A refined version of the integro-local Stone theorem

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 ${\bf P} \bigl(S_n\in [x,x+\Delta)\bigr)$with$x\in\mathbb R,$ $\mathrm{\Delta }>0,$ as $n\to \mathrm{\infty }$ 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 $\mathbf{E}\mathrm{\mid }X{\mathrm{\mid }}^r<\mathrm{\infty }$ for some $r\ge 3$.