Separation of jets and a generalization to the non-homogeneous case of a theorem on decomposition of a finite homogeneous Markov chain into ergodic components.

*(English. Russian original)*Zbl 0677.60074
Sov. Math., Dokl. 39, No. 1, 27-31 (1989); translation from Dokl. Akad. Nauk SSSR 304, No. 1, 36-40 (1989).

The author treats three problems that seem not very closely related at first glance, but turn out to be united by a deep intrinsic connection.

Problem 1 concerns the possibility of separating a nonhomogeneous fluid flow in discrete time and space into individual “jets” with certain desired properties concerning asymptotic homogeneity and finite exchange of solutions between jets. As it turns out, Problem 1 has a probabilistic interpretation in terms of the asymptotic behaviour of certain nonhomogeneous Markov chains. In fact, the solution to Problem 1 leads to a generalization of the well-known results of Kolmogorov and Doeblin on decomposition of a finite homogeneous Markov chain into ergodic components.

Problem 2 concerns classes of random sequences \((X_ n)\) with coinciding two-dimensional distributions for \((X_ n,X_{n+1})\). The problem is to determine which random sequences in a class have asymptotically maximal scatter, i.e. have the maximal probability of being outside an arbitrary specified sequence \((D_ n)\) of subsets of the range spaces infinitely often. It is shown that a nonhomogeneous Markov chain within the class always has this property.

It is finally demonstrated that the solution of Problems 1 and 2 depends essentially on the answer to the following question:

Problem 3: For a bounded submartingale \((X_ n)\) in reverse time with finitely many values, does there exist a numerical sequence \((d_ n)\) that is in a given interval (a,b) and such that the average number of times the sample paths of \((X_ n)\) cross \((d_ n)\) on the infinite time interval is finite ?

Now the answer is yes, as follows by the stated Theorem 4. This theorem, which is therefore the clue to the solution of all the problems, follows from a lemma in the author’s paper, Stochastics 21, 231-249 (1987; Zbl 0626.60070).

Problem 1 concerns the possibility of separating a nonhomogeneous fluid flow in discrete time and space into individual “jets” with certain desired properties concerning asymptotic homogeneity and finite exchange of solutions between jets. As it turns out, Problem 1 has a probabilistic interpretation in terms of the asymptotic behaviour of certain nonhomogeneous Markov chains. In fact, the solution to Problem 1 leads to a generalization of the well-known results of Kolmogorov and Doeblin on decomposition of a finite homogeneous Markov chain into ergodic components.

Problem 2 concerns classes of random sequences \((X_ n)\) with coinciding two-dimensional distributions for \((X_ n,X_{n+1})\). The problem is to determine which random sequences in a class have asymptotically maximal scatter, i.e. have the maximal probability of being outside an arbitrary specified sequence \((D_ n)\) of subsets of the range spaces infinitely often. It is shown that a nonhomogeneous Markov chain within the class always has this property.

It is finally demonstrated that the solution of Problems 1 and 2 depends essentially on the answer to the following question:

Problem 3: For a bounded submartingale \((X_ n)\) in reverse time with finitely many values, does there exist a numerical sequence \((d_ n)\) that is in a given interval (a,b) and such that the average number of times the sample paths of \((X_ n)\) cross \((d_ n)\) on the infinite time interval is finite ?

Now the answer is yes, as follows by the stated Theorem 4. This theorem, which is therefore the clue to the solution of all the problems, follows from a lemma in the author’s paper, Stochastics 21, 231-249 (1987; Zbl 0626.60070).

Reviewer: B.H.Lindqvist

##### MSC:

60J10 | Markov chains (discrete-time Markov processes on discrete state spaces) |

76D25 | Wakes and jets |