A refined version of the integro-local Stone theorem

Alexander A. Borovkov and Konstantin A. Borovkov

Statistics & Probability Letters, 2017, vol. 123, issue C, 153-159

Abstract: 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.

Keywords: Integro-local Stone theorem; Asymptotic expansion; Random walk; Central limit theorem; Independent identically distributed random variables (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1016/j.spl.2016.12.004

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