A de Finetti theorem for a class of pairwise independent stationary processes. (English) Zbl 0636.60033

Robertson, James B.; Simons, Stephen

Ann. Probab. 16, No. 1, 344-354 (1988).

Let \(X_ n\), \(n=1,2,...\), be a stochastic process such that \(P\{X_ n=0\}=P\{X_ n=1\}=1/2\). A one-parameter family of distributions for pairwise independent, ergodic stationary stochastic processes of this kind was constructed as well as sufficient statistics for this parameter were calculated by the first author [Math. Sci. 10, 109-117 (1985; Zbl 0583.60031)].
The authors consider generalizations of these sufficient statistics and, among other results, describe all the processes with these sufficient statistics. Pairwise independent and ergodic processes are identified. Extreme point methods for convex sets are applied, especially the authors make use of a representation theorem by P. Ressel [Ann. Probab. 13, 898-922 (1985; Zbl 0579.60012)].

Reviewer: H.Niemi

MSC:

60G10	Stationary stochastic processes
28D05	Measure-preserving transformations

Keywords:

de Finetti theorem; ergodic stationary stochastic processes; Extreme point methods for convex sets

Citations:

Zbl 0583.60031; Zbl 0579.60012