Theory of Probability &Its Applications, Volume 67, Issue 2, Page 327-334, August 2022. This paper presents abstracts of talks given at the General Seminar of the Department of Probability, Faculty of Mathematics and Mechanics, Lomonosov Moscow State University, held in Moscow in 2021. Current information about the seminar is available at http://new.math.msu.su/department/probab/seminar.html.

Theory of Probability &Its Applications, Volume 67, Issue 2, Page 318-326, August 2022. We consider a stochastic modification of the Colonel Blotto game, also called the gladiator game. Each of two players has a given amount of resources (strengths), which can be arbitrarily distributed between a given number of gladiators. Once the strengths are distributed, the teams begin a battle consisting of

Theory of Probability &Its Applications, Volume 67, Issue 2, Page 310-317, August 2022. The paper continues the author's long-term studies on extremes of random particle scores in branching processes. It is assumed that multiplication of particles is described by an immortal supercritical discrete-time branching process, the particle scores are dependent due to general heredity, and this dependence

Theory of Probability &Its Applications, Volume 67, Issue 2, Page 294-309, August 2022. This note is devoted to the study of the maximum of the excursion of a random walk with negative drift and light-tailed increments. More precisely, we determine the local asymptotics of the joint distribution of the length, the maximum, and the time at which this maximum is achieved. This result allows one to obtain

Theory of Probability &Its Applications, Volume 67, Issue 2, Page 282-293, August 2022. We establish some asymptotic estimations for the extrema of the generalized allocation scheme driven by $m$-dependent random variables, which extend some known results of Chuprunov and Fazekash [Discrete Math. Appl., 22 (2012), pp. 307--314] from the independent case to the $m$-dependent case.

Theory of Probability &Its Applications, Volume 67, Issue 2, Page 261-281, August 2022. In this paper, we investigate the complete $f$-moment convergence for randomly weighted sums of extended negatively dependent (END for short) random variables. Some results obtained in this paper extend and improve the corresponding ones of P. Li, X. Li, and K. Wu [J. Inequal. Appl., 2017 (2017), 182]. As an application

Theory of Probability &Its Applications, Volume 67, Issue 2, Page 246-260, August 2022. This paper establishes the LAN property for the curved normal families and the simultaneous equation systems. In addition, we show that one-way random ANOVA models fail to have the LAN property. We consider the two cases when the variance of random effect lies in the interior and boundary of parameter space. In

Theory of Probability &Its Applications, Volume 67, Issue 2, Page 229-245, August 2022. This paper addresses the log-optimal portfolio, which is the portfolio with finite expected log-utility that maximizes the expected logarithm utility from terminal wealth, for an arbitrary general semimartingale model. The most advanced literature on this topic elaborates existence and characterization of this portfolio

Theory of Probability &Its Applications, Volume 67, Issue 2, Page 208-228, August 2022. In this paper, we study the one-dimensional Hua--Pickrell diffusion. We start by revisiting the stationary case considered by E. Wong for which we supply omitted details and write down a unified expression of its semigroup density through the associated Legendre function in the cut. Next, we focus on the general

Theory of Probability &Its Applications, Volume 67, Issue 2, Page 194-207, August 2022. A two-dimensional diffusion process is considered. The distribution of the first exit point of such a process from an arbitrary domain of its values is determined, as a function of the initial point of the process, by an elliptic second-order differential equation, and corresponds to the solution of the Dirichlet

Theory of Probability &Its Applications, Volume 67, Issue 2, Page 175-193, August 2022. The limit concentration of the values of the chromatic number of the random hypergraph $H(n,k,p)$ in the binomial model is studied. It is proved that, for a fixed $k\ge 3$ and with not too rapidly increasing $n^{k-1}p$, the chromatic number of the hypergraph $H(n,k,p)$ lies, with probability tending to 1, in the

Theory of Probability &Its Applications, Volume 67, Issue 2, Page 163-174, August 2022. A new estimator for a Bernoulli regression function based on Bernstein polynomials is constructed. Its consistency and asymptotic normality are studied. A criterion for testing the hypothesis on the form of a Bernoulli regression function and a criterion for testing the hypothesis on the equality of two Bernoulli

Theory of Probability &Its Applications, Volume 67, Issue 1, Page 161-162, May 2022. A celebration of the life and accomplishments of internationally known mathematician Doctor Hans-Jürgen Engelbert, who passed away on May 23, 2021.

