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Stochastic processes with regeneration

Work number - M 38 AWARDED


PhD, associate professor at Operations Research department, Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv

Alexander Marynych


Proposed by Faculty of Computer Science and Cybernetics of Taras Shevchenko National University of Kyiv

The series of paper consists of one monograph and 22 articles published during the last 7 years.

The series of papers is devoted to the analysis of random regenerative structures and random processes with regeneration. A regenerative random structure is a random structure or a family of random structures with an appropriately defined notion of size, such that distributional properties of structures of different sizes are consistent and invariant under a fixed operation that deletes a part of the structure. A random process with regeneration is a stochastic process defined on such a structure and indexed by a discrete or continuous variable representing its size.

          The notion of random process with immigration at the epochs of a renewal process is proposed and a classification of the modes of weak convergence of such processes is constructed. We obtain conditions for the weak convergence to a stationary process with immigration; prove limit theorems for renewal shot noise processes with eventually decreasing response functions in the cases of regular and slow variation of the normalization; derive limit theorems for random processes with immigration in case of the regularly varying normalization. As a by-product, limit theorems for several functionals on perturbed random walks are proved. For regenerative random compositions, we derive a number of limit theorems for various functionals. In particular, a functional limit theorem for the number of blocks in regenerative compositions derived from a compound Poisson processes (the Bernoulli sieve) is proved. We introduce a coupling of regenerative random compositions and coalescents with multiple collisions and apply it to prove several asymptotic results for coalescents with dust component. We propose and analyse a new stochastic operation of a random sieving. A connection of this operation with classical Galton-Watson processes is established. The notion of stability of point processes with respect to sieving is proposed, and a characterization of point processes which are stable with respect to sieving by random walks is derived.

The results presented in the series of papers are published in 1 monograph, 22 articles (including 18 articles in international journals). Author’s papers are cited in 64 papers according to Scopus (239 according to Google Scholar), h-index=5 according to Scopus (10 according to Google Scholar).