A refined version of the integro-local Stone theorem

Let X,X1,X2,… be a sequence of non-lattice i.i.d. random variables with EX=0,EX=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+Δ)), x∈R,Δ>0, and establishing uniform in x and Δ 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.