A de Finetti theorem for a class of pairwise independent stationary processes (Q1098157)

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 \textit{P. Ressel} [Ann. Probab. 13, 898-922 (1985; Zbl 0579.60012)].