# Vol. 52 No. SI (2023): Recent Trends in Probability and Statistics in Ukraine

The special issue is devoted to Recent Trends in Probability and Statistics in Ukraine, with an emphasis on the presentation of research by young scientists. This collection of papers represents some latest results of researchers from Taras Shevchenko National University of Kyiv, National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute” and Kyiv School of Economics.

The topics covered is a mixture of classical and modern ones. Some classical and new stochastic models and problems of statistical inference have received in the papers a modern treatment, that warrants addressing recent challenges from both theoretical and practical viewpoints. We believe the issue will be interesting for a wide audience.

O. Braganets and A. Iksanov investigate a nested occupancy scheme in a random environment which can be thought of as a generalization of the classical Karlin occupancy scheme. The model is given by a nested hierarchy of boxes with hitting probabilities of boxes defined in terms of iterated fragmentation of a unit mass. Previous research resulted in a multivariate functional central limit theorem with centering for the cumulative occupancy counts as the number of balls becomes large. In the present paper a counterpart of that result is obtained, in which centering is not needed and the limit processes are no longer Gaussian.

C. Dong, O. Marynych and V. Melnykov work out a universal approach to the analysis of random sieves and generalized leader-election procedures. A random sieve of the set of positive integers by a random set R is a nested sequence of subsets such that every set in the sequence is obtained by removing elements of the previous set with indices lying outside an independent copy of R. This model has been previously studied in two particular cases only: (a) R is a range of an increasing random walk on positive integers; (b) R is the set of record times in an infinite sample from a continuous distribution. Using a martingale approach the authors prove various limit theorems for several functionals that characterise the speed of sieving.

I. Samoilenko, G. Verovkina and T. Samoilenko are concerned with a model of particle evolution on a complex plane, which is a generalization of the classical Goldstein-Kac model. The authors obtain a telegraph-type equation for some functionals of the evolution and construct solutions to the corresponding Cauchy problem with complex-analytic initial conditions. The method is based on reconstruction of complex-analytic functions by combination of power functions, for which the corresponding solutions are the moments of the evolution process. This approach enables avoiding analytic difficulties of the classical Riemann method for the telegraph equation. The solutions constructed in the present paper explicitly contain regular and boundary-layer components that may be useful for calculating approximate solutions.

G. Shevchenko and A. Yaroshevskiy investigate continuous-time lattice random walks in a stationary random environment and prove a limit theorem for these walks, which is similar to that in the nonlattice case but stated under less restrictive assumptions on the distribution of jumps and under very general conditions on a random environment.

The paper by V. Golomoziy and O. Moskanova is related to stability theory which belongs to a classical part of the theory of Markov chains. The authors are interested in recurrence properties of a time-inhomogeneous Markov chain and demonstrate that such a chain can be polynomially recurrent while exhibiting different dynamics in comparison to its homogeneous counterpart.

A. Dzhoha and I. Rozora investigate the problem of design of clinical trials by using the multi-armed bandit problem, which is a classical example of the exploration-exploitation trade- off suited to model sequential resource allocation under uncertainty. Since the response to a procedure in clinical trials is not immediate, the authors justify the importance of adaptation of multi-armed bandit policies to delays. The Upper Confidence Bound policy is analyzed by applying such a classical tool as sub-Gaussian concentration inequalities.

The paper by O. Hopkalo, L. Sakhno and O. Vasylyk investigates sample paths properties of random fields from the spaces of φ-sub-Gaussian random variables, which generalize Gaussian and sub-Gaussian ones. By using the entropy approach, the authors point out some bounds for the distribution tails of suprema of φ-sub-Gaussian random fields under different conditions imposed on their increments. This work is motivated and illustrated by applications to random solutions of partial differential equations.

Tykhonenko D. and Yamnenko R. are focused on several particular classes of random processes from Orlicz spaces of exponential type and derive some estimates for the distribution of supremum of a weighted sum of such processes deviated by a continuous monotone function. Weighted sums of sub-Gaussian Wiener and fractional Brownian motion processes are considered as examples.

A. Ivanov and V. Hladun analyze the statistical inference problem for a time continuous statistical model of multiple chirp signal observed against the background of strongly or weakly dependent stationary Gaussian noise. In this special trigonometric regression model frequencies vary with time in a non-linear fashion like quadratic functions. The main result of the paper states the strong consistency of the least squares estimates for the model parameters.

The paper by K. Ralchenko and M. Yakovliev is devoted to the estimation of parameters of a mixed fractional Brownian motion with a linear trend. The model is driven by both a standard Brownian motion and a fractional Brownian motion. The authors consider strongly consistent estimators of unknown model parameters, which were derived in an earlier work, and prove their joint asymptotic normality. A behavior of the estimators is also analyzed numerically.

S. Shklyar considers a classical generalized linear model and its generalizations to cover various forms of errors and incomplete data. The base model is a simple exponential regression, in which the rate parameter of the response variable linearly depends on the explanatory variable. Estimates are presented for the cases where the base model becomes more complicated by adding the censoring of the response variable and/or measurement errors in the explanatory variable. The performance of estimates is verified by simulation.

We would like to introduce the young researchers, who contributed to the issue. Taras Shevchenko National University of Kyiv is represented by: PhD students Oksana Braganets, Andrii Dzhoha and Viacheslav Melnykov from Faculty of Computer Science and Cybernetics; PhD students Dmytro Tykhonenko and Mykyta Yakovliev and Master Degree student Olga Moskanova from Faculty of Mechanics and Mathematics; Dr. Olga Hopkalo, who received her PhD degree in 2021 and is now an Assistant Professor at Faculty of Economics. Andriy Yaroshevskiy received his Master Degree from Faculty of Mechanics and Mathematics in 2021. The Master Degree student Viktor Hladun is from Igor Sikorsky Kyiv Polytechnic Institute.

We thank all the authors of this issue. Also, our special thanks go to the reviewers for their valuable remarks and suggestions.

We are very grateful to the Editor of the Austrian Journal of Statistics Professor Matthias Templ, who initiated the special issue to express solidarity with Ukraine and to support Ukrainian scientists. His help with the issue preparation is highly appreciated.

This issue is published in a very hard time for Ukraine and the Ukrainian people. Ukraine is fighting and resisting the devastating Russian invasion that began on February 24, 2022, in a war that Russia has been waging since 2014. The damage to Ukraine is impossible to describe and grasp. In all areas, including educational and research institutions and science. We greatly appreciate all the help from all over the world and all the steps taken so far to provide support to Ukraine at all levels. To respond to the severity of the consequences that have been inflicted on Ukraine and to prevent further threats, we believe that comprehensive further steps against this war, which is contrary to all the values of our civilization, are necessary and urgent.

Lyudmyla Sakhno and Alexander Iksanov

Taras Shevchenko National University of Kyiv