## The Annals of Probability

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

#### 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

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