Theory of Probability &Its Applications, Volume 67, Issue 1, Page 158-160, May 2022. Bertrand's Paradox is classical in the theory of probability. Its point of contention is the existence of three distinct solutions to a seemingly identical required probability, with each solution obtained through a different method. This paper depicts yet another solution, a novel approach originating from diametric

Theory of Probability &Its Applications, Volume 67, Issue 1, Page 154-157, May 2022. Representations are put forward for the moments and the truncated Senatov quasi-moments of normalized sums of random variables (r.v.'s) in terms of the Senatov moments of the original distribution. These representations make possible the direct transition from new asymptotic expansions in the central limit theorem

Theory of Probability &Its Applications, Volume 67, Issue 1, Page 141-153, May 2022. This article is an essay, both expository and argumentative, on the Galton--Watson process as a tool in the domain of branching processes. It is at the same time the author's way of honoring two distinguished scientists in this domain, both from the Russian Academy of Sciences, and congratulating them on their special

Theory of Probability &Its Applications, Volume 67, Issue 1, Page 140-140, May 2022. Recognition of the 75th birthday of Andrei Mikhailovich Zubkov and the 70th birthday of Vladimir Alekseevich Vatutin.

Theory of Probability &Its Applications, Volume 67, Issue 1, Page 118-139, May 2022. The aim of this paper is to present a new proof of an explicit version of the Berry--Esseen type inequality of Bolthausen [Z. Wahrsch. Verw. Gebiete, 66 (1984), pp. 379--386]. The literature already provides several proofs using variants of Stein's method. The characteristic function method has also been applied but

Theory of Probability &Its Applications, Volume 67, Issue 1, Page 105-117, May 2022. We identify the background driving distribution functions (BDDF) for self-decomposable distributions (random variables). For log-gamma variables and their background driving variables, we find their series representations. An innovation variable for Bessel-K distribution is given as a compound Poisson variable.

Theory of Probability &Its Applications, Volume 67, Issue 1, Page 89-104, May 2022. Uniform integrability is an interesting concept in probability theory and functional analysis since it plays an important role in establishing laws of large numbers. In the literature, there are several versions of uniform integrability. Some are defined with the help of matrix summability methods, such as the Cesàro

Theory of Probability &Its Applications, Volume 67, Issue 1, Page 77-88, May 2022. We consider the persistence probability for the integrated fractional Brownian motion and the fractionally integrated Brownian motion with parameter $H$, respectively. For the integrated fractional Brownian motion, we discuss a conjecture of Molchan and Khokhlov and determine the asymptotic behavior of the persistence

Theory of Probability &Its Applications, Volume 67, Issue 1, Page 62-76, May 2022. In this paper, we construct a probabilistic approximation of the evolution operator $\exp\bigl(t\bigl({\frac{(-1)^{m+1}}{(2m)!}\,\frac{d^{2m}}{dx^{2m}}+V}\bigr)\bigr)$ in the form of expectations of functionals of a point random field. This approximation can be considered as a generalization of the Feynman--Kac formula

Theory of Probability &Its Applications, Volume 67, Issue 1, Page 44-61, May 2022. We propose a sequence of accompanying laws in the B.,V. Gnedenko limit theorem for maxima of independent random variables with distributions lying in the Gumbel max domain of attraction. We show that this sequence provides a power-law convergence rate, whereas the Gumbel distribution provides only the logarithmic rate

Theory of Probability &Its Applications, Volume 67, Issue 1, Page 28-43, May 2022. We consider an optimal linear-quadratic control problem for a control system where the matrices corresponding to the state in the controlled process equation and the cost functional are absolutely integrable over an infinite time interval. The integral quadratic performance index includes two mutually inversely proportional

Theory of Probability &Its Applications, Volume 67, Issue 1, Page 17-27, May 2022. We consider special one-dimensional Markov processes, namely, asymmetric jump Lévy processes, which have values in a given interval and reflect from the boundary points. We show that in this case, in addition to the standard semigroup of operators generated by the Markov process, there also appears the family of “boundary”

