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(Created page with " {| class="wikitable" !Copyright notice <!-- don't remove! --> |- | This article ''Strong Mixing Conditions'' was adapted from an original article by Richard Crane Bradle...")
 
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 +
<center>'''Strong Mixing Conditions'''</center>
  
<center>'''Strong Mixing Conditions'''</center>
+
<center>
 +
Richard C. Bradley
  
<!-- \noindent -->
 
Richard C. Bradley \hfil\break
 
<!-- \noindent -->
 
 
Department of Mathematics, Indiana University,
 
Department of Mathematics, Indiana University,
 
Bloomington, Indiana, USA
 
Bloomington, Indiana, USA
 
+
</center>
  
 
There has been much research on stochastic models
 
There has been much research on stochastic models
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"strong mixing conditions" to handle such situations.
 
"strong mixing conditions" to handle such situations.
 
This note is a brief description of that theory.
 
This note is a brief description of that theory.
\smallskip
+
<br>
 +
 
  
 
The field of strong mixing conditions is a vast area,
 
The field of strong mixing conditions is a vast area,
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All that can be done here is to give a narrow snapshot
 
All that can be done here is to give a narrow snapshot
 
of part of the field.
 
of part of the field.
\hfil\break
 
  
'''The strong mixing ($\alpha$-mixing) condition.''' \ Suppose
+
 
 +
'''The strong mixing ($\alpha$-mixing) condition.''' Suppose
 
$X := (X_k, k \in {\mathbf Z})$ is a sequence of
 
$X := (X_k, k \in {\mathbf Z})$ is a sequence of
 
random variables on a given probability space
 
random variables on a given probability space
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or "$\alpha$-mixing", if $\alpha(n) \to 0$ as
 
or "$\alpha$-mixing", if $\alpha(n) \to 0$ as
 
$n \to \infty$.
 
$n \to \infty$.
This condition was introduced in 1956 by Rosenblatt [Ro1],
+
This condition was introduced in 1956 by Rosenblatt {{Cite|Ro1}},
 
and was used in that paper in the proof of a central limit
 
and was used in that paper in the proof of a central limit
 
theorem.
 
theorem.
 
(The phrase "central limit theorem" will henceforth
 
(The phrase "central limit theorem" will henceforth
 
be abbreviated CLT.)
 
be abbreviated CLT.)
\smallskip
+
<br>
 +
 
  
 
In the case where the given sequence $X$ is strictly
 
In the case where the given sequence $X$ is strictly
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\tag{3} $$
 
\tag{3} $$
 
For simplicity, ''in the rest of this note,
 
For simplicity, ''in the rest of this note,
we shall restrict to strictly stationary sequences\/''.
+
we shall restrict to strictly stationary sequences''.
 
(Some comments below will have obvious adaptations to
 
(Some comments below will have obvious adaptations to
nonstationary processes.) \smallskip
+
nonstationary processes.) <br>
 +
 
  
 
In particular, for strictly stationary sequences,
 
In particular, for strictly stationary sequences,
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which in turn implies "mixing" (in the ergodic-theoretic
 
which in turn implies "mixing" (in the ergodic-theoretic
 
sense), which in turn implies ergodicity.
 
sense), which in turn implies ergodicity.
(None of the converse implications holds.) \
+
(None of the converse implications holds.)
 
For further related information, see
 
For further related information, see
e.g. [Br, v1, Chapter 2].
+
e.g. {{Cite|Br| v1, Chapter 2}}.
\hfil\break
 
  
'''Comments on limit theory under
+
 
$\alpha$-mixing.''' \
+
'''Comments on limit theory under $\alpha$-mixing.'''
 
Under $\alpha$-mixing and other similar conditions
 
Under $\alpha$-mixing and other similar conditions
 
(including ones reviewed below), there has been a vast development of limit theory --- for example,
 
(including ones reviewed below), there has been a vast development of limit theory --- for example,
Line 108: Line 109:
 
principles, and rates of convergence in the strong law of
 
principles, and rates of convergence in the strong law of
 
large numbers.
 
large numbers.
For example, the CLT in [Ro1] evolved through
+
For example, the CLT in {{Cite|Ro1}} evolved through
 
subsequent refinements by several researchers
 
subsequent refinements by several researchers
 
into the following "canonical" form.
 
into the following "canonical" form.
 
