Asymptotic expansion for the density function of the series scheme of a random variables in the large deviation in Cramer zone

Abstract

In this work, the expansion of the density function of series schemes of independent variables ξ⁽ⁿ⁾₁,ξ⁽ⁿ⁾₂ ,..., ξ⁽ⁿ⁾j,   with means Eξ⁽ⁿ⁾j = 0, and dispersions σ⁽ⁿ⁾²_j = Eξ⁽ⁿ⁾²j   has been obtained in the Cramer zone of large deviations. The result was obtained, based on General Lemma 6.1 [2] by joining the methods of characteristic functions and cumulants. The work broadens theory of sums of random variables [1] and in special case improves S.A. Book [5] results of sums of random variables with weights.