Theory of Probability &Its Applications, Volume 67, Issue 1, Page 1-16, May 2022. The aim of the present work is to show that our recent results on the approximation of distributions of sums of independent summands by the infinitely divisible laws on convex polyhedra can be obtained via an alternative class of approximating infinitely divisible distributions. We will also generalize the results to

Theory of Probability &Its Applications, Volume 66, Issue 4, Page 729-741, February 2022. Based on the law of large numbers and the central limit theorem under nonlinear expectation, we introduce a new method of using ${G}$-normal distribution to measure financial risks. Applying max-mean estimators and a small windows method, we establish autoregressive models to determine the parameters of ${G}$-normal

Theory of Probability &Its Applications, Volume 66, Issue 4, Page 713-728, February 2022. We derive moment and exponential inequalities for the maximum of a generalized Ornstein--Uhlenbeck process under some assumptions on tail distributions of a jump component.

Theory of Probability &Its Applications, Volume 66, Issue 4, Page 708-712, February 2022. Suppose that $n$ different pairs of socks are put in a tumble dryer. When the dryer is finished, socks are taken out one by one. If a sock matches one of the socks on the sorting table, both are removed; otherwise, it is put on the table until its partner emerges from the dryer. We note the number of socks on

Theory of Probability &Its Applications, Volume 66, Issue 4, Page 668-707, February 2022. We revisit the variational characterization of conservative diffusion as entropic gradient flow and provide for it a probabilistic interpretation based on stochastic calculus. It was shown by Jordan, Kinderlehrer, and Otto that, for diffusions of Langevin--Smoluchowski type, the Fokker--Planck probability density

Theory of Probability &Its Applications, Volume 66, Issue 4, Page 640-667, February 2022. Based on the method of subordinating functions we prove bounds for the minimal error of approximations of $n$-fold convolutions of probability measures by free infinitely divisible probability measures.

Theory of Probability &Its Applications, Volume 66, Issue 4, Page 613-639, February 2022. We consider a mean field game with common noise in which the diffusion coefficients may be controlled. We prove existence of a weak relaxed solution under some continuity conditions on the coefficients. We then show that, when there is no common noise, the solution of this mean field game is characterized by a

Theory of Probability &Its Applications, Volume 66, Issue 4, Page 601-612, February 2022. The purpose of this paper is twofold. First, we extend the well-known relation between optimal stopping and randomized stopping of a given stochastic process to a situation where the available information flow is a filtration with no a priori assumed relation to the filtration of the process. We call these problems

Theory of Probability &Its Applications, Volume 66, Issue 4, Page 582-600, February 2022. This paper describes the structure of solutions to Kolmogorov's equations for nonhomogeneous jump Markov processes and applications of these results to control of jump stochastic systems. These equations were studied by Feller [Trans. Amer. Math. Soc., 48 (1940), pp. 488--515], who clarified in 1945 in the errata

Theory of Probability &Its Applications, Volume 66, Issue 4, Page 570-581, February 2022. Let $\xi_1,\xi_2,\dots$ be a sequence of independent copies of a random variable (r.v.) $\xi$, ${S_n=\sum_{j=1}^n\xi_j}$, $A(\lambda)=\ln\mathbf{E}e^{\lambda\xi}$, $\Lambda(\alpha)=\sup_\lambda(\alpha\lambda-A(\lambda))$ is the Legendre transform of $A(\lambda)$. In this paper, which is partially a review to some

Theory of Probability &Its Applications, Volume 66, Issue 4, Page 550-569, February 2022. \bad This paper gives a survey of several directions of research connected with Chebyshev--Hermite polynomials on finite-dimensional and infinite-dimensional spaces, in particular, of approaches using the Malliavin calculus and other methods of investigation of distributions of polynomials in Gaussian random variables

Theory of Probability &Its Applications, Volume 66, Issue 4, Page 537-549, February 2022. We give a short overview of the results related to the refined forms of the central limit theorem, with a focus on independent integer-valued random variables (r.v.'s). In the independent and non-identically distributed (non-i.i.d.) case, an approximation is then developed for the distribution of the sum by means

