##
**The central limit theorem and Markov sequences.**
*(English)*
Zbl 0787.60080

Cohn, Harry (ed.), Doeblin and modern probability. Proceedings of the Doeblin conference ‘50 years after Doeblin: development in the theory of Markov chains, Markov processes, and sums of random variables’ held November 2-7, 1991 at the University of Tübingen’s Heinrich Fabri Institut, Blaubeuren, Germany. Providence, RI: American Mathematical Society. Contemp. Math. 149, 171-177 (1993).

Developments and connections in the theory of Markov sequences close related to Doeblin’s conditions D and \(\text{D}_ 0\) are presented. Characterizing the conditions D and \(\text{D}_ 0\) as mixing conditions shows the current meaning of the original ideas. So it is possible to treat stationary sequences and stationary random fields under this points of view. Mixing conditions are given under which the central limit theorem for stationary sequences holds. For stationary sequences also the following representation problem of N. Wiener is considered: When one could generate such a sequence \(X=(X_ n)\) by means of a sequence of independent uniformly on [0,1] distributed random variables \(\xi=(\xi_ n)\) and an appropriate function \(g\), \(X_ n=g(\tau^ n\xi)\) \((\tau\) the shift operator)?

For the entire collection see [Zbl 0777.00028].

For the entire collection see [Zbl 0777.00028].

Reviewer: W.Schenk (Dresden)

### MSC:

60J05 | Discrete-time Markov processes on general state spaces |

60F05 | Central limit and other weak theorems |

60G60 | Random fields |

60G10 | Stationary stochastic processes |