Theory of Probability & Its Applications

We study the rate of convergence of the distribution of the normalized sojourn time of a classical random walk above some nonnegative level to its limit law with unboundedly growing observation time.

The consistency of Nadaraya--Watson estimators in nonparametric regression is proved without using traditional conditions for dependence of design elements (regressors). A design can be either fixed and not necessarily regular, or random and not necessarily satisfying classical correlation conditions for the design elements. A new characterization of the dependence of design elements is proposed, in terms of which sufficient conditions are formulated for both pointwise and uniform consistency of Nadaraya--Watson estimators.

In this paper, we develop asymptotic Asian option hedging methods for the Black--Scholes markets with transaction costs. We first construct classical replication strategies and then, using the Leland approach, propose corresponding modifications for the financial markets with proportional transaction costs. Sufficient conditions are found on the transaction costs implying the asymptotic hedging for the constructed strategies. The pricing problem is also considered. Three cases are studied: the case where the option price is the same as for the hedging problem without transaction costs, the case of increasing volatility, and the case where the option price equals the option price of the “buy and hold” strategy for European call options.

We consider a continuous-time supercritical symmetric branching random walk on a multidimensional graph with periodic particle generation sources. A logarithmic asymptotic formula is obtained for the moments of population sizes of particles at each vertex of the graph as ${t\to\infty}$.

Motivated by an application in neuroimaging, we consider the problem of establishing global minimax lower bound in a high order tensor model. In particular, the methodology we describe provides the global minimax bound for the integral curve estimator proposed in [O. Carmichael and L. Sakhanenko, Linear Algebra Appl., 473 (2015), pp. 377--403] under a semiparametric estimation setting. The theoretical results in this paper guarantee that the estimator used for tracing the complex fiber structure inside a live human brain obtained from high angular resolution diffusion imaging (HARDI) data is not only optimal locally but also optimal globally. The global minimax bound on the asymptotic risk of the estimators thus will provide a quantification of uncertainty for the estimation method in the whole domain of the imaging field. In addition to theoretical results, we also provide a detailed simulation study in order to find the optimal number of gradient directions for the imaging protocols, which we further illustrate with a real data analysis of a live human brain scan to showcase the uncertainty quantification of the estimation method in [O. Carmichael and L. Sakhanenko, Linear Algebra Appl., 473 (2015), pp. 377--403]. Furthermore, based on the global minimax bound, we propose a method for comparing the relative accuracy of several commonly used neuroimaging protocols in diffusion tensor imaging (DTI).

We consider the small deviation probability for a random walk with random environment in time. Compared to [A. A. Mogul'skii, Theory Probab. Appl., 19 (1975), pp. 726--736], for the independent and identically distributed (i.i.d.) random walk, the rate is smaller (due to the random environment), which is specified in terms of the quenched and annealed variance.

We study complete and complete integration convergence for arrays of rowwise extended negatively dependent random variables under sublinear expectations. Our results generalize complete moment convergence results of [T.-C. Hu, K.-L. Wang, and A. Rosalsky, Sankhya A, 77 (2015), pp. 1--29] and [Y. Wu, M. Ordón͂ez Cabrera, and A. Volodin, Glas. Mat. Ser. III, 49(69) (2014), pp. 447--466] from classical probability spaces to spaces with sublinear expectation.

This article looks back on the life of world-renowned probabilist Sergey Viktorovich Nagaev, who celebrated his 90th birthday on December 11, 2022. At present, he continues to be actively involved in various fields of probability.

We consider a queueing system $M|G|1$ with possible vacations in server operations for principal customers (for example, if a server is leased). A cost optimization problem is solved. As control parameters, we use the probability $\alpha$ of the vacation and its duration. Under fairly general assumptions about the system behavior during vacations, we show that the optimal value of the probability $\alpha$ is either 0 or 1. We also give necessary and sufficient conditions for a vacation to be carried out, i.e., $\alpha=1$. With constant vacation durations, we find conditions such that $\alpha=1$, and the vacation duration is optimal. Two examples are considered. In the first example, the revenue from the vacation is a linear function of its duration, and, in the second example, the revenue is a quadratic function.

In the same context as in the seminal paper [P. Cheridito, Bernoulli, 7 (2001), pp. 913--934], we are concerned with the semimartingale property of the sum of some Gaussian martingale and an independent fractional Brownian motion with Hurst parameter $H \in (0,1)$. At the same time, we emphasize that the Markov property is lost even if the martingale owns it.

The statistical inference on missing mass aims to estimate the weight of elements not observed during sampling. Since the pioneer work of Good and Turing, the problem has been studied in many areas, including statistical linguistics, ecology, and machine learning. Proving the sub-Gaussian behavior of the missing mass has been notoriously hard, and a number of complicated arguments have been proposed: logarithmic Sobolev inequalities, thermodynamic approaches, and information-theoretic transportation methods. Prior works have argued that the difficulty is inherent, and classical tools are inadequate. We show that this common belief is false, and all that we need to establish the sub-Gaussian concentration is the classical inequality of Bernstein. The strong educational value of our work is in its demonstration of this inequality in its full generality, an aspect not well recognized by researchers.

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 2022. Current information about the seminar is available at http://new.math.msu.su/department/probab/seminar.html.

This note presents information about four remote seminars in probability that are currently ongoing and six in-person conferences that were held in 2022. Also given are synopses of two probability textbooks published in 2022.

This note presents information about three workshops on financial and actuarial mathematics held in 2022 by The Vega Institute Science Promotion Foundation. The goal of the Foundation is to support and develop projects for training highly qualified personnel in the field of financial and actuarial mathematics in parallel with its economic and numerical components.