Theory of Probability &Its Applications, Volume 66, Issue 1, Page 160-168, January 2021. We give an equivalent definition of the conditional quantile reproducibility property for multivariate probability distributions using the concept of copula. Based on this definition, we establish a connection between the conditional quantile reproducibility property and the simplifying assumption from the simplified

Theory of Probability &Its Applications, Volume 66, Issue 1, Page 142-159, January 2021. In [J. Lee, ALEA Lat. Am. J. Probab. Math. Stat., 15 (2018), pp. 837--849] it is proved that we can have a continuous first-passage-time density function of one-dimensional standard Brownian motion when the boundary is Hölder continuous with exponent greater than 1/2. For the purpose of extending the results of

Theory of Probability &Its Applications, Volume 66, Issue 1, Page 121-141, January 2021. In this paper we modify the Stein method and the auxiliary technique of distributional transformations of random variables. This enables us to estimate the convergence rate of distributions of normalized geometric sums to the Laplace law. For independent summands, the developed approach provides an optimal estimate

Theory of Probability &Its Applications, Volume 66, Issue 1, Page 105-120, January 2021. We consider a communication network with a fixed set of links that are shared by a large number of users. The resource allocation is performed on the basis of the aggregate utility maximization in accordance with the popular approach proposed by F. Kelly and coauthors in 1998. The problem is to construct a pricing

Theory of Probability &Its Applications, Volume 66, Issue 1, Page 89-104, January 2021. An upper estimate for the absolute value of the sum of the Spitzer series is obtained. This estimate depends explicitly on the distribution in terms of which the Spitzer series is defined.

Theory of Probability &Its Applications, Volume 66, Issue 1, Page 75-88, January 2021. We find the exact asymptotics of the distribution of the time when the trajectory of the process $Y(t)=at-\nu_+(pt)+\nu_-(-qt)$, $t\in(-\infty,\infty)$ attains its maximum, where $\nu_{\pm}(t)$ are independent standard Poisson processes extended by zero on the negative semiaxis. The parameters $a$, $p$, $q$ are assumed

Theory of Probability &Its Applications, Volume 66, Issue 1, Page 58-74, January 2021. A sequence of compound Poisson processes constructed from sums of identically distributed random variables that weakly converges to a Wiener process is considered. Certain functionals of these processes are shown to converge in distribution to the local time of a Wiener process.

Theory of Probability &Its Applications, Volume 66, Issue 1, Page 44-57, January 2021. In this paper, we introduce several natural definitions of asymptotic independence of two sequences of random elements. We discuss their basic properties, some simple connections between them, and connections with properties of weak dependence. In particular, the case of tight sequences is considered in detail. Finally

Theory of Probability &Its Applications, Volume 66, Issue 1, Page 15-43, January 2021. A probabilistic approach to construction of the solution to the Cauchy problem for systems of nonlinear parabolic equations is developed. The systems under consideration can be subdivided into two classes: the systems of the first class can be interpreted, after a simple transformation, as systems of nonlinear backward

Theory of Probability &Its Applications, Volume 66, Issue 1, Page 1-14, January 2021. We consider a single-channel system with service vacations, a Poisson input flow, and an arbitrarily distributed service time. Interruptions in service can mean either a complete shutdown of the server for a random period of time or a transition to a different (nonstandard) regime---they can occur either at the end

Theory of Probability &Its Applications, Volume 65, Issue 4, Page 665-672, January 2021. We consider the sequential $r$-stage chi-square test. For $r=2$, we study the asymptotic properties of the error probabilities as a function of the sizes of the rectangular critical domain, which via the Bonferroni inequality makes it possible to derive asymptotic properties of the error probability for an arbitrary

Theory of Probability &Its Applications, Volume 65, Issue 4, Page 656-664, January 2021. A random walk of a particle in ${R}^d$ is considered. The weak convergence of various transformations of trajectories of random flights with Poisson switching times was studied by Davydov and Konakov in [Random walks in nonhomogeneous Poisson environment, in Modern Problems of Stochastic Analysis and Statistics