(For its history and a generously detailed presentation
 
(For its history and a generously detailed presentation
of its proof, see e.g. [Br, v1,
+
of its proof, see e.g. {{Cite|Br| v1, Theorems 1.19 and 10.2}}.)
Theorems 1.19 and 10.2].)
 
  
  
'''Theorem 1.''' \ {\sl Suppose $(X_k, k \in {\mathbf Z})$
+
'''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:''
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:
 
  
<!-- \noindent -->
+
''(A) The family of random variables $(S_n^2/\sigma_n^2, n \in {\mathbf N})$ is uniformly integrable.''
(A) The family of random variables
 
$(S_n^2/\sigma_n^2, n \in {\mathbf N})$ is uniformly
 
integrable.
 
  
<!-- \noindent -->
+
''(B) $S_n/\sigma_n \Rightarrow N(0,1)$ as $n \to \infty$.''
(B) $S_n/\sigma_n \Rightarrow N(0,1)$ as
 
$n \to \infty$.
 
  
If (the hypothesis and) these two equivalent
+
''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 $t \to \infty$.''
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\
+
Here $S_n := X_1 + X_2 + \dots + X_n$; and  
 
$\Rightarrow$ denotes convergence in distribution.
 
$\Rightarrow$ denotes convergence in distribution.
 
The assumption $ES_n^2 \to \infty$ is needed here in
 
The assumption $ES_n^2 \to \infty$ is needed here in
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"growing" (in probability) and becoming asymptotically
 
"growing" (in probability) and becoming asymptotically
 
normal.
 
normal.
\hfil\break
+
 
  
 
In the context of Theorem 1, if one wants to obtain asymptotic normality of the
 
In the context of Theorem 1, if one wants to obtain asymptotic normality of the
Line 176: Line 160:
 
distribution of $X_0$ (rather than just moments).
 
distribution of $X_0$ (rather than just moments).
 
For a generously detailed exposition of such CLTs,
 
For a generously detailed exposition of such CLTs,
see [Br, v1, Chapter 10]; and for further
+
see {{Cite|Br| v1, Chapter 10}}; and for further
related results, see also Rio [Ri].
+
related results, see also Rio {{Cite|Ri}}.
\smallskip
+
<br>
 +
 
  
 
Under the hypothesis (first sentence) of Theorem 1
 
Under the hypothesis (first sentence) of Theorem 1
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counterexamples, one by the author and the other by
 
counterexamples, one by the author and the other by
 
Herrndorf.
 
Herrndorf.
See [Br, v1&3, Theorem 10.25 and Chapter 31].
+
See {{Cite|Br| v1&3, Theorem 10.25 and Chapter 31}}.
\hfil\break
 
  
'''Several other classic strong mixing conditions.''' \
+
 
 +
'''Several other classic strong mixing conditions.'''
 
As indicated above, the terms "$\alpha$-mixing" and
 
As indicated above, the terms "$\alpha$-mixing" and
 
"strong mixing condition" (singular) both refer to the condition $\alpha(n) \to 0$.
 
"strong mixing condition" (singular) both refer to the condition $\alpha(n) \to 0$.
Line 199: Line 184:
 
"mixing in the ergodic-theoretic sense",
 
"mixing in the ergodic-theoretic sense",
 
which is weaker than
 
which is weaker than
$\alpha$-mixing as noted earlier.) \
+
$\alpha$-mixing as noted earlier.)
 
The term "strong mixing conditions" (plural) can
 
The term "strong mixing conditions" (plural) can
 
reasonably be thought of as referring
 
reasonably be thought of as referring
Line 208: Line 193:
 
$\alpha$-mixing itself and four others that will be
 
$\alpha$-mixing itself and four others that will be
 
defined here.
 
defined here.
\smallskip
+
<br>
 +
 
  
 
Recall our probability space $(\Omega, {\cal F}, P)$.
 
