Marynych O. Limit theorems for random processes with regeneration.

The thesis 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. We study asymptotic properties of particular classes of random processes with regeneration, including random processes with immigration and renewal shot noise processes; regenerative random compositions and permutations; coalescents with multiple collisions; random sieves and leader election procedures. 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 byproduct, limit theorems for several functionals on perturbed random walks are proved. For regenerative random compositions we derive a number of limit theorems for different 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. The notion of regenerative random permutation is proposed and limit theorems for the order of such permutations are 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, including limit theorems for the number of collisions and the absorption time. For exchangeable coalescents without dust component analogous results are proved using the technique of probability distances. The latter method is also applied to derive a central limit theorem for the number of zero increments in a random walk with a barrier. We propose and analyze a new stochastic operation of a random sieving. A connection of this operation with classical Galton-Watson processes and exchangeable coalescents 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. A generalized leader-election procedures are discussed and a number of limit theorems for different characteristics of these procedures are proved. A limit theorem for the number of collisions in the Poisson-Dirichlet coalescent is established.