A refined version of the integro-local Stone theorem

作者：Borovkov Alexander A; Borovkov Konstantin A^*

来源：Statistics & Probability Letters, 2017, 123: 153-159.

DOI：10.1016/j.spl.2016.12.004

• Integro-local Stone theorem
• Asymptotic expansion
• Random walk
• Central limit theorem
• Independent identically distributed random variables

Let X, X-1, X-2,... be a sequence of non-lattice i.i.d. random variables with EX = 0, EX = 1, and let S-n, := X-1 +...+X-n, n >= 1. We refine Stone's integro-local theorem by deriving the first term in the asymptotic expansion, as n -> infinity, for the probability P(S-n is an element of [chi, chi + Delta)), chi is an element of R, Delta > 0, and establishing uniform in chi and Delta bounds for the remainder term, under the assumption that the distribution of X satisfies Cramer's strong non-lattice condition and E vertical bar X vertical bar(r) < infinity for some r >= 3.

• 出版日期2017-4