## The Annals of Statistics

- Ann. Statist.
- Volume 10, Number 2 (1982), 630-633.

### A Characterization Problem in Stationary Time Series

#### Abstract

If a strictly stationary process $\{Z_k\}$ has residuals $Z_{k+1} - \sum^k_{j=1} a_{k,j}Z_j$ independent of $(Z_1, \cdots, Z_k)$ for all $k \geq m$, it is shown that the process is Gaussian or degenerate or $m$-step Markovian. Generalized (nonlinear) autoregressive stationary processes are defined and partially characterized.

#### Article information

**Source**

Ann. Statist., Volume 10, Number 2 (1982), 630-633.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176345805

**Digital Object Identifier**

doi:10.1214/aos/1176345805

**Mathematical Reviews number (MathSciNet)**

MR653539

**Zentralblatt MATH identifier**

0488.62066

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62E10: Characterization and structure theory

Secondary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 60G10: Stationary processes 60E10: Characteristic functions; other transforms

**Keywords**

Stationary time series generalized autoregressive process characterization problem nonlinear prediction

#### Citation

Slud, Eric V. A Characterization Problem in Stationary Time Series. Ann. Statist. 10 (1982), no. 2, 630--633. doi:10.1214/aos/1176345805. https://projecteuclid.org/euclid.aos/1176345805