A refined version of the integro-local Stone theorem

Abstract

Let $X,X_1,X_2,\dots$ be a sequence of non-lattice i.i.d. random variables with $\mathbf{E}X=0,$ $\mathbf{E}X=1,$ 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 $\mathbf{P}(S_n\in [x,x+\Delta ))$ 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 Cramér's strong non-lattice condition and $\mathbf{E}|X{|}^r<\infty$ for some $r\ge 3$.

arXiv comment11 pages. Fixed two obvious typos in the statement of Theorem 1 (the upper bounds there are for the absolute value |R_n| of the reminder term, not just for R_n). We also added two remarks in Section 1 commenting on the main result of the paper