Mathematics & Physics 2016, 9(1), 17–29 УДК 519.21 On Limit Distribution of Sums of Random Variables Sergey V.Chebotarev∗ Altai state pedagogical university Molodezhnaya, 55, Barnaul, 656015 Russia Received 24.06.2015, received in revised form 29.12.2015, accepted 12.01.2016 Centered Rademacher sequences and centered sequences of lattice random variables with a non-trivial weak limit of the sums 1 √n ∑ i=1 n ξi are considered in the article. <...> A general form of limit distribution is found for these sequences. <...> It is shown that the form of limit distribution depends only on the average mixed moments of the first order characterizing random variables of the sequence. <...> In the case of lattice random variables we mean a sequence of Rademacher random variables in which we can distribute the elements of the given sequence. <...> Keywords: sequences of random variables, sum of random variables, sum of dependent random variables, limit distribution. <...> It is assumed that random variables are defined discussed problems [1]. <...> We study the sequences of random variables ξ = (ξt)t∈I, defined on probability space on similar spaces of elementary events Ωt = Ω, t ∈ I, and ΩI = Ω1 Ч Ω2 Ч . . . <...> The main result presented in the paper is Theorem 3. 1. <...> There are no additional limits for the k=0 s Sergey V.Chebotarev On Limit Distribution of Sums of Random Variables takes place. <...> Let us consider the subsequences of the finite sequence ξ(n). <...> Two kinds of average moments of order m are used: v˙m = vm Cm n and n is a binominal coefficient. <...> When total mixed moment vm or some average mixed moments ˙vm, ¨ write vm(ξ) or ˙vm(ξ), ¨ Here Cm vm are defined for the sequence ξ we vm(ξ), respectively. <...> Sergey V.Chebotarev On Limit Distribution of Sums of Random Variables and when m = 0 we assume v0(π(n)) = v0(ξ(n)) = 1. <...> We also introduce mk(η) = ∫ ∞ −∞ = θm˙vm(ξ(n)) and ¨ √Cm = θm¨ n Let η be some absolutely continuous random variable with probability density function µη. <...> Here Hm is the orthogonal and hm is <...>