## The Annals of Probability

- Ann. Probab.
- Volume 16, Number 1 (1988), 344-354.

### A De Finetti Theorem for a Class of Pairwise Independent Stationary Processes

James B. Robertson and Stephen Simons

#### Abstract

Consider a $\{0, 1\}$-valued strictly stationary stochastic process $\{X_1,X_2,\ldots\}$. Let $k$ and $l$ be natural numbers and define $y_i = 0$ or 1 according as $x_1 + \cdots + x_{i+k-1}$ is even or odd. Then, for $1 \leq j \leq l$ set $S_j(x_1 \cdots x_n) = \sum_{0\leq i \leq m - 1}y_{j+il}$. We consider all processes that have $(S_1,\ldots,S_l)$ as sufficient statistics. We obtain explicit formulas for the distributions of the processes that are extreme points. We also represent these processes as finitary processes and use this representation to investigate their pairwise independence, ergodicity and mixing properties.

#### Article information

**Source**

Ann. Probab., Volume 16, Number 1 (1988), 344-354.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176991906

**Digital Object Identifier**

doi:10.1214/aop/1176991906

**Mathematical Reviews number (MathSciNet)**

MR920276

**Zentralblatt MATH identifier**

0636.60033

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G10: Stationary processes

Secondary: 28D05: Measure-preserving transformations

**Keywords**

de Finetti theorem stationary stochastic process finitary process pairwise independence ergodic weakly mixing extreme point of compact convex set

#### Citation

Robertson, James B.; Simons, Stephen. A De Finetti Theorem for a Class of Pairwise Independent Stationary Processes. Ann. Probab. 16 (1988), no. 1, 344--354. doi:10.1214/aop/1176991906. https://projecteuclid.org/euclid.aop/1176991906