Recall our probability space $(\Omega, {\cal F}, P)$.
Line 236: Line 222:
 
(equivalence classes of) square-integrable,
 
(equivalence classes of) square-integrable,
 
${\cal D}$-measurable random variables.
 
${\cal D}$-measurable random variables.
\smallskip
+
<br>
 +
 
  
 
Now suppose $X := (X_k, k \in {\mathbf Z})$ is a strictly
 
Now suppose $X := (X_k, k \in {\mathbf Z})$ is a strictly
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coefficients is trivially nonincreasing.
 
coefficients is trivially nonincreasing.
 
The (strictly stationary) sequence $X$ is said to be
 
The (strictly stationary) sequence $X$ is said to be
\hfil\break
+
 
 
"$\phi$-mixing" if $\phi(n) \to 0$ as $n \to \infty$;
 
"$\phi$-mixing" if $\phi(n) \to 0$ as $n \to \infty$;
\hfil\break
+
 
 
"$\psi$-mixing" if $\psi(n) \to 0$ as $n \to \infty$;
 
"$\psi$-mixing" if $\psi(n) \to 0$ as $n \to \infty$;
\hfil\break
+
 
 
"$\rho$-mixing" if $\rho(n) \to 0$ as $n \to \infty$;
 
"$\rho$-mixing" if $\rho(n) \to 0$ as $n \to \infty$;
 
and
 
and
\hfil\break
+
 
 
"absolutely regular", or "$\beta$-mixing", if $\beta(n) \to 0$ as $n \to \infty$.
 
"absolutely regular", or "$\beta$-mixing", if $\beta(n) \to 0$ as $n \to \infty$.
\smallskip
+
<br>
 +
 
  
 
The $\phi$-mixing condition was introduced by
 
The $\phi$-mixing condition was introduced by
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The $\psi$-mixing condition evolved through papers of Blum,
 
The $\psi$-mixing condition evolved through papers of Blum,
 
Hanson, and Koopmans in 1963 and Philipp in 1969; and
 
Hanson, and Koopmans in 1963 and Philipp in 1969; and
(see e.g. [Io]) it was also implicitly present
+
(see e.g. {{Cite|Io}}) it was also implicitly present
 
in earlier work of Doeblin in 1940 involving the metric
 
in earlier work of Doeblin in 1940 involving the metric
 
theory of continued fractions.
 
theory of continued fractions.
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$\rho({\cal A}, {\cal B})$ itself was first studied by
 
$\rho({\cal A}, {\cal B})$ itself was first studied by
 
Hirschfeld in 1935 in a statistical context that had
 
Hirschfeld in 1935 in a statistical context that had
no particular connection with "stochastic processes".) \
+
no particular connection with "stochastic processes".)
 
The absolute regularity ($\beta$-mixing) condition was introduced by Volkonskii and Rozanov in 1959, and
 
The absolute regularity ($\beta$-mixing) condition was introduced by Volkonskii and Rozanov in 1959, and
 
in the ergodic theory literature it
 
in the ergodic theory literature it
 
is also called the "weak Bernoulli" condition.
 
is also called the "weak Bernoulli" condition.
\smallskip
+
<br>
 +
 
  
 
For the five measures of dependence in (1) and (4)--(7),
 
For the five measures of dependence in (1) and (4)--(7),
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2 [\phi({\cal A}, {\cal B})]^{1/2}. \cr
 
2 [\phi({\cal A}, {\cal B})]^{1/2}. \cr
 
} $$
 
} $$
For a history and proof of these inequalities, see e.g.\
+
For a history and proof of these inequalities, see e.g.
 
[Br, v1, Theorem 3.11].
 
[Br, v1, Theorem 3.11].
 