Theory of Probability &Its Applications, Volume 65, Issue 4, Page 652-655, January 2021. We prove that the set of laws of stochastic integrals $H\,{\cdot}\, W$, where $W$ is a multidimensional Wiener process and $H^2$ takes values in a compact convex subset $D$ of the set of symmetric positive semidefinite matrices, is weakly dense in the set of laws of martingales $M$ with $d\langle M \rangle/dt$

Theory of Probability &Its Applications, Volume 65, Issue 4, Page 648-651, January 2021. A modification of the Lindeberg and Rotar' conditions was considered in the papers by Presman and Formanov [Dokl. Math., 99 (2019), pp. 204--207] and [Dokl. Ross. Akad. Nauk Ser. Mat., 485 (2019), pp. 548--552 (in Russian)]. This modification was concerned with the sums of absolute (respectively, difference) moments

Theory of Probability &Its Applications, Volume 65, Issue 4, Page 640-647, January 2021. We give conditions such that solutions exist of the stochastic equation with the so-called osmotic velocities (the so-called Nelson's antisymmetric mean derivatives). We show that the solutions may not exist. Specifically, initial conditions are determined by the right-hand sides; however, some right-hand sides

Theory of Probability &Its Applications, Volume 65, Issue 4, Page 639-639, January 2021. Recognition of the 75th birthday of prominent scientist Professor Aleksander Aleksandrovich Novikov, who has made important contributions to probability, mathematical statistics, the theory of random processes, and the theory of Brownian motion. His work is well known and constantly cited.

Theory of Probability &Its Applications, Volume 65, Issue 4, Page 616-638, January 2021. Prokhorov distances under sublinear expectations are presented in the CLT and the functional CLT, and the convergence rates for them are obtained by the Lindeberg method. In particular, the obtained estimate in the functional CLT yields the known Borovkov estimate in the classical functional CLT with an explicit

Theory of Probability &Its Applications, Volume 65, Issue 4, Page 588-615, January 2021. In this paper, we consider a strongly repelling model of $n$ ordered particles $\{e^{i \theta_j}\}_{j=0}^{n-1}$ with the density $p({\theta_0},\dots, \theta_{n-1})=\frac{1}{Z_n} \exp \big\{-\frac{\beta}{2}\sum_{j \ne k} \sin^{-2} \big( \frac{\theta_j-\theta_k}{2}\big)\big\}$, $\beta>0$. Let $\theta_j=2 \pi j/n+x_j/n^2+{const}$

Theory of Probability &Its Applications, Volume 65, Issue 4, Page 570-587, January 2021. Let $X_t=\sum_{j=-\infty}^{\infty}A_j\varepsilon_{t-j}$ be a dependent linear process, where the $\{\varepsilon_n,\, n\in {Z}\}$ is a sequence of zero mean $m$-extended negatively dependent ($m$-END, for short) random variables which is stochastically dominated by a random variable $\varepsilon$, and $\{A_n,\,

Theory of Probability &Its Applications, Volume 65, Issue 4, Page 558-569, January 2021. We propose two methods of approximation of the solution of the Cauchy problem for the higher-order Schrödinger equation. In the first method, the expectation of a functional of some random point field is used, and in the second, the expectation of a functional of the normed sums of independent and identically distributed

Theory of Probability &Its Applications, Volume 65, Issue 4, Page 545-557, January 2021. We consider the family of converging max-continuous local submartingales starting from zero. An equivalence relation for processes of this family is introduced so that two processes are called equivalent if their joint laws of the terminal value and the maximum are the same. We single out a subfamily of processes

Theory of Probability &Its Applications, Volume 65, Issue 4, Page 527-544, January 2021. We consider a subcritical branching process in an independent and identically distributed (i.i.d.) random environment, where one immigrant arrives at each generation. We consider the event $\mathcal{A}_{i}(n)$ in which all individuals alive at time $n$ are descendants of the immigrant, who joined the population

Theory of Probability &Its Applications, Volume 65, Issue 4, Page 511-526, January 2021. The invariance principle for compound renewal processes is extended (in the sense of asymptotic equivalence) to the zone of moderately large and small deviations. It is assumed that the vector $(\tau,\zeta)$, which “governs” the process, satisfies certain moment conditions (for example, the Cramér condition), and

