A refined version of the integro-local Stone theorem

Abstract

Let X, X1, X2, . . . be a sequence of non-lattice i.i.d. random variables with E X = 0, E X = 1, and let Sn := X1 + · · · +Xn, n ≥ 1. We refine Stone’s integro-local theorem by deriving the first term in the asymptotic expansion, as n → ∞, for the probability P(Sn ∈ [x, x + ∆)), x ∈ R, ∆ > 0, and establishing uniform in x and ∆ bounds for the remainder term, under the assumption that the distribution of X satisfies Cramér’s strong non-lattice condition and E |X|r < ∞ for some r ≥ 3.