Limit theorems for the maxima of functions of Gaussian time series are studied. The limit behavior of the normalized sequence of maxima is examined under the condition that the correlation function of the process under consideration decays strictly logarithmically. Under some reasonable constraints on the function under consideration, the distribution is shown to be a modification of the corresponding distribution from Gnedenko's theorem. In addition, we derive a limit theorem for the reliability index of the vector function of a dependent vector of standard normal random variables, in which each component has the distribution function from the attraction domain of the Frechet distribution. This result is obtained under the assumption that the correlation function of each component of the Gaussian vector decays at least logarithmically.
We consider two nontransitive Trybula triplets. The first one features the maximum possible nontransitivity strength for a nontransitive cycle of three random variables. The second one is the parametric triplet with equal expectations and variances. For both triplets, we study the stability of nontransitivity under taking the sum and the maximum of two independent copies of random variables. We show that the first one is stable under taking the sums (in the opposite direction of stochastic precedence) and ceases to be nontransitive under taking the maximum. For the second triplet, we show that the nontransitivity is preserved on a certain subinterval of the original interval of values of the parameter epsilon (under taking the sums, also in the opposite direction of stochastic precedence). For both transformations, we obtain polynomial equations, whose roots epsilon(cr) define the boundaries of the stability intervals.
We consider the problem of construction of energy-efficient Markov approximations (pursuit strategies) of a multidimensional Wiener process. In a certain natural class of pursuit strategies, we find an optimal strategy and obtain the asymptotics of the minimal pursuit energy on large time intervals. We also establish a correspondence between a Wiener process contained in the unit ball and an energy-efficient approximation. A relation to the minimization problem of Fisher information is revealed.
We consider the model of symmetric branching continuous-time random walks on the lattice Zd with n types of particles and periodically located branching sources. It is assumed that initially there is only one particle of type Ts at some point. For this process, we construct a periodic operator describing the evolution of the mean number of particles of type Tj and study its spectral properties. We also obtain the asymptotics of the mean number of particles of type Tj at a fixed point of the lattice as t -> infinity.
The paper describes the 17th--20th Kolmogorov Student Olympiads in Probability held from 2022 to 2025 at the Department of Probability of the Faculty of Mechanics and Mathematics at Lomonosov Moscow State University.
The paper looks back on the life of leading Russian mathematician Andrey Mikhailovich Zubkov, who passed away on August 6, 2025.
We study universal locally constant kernel estimators in the classical nonparametric regression problem, where a multivariate regression function should be recovered from observations of its noisy values in some available tuple of fixed or random points (a tuple of regressors). Earlier these kernel estimators were studied only in the case of continuous multivariate regression functions. A distinctive feature of universal nuclear estimators is the presence of quite weak, fairly simple, and minimal (in a sense) conditions on the regressors which are universal relative to the stochastic nature of these quantities. In particular, in the case of a continuous regression function, for the uniform consistency of these kernel estimators, it is sufficient to require only the property of asymptotically (with increasing volume of observations) dense filling of the domain of the regression function by the regressors. We show that, under the additional smoothness assumption on the function, the accuracy of uniform approximation can be improved, where, as above, the regressors should only satisfy the above fairly general condition in terms of data density.
This article recognizes the 80th birthday of world-renowned scientist Vladimir Il'ich Piterbarg, whose works have become classics of stochastic analysis.
We consider the model of an N-vertex configuration graph, where the degrees of vertices are independent and identically distributed random variables, and the distribution of the random variable eta, which is the degree of each vertex, satisfies the condition pk = P{eta = k} similar to h(k)/kg, 2 < g < 3, k -> infinity, where h(x) is a function which slowly varies at infinity. We prove the limit theorem for the maximal degree of vertices in this configuration graph provided that the number n of edges in the graph is known, N, n -> infinity, and h(N)n(2)N((4-3g)/(g-1)) >= C > 0.
