|^{2+r/2}<\infty$ if $r < 4$, $\mathbf{E}|X|^r<\infty$ if $r>4$, and $\mathbf{E} X^4 \ln{(1+|X|)}<\infty$ if $r=4$. Here, $\xi$ is a standard Gaussian r.v.
On Complete Convergence of Moments in Exact Asymptotics under Normal Approximation
10.1137/S0040585X97T991660
Theory of Probability & Its Applications
2024-02-07T08:00:00Z
© 2024, Society for Industrial and Applied Mathematics
L. V. Rozovsky
On Complete Convergence of Moments in Exact Asymptotics under Normal Approximation
68
4
622
629
2024-02-06T08:00:00Z
2024-02-06T08:00:00Z
10.1137/S0040585X97T991660
https://epubs.siam.org/doi/abs/10.1137/S0040585X97T991660?ai=s4&mi=3csssi&af=R
© 2024, Society for Industrial and Applied Mathematics
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One Limit Theorem for Branching Random Walks
https://epubs.siam.org/doi/abs/10.1137/S0040585X97T991672?ai=s4&mi=3csssi&af=R
Theory of Probability &Its Applications, <a href="https://epubs.siam.org/toc/tprbau/68/4">Volume 68, Issue 4</a>, Page 630-642, February 2024. <br/> 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 right-hand 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 mean-square convergence of the normalized number of particles at an arbitrary fixed point of the lattice as $t\to\infty$.
Theory of Probability & Its Applications, Volume 68, Issue 4, Page 630-642, February 2024. <br/> 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 right-hand 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 mean-square convergence of the normalized number of particles at an arbitrary fixed point of the lattice as $t\to\infty$.
One Limit Theorem for Branching Random Walks
10.1137/S0040585X97T991672
Theory of Probability & Its Applications
2024-02-07T08:00:00Z
© 2024, Society for Industrial and Applied Mathematics
N. V. Smorodina
E. B. Yarovaya
One Limit Theorem for Branching Random Walks
68
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630
642
2024-02-06T08:00:00Z
2024-02-06T08:00:00Z
10.1137/S0040585X97T991672
https://epubs.siam.org/doi/abs/10.1137/S0040585X97T991672?ai=s4&mi=3csssi&af=R
© 2024, Society for Industrial and Applied Mathematics
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On Forward and Backward Kolmogorov Equations for Pure Jump Markov Processes and Their Generalizations
https://epubs.siam.org/doi/abs/10.1137/S0040585X97T991684?ai=s4&mi=3csssi&af=R
Theory of Probability &Its Applications, <a href="https://epubs.siam.org/toc/tprbau/68/4">Volume 68, Issue 4</a>, Page 643-656, February 2024. <br/> 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.
Theory of Probability & Its Applications, Volume 68, Issue 4, Page 643-656, February 2024. <br/> 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.
On Forward and Backward Kolmogorov Equations for Pure Jump Markov Processes and Their Generalizations
10.1137/S0040585X97T991684
Theory of Probability & Its Applications
2024-02-07T08:00:00Z
© 2024, Society for Industrial and Applied Mathematics
E. A. Feinberg
A. N. Shiryaev
On Forward and Backward Kolmogorov Equations for Pure Jump Markov Processes and Their Generalizations
68
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643
656
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2024-02-06T08:00:00Z
10.1137/S0040585X97T991684
https://epubs.siam.org/doi/abs/10.1137/S0040585X97T991684?ai=s4&mi=3csssi&af=R
© 2024, Society for Industrial and Applied Mathematics
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On Sufficient Conditions in the Marchenko--Pastur Theorem
https://epubs.siam.org/doi/abs/10.1137/S0040585X97T991696?ai=s4&mi=3csssi&af=R
Theory of Probability &Its Applications, <a href="https://epubs.siam.org/toc/tprbau/68/4">Volume 68, Issue 4</a>, Page 657-673, February 2024. <br/> We find general sufficient conditions in the Marchenko--Pastur theorem for high-dimensional 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.
Theory of Probability & Its Applications, Volume 68, Issue 4, Page 657-673, February 2024. <br/> We find general sufficient conditions in the Marchenko--Pastur theorem for high-dimensional 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.
On Sufficient Conditions in the Marchenko--Pastur Theorem
10.1137/S0040585X97T991696
Theory of Probability & Its Applications
2024-02-07T08:00:00Z
© 2024, Society for Industrial and Applied Mathematics
P. A. Yaskov
On Sufficient Conditions in the Marchenko--Pastur Theorem
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657
673
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2024-02-06T08:00:00Z
10.1137/S0040585X97T991696
https://epubs.siam.org/doi/abs/10.1137/S0040585X97T991696?ai=s4&mi=3csssi&af=R
© 2024, Society for Industrial and Applied Mathematics
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Abstracts of Talks Given at the 8th International Conference on Stochastic Methods
https://epubs.siam.org/doi/abs/10.1137/S0040585X97T991702?ai=s4&mi=3csssi&af=R
Theory of Probability &Its Applications, <a href="https://epubs.siam.org/toc/tprbau/68/4">Volume 68, Issue 4</a>, Page 674-711, February 2024. <br/> This paper presents abstracts of talks given at the 8th International Conference on Stochastic Methods (ICSM-8), held June 1--8, 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.
Theory of Probability & Its Applications, Volume 68, Issue 4, Page 674-711, February 2024. <br/> This paper presents abstracts of talks given at the 8th International Conference on Stochastic Methods (ICSM-8), held June 1--8, 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.
Abstracts of Talks Given at the 8th International Conference on Stochastic Methods
10.1137/S0040585X97T991702
Theory of Probability & Its Applications
2024-02-07T08:00:00Z
© 2024, Society for Industrial and Applied Mathematics
A. N. Shiryaev
I. V. Pavlov
P. A. Yaskov
T. A. Volosatova
Abstracts of Talks Given at the 8th International Conference on Stochastic Methods
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10.1137/S0040585X97T991702
https://epubs.siam.org/doi/abs/10.1137/S0040585X97T991702?ai=s4&mi=3csssi&af=R
© 2024, Society for Industrial and Applied Mathematics