Theory of Probability &Its Applications, Volume 66, Issue 4, Page 522-536, February 2022. We consider critical symmetric branching random walks on a multidimensional lattice with continuous time and with the source of particle birth and death at the origin. We prove limit theorems on the distribution of the sojourn time of the underlying random walk at a point depending on the lattice dimension under

Theory of Probability &Its Applications, Volume 66, Issue 4, Page 506-521, February 2022. We survey briefly the life and work of P. L. Chebyshev and his ongoing influence. We discuss his contributions to probability, number theory, and mechanics; his pupils and mathematical descendants; and his role as the founding father of Russian mathematics in general and of the Russian school of probability in

Theory of Probability &Its Applications, Volume 66, Issue 4, Page 497-505, February 2022. This article is an extended version of the address given at the International Conference “Theory of Probability and Its Applications: P.L. Chebyshev--200” (Moscow, May 17--22, 2021).

Theory of Probability &Its Applications, Volume 66, Issue 3, Page 495-496, January 2021. This article celebrates the life and professional accomplishments of outstanding mathematician Vladimir Vasil'evich Senatov, who passed away on June 22, 2021. During his long career, he made essential contributions toward the solution of several central problems in probability theory.

Theory of Probability &Its Applications, Volume 66, Issue 3, Page 488-494, January 2021. This paper gives the explicit expressions for moments of the $k$th record values from normal distribution and relations between the $r$th, $(r-2)$nd single, and product moments. These relations allow us to give skewness and kurtosis of record values and present a new characterization of normal distribution.

Theory of Probability &Its Applications, Volume 66, Issue 3, Page 482-487, January 2021. Let $X_1, \dots, X_n$ be independent random variables taking values in the alphabet $\{0, 1, \dots, r\}$, and let $S_n=\sum_{i=1}^n X_i$. The Shepp--Olkin theorem states that in the binary case (${r=1}$), the Shannon entropy of $S_n$ is maximized when all the $X_i$'s are uniformly distributed, i.e., Bernoulli(1/2)

Theory of Probability &Its Applications, Volume 66, Issue 3, Page 474-481, January 2021. Local limit theorems for compound discrete distributions (a normal theorem and a large deviation theorem) are put forward. Some examples of such distributions, including compound Poisson, compound binomial, and negative binomial distributions, are considered. Our analysis is based on elementary asymptotic methods

Theory of Probability &Its Applications, Volume 66, Issue 3, Page 469-473, January 2021. Pathwise uniqueness for the multidimensional stochastic McKean--Vlasov equation is established under moderate regularity conditions on the drift and diffusion coefficients. Both drift and diffusion depend on the marginal measure of the solution. It is assumed that both coefficients are bounded, and, moreover, the

Theory of Probability &Its Applications, Volume 66, Issue 3, Page 455-468, January 2021. Let $\widehat F_n$ be the smooth empirical estimator obtained by integrating a kernel type density estimator based on a random sample of size $n$ from continuous distribution function $F$. The almost sure deviation between smooth empirical and smooth quantile processes is investigated under $\phi$-mixing and strong

Theory of Probability &Its Applications, Volume 66, Issue 3, Page 445-454, January 2021. Let $p\in [1,2)$. We show that if $(X_n)_{n=1}^\infty$ is a sequence of pairwise i.i.d. random variables with ${E}|X_1|^p<\infty$, then ${P}\{\sup_n|{S_n}/{n^{1/p}}|> \alpha\}\le {C_p\,{E}|X_1|^p}/{\alpha^p}$ for every $\alpha>0$ for some constant $C_p$ depending only on $p$, where $S_n:=\sum_{i=1}^n(X_i-{E}X_i)$

Theory of Probability &Its Applications, Volume 66, Issue 3, Page 430-444, January 2021. For a given set of independent random variables (r.v.'s) $X_1,\dots,X_d$ belonging to a given Meixner class, we seek r.v.'s $Y_1,\dots,Y_d$ such that the marginal laws and the laws of the sums match: $Y_i\stackrel{d}{=} X_i$ and $\sum_iY_i\stackrel{d}{=}\sum_iX_i$. We give a full characterization of the r.v.'s

