A refined version of the integro-local Stone theorem

TL;DR

This paper refines the integro-local Stone theorem by providing an asymptotic expansion for the probability that the sum of non-lattice i.i.d. random variables falls within a small interval, under certain moment and non-lattice conditions.

Contribution

It derives the first term in the asymptotic expansion for the probability of the sum's interval and establishes uniform bounds for the remainder, refining previous results.

Findings

First asymptotic term for probability in interval

Uniform bounds for the remainder term

Conditions include Cramér's non-lattice and finite moments

Abstract

Let $X, X_1, X_2,\ldots $ be a sequence of non-lattice i.i.d. random variables with ${\bf E} X=0,$ ${\bf E} X=1,$ and let $S_n:= X_1+ \cdots+ X_n$, $n\ge 1.$ We refine Stone's integro-local theorem by deriving the first term in the asymptotic expansion for the probability ${\bf P} \bigl(S_n\in [x,x+\Delta)\bigr)$ with $x\in\mathbb R,$ $\Delta >0,$ as $n\to\infty$ and establishing uniform bounds for the remainder term, under the assumption that the distribution of $X$ satisfies Cram\'er's strong non-lattice condition and ${\bf E} |X|^r<\infty$ for some $r\ge 3$.