The novelty and originality of the work consists in the fact that in order to obtain asymptotic expansions with optimal values of the remainder terms in the zone of large deviations, along with the cumulant method the classical method of characteristic functions has to be used. In addition, when solving the problems stated in the work, other than the well known results in the problems of limit theorems of the probability theory and mathematical statistics, we have to estimate constants. Technically it is frequently rather a complicated task. The results obtained in the work have good opportunities to be applied in probability theory, mathematical statistics, econometric, etc. That is illustrated in the last section of the work in which theorems of large deviations are proved in the summation of weighted random variables with weights as well as discounted limit theorems.

Nagrinėjama silpnai priklausomų atsitiktinių dydžių statistika. Šio magistrinio darbo tikslas, turimą sumą, suvesti į nepriklausomų atsitiktinių dydžių sumą, kuri leistų tolimesniam tyrimui, pritaikyti žinomas teoremas, skaičiavimus bei rezultatus iš nepriklausomų atsitiktinių dydžių teorijos. / We analyze of weakly dependent random variables statistics. The objective of this master thesis is to deduce sum to independent random variables sum, which will be useful for applaying known theorems, calculations and results from the theory in independent random variables.