Adityviųjų funkcijų, apibrėžtų atsitiktinių ansamblių aibėje, momentai

The doctoral dissertation deals with additive functions defined on combinatorial structures. The problem is to estimate the moments of such functions, if a structure is taken at random. We establish analogues of the well-known Turan-Kubilius inequality and other power moment estimates. The main mathematical results presented in the dissertation are as follows: 1. Variance of additive functions with respect to a generalized Ewens probability. Chapter 1 deals with additive functions defined on the symmetric group, where a permutation is taken according to a generalized Ewens probability. We establish an upper bound of its variance via a sum of variances of the summands. 2. Variance of additive functions defined on random assemblies. Chapter 2 presents an analogue of the Turan-Kubilius inequality for an additive function defined on random decomposable structures, called assemblies. 3. Moments of additive functions with respect to the Ewens Sampling Formula. Chapter 3 explores the additive semigroup of vectors with non-negative integer coordinates endowed with the Ewens Probability Measure. We obtain upper estimates of the power moments of additive statistics defined on the semigroup. In order to solve these mathematical problems, we used combinatorial, probabilistic and analytical methods. The technical approaches applied in probabilistic number theory are adopted and further enriched