The Annals of Probability

Sufficient Statistics and Extreme Points

E. B. Dynkin

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A convex set $M$ is called a simplex if there exists a subset $M_e$ of $M$ such that every $P \in M$ is the barycentre of one and only one probability measure $\mu$ concentrated on $M_e$. Elements of $M_e$ are called extreme points of $M$. To prove that a set of functions or measures is a simplex, usually the Choquet theorem on extreme points of convex sets in linear topological spaces is cited. We prove a simpler theorem which is more convenient for many applications. Instead of topological considerations, this theorem makes use of the concept of sufficient statistics.

Article information

Ann. Probab., Volume 6, Number 5 (1978), 705-730.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60J50: Boundary theory
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82A25 28A65

60-02 Extreme points sufficient statistics Gibbs states ergodic decomposition of an invariant measure symmetric measures entrance and exit laws excessive measures and functions


Dynkin, E. B. Sufficient Statistics and Extreme Points. Ann. Probab. 6 (1978), no. 5, 705--730. doi:10.1214/aop/1176995424.

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