Theory of Probability &Its Applications, Volume 65, Issue 3, Page 497-509, January 2020. We have analyzed some conditions which are essentially involved in deciding whether or not a probability distribution is unique (moment-determinate) or nonunique (moment-indeterminate) by its moments. We suggest new conditions concerning both absolutely continuous and discrete distributions. By using the new conditions

Theory of Probability &Its Applications, Volume 65, Issue 3, Page 482-496, January 2020. For $q\in (0,1)$, $p_1,p_2,p\in \mathbb{R}_+$, we characterize all the solutions of the $q$-functional equations $(1+p_2q^{1/2}x)f(qx)=q^{\beta-1/2}(x+p_1q^{-1/2})f(x)$ and $f(qx)=q^{\beta- 1}(x^2+p^2q^{-1})f(x)$, $x>0$, $\beta\in \mathbb{R}$, and we also show that these solutions solve corresponding indetermined

Theory of Probability &Its Applications, Volume 65, Issue 3, Page 470-481, January 2020. In this article, we show that the centered relay correlation function is a sub-Gaussian random variable. This is done by a careful analysis of its Laplace transform and by estimating the sub-Gaussian standard of the $r$-correlograms.

Theory of Probability &Its Applications, Volume 65, Issue 3, Page 454-469, January 2020. We consider an explosive Cox--Ingersoll--Ross process. We establish a sharp large deviation principle for the maximum likelihood estimator of its drift parameter.

Theory of Probability &Its Applications, Volume 65, Issue 3, Page 418-453, January 2020. It has been understood that the “local” existence of the Markowitz optimal portfolio or the solution to the local-risk minimization problem is guaranteed by some specific mathematical structures on the underlying assets' price processes known in the literature as “structure conditions.” In this paper, we consider

Theory of Probability &Its Applications, Volume 65, Issue 3, Page 405-417, January 2020. In Nevzorov's $F^\alpha$-scheme, one deals with a sequence of independent random variables with distribution functions that are powers of a common continuous distribution function. A key property of the $F^\alpha$-scheme is that the record indicators for such a sequence are independent. This allows one to obtain

Theory of Probability &Its Applications, Volume 65, Issue 3, Page 388-404, January 2020. We present a surprisingly simple version of the fundamental theorem of asset pricing (FTAP) for continuous time large financial markets with two filtrations in an $L^p$-setting for ${1 \leq p < \infty}$. This extends the results of Kabanov and Stricker in [“The Dalang--Morton--Willinger theorem under delayed and

Theory of Probability &Its Applications, Volume 65, Issue 3, Page 375-387, January 2020. We introduce the concept of $\varepsilon$-complexity of an individual continuous finite-dimensional map. This concept is in good accord with the principle of A.N. Kolmogorov's idea of measuring complexity of objects. It is shown that the $\varepsilon$-complexity of an “almost all” Hölder map can be effectively

Theory of Probability &Its Applications, Volume 65, Issue 3, Page 359-374, January 2020. Let $(p_i,q_i)$, $i\in {Z}$, be a sequence of independent identically distributed random vectors such that $p_i,q_i>0$ and $p_i+q_i$ $=1$ a.s. for $i\in {Z}$. We consider a random walk in the random environment $\{(p_i,q_i)$, $i\in {Z}\}$. It is assumed that ${E}\ln (p_0/q_0)=0$ and $0<{E}\ln^{2}(q_0/p_0)<+\infty$

Theory of Probability &Its Applications, Volume 65, Issue 3, Page 341-358, January 2020. This paper is concerned with the optimal stopping problem for Itô diffusion processes over a class of stopping times. Necessary and sufficient optimality conditions are studied for a parametrically specified class of stopping times. A detailed analysis is given for the case of one-dimensional diffusion processes

Theory of Probability &Its Applications, Volume 65, Issue 2, Page 338-339, January 2020. This note takes a retrospective look at the life and accomplishments of mathematician Sagdy Khasanovich Sirazhdinov on the centenary of his birth.

