Asymptotic formulas for the left truncated moments of sums with consistently varying distributed increments

Abstract

In this paper, we consider the sum S_n^ξ = ξ₁ + ... + ξ_n of possibly dependent and nonidentically distributed real-valued random variables ξ₁, ... , ξ_n with consistently varying distributions. By assuming that collection {ξ₁, ... , ξ_n} follows the dependence structure, similar to the asymptotic independence, we obtain the asymptotic relations for E((S_nξ)^α1(S_n^ξ > x)) and E((S_n^ξ – x)⁺)^α, where α is an arbitrary nonnegative real number. The obtained results have applications in various fields of applied probability, including risk theory and random walks.