Strong mixing conditions
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2020 Mathematics Subject Classification: Primary: 60G10 Secondary: 60G99 [MSN][ZBL]
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
$X := (X_k, k \in {\mathbf Z})$ 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
$X_k, j \leq k \leq \ell (k \in {\mathbf Z})$.
For any two $\sigma$-fields ${\cal A}$ and
${\cal B} \subset {\cal F}$, define the "measure of
dependence"
$$ \alpha({\cal A}, {\cal B}) :=
\sup_{A \in {\cal A}, B \in {\cal B}}
|P(A \cap B) - P(A)P(B)|. \tag{1} $$
For the given random sequence $X$, for any positive
integer $n$, define the dependence coefficient
$$\alpha(n) = \alpha(X,n) :=
\sup_{j \in '''Z'''}
\alpha({\cal F}_{-\infty}^j, {\cal F}_{j + n}^{\infty}).
\tag{2} $$
By a trivial argument, the sequence of numbers
$(\alpha(n), n \in {\mathbf N})$ 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
$$\alpha(n) = \alpha(X,n) :=
\alpha({\cal F}_{-\infty}^0, {\cal F}_n^{\infty}).
\tag{3} $$
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 $(X_k, k \in {\mathbf Z})$ 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 $(S_n^2/\sigma_n^2, n \in {\mathbf N})$ 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":
$$ \eqalignno{
\phi({\cal A}, {\cal B}) &:=
\sup_{A \in {\cal A}, B \in {\cal B}, P(A) > 0}
|P(B|A) - P(B)|; & (4) \cr
\psi({\cal A}, {\cal B}) &:=
\sup_{A \in {\cal A}, B \in {\cal B}, P(A) > 0, P(B) > 0}
|P(B \cap A)/[P(A)P(B)]\thinspace -\thinspace 1|; & (5) \cr
\rho({\cal A}, {\cal B}) &:=
\sup_{f \in {\cal L}^2({\cal A}),\thinspace g \in {\cal L}^2({\cal B})}
|{\rm Corr}(f,g)|; \quad {\rm and} & (6) \cr
\beta ({\cal A}, {\cal B}) &:= \sup (1/2)
\sum_{i=1}^I \sum_{j=1}^J |P(A_i \cap B_j) - P(A_i)P(B_j)|
& (7) \cr } $$
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.
Now suppose $X := (X_k, k \in {\mathbf Z})$ is a strictly
stationary sequence of random variables on
$(\Omega, {\cal F}, P)$.
For any positive integer $n$, analogously to (3), define
the dependence coefficient
$$\phi(n) = \phi(X,n) :=
\phi({\cal F}_{-\infty}^0, {\cal F}_n^{\infty}),
\tag{8} $$
and define analogously the dependence coefficients
$\psi(n)$, $\rho(n)$, and $\beta(n)$.
Each of these four sequences of dependence
coefficients is trivially nonincreasing.
The (strictly stationary) sequence $X$ is said to be
"$\phi$-mixing" if $\phi(n) \to 0$ as $n \to \infty$;
"$\psi$-mixing" if $\psi(n) \to 0$ as $n \to \infty$;
"$\rho$-mixing" if $\rho(n) \to 0$ as $n \to \infty$; and
"absolutely regular", or "$\beta$-mixing", if $\beta(n) \to 0$ as $n \to \infty$.
The $\phi$-mixing condition was introduced by
Ibragimov in 1959 and was also studied by Cogburn in 1960.
The $\psi$-mixing condition evolved through papers of Blum,
Hanson, and Koopmans in 1963 and Philipp in 1969; and
(see e.g. [Io]) it was also implicitly present
in earlier work of Doeblin in 1940 involving the metric
theory of continued fractions.
The $\rho$-mixing condition was introduced by
Kolmogorov and Rozanov 1960.
(The "maximal correlation coefficient"
$\rho({\cal A}, {\cal B})$ itself was first studied by
Hirschfeld in 1935 in a statistical context that had
no particular connection with "stochastic processes".)
The absolute regularity ($\beta$-mixing) condition was introduced by Volkonskii and Rozanov in 1959, and
in the ergodic theory literature it
is also called the "weak Bernoulli" condition.
For the five measures of dependence in (1) and (4)--(7),
one has the following well known inequalities:
$$ \eqalignno{
2\alpha({\cal A}, {\cal B}) \thinspace & \leq \thinspace
\beta({\cal A}, {\cal B}) \thinspace \leq \thinspace
\phi({\cal A}, {\cal B}) \thinspace \leq \thinspace
(1/2) \psi({\cal A}, {\cal B}); \cr
4 \alpha({\cal A}, {\cal B})\thinspace &\leq \thinspace
\rho({\cal A}, {\cal B}) \thinspace \leq \thinspace
\psi({\cal A}, {\cal B}); \quad {\rm and} \cr
\rho({\cal A}, {\cal B}) \thinspace &\leq \thinspace
2 [\phi({\cal A}, {\cal B})]^{1/2}
[\phi({\cal B}, {\cal A})]^{1/2} \thinspace \leq
\thinspace
2 [\phi({\cal A}, {\cal B})]^{1/2}. \cr
} $$
For a history and proof of these inequalities, see e.g.