In this paper, mixed two- and four-level designs are transformed to two-level designs via two types of quaternary coding mapping, and the uniformity measured by wrap-around L2-discrepancy of mixed two- and four-level designs is studied, respectively. Some lower bounds of wrap-around L2-discrepancy for mixed two- and four-level designs are obtained, three numerical examples are provided to illustrate the theoretical results, and it is shown that these lower bounds are tight, which can serve as a benchmark in the construction of uniform mixed two- and four-level designs.
This paper acknowledges the 90th birthday of eminent Russian mathematician Yakov Grigor'evich Sinai.
This paper acknowledges the 90th birthday of Yakov Grigor'evich Sinai, outstanding contributor to the development of modern ergodic theory and statistical mechanics, and presents a short history of his mathematical accomplishments.
We consider the numerical approximation by means of the Euler scheme of the unique solution to a class of stochastic differential equations (SDEs) driven by fractional Brownian motion (fBm) with Hurst parameter H is an element of (1/2, 1) and a Markov switching (MS). We first study the d-dimensional additive case, followed by a one-dimensional equation with multiplicative noise. The strong convergence of the scheme in a finite time interval is studied and a convergence rate is obtained. Some simulations are provided to show the application of the theoretical results.
The standard small-time functional central limit theorem of semimartingales has been established in [S. Gerhold et al., Stochastics, 87 (2015), pp. 723-746], proving that the scaling limit law of a large class of stochastic processes in increasingly small time scales is that of a Brownian motion with a possibly nontrivial variance-covariance matrix. In this paper, we focus on the time-homogeneous diffusion processes described by Ito SDEs. Instead of the simple time scaling 1/n of [S. Gerhold et al., Stochastics, 87 (2015), pp. 723-746], we consider the scaled processes stopped at the first exit times from the balls of decreasing radius n-1/2 without scaling time itself. To the best of our knowledge, this particular scaling has not been investigated in the literature. We prove that this is a nontrivial example of a sequence of processes which converges in the sense of finite-dimensional distributions over a dense subset of [0, infinity), but it does not converge weakly in the sense of laws of ca`dla`g processes. We also characterize the limit law of the scaled processes evaluated at their respective first exit times.
We examine the applicability of the Skorokhod representation theorem on filtrated probability spaces to the utility maximization problem in the Kabanov conic model of multiasset markets with proportional transaction costs. Filtrations on different stochastic bases are generally not related to one another under Skorokhod transitions, and hence the corresponding strategies may cease to be adapted. Consequently, the solutions obtained on the new probability space may not correspond to those on the original space. We show that, under fairly general conditions, the Bellman function of the control problem is preserved under changes of the underlying probability space.
We consider the ruin problem with random premiums whose densities have rational Laplace transforms, and investments in a risky asset whose price follows a geometric Brownian motion. The asymptotic behavior of the ruin probability for large initial capital values is also studied.
Moment inequalities play important roles in probability limit theory and mathematical statistics. In this work, the von Bahr-Esseen type inequality for extended negatively dependent random variables under sub-linear expectations is established successfully. By virtue of the inequality, we further obtain the Kolmogorov type weak law of large numbers for partial sums and the complete convergence for weighted sums, which extend and improve corresponding results in sub-linear expectation space.
This paper presents abstracts of talks given at the 10th International Conference on Stochastic Methods (ICSM-10), held May 31--June 6, 2025 in Divnomorskoe (near the town of Gelendzhik) at the Raduga sports and fitness center of Don State Technical University. The conference was dedicated to the 90th anniversary of its founding by A.N. Kolmogorov of the Department of Probability Theory at Lomonosov Moscow State University.
This paper gives summaries of the talks presented during the 2024 fall term and 2025 spring term 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 from the Department of Probability Theory website.
This article presents an overview of historical events and research that laid the foundation for the creation of the Department of Probability Theory in the Mathematical Department of the Faculty of Mechanics and Mathematics at Lomonosov Moscow State University. The department was founded in 1935, and Andrey Nikolaevich Kolmogorov served as its first chair.