Theory of Probability &Its Applications, Volume 66, Issue 3, Page 408-429, January 2021. We establish an exponential inequality for degenerated $U$-statistics of order $r$ of independent and identically distributed (i.i.d.) data. This inequality gives a control of the tail of the maxima absolute values of the $U$-statistic by the sum of the two terms: an exponential term and one involving the tail

Theory of Probability &Its Applications, Volume 66, Issue 3, Page 391-407, January 2021. The paper is concerned with the problem of construction of characterizing statistics for some types of distributions within certain classes. A characterizing statistic for multivariate normal distribution is constructed with the use of the theory of random matrices. Properties and characteristics of spherically

Theory of Probability &Its Applications, Volume 66, Issue 3, Page 376-390, January 2021. We consider the model of an $N$-vertex configuration graph where the number of edges is at most $n$ and the degrees of vertices are independent and identically distributed (i.i.d.) random variables (r.v.'s). The distribution of the r.v. $\xi$, which is defined as the degree of any given vertex, is assumed to satisfy

Theory of Probability &Its Applications, Volume 66, Issue 3, Page 364-375, January 2021. As an alternative to the splitting technique of Athreya--Ney and Nummelin, we propose a new method for the proof of ergodic theorems for Markov chains with arbitrary state space. Under our approach, the expansion of the original state space, which, in our opinion, is an ingenious but still artificial technique

Theory of Probability &Its Applications, Volume 66, Issue 3, Page 348-363, January 2021. We propose a test procedure for distinguishing between two separable classes of arbitrary distribution tails and, moreover, prove its consistency. The method is based on the goodness-of-fit test for an arbitrary distribution tail, and properties of this test are also discussed. This is the first goodness-of-fit

Theory of Probability &Its Applications, Volume 66, Issue 3, Page 337-347, January 2021. For a class of Gaussian stationary processes, we prove a limit theorem on the convergence of the distributions of the scaled last exit time over a slowly growing linear boundary. The limit is a double exponential (Gumbel) distribution.

Theory of Probability &Its Applications, Volume 66, Issue 2, Page 326-335, January 2021. We propose and study a simple model for microcredit using two sums, with a random number of terms, of identically distributed random variables, the number of terms being Poisson distributed; the first sum accounts for the payments---the payables---made to the collective vault by the participants, and the second

Theory of Probability &Its Applications, Volume 66, Issue 2, Page 318-325, January 2021. Extensions of S.N. Bernstein's example of three dependent random events of which any two are independent are considered. A complete description of such examples is given in the framework of the symmetric probability model.

Theory of Probability &Its Applications, Volume 66, Issue 2, Page 299-317, January 2021. We prove characterizations of positive dependence for a general class of time-inhomogeneous Markov processes called Feller evolution processes (FEPs) and for jump-FEPs. General FEPs can be analyzed through their time and state-space dependent (extended) generators. We will use the time and state-space dependent

Theory of Probability &Its Applications, Volume 66, Issue 2, Page 276-298, January 2021. Under natural assumptions, we prove the ergodicities and exponential ergodicities in Wasserstein and total variation distances of Dawson--Watanabe superprocesses without or with immigration. The strong Feller property in the total variation distance is derived as a by-product. The key of the approach is a set of

Theory of Probability &Its Applications, Volume 66, Issue 2, Page 263-275, January 2021. Using the approach of Etemadi for the strong law of large numbers [Z. Wahrsch. Verw. Gebiete, 55 (1981), pp. 119--122] and its elaboration by Csörgö, Tandori, and Totik [Acta Math. Hungar., 42 (1983), pp. 319--330], we give weaker conditions under which the strong law of large numbers still holds, namely for pairwise

Theory of Probability &Its Applications, Volume 66, Issue 2, Page 245-262, January 2021. We present a backward diffusion flow (i.e., a backward-in-time stochastic differential equation) whose marginal distribution at any (earlier) time is equal to the smoothing distribution when the terminal state (at a later time) is distributed according to the filtering distribution. This is a novel interpretation