On Exact Asymptotics in the Weak Law of Large Numbers for Sums of Independent Random Variables with a Common Distribution Function from the Domain of Attraction of a Stable Law

摘要

Let us consider independent identically distributed random variables X_1, X_2, ..., such that U_n=S_n/B_n-n a_n→ξ_α weakly as n → ∞, where S_n = X_1 + ... + X_n, B_n>0, a_n are some numbers (n ≥ 1), and a random variable ξ_α has a stable distribution with characteristic exponent α ∈ (0, 2). Our basic purpose is to find conditions under which Σ_n f_n P{|U_n|≥εφ_n}～Σ_n f_n P{|ξ_α|≥εφ_n}, ε↓0, with a positive sequence φ_n, which tends to infinity and satisfies mild additional restrictions, and with a nonnegative sequence f_n such that Σ_n f_n = ∞.