EXPLORING STOCHASTIC LAWS : festschrift in honour of the 70th birthday of academician vladimir ... semenovich korolyuk

## Preface This collection of articles gives an opportunity for the friends and collegues, the pupils and the pupils of the pupils to congratulate Vladimir Korolyuk on his seventieth birthday. Present day troubles and conflicts have broken off many contacts between countries and people from around the world. I believe there are people who would have liked to have contributed to this book, but who could not because of the lack of information and communication. And there are our old friends who we shall never forget, who are sadly not with us today. I remember clearly our first meeting forty five years ago at a University track and field athletics competition. He carried out his performance quickly, efficiently and excellently and left us as soon as possible to work on some scientific problems. The qualities Korolyuk possesses, to formulate problems exactly, to find the appropriate methods, and to approach quietly but confidently to the solution were always inherent in his mathematical studies. He discovered his famous results in such a difficult branch of Probability, as asymptotic expansions for distributions in boundary problems of statistics, primarily because of his purposefulness and persistence. This was the start of a glorious journey to the summit of scientific excellence. In all of his studies Korolyuk always saw clear goals, the best ways to reach them, and had the ability to find or to create the tools for the achievement of these goals. Sometimes it seemed that all the possibilities had been tried and the goal was totally inaccessible, but Korolyuk persistently tried new methods until the problem was conquered. For example, in the problem of merging in the theory of semi-Markov processes, up until perturbations on the spectrum were used. Korolyuk was never the worshipper of one idea only. He took great interest in Cybernetics and applied Mathematics. Even in Probability he often changed the object of his attention and began to work in a new field with immutable success, introducing new methods. His pupils were greatly influenced by him, and would often follow him. Many of them are now in various cities and countries around the world, with their own XVI Preface students. His ability to work in new fields is preserved in him even now, and his new article in this collection supports this. I believe our long friendship allows me to say, "Dear Volodya, your friends and pupils are happy that we were beside you all that time. We are waiting for your further successes and victories on the path that you have been working on for many fruitful years. We wish you a happy seventieth birthday. Many happy returns of the day!" CONTENTS Preface Vladimir Semenovich Korolyuk: A Short Biography Memories of my youth "Forward, and only forward!" PAPERS Convergence of branching processes with several types of particles to Jirina processes Diffusion approximation in switching stochastic models and its applications Approximation of generalized U-statistics Largest-fit selection of random sizes under a sum constraint: comparisons by weak convergence Limit theorems for some characteristics of system G I / G I / l Continuity of vector-valued Gaussian Markov processes Both an n-dimensional Wiener process and its continuous one-dimensional image generate the same filtration Formal stochastic calculus and admissible transformations Stability of periodic solutions of operator equations with perturbation coefficients The number of crossings through a hyperplane by some generalized diffusion process Bounds for absolute moments of order statistics Fundamental identities for boundary functionals of additive sequences A note on stochastic variational equations Summability method in some ergodic type theorems On the Laplace transform of finite-dimensional distribution functions of semi-continuous random processes with reflecting and delaying screens Large deviation probabilities for U-statistics based on samples from finite populations Spatial quantiles and L-estimators Asymptotic inference for nearly nonstationary complex-valued AR(1) processes Average and stability of dynamical systems with rapid stochastic switchings Uniform exponential bounds for the pointwise availability of a repairable system Simple proof of Hida distribution characterization theorem Asymptotic Normality of an Estimator of an Infinite-dimensional Parameter in a Model with a C 1 smooth Regression Function On the efficiency of the least squares estimator of regression coefficients of a random field observed on a sphere Large deviation theorems in hypotheses testing problems Quasi-diffusion measure-valued processes and a limit theorem for jump measure-valued Markov processes Asymptotic behavior of solutions of stochastic equations with periodic coefficients Trajectories of homogeneous and isotropic random flows have no fractal properties The weak convergence of point fields on a plane to a point field with conditionally independent increments Interpolation of vector-valued stochastic processes Commutation Relations for Mixtures of Gaussian Measures A characterization of an exponential distribution based on renewals Planar random evolution with three directions Applications of null-additive set functions in mathematics Gaussian processes via independence of linear forms On the irregular-derivative sampling with uniformly dense sample points for bandlimited stochastic processes Fractal, superfractal and anomalously fractal distributions of random variables with a fixed infinite set of independent n-adic digits Semi-Markov random evolutions: some ideas, methods and results On the limit distribution of random permanents A simple approach to classical extreme value theory Averaging and stability of cocycles under dynamical systems with rapid Markov switching The bootstrap estimator for the asymptotic variance of (/-quantiles

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