As a consequence of these inequalities and some
 
As a consequence of these inequalities and some
 
well known examples, one has the following "hierarchy"
 
well known examples, one has the following "hierarchy"
of the five strong mixing conditions here: \hfil\break
+
of the five strong mixing conditions here: <br>
\indent (i) $\psi$-mixing implies $\phi$-mixing. \hfil\break
+
 
\indent (ii) $\phi$-mixing implies both $\rho$-mixing and
+
(i) $\psi$-mixing implies $\phi$-mixing. <br>
$\beta$-mixing (absolute regularity). \hfil\break
+
 
\indent (iii) $\rho$-mixing and $\beta$-mixing each imply
+
(ii) $\phi$-mixing implies both $\rho$-mixing and $\beta$-mixing (absolute regularity). <br>
$\alpha$-mixing (strong mixing). \hfil\break
+
 
\indent (iv) Aside from "transitivity", there are in
+
(iii) $\rho$-mixing and $\beta$-mixing each imply $\alpha$-mixing (strong mixing). <br>
general
+
 
no other implications between these five mixing conditions.
+
(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.
In particular, neither of the conditions $\rho$-mixing
+
 
and $\beta$-mixing implies the other. \smallskip
+
For all of these mixing conditions, the “mixing rates” can be essentially arbitrary,
  
 
For all of these mixing conditions, the
 
For all of these mixing conditions, the
Line 316: Line 305:
 
That general principle was established by Kesten and
 
That general principle was established by Kesten and
 
O'Brien in 1976 with several classes of examples.
 
O'Brien in 1976 with several classes of examples.
For further details, see e.g. [Br, v3, Chapter 26].
+
For further details, see e.g. {{Cite|Br| v3, Chapter 26}}.
\smallskip
+
<br>
 +
 
  
 
The various strong mixing conditions above have been
 
The various strong mixing conditions above have been
 
used extensively in statistical inference for weakly
 
used extensively in statistical inference for weakly
 
dependent data.
 
dependent data.
See e.g. [DDLLLP], [DMS], [Ro3], or [\v Zu].
+
See e.g. {{Cite|DDLLLP}}, {{Cite|DMS}}, {{Cite|Ro3}}, or {{Cite|&#381;u}}.
\hfil\break
 
  
'''Ibragimov's conjecture and related material.''' \
+
 
 +
'''Ibragimov's conjecture and related material.'''
 
Suppose (as in Theorem 1) $X := (X_k, k \in {\mathbf Z})$
 
Suppose (as in Theorem 1) $X := (X_k, k \in {\mathbf Z})$
 
is a strictly stationary
 
is a strictly stationary
 
sequence of random variables such that
 
sequence of random variables such that
$$ EX_0 = 0, \ EX_0^2 < \infty, \ {\rm and} \
+
$$ EX_0 = 0, \ EX_0^2 < \infty, \ {\ \rm and\ }
ES_n^2 \to \infty {\rm as} n \to \infty. \tag{9} $$
+
ES_n^2 \to \infty {\ \rm as\ } n \to \infty. \tag{9} $$
  
 
In the 1960s, I.A. Ibragimov conjectured that
 
In the 1960s, I.A. Ibragimov conjectured that
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stronger "growth" assumption
 
stronger "growth" assumption
 
$\liminf_{n \to \infty} n^{-1} ES_n^2 > 0$.
 
$\liminf_{n \to \infty} n^{-1} ES_n^2 > 0$.
(See e.g. [Br, v2, Theorem 17.7].)
+
(See e.g. {{Cite|Br| v2, Theorem 17.7}}.)
\smallskip
+
<br>
 +
 
  
 
Under (9) and $\rho$-mixing (which is weaker
 
Under (9) and $\rho$-mixing (which is weaker
 
than $\phi$-mixing), a CLT need not hold (see
 
than $\phi$-mixing), a CLT need not hold (see
[Br, v3, Chapter 34] for counterexamples).
+
{{Cite|Br| v3, Chapter 34}} for counterexamples).
 
However, if one also imposes either the stronger
 
However, if one also imposes either the stronger
 
moment condition $E|X_0|^{2 + \delta} < \infty$ for
 
moment condition $E|X_0|^{2 + \delta} < \infty$ for
Line 353: Line 344:
 
Ibragimov in 1975).