Theory of Probability &Its Applications, Volume 65, Issue 2, Page 330-337, January 2020. We treat a fairly broad class of financial models that includes markets with proportional transaction costs. We consider an investor with cumulative prospect theory preferences and a nonnegativity constraint on portfolio wealth. The existence of an optimal strategy is shown in this context for a class of generalized

Theory of Probability &Its Applications, Volume 65, Issue 2, Page 322-329, January 2020. We provide an equivalent characterization of the absence of arbitrage opportunity for the bid and ask financial market model. This result, which is an analogue of the Dalang--Morton--Willinger theorem formulated for discrete-time financial market models without friction, supplements and improves the Grigoriev theorem

Theory of Probability &Its Applications, Volume 65, Issue 2, Page 292-321, January 2020. We consider the problem of eliciting truthful responses to a survey question when the respondents share a common prior that the survey planner is agnostic about. The planner would therefore like to have a ``universal” mechanism, which would induce honest answers for all possible priors. If the planner also requires

Theory of Probability &Its Applications, Volume 65, Issue 2, Page 270-291, January 2020. The classical Fatou lemma states that the lower limit of a sequence of integrals of functions is greater than or equal to the integral of the lower limit. It is known that Fatou's lemma for a sequence of weakly converging measures states a weaker inequality because the integral of the lower limit is replaced with

Theory of Probability &Its Applications, Volume 65, Issue 2, Page 249-269, January 2020. In this paper, we study the ruin problem with investment in a general framework where the business part $X$ is a Lévy process and the return on investment $R$ is a semimartingale. Under some conditions, we obtain upper and lower bounds on the finite and infinite time ruin probabilities as well as the logarithmic

Theory of Probability &Its Applications, Volume 65, Issue 2, Page 224-248, January 2020. We study a problem of option replication under constant proportional transaction costs in models where stochastic volatility and jumps are combined to capture the market's important features. Assuming some mild condition on the jump size distribution, we show that transaction costs can be approximately compensated

Theory of Probability &Its Applications, Volume 65, Issue 2, Page 191-223, January 2020. For a large financial market (which is a sequence of usual, “small” financial markets), we introduce and study a concept of no asymptotic arbitrage (of the first kind), which is invariant under discounting. We give two dual characterizations of this property in terms of (1) martingale-like properties for each small

Theory of Probability &Its Applications, Volume 65, Issue 2, Page 179-190, January 2020. It is shown that a recently introduced lower cone distribution function, together with the set-valued multivariate quantile, generates a Galois connection between a complete lattice of closed convex sets and the interval [0,1]. This generalizes the corresponding univariate result. It is also shown that an extension

Theory of Probability &Its Applications, Volume 65, Issue 1, Page 175-178, January 2020. Obituary of outstanding mathematician Vladimir Mikhailovich Zolotarev, who passed away November 7, 2019. He is remembered as a supreme master of both modern and classical techniques of analysis who enriched the store of probability theory through his contributions of novel analytical methods and tools.

Theory of Probability &Its Applications, Volume 65, Issue 1, Page 173-174, January 2020. An obituary of Dmitrii Mikhailovich Chibisov, the former deputy editor-in-chief of Theory of Probability and Its Applications, who passed away on November 20, 2019.

Theory of Probability &Its Applications, Volume 65, Issue 1, Page 121-172, January 2020. This paper presents abstracts of the talks presented at The Fourth International Conference on Stochastic Methods (ICSM-4), which was held June 2--9, 2019 at Divnomorskoe (near the town of Gelendzhik) at the Raduga sports and fitness center of the Don State Technical University. Participants included many leading

Theory of Probability &Its Applications, Volume 65, Issue 1, Page 114-120, January 2020. We consider Boolean functions $f$ defined on Boolean cube $\{-1,1\}^n$ of half-spaces, i.e., functions of the form $f(x)=\operatorname{sign}(\omega\cdot x-\theta)$. Half-space functions are often called linear threshold functions. We assume that the Boolean cube $\{-1,1\}^n$ is equipped with a uniform measure.

