## The Annals of Statistics

- Ann. Statist.
- Volume 22, Number 4 (1994), 2062-2088.

### Asymptotic Distribution of Statistics in Time Series

#### Abstract

Verifiable conditions are given for the validity of formal Edgeworth expansions for the distribution of sums $X_1 + \cdots + X_n$, where $X_i = F(Z_i, \ldots, Z_{i + p - 1})$ and $Z_1, Z_2, \ldots$ is a strict sense stationary sequence that can be written as $Z_j = g(\varepsilon_{j - k}: k \geq 0)$ with an $\operatorname{iid}$ sequence $(\varepsilon_i)$ of innovations. These models include nonlinear functions of ARMA processes $(Z_i)$ as well as certain nonlinear AR processes. The results apply to many statistics in (nonlinear) time series models.

#### Article information

**Source**

Ann. Statist., Volume 22, Number 4 (1994), 2062-2088.

**Dates**

First available in Project Euclid: 11 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176325772

**Digital Object Identifier**

doi:10.1214/aos/1176325772

**Mathematical Reviews number (MathSciNet)**

MR1329183

**Zentralblatt MATH identifier**

0827.62015

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62E20: Asymptotic distribution theory

Secondary: 60F05: Central limit and other weak theorems

**Keywords**

Edgeworth expansions statistics in time series models

#### Citation

Gotze, F.; Hipp, C. Asymptotic Distribution of Statistics in Time Series. Ann. Statist. 22 (1994), no. 4, 2062--2088. doi:10.1214/aos/1176325772. https://projecteuclid.org/euclid.aos/1176325772