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Some remarks on a mixing condition. (English) Zbl 0925.60052
Ann. Probab. 7, No. 1, 170-172 (1979); corrections ibid. No. 6, 1097 (1979).
Let \({\mathcal B}_s^t\), \(s\leq t\), denote the \(sigma\)-algebra generated by the random variables \(X_k\), \(s\leq k\leq t\), and consider the quantity \(\gamma (s, 3t)= \sup| P(A| B\cap C) - P(A| B)|\), where the upper bound is taken over all \(A \in {\mathcal B}_t^t\), \(B\in {\mathcal B}_{s+1}^{t - 1}\), \(C\in{\mathcal b}B_s^s\). The condition \(gamma(s, 3t)\to 0\) as \(t - s\to\infty\) is referred to as a Markov-type regularity condition (see, e.g., V. A. Statulevicius, Multivariate analysis IV, Proc. 4th Int. Symp., Dayton 1975 325-337 (1977; Zbl 0445.60019); I. G. Zhurbenko, Stoch. Processes Relat. Top., Vol 1 Proc. Summer Res. Inst., Bloomington 1974, 259-265 (1975; Zbl 0349.60031)). It is shown that the only stationary countable-state Markov chains satisfying the above regularity condition are sequences of independent random variables. (In the erratum the author replaces the word “nominal” in the last sentence of the paper by “nontrivial”).

MSC:
60G99 Stochastic processes
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