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December, 1977 Edgeworth Expansions for Integrals of Smooth Functions
C. Hipp
Ann. Probab. 5(6): 1004-1011 (December, 1977). DOI: 10.1214/aop/1176995667


Let $X_1, X_2,\cdots$ be a sequence of independent, identically distributed random variables with $E(X_1) = 0, E(X_1^2) = 1$, and $E(X_1^4) < \infty$, and for $n = 1,2,\cdots$ let $P_n$ be the distribution of $n^-\frac{1}{2} \sum^n_{i=1} X_i$. If $f$ is a function with bounded uniformly continuous derivative of order 4, then $\int f dP_n$ has an asymptotic expansion in terms of $n^{-\frac{1}{2}}$ with a remainder term of $o(n^{-1})$. This remains true even if $P_1$ is purely discrete and nonlattice.


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C. Hipp. "Edgeworth Expansions for Integrals of Smooth Functions." Ann. Probab. 5 (6) 1004 - 1011, December, 1977.


Published: December, 1977
First available in Project Euclid: 19 April 2007

zbMATH: 0375.60032
MathSciNet: MR455076
Digital Object Identifier: 10.1214/aop/1176995667

Primary: 60F05
Secondary: 60G50

Keywords: Edgeworth expansions , Sums of independent random variables

Rights: Copyright © 1977 Institute of Mathematical Statistics

Vol.5 • No. 6 • December, 1977
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