Theory of Probability &Its Applications, Volume 65, Issue 1, Page 111-113, January 2020. We study the form of excursions of a Gaussian stationary process intersecting a high level $u$. We show that the trajectories fluctuate in this case in a narrow tube around the expected motion. We also find an upper bound for the probability that the trajectory intersects the boundary of this tube as $u$ goes off

Theory of Probability &Its Applications, Volume 65, Issue 1, Page 102-110, January 2020. We consider a stationary linear $\operatorname{AR}(p)$-model with observations subject to gross errors (outliers). The autoregression parameters and the distribution of innovations are unknown. Based on the residuals from the parameter estimators, we construct an analogue of an empirical distribution function and

Theory of Probability &Its Applications, Volume 65, Issue 1, Page 82-101, January 2020. We derive sufficient conditions for the differentiability of all orders for the flow of stochastic differential equations with jumps and prove related $L^p$-integrability results for all orders. Our results extend similar results obtained by H. Kunita [Stochastic differential equations based on Lévy processes and

Theory of Probability &Its Applications, Volume 65, Issue 1, Page 62-81, January 2020. We explore the asymptotic behavior of densities of sums of independent random variables that are convoluted with a small continuous noise.

Theory of Probability &Its Applications, Volume 65, Issue 1, Page 49-61, January 2020. We consider the problem of constructing guarantee procedures of statistical inference with fixed minimal observation number $n^*$ for discrimination of two hypotheses $H_0\colon\theta\in[\theta_1,\theta_2]$ and $H_1\colon\theta\notin[\theta_1,\theta_2]$ with a one-dimensional parameter $\theta$ under the so-called

Theory of Probability &Its Applications, Volume 65, Issue 1, Page 32-48, January 2020. Sample statistics of samples from a fractional Brownian motion with Hurst exponent $H$, and in particular, autocovariance statistics, are considered. Two statistics characterizing the covariate dependence between the increments of this process are studied; in particular, their asymptotic properties and the limit

Theory of Probability &Its Applications, Volume 65, Issue 1, Page 17-31, January 2020. We examine the asymptotic behavior of integral quadratic functionals defined on time-varying Ornstein--Uhlenbeck processes. We find an upper function that majorizes with probability $1$ the deviation of the integral from its expected value as time increases. The results obtained are applied to evaluate the control

Theory of Probability &Its Applications, Volume 65, Issue 1, Page 1-16, January 2020. We consider statistics of the form $T =\sum_{j=1}^n \xi_{j} f_{j}+ \mathcal R $, where $\xi_j, f_j$, $j=1, \dots, n$, and $\mathcal R$ are $\mathfrak M$-measurable random variables for some $\sigma$-algebra $ \mathfrak M$. Assume that there exist $\sigma$-algebras $\mathfrak M^{(1)}, \dots, \mathfrak M^{(n)}$, $ \mathfrak

Theory of Probability &Its Applications, Volume 64, Issue 4, Page 656-656, January 2020. A remembrance of the life and accomplishments of Vadim Rolandovich Fatalov, who passed away on June 4, 2019.

Theory of Probability &Its Applications, Volume 64, Issue 4, Page 646-655, January 2020. In this paper a two-dimensional Brownian motion (modeling the endowment of two companies), absorbed at the boundary of the positive quadrant, with controlled drift, is considered. We allow that both drifts add up to the maximal value of one. Our target is to choose the strategy in a way s.t. the expected value

Theory of Probability &Its Applications, Volume 64, Issue 4, Page 636-645, January 2020. When collecting samples, sometimes failures of observations occur and consequently missing data. This can have an impact on the analysis and subsequent inference, especially if the study focuses on the extreme values where the data is more scarce. In this work, we analyze the effect of different types of failures

Theory of Probability &Its Applications, Volume 64, Issue 4, Page 631-635, January 2020. It is known that the values of functions that depend on a large number of similar variables are almost constant from the point of view of an observer evaluating their values at random points of the domains of their definition (the “nonlinear law of large numbers''). We show that, under certain normalization conditions

Theory of Probability &Its Applications, Volume 64, Issue 4, Page 615-630, January 2020. This paper describes Fatou's lemma for a sequence of measures converging weakly to a finite measure and for a sequence of functions whose negative parts are uniformly integrable with respect to these measures. The paper also provides new formulations of uniform Fatou's lemma, uniform Lebesgue's convergence theorem

Theory of Probability &Its Applications, Volume 64, Issue 4, Page 595-614, January 2020. Slicing a Voronoi tessellation in ${R}^n$ with a $k$-plane gives a $k$-dimensional weighted Voronoi tessellation, also known as a power diagram or Laguerre tessellation. Mapping every simplex of the dual weighted Delaunay mosaic to the radius of the smallest empty circumscribed sphere whose center lies in the $k$-plane