## The Annals of Probability

### Edgeworth Expansions for Integrals of Smooth Functions

C. Hipp

#### Abstract

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.

#### Article information

Source
Ann. Probab., Volume 5, Number 6 (1977), 1004-1011.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176995667

Digital Object Identifier
doi:10.1214/aop/1176995667

Mathematical Reviews number (MathSciNet)
MR455076

Zentralblatt MATH identifier
0375.60032

JSTOR