Mixing properties of Harris chains and autoregressive processes. (English) Zbl 0623.60087

Athreya, Krishna B.; Pantula, Sastry G.

J. Appl. Probab. 23, 880-892 (1986).

The authors consider criteria for Markov processes (with stationary transitions, in discrete time, general state space) to be strongly mixing (\(\alpha\)-mixing in usual terminology) or uniformly mixing (i.e., \(\phi\)- mixing). They prove the very convenient result that a Harris-recurrent process which admits a nontrivial invariant measure is necessarily strongly mixing for any initial distribution. An example shows that the requirement of existence of an invariant measure cannot be dropped. It is also shown for the first-order autoregressive process \(Y_{n+1}=\rho Y_ n+e_{n+1}\), where \(| \rho | <1\) and \(\{e_ n\}\) is i.i.d. independent of \(Y_ 0\) such that E log\({}^+| e_ 1| <\infty\) and \(\sum^{n_ 0}_{j=1}\rho^ je_ j\) has an absolutely continuous component, that if \(Y_ 0\) is essentially bounded then \(\{Y_ n\}\) is uniformly mixing if and only if \(e_ 1\) is essentially bounded.

Reviewer: E.Slud

MSC:

60J05	Discrete-time Markov processes on general state spaces
60G10	Stationary stochastic processes

Keywords:

strongly mixing; Harris-recurrent process; invariant measure; first-order autoregressive process