ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

12. Probability and Statistics

1991 MS Classification: 60 Sonin, Isaac M., University of North Carolina at Charlotte, Charlotte, USA 0-1 Laws for Nonhomogeneous Markov Chains We present a very simple but useful generalization of the classical Kolmogorov’s 0-1 Law for a sequence of independent random variables. Let (Z_{n}) be a nonhomogeneous Markov chain with values in countable state space M, Φ be the corresponding “tail” σ-algebra, i. e. Φ = ∩_{n}F_{n∞}, where F_{n∞} = σ(Z_{n}, Z_{n+1}, ...) and event A ∈ Φ. Obviously both P (A) and P (A|(Z_{n} = i) may take in general any values between 0 and 1. But for large n the trajectories of (Z_{n}) are in states where P (A|(Z_{n} = i) is near 0 or near 1. The precise formulation is as follows. Denote by M_{n}(p, q) = {i ∈ M : p ≤ P (A|(Z_{n} = i) ≤ q}, n ∈ N, 0 ≤ p ≤ q ≤ 1. Theorem 1. For any A ∈ Φ and any 0 < p < q < 1

a) lim_{n} P (Z_{n} ∈ M_{n}(q, 1)) = lim_{n} P ((Z_{n} ∈ M_{n}(q, 1)) ∩ A) = P (A),

b) lim_{n} P (Z_{n} ∈ M_{n}(0, p)) = 1 − P (A),

c) lim_{n} P (Z_{n} ∈ M_{n}(p, q) = 0. Theorem 1 was used in the proof of the Decomposition-Separation theorem describing the asymptotic behaviour of general finite nonhomogeneous Markov chain. [I. Sonin, IMS, Lect. Notes-Monograph Series, v. 30, (1996), Stat., Prob. and Game Th., papers in Honor of David Blackwell, eds. T. S. Ferguson, L. S. Shapley and J. B. MacQueen, 337-346].