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Strong Mixing Conditions

Richard C. Bradley

Department of Mathematics, Indiana University, Bloomington, Indiana, USA

There has been much research on stochastic models that have a well defined, specific structure — for example, Markov chains, Gaussian processes, or linear models, including ARMA (autoregressive – moving average) models. However, it became clear in the middle of the last century that there was a need for a theory of statistical inference (e.g. central limit theory) that could be used in the analysis of time series that did not seem to "fit" any such specific structure but which did seem to have some "asymptotic independence" properties. That motivated the development of a broad theory of "strong mixing conditions" to handle such situations. This note is a brief description of that theory.

The field of strong mixing conditions is a vast area, and a short note such as this cannot even begin to do justice to it. Journal articles (with one exception) will not be cited; and many researchers who made important contributions to this field will not be mentioned here. All that can be done here is to give a narrow snapshot of part of the field.

The strong mixing ($\alpha$-mixing) condition. Suppose $X := (X_k, k \in {\bf Z})$ is a sequence of random variables on a given probability space $(\Omega, {\cal F}, P)$. For $-\infty \leq j \leq \ell \leq \infty$, let ${\cal F}_j^\ell$ denote the $\sigma$-field of events generated by the random variables $X_k,\ j \le k \leq \ell\ (k \in {\bf Z})$. For any two $\sigma$-fields ${\cal A}$ and ${\cal B} \subset {\cal F}$, define the "measure of dependence" \begin{equation} \alpha({\cal A}, {\cal B}) := \sup_{A \in {\cal A}, B \in {\cal B}} |P(A \cap B) - P(A)P(B)|. \end{equation} For the given random sequence $X$, for any positive integer $n$, define the dependence coefficient \begin{equation}\alpha(n) = \alpha(X,n) := \sup_{j \in {\bf Z}} \alpha({\cal F}_{-\infty}^j, {\cal F}_{j + n}^{\infty}). \end{equation} By a trivial argument, the sequence of numbers $(\alpha(n), n \in {\bf N})$ is nonincreasing. The random sequence $X$ is said to be "strongly mixing", or "$\alpha$-mixing", if $\alpha(n) \to 0$ as $n \to \infty$. This condition was introduced in 1956 by Rosenblatt [Ro1], and was used in that paper in the proof of a central limit theorem. (The phrase "central limit theorem" will henceforth be abbreviated CLT.)

In the case where the given sequence $X$ is strictly stationary (i.e. its distribution is invariant under a shift of the indices), eq. (2) also has the simpler form \begin{equation}\alpha(n) = \alpha(X,n) := \alpha({\cal F}_{-\infty}^0, {\cal F}_n^{\infty}). \end{equation} For simplicity, in the rest of this note, we shall restrict to strictly stationary sequences. (Some comments below will have obvious adaptations to nonstationary processes.)

In particular, for strictly stationary sequences, the strong mixing ($\alpha$-mixing) condition implies Kolmogorov regularity (a trivial "past tail" $\sigma$-field), which in turn implies "mixing" (in the ergodic-theoretic sense), which in turn implies ergodicity. (None of the converse implications holds.) For further related information, see e.g. [Br, v1, Chapter 2].