[Br, v1, Theorem 3.11].
As a consequence of these inequalities and some
well known examples, one has the following "hierarchy"
of the five strong mixing conditions here:
(i) $\psi$-mixing implies $\phi$-mixing.
(ii) $\phi$-mixing implies both $\rho$-mixing and $\beta$-mixing (absolute regularity).
(iii) $\rho$-mixing and $\beta$-mixing each imply $\alpha$-mixing (strong mixing).
(iv) Aside from “transitivity”, there are in general no other implications between these five mixing conditions. In particular, neither of the conditions $\rho$-mixing and $\beta$-mixing implies the other.
For all of these mixing conditions, the “mixing rates” can be essentially arbitrary,
For all of these mixing conditions, the
"mixing rates" can be essentially arbitrary, and in particular, arbitrarily slow.
That general principle was established by Kesten and
O'Brien in 1976 with several classes of examples.
For further details, see e.g. [Br, v3, Chapter 26].
The various strong mixing conditions above have been
used extensively in statistical inference for weakly
dependent data.
See e.g. [DDLLLP], [DMS], [Ro3], or [Žu].
Ibragimov's conjecture and related material.
Suppose (as in Theorem 1) $X := (X_k, k \in {\mathbf Z})$
is a strictly stationary
sequence of random variables such that
$$ EX_0 = 0, \ EX_0^2 < \infty, \ {\ \rm and\ }
ES_n^2 \to \infty {\ \rm as\ } n \to \infty. \tag{9} $$
In the 1960s, I.A. Ibragimov conjectured that
under these assumptions, if also $X$ is $\phi$-mixing,
then a CLT holds.
Technically, this conjecture remains unsolved.
Peligrad showed in 1985 that it holds under the
stronger "growth" assumption
$\liminf_{n \to \infty} n^{-1} ES_n^2 > 0$.
(See e.g. [Br, v2, Theorem 17.7].)
Under (9) and $\rho$-mixing (which is weaker
than $\phi$-mixing), a CLT need not hold (see
[Br, v3, Chapter 34] for counterexamples).
However, if one also imposes either the stronger
moment condition $E|X_0|^{2 + \delta} < \infty$ for
some $\delta > 0$, or else the "logarithmic"
mixing rate assumption
$\sum_{n=1}^\infty \rho(2^n) < \infty$,
then a CLT does hold (results of
Ibragimov in 1975).
For further limit theory under $\rho$-mixing,
see e.g. [LL] or [Br, v1, Chapter 11].
Under (9) and an "interlaced" variant of the
$\rho$-mixing condition (i.e. with the two index sets
allowed to be "interlaced" instead of just "past" and
"future"), a CLT does hold.
For this and related material, see e.g. [Br, v1, Sections 11.18-11.28].
There is a vast literature on central limit theory for
random fields satisfying various strong mixing conditions.
See e.g. [Ro3], [Žu], [Do], and [Br, v3].
In the formulation of mixing conditions for random fields
--- and also "interlaced" mixing conditions for random
sequences --- some caution is needed; see e.g.
[Br, v1&3, Theorems 5.11, 5.13, 29.9, and 29.12].
Connections with specific types of models.
Now let us return briefly to a theme from the beginning of this write-up: the connection between strong mixing
conditions and specific structures.
Markov chains. Suppose
$X := (X_k, k \in {\mathbf Z})$ is a strictly stationary
Markov chain.
In the case where $X$ has finite state space and is irreducible and aperiodic, it is $\psi$-mixing,
with at least exponentially fast mixing rate.
In the case where $X$ has countable (but not
necessarily finite) state space and is irreducible
and aperiodic, it satisfies $\beta$-mixing, but the mixing rate can be arbitrarily slow.
In the case where $X$ has (say) real (but not necessarily
countable) state space, (i) Harris recurrence and
"aperiodicity" (suitably defined) together are equivalent
to $\beta$-mixing, (ii) the "geometric ergodicity"
condition is equivalent to $\beta$-mixing with
at least exponentially fast mixing rate, and
(iii) one particular version of
"Doeblin's condition" is equivalent to $\phi$-mixing
(and the mixing rate will then be at least exponentially
fast).
There exist strictly stationary, countable-state
Markov chains that are $\phi$-mixing but not
"time reversed" $\phi$-mixing (note the asymmetry in the
definition of $\phi({\cal A}, {\cal B})$ in (4)).
For this and other information on strong mixing
conditions for Markov chains,
see e.g. [Ro2, Chapter 7], [Do], [MT], and
[Br, v1&2, Chapters 7 and 21].
Stationary Gaussian sequences. For
stationary Gaussian sequences
$X := (X_k, k \in {\mathbf Z})$, Ibragimov and Rozanov [IR]
give characterizations of various strong mixing
conditions in terms of properties of spectral density
functions.
Here are just a couple of comments:
For stationary Gaussian sequences, the $\alpha$- and
$\rho$-mixing conditions are equivalent to each
other, and the $\phi$- and $\psi$-mixing conditions
are each equivalent to $m$-dependence.
If a stationary Gaussian sequence has a continuous
positive spectral density function, then it is
$\rho$-mixing.
For some further closely related information on
stationary Gaussian sequences, see also
[Br, v1&3, Chapters 9 and 27].
Dynamical systems. Many dynamical systems
have strong mixing properties.
Certain one-dimensional "Gibbs states"
processes are $\psi$-mixing with at least exponentially
fast mixing rate.
A well known standard "continued fraction" process
is $\psi$-mixing with at least exponentially fast
mixing rate (see [Io]).
For certain stationary finite-state stochastic processes
built on piecewise expanding mappings of the
unit interval onto itself,
the absolute regularity condition holds
with at least exponentially fast mixing rate.
For more detains on the mixing properties of these and
other dynamical systems, see e.g. Denker [De].
Linear and related processes. There is
a large literature on strong mixing properties of
strictly stationary linear processes (including strictly
stationary ARMA
processes and also "non-causal" linear processes
and linear random fields) and also of some other related processes such as bilinear, ARCH, or GARCH models.
For details on strong mixing properties of these and other related processes,
see e.g. Doukhan [Do, Chapter 2].
However, many strictly stationary linear
processes fail to be $\alpha$-mixing.
A well known classic example is the
strictly stationary AR(1) process
(autoregressive process of order 1)
$X := (X_k, k \in {\mathbf Z})$ of the form
$X_k = (1/2)X_{k-1} + \xi_k$ where
$(\xi_k, k \in {\mathbf Z})$ is a sequence of independent,
identically distributed random variables, each taking
the values 0 and 1 with probability 1/2 each.
It has long been well known that this random sequence $X$
is not $\alpha$-mixing.
For more on this example, see e.g.
[Br, v1, Example 2.15] or [Do, Section 2.3.1].
Further related developments. The AR(1)
example spelled out above, together with many other
examples that are not $\alpha$-mixing but seem to
have some similar "weak dependence" quality,
have motivated the development of more general conditions
of weak dependence that have the "spirit" of, and most
of the advantages of, strong mixing conditions, but are
less restrictive, i.e. applicable to a much broader class of models (including the AR(1) example above).
There is a substantial development of central limit theory
for strictly stationary sequences under weak dependence assumptions explicitly involving characteristic functions
in connection with "block sums"; much of that theory
is codified in [Ja].
There is a substantial development of limit theory of
various kinds under weak dependence assumptions that involve
covariances of certain multivariate Lipschitz functions of random variables from the "past" and "future"
(in the spirit of, but much less restrictive than, say,
the dependence coefficient $\rho(n)$ defined analogously
to (3) and (8)); see e.g. [DDLLLP].
There is a substantial development of limit theory under
weak dependence assumptions that involve dependence
coefficients similar to $\alpha(n)$ in (3) but in
which the classes of events are restricted to
intersections of finitely many events of the form
$\{X_k > c\}$ for appropriate indices $k$ and
appropriate real numbers $c$; for the use of such
conditions in extreme value theory, see e.g. [LLR].
In recent years, there has been a considerable
development of central limit theory under "projective"
criteria related to martingale theory (motivated
by Gordin's martingale-approximation
technique --- see [HH]); for details,
see e.g. [Pe].
There are far too many other types of weak dependence
conditions, of the general spirit of strong mixing
conditions but less restrictive, to describe here;
for more details, see
e.g. [DDLLLP] or [Br, v1, Chapter 13].
References
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[Pe] | M. Peligrad. Conditional central limit theorem via martingale approximation. In: Dependence in Probability, Analysis and Number Theory, (I. Berkes, R.C. Bradley, H. Dehling, M. Peligrad, and R. Tichy, eds.), pp. 295-309. Kendrick Press, Heber City (Utah), 2010. |
[Ri] | E. Rio. Théorie Asymptotique des Processus Aléatoires Faiblement Dépendants. Mathématiques & Applications 31. Springer, Paris, 2000. |
[Ro1] | M. Rosenblatt. A central limit theorem and a strong mixing condition. Proc. Natl. Acad. Sci. USA 42 (1956) 43-47. |
[Ro2] | M. Rosenblatt. Markov Processes, Structure and Asymptotic Behavior. Springer-Verlag, New York, 1971. |
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[Žu] | I.G. Žurbenko. The Spectral Analysis of Time Series. North-Holland, Amsterdam, 1986. |
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