Constructive Markov chains indexed by \(\mathbb Z\). (Chaînes de Markov constructives indexées par \(\mathbb Z\).) (French) Zbl 1123.60060

Brossard, Jean; Leuridan, Christophe

Ann. Probab. 35, No. 2, 715-731 (2007).

The authors are concerned with general state space Markov processes \((X_{n})_{n\in \mathbb{Z}}\), governed by recurrence relations of the form \(X_{n+1}=f_{n}( X_{n},V_{n+1}) \), where \( (V_{n})_{n\in \mathbb{Z}}\) is a doubly infinite sequence of i.i.d. random variables such that \(V_{n+1}\) is independent of the sequence \(((X_{k},V_{k}))_{k\leq n}\). Three cases are considered: the general one just described, the homogeneous case \(f_{n}=f\), \(n\in \mathbb{Z}\), for a countable state space, and the homogeneous case for a finite state space. In the second case, the authors give a necessary and sufficient condition under which the sequence \((V_{n})_{n\in \mathbb{Z}}\) fully determines the process \((X_{n})_{n\in \mathbb{Z}}\). Moreover, in the third case, they describe the missing information when this does not happen. A result of M. Rosenblatt [J. Math. Mech. 8, 665–681 (1959; Zbl 0092.33601)] is generalized and improved.

Reviewer: Marius Iosifescu (Bucureşti)

MSC:

60J05	Discrete-time Markov processes on general state spaces
60J10	Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords:

Recursive Markov chain; filtration

Citations:

Zbl 0092.33601

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