Society for Industrial and Applied Mathematics: Theory of Probability & Its Applications: Table of Contents

Approximation of Smooth Multivariate Regression Functions by Universal Kernel Estimators

Yu. Yu. Linke — 2026-05-07T07:00:00Z

Theory of Probability & Its Applications, Volume 71, Issue 1, Page 1-12, May 2026.
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.

Periodic Multitype Symmetric Branching Random Walks on $\mathbf{Z}^{{d}}$

I. I. Lukashova — 2026-05-07T07:00:00Z

Theory of Probability & Its Applications, Volume 71, Issue 1, Page 13-28, May 2026.
We consider the model of symmetric branching continuous-time random walks on the lattice $\mathbf Z^d$ with $n$ types of particles and periodically located branching sources. It is assumed that initially there is only one particle of type $\mathcal T_s$ at some point. For this process, we construct a periodic operator describing the evolution of the mean number of particles of type $\mathcal T_j$ and study its spectral properties. We also obtain the asymptotics of the mean number of particles of type $\mathcal T_j$ at a fixed point of the lattice as $t \to\infty$.

A Limit Theorem for Maxima of Functions of Gaussian Processes with Logarithmic Decay of Correlation

A. V. Savich — 2026-05-07T07:00:00Z

Theory of Probability & Its Applications, Volume 71, Issue 1, Page 29-48, May 2026.
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 Fréchet distribution. This result is obtained under the assumption that the correlation function of each component of the Gaussian vector decays at least logarithmically.

Skorokhod Transition in the Conic Market Model

A. P. Sidorenko — 2026-05-07T07:00:00Z

Theory of Probability & Its Applications, Volume 71, Issue 1, Page 49-58, May 2026.
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.

On the Maximal Degree of a Vertex of a Conditional Configuration Graph

I. A. Cheplyukova — 2026-05-07T07:00:00Z

Theory of Probability & Its Applications, Volume 71, Issue 1, Page 59-72, May 2026.
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 $p_k=\mathbf{P}\{\eta=k\}\sim \frac{h(k)}{k^g}$, $20$.

Euler Scheme for Some SDEs with Fractional Noise and Markov Switching

H. Araya — 2026-05-07T07:00:00Z

Theory of Probability & Its Applications, Volume 71, Issue 1, Page 73-87, May 2026.
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 $H \in (1/2, 1)$ 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.

Uniformity of Mixed Two- and Four-Level Designs under Quaternary Codes

Xiangyu Fang — 2026-05-07T07:00:00Z

Theory of Probability & Its Applications, Volume 71, Issue 1, Page 88-104, May 2026.
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 $L_2$-discrepancy of mixed two- and four-level designs is studied, respectively. Some lower bounds of wrap-around $L_2$-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.

The von Bahr--Esseen Type Inequality under Sublinear Expectations and Applications

Yi Wu — 2026-05-07T07:00:00Z

Theory of Probability & Its Applications, Volume 71, Issue 1, Page 105-119, May 2026.
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 sublinear expectations is established. 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 sublinear expectation space.

On the 80th Birthday of V. I. Piterbarg

A.N. Shiryaev — 2026-05-07T07:00:00Z

Theory of Probability & Its Applications, Volume 71, Issue 1, Page 120-121, May 2026.
This article recognizes the 80th birthday of world-renowned scientist Vladimir Il'ich Piterbarg, whose works have become classics of stochastic analysis.

An Application of the Laplace Transform to Ruin Problems with Random Insurance Payments and Risky Asset Investments

V. A. Antipov — 2026-05-07T07:00:00Z

Theory of Probability & Its Applications, Volume 71, Issue 1, Page 122-128, May 2026.
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.

Energy-Efficient Pursuit of Multidimensional Wiener Processes

I. M. Lialinov — 2026-05-07T07:00:00Z

Theory of Probability & Its Applications, Volume 71, Issue 1, Page 129-136, May 2026.
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.

Stability of Nontransitive Trybuła Triplets under Taking Sums and Maxima

A. N. Yakusheva — 2026-05-07T07:00:00Z

Theory of Probability & Its Applications, Volume 71, Issue 1, Page 137-145, May 2026.
We consider two nontransitive Trybuła 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 $\varepsilon$ (under taking the sums, also in the opposite direction of stochastic precedence). For both transformations, we obtain polynomial equations, whose roots $\varepsilon_{\mathrm{cr}}$ define the boundaries of the stability intervals.

A Counterexample to Small-Time Limit Theorems for Stochastic Processes

P. M. Sparago — 2026-05-07T07:00:00Z

Theory of Probability & Its Applications, Volume 71, Issue 1, Page 146-153, May 2026.
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 Itô 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 $[0,\infty)$, but it does not converge weakly in the sense of laws of càdlàg processes. We also characterize the limit law of the scaled processes evaluated at their respective first exit times.

Kolmogorov Student Olympiads in Probability (2022--2025)

E. E. Bashtova — 2026-05-07T07:00:00Z

Theory of Probability & Its Applications, Volume 71, Issue 1, Page 154-162, May 2026.
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.

In Memory of A. M. Zubkov (12.30.1946--8.6.2025)

V. A. Vatutin — 2026-05-07T07:00:00Z

Theory of Probability & Its Applications, Volume 71, Issue 1, Page 163-164, May 2026.
The paper looks back on the life of leading Russian mathematician Andrey Mikhailovich Zubkov, who passed away on August 6, 2025.

On the 90th Birthday of Ya. G. Sinai

A. N. Shiryaev — 2026-02-19T08:00:00Z

Theory of Probability & Its Applications, Volume 70, Issue 4, Page 511-511, February 2026.
This paper acknowledges the 90th birthday of eminent Russian mathematician Yakov Grigor'evich Sinai.

A Life in Mathematics: On the 90th Birthday of Yakov Grigor'evich Sinai

M. L. Blank — 2026-02-19T08:00:00Z

Theory of Probability & Its Applications, Volume 70, Issue 4, Page 512-515, February 2026.
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.

Local Dynamical Entropies and Their Applications in Number Theory

M. L. Blank — 2026-02-19T08:00:00Z

Theory of Probability & Its Applications, Volume 70, Issue 4, Page 516-539, February 2026.
The concept of entropy of a dynamical system, which was introduced by Ya.,G. Sinai and A.,N. Kolmogorov, describes the degree of randomness/complexity of its trajectories. The construction of the entropy theory of dynamical systems is an important step in the study of their ergodic properties, but it is, in a sense, an averaged characterization. Complexity/chaoticity of separate trajectories varies extremely strongly---from the trivial behavior in the case of periodic trajectories to the highly involved behavior for “generic” trajectories of the same chaotic systems. We propose a new approach to the construction of local dynamical entropies, which are applicable to characterizing the complexity of individual trajectories, in turn, enabling us to close this gap. Moreover, this approach can be applied not only to trajectories but also to arbitrary sequences. In particular, we employ it to estimate the complexity of various concepts in number theory (prime numbers, quadratic residues, etc.).

Uniform Continuity for Borel Transformations of Stochastic Fields

A.I. Bufetov — 2026-02-19T08:00:00Z

Theory of Probability & Its Applications, Volume 70, Issue 4, Page 540-547, February 2026.
We give simple sufficient conditions for uniform continuity of Borel transformations of random processes and fields.

Correlations and Convergence Rates in General Ergodic Theorems

A. G. Kachurovskii — 2026-02-19T08:00:00Z

Theory of Probability & Its Applications, Volume 70, Issue 4, Page 548-562, February 2026.
In the ergodic theorem for unitary actions of Hausdorff locally compact second-countable groups, a formula is obtained for calculation of the norms of deviations of ergodic averages from their limit using known correlations. Illustrative examples are given. We also examine the inverse problem for recovery of sums of some sets of correlations from the known values of the norms of these deviations.

On Basic Context-Dependent Concepts of Information Theory and Statistics

M. Ya. Kelbert — 2026-02-19T08:00:00Z

Theory of Probability & Its Applications, Volume 70, Issue 4, Page 563-583, February 2026.
We discuss new and well-known results related to the weighted entropies and their applications in the context-dependent versions of information theory and statistical inference.

On the Recurrence of Birkhoff Sums for a Circle Rotation and a Hölder Function

A. V. Kochergin — 2026-02-19T08:00:00Z

Theory of Probability & Its Applications, Volume 70, Issue 4, Page 584-592, February 2026.
We consider a zero-mean Hölder function on the circle. If an irrational rotation of the circle is sufficiently well approximable by rational rotations, then the sequence of values of the Birkhoff sums remains finite at each point of the circle.

On Cocycles over Ergodic Transformations and Barycenters

M. E. Lipatov — 2026-02-19T08:00:00Z

Theory of Probability & Its Applications, Volume 70, Issue 4, Page 593-601, February 2026.
The barycenters of measures on boundaries of symmetric spaces are applied for classification of linear cocycles over ergodic transformations. We also consider the case of semisimple Lie groups of rank $1$ and recurrence of stationary random walks on them.

On Existence of Invariant Measures of a Continuous Contact Model in a Periodic Medium

S. A. Pirogov — 2026-02-19T08:00:00Z

Theory of Probability & Its Applications, Volume 70, Issue 4, Page 602-613, February 2026.
In their previous studies, the authors of the present paper considered contact models in homogeneous and inhomogeneous, deterministic or random media, on graphs and on the Lobachevsky plane. However, the case of contact model with spatially periodic birth rates has not been considered. Here, we study this class of random processes and formulate conditions on the birth and death rates under which there exists a family of invariant measures of a contact model in a periodic medium.

Multiple Mixing, Local Rank, and Joinings of Automorphisms of a Probability Space

V. V. Ryzhikov — 2026-02-19T08:00:00Z

Theory of Probability & Its Applications, Volume 70, Issue 4, Page 614-622, February 2026.
We discuss selected unsolved problems on multiple mixing, structures of self-joinings, and the local rank of automorphisms of a probability space. For an account of the spectrum and entropy of the dynamical system that we use, see, for example, [I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinai, Ergodic Theory, Springer, 1982].

Inducing Countable Lebesgue Spectrum

F. Abdedou — 2026-02-19T08:00:00Z

Theory of Probability & Its Applications, Volume 70, Issue 4, Page 623-631, February 2026.
We show that any ergodic dynamical system generates a system with pure Lebesgue spectrum of infinite multiplicity.

Quantization as a Universal Probabilistic Phenomenon

L. Accardi — 2026-02-19T08:00:00Z

Theory of Probability & Its Applications, Volume 70, Issue 4, Page 632-649, February 2026.
We survey some results obtained in the last 25 years to illustrate the following probabilistic quantization statement: quantum probability is not a generalization of classical probability but a deeper level of it. Generalizing the classical theory of orthogonal polynomials, one can show that any random field $X$ admitting all the moments can be represented as the sum of three operators, which are natural extensions of the creation, annihilation, and preservation operators in the usual boson Fock quantum theory. These operators generate a noncommutative $*$-algebra on which the quantum extension of the expected value relative to the probability distribution of the field $X$ induces a quantum state. So, any classical algebraic probability space generates a quantum space with its own commutation relations. The Heisenberg commutation relation characterizes the classical fields, while the new type of commutation relations (of type II) appears in non-Gaussian cases. The same machinery, but applied to Bernoulli fields, leads to Fermi--Dirac anticommutation relations (see the introduction).

Finite Time Dynamics and Finite Time Predictions for Stochastic and Deterministic Chaotic Systems

L. A. Bunimovich — 2026-02-19T08:00:00Z

Theory of Probability & Its Applications, Volume 70, Issue 4, Page 650-658, February 2026.
Probability theory deals with limit theorems, which consider limits (when time tends to infinity) of some functions (observables) on a sample space, or averages of these observables over an infinite time interval. But what is happening in a finite time or over finite time intervals? Such questions, important for virtually all applications, seem to be intractable mathematically (and generally sound unreasonable). For instance, equilibrium statistical mechanics deals with phase transitions (a number of equilibrium probability distributions/states) rather than time evolution, while nonequilibrium statistical mechanics is concerned with convergence of nonequilibrium states to equilibrium ones. Again, such processes occur on infinite time intervals. It turns out, however, that there are natural and reasonable questions about finite time dynamics of random and deterministic chaotic systems, which can be answered and, moreover, rigorously answered. This allows one to make predictions about a finite time evolution of such systems.

Irregular Sets in Dynamical Systems: A Survey

S. Burgos — 2026-02-19T08:00:00Z

Theory of Probability & Its Applications, Volume 70, Issue 4, Page 659-664, February 2026.
We survey some foundational and recent results on the size and structure of irregular sets in dynamical systems, that is, sets of points for which ergodic averages of continuous functions fail to converge. While these sets are negligible from the measure-theoretic point of view, they can be “large” when other characteristics are considered: they may carry full topological entropy, full topological pressure, or full Hausdorff dimension. We discuss some recent key developments in the study of irregular sets in the setting of symbolic dynamics and more general dynamical systems, emphasizing the main ideas behind the constructions and the mechanisms that lead to irregular behavior. We also describe a recent result on dichotomy for Lyapunov exponents in linear cocycles, where the failure of complete regularity leads to residual irregular sets. Throughout, we aim to provide an accessible overview while minimizing technical details.

Inverse and Direct Spectral Problems for a Class of Lattice Hamiltonians

V. A. Chulaevsky — 2026-02-18T08:00:00Z

Theory of Probability & Its Applications, Volume 70, Issue 4, Page 665-683, February 2026.
We consider the solvability of the inverse and direct spectral problems for a class of limit-periodic operators on a lattice $\mathbb{Z}^d$ for any $d\ge 1$, generalizing the lattice Schrödinger operator $H = \Delta + U$ in $\ell^2(\mathbb{Z}^d)$ with an external potential $U$. This problem was intensively studied in the 1980s where the solutions were shown to exist under the assumption of subexponential decay rate of the so-called small denominators. Since then, this assumption has appeared in a number of mathematical works. We show that it can be relaxed to a weaker assumption of exponential decay of small denominators for any arbitrarily large positive decay exponent. As in many prior works, we prove that the admissible limit-periodic operators have an eigenbasis formed by exponentially decaying lattice functions, thus featuring the uniform exponential Anderson localization. To this end, we improve the existing techniques and, to a certain extent, simplify them, rendering the proofs more transparent. Some classes of operators are discussed in an explicit manner.

On Intermediate Factors of a Product of Disjoint Systems

E. Glasner — 2026-02-19T08:00:00Z

Theory of Probability & Its Applications, Volume 70, Issue 4, Page 684-692, February 2026.
We consider an intermediate factor situation in two categories: probability measure preserving ergodic theory and compact topological dynamics. In the first, we prove a master-key theorem and examine a wide range of applications. In the second, we treat the case when one of the systems is distal and then provide some counterexamples.