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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].

Comments on limit theory under $\alpha$-mixing. Under $\alpha$-mixing and other similar conditions (including ones reviewed below), there has been a vast development of limit theory — for example, CLTs, weak invariance principles, laws of the iterated logarithm, almost sure invariance principles, and rates of convergence in the strong law of large numbers. For example, the CLT in [Ro1] evolved through subsequent refinements by several researchers into the following "canonical" form. (For its history and a generously detailed presentation of its proof, see e.g. [Br, v1, Theorems 1.19 and 10.2].)

Theorem 1. Suppose $(X_k, k \in {\bf Z})$ is a strictly stationary sequence of random variables such that $EX_0 = 0$, $EX_0^2 < \infty$, $\sigma_n^2 := ES_n^2 \to \infty$ as $n \to \infty$, and $\alpha(n) \to 0$ as $n \to \infty$. Then the following two conditions (A) and (B) are equivalent:

(A) The family of random variables $(S_n^2/\sigma_n^2, n \in {\bf N})$ is uniformly integrable.

(B) $S_n/\sigma_n \Rightarrow N(0,1)$ as $n \to \infty$.

If (the hypothesis and) these two equivalent conditions (A) and (B) hold, then $\sigma_n^2 = n \cdot h(n)$ for some function $h(t),\ t \in (0, \infty)$ which is slowly varying as $t \to \infty$.

Here $S_n := X_1 + X_2 + \dots + X_n$; and $\Rightarrow$ denotes convergence in distribution. The assumption $ES_n^2 \to \infty$ is needed here in order to avoid trivial $\alpha$-mixing (or even 1-dependent) counterexamples in which a kind of "cancellation" prevents the partial sums $S_n$ from "growing" (in probability) and becoming asymptotically normal.

In the context of Theorem 1, if one wants to obtain asymptotic normality of the partial sums (as in condition (B)) without an explicit uniform integrability assumption on the partial sums (as in condition (A)), then as an alternative, one can impose a combination of assumptions on, say, (i) the (marginal) distribution of $X_0$ and (ii) the rate of decay of the numbers $\alpha(n)$ to 0 (the "mixing rate"). This involves a "trade-off"; the weaker one assumption is, the stronger the other has to be. One such CLT of Ibragimov in 1962 involved such a "trade-off" in which it is assumed that for some $\delta > 0$, $E|X_0|^{2 + \delta} < \infty$ and $\sum_{n=1}^\infty [\alpha(n)]^{\delta/(2 + \delta)} < \infty$. Counterexamples of Davydov in 1973 (with just slightly weaker properties) showed that that result is quite sharp. However, it is not at the exact "borderline". From a covariance inequality of Rio in 1993 and a CLT (in fact a weak invariance principle) of Doukhan, Massart, and Rio in 1994, it became clear that the "exact borderline" CLTs of this kind have to involve quantiles of the (marginal) distribution of $X_0$ (rather than just moments). For a generously detailed exposition of such CLTs, see [Br, v1, Chapter 10]; and for further related results, see also Rio [Ri].

Under the hypothesis (first sentence) of Theorem 1 (with just finite second moments), there is no mixing rate, no matter how fast (short of $m$-dependence), that can insure that a CLT holds. That was shown in 1983 with two different counterexamples, one by the author and the other by Herrndorf. See [Br, v1\&3, Theorem 10.25 and Chapter 31].

Several other classic strong mixing conditions.

As indicated above, the terms "$\alpha$-mixing" and "strong mixing condition" (singular) both refer to the condition $\alpha(n) \to 0$. (A little caution is in order; in ergodic theory, the term "strong mixing" is often used to refer to the condition of "mixing in the ergodic-theoretic sense", which is weaker than $\alpha$-mixing as noted earlier.) The term "strong mixing conditions" (plural) can reasonably be thought of as referring to all conditions that are at least as strong as (i.e. that imply) $\alpha$-mixing. In the classical theory, five strong mixing conditions (again, plural) have emerged as the most prominent ones: $\alpha$-mixing itself and four others that will be defined here.

Recall our probability space $(\Omega, {\cal F}, P)$. For any two $\sigma$-fields ${\cal A}$ and ${\cal B} \subset {\cal F}$, define the following four "measures of dependence": \begin{eqnarray} \phi({\cal A}, {\cal B}) &:= & \sup_{A \in {\cal A}, B \in {\cal B}, P(A) > 0} |P(B|A) - P(B)|; \\ \psi({\cal A}, {\cal B}) &:= & \sup_{A \in {\cal A}, B \in {\cal B}, P(A) > 0, P(B) > 0} |P(B \cap A)/[P(A)P(B)]\thinspace -\thinspace 1|; \\ \rho({\cal A}, {\cal B}) &:= & \sup_{f \in {\cal L}^2({\cal A}),\thinspace g \in {\cal L}^2({\cal B})} |{\rm Corr}(f,g)|; \quad {\rm and} \\ \beta ({\cal A}, {\cal B}) &:=& \sup\ (1/2) \sum_{i=1}^I \sum_{j=1}^J |P(A_i \cap B_j) - P(A_i)P(B_j)| \end{eqnarray} where the latter supremum is taken over all pairs of finite partitions $(A_1, A_2, \dots, A_I)$ and $(B_1, B_2, \dots, B_J)$ of $\Omega$ such that $A_i \in {\cal A}$ for each $i$ and $B_j \in {\cal B}$ for each $j$. In (6), for a given $\sigma$-field ${\cal D} \subset {\cal F}$, the notation ${\cal L}^2({\cal D})$ refers to the space of (equivalence classes of) square-integrable, ${\cal D}$-measurable random variables.