Strong stationarity and de Finetti's theorem
The classical result of \textit{B. de Finetti} [Ann. Inst. Henri Poincaré 7, 1-68 (1937; Zbl 0017.07602)] states that any exchangeable process must be a mixture of independent and i.i.d. processes; if such a process is ergodic it must therefore be i.i.d. In this paper a notion weaker than exchangeability is introduced. A stationary process \(\{X_i\}\) is called strongly stationary if restricting the index \(i\) to any arithmetic progression \(\{X_{ri+b}\}\) gives a process with the same distribution. The main result is that any ergodic strongly stationary process is i.i.d. The methods used come from ergodic theory (analysing the eigenvalues of a measure-preserving dynamical system generating the process) and from the notion of IP-systems in the work of \textit{H. Furstenberg} and \textit{Y. Katznelson} [J. Anal. Math. 45, 117-168 (1985; Zbl 0605.28012)].
scientific article; zbMATH DE number 1146559