On the existence of exchangeable or stationary sequences of \(s\) by \(s\) independent random variables. (Sur l’existence des suites de variables aléatoires \(s\) à \(s\) indépendantes échangeables ou stationnaires.) (French) Zbl 0819.60035

Bretagnolle, Jean; Klopotowski, Andrzej

Ann. Inst. Henri Poincaré, Probab. Stat. 31, No. 2, 325-350 (1995).

D. J. Aldous [in: École d’été de probabilités de Saint- Flour XIII–1983, Lect. Notes Math. 1117, 1-198 (1985; Zbl 0562.60042)] noted that if an infinite sequence of exchangeable random variables is pairwise independent, then the component random variables must be mutually independent. However it is possible to have finite sequences of exchangeable random variables that are only pairwise independent. This paper extends this idea and considers finite exchangeable sequences \(\{X_ 1, X_ 2, \dots, X_ n\}\) which satisfy the condition that for each \(i_ 1 < i_ 2 < \cdots < i_ s\) the variables \(X_{i_ k}\), \(1 \leq k \leq s\), are independent. Examples are given of such sequences where \(n\) cannot exceed some value that can be explicitly computed. Further some new examples of infinite pairwise independent, stationary sequences are given.

Reviewer: N.Weber (Sydney)

MSC:

60G09	Exchangeability for stochastic processes
60G10	Stationary stochastic processes
60A99	Foundations of probability theory

Keywords:

pairwise independent; exchangeable random variables; mutually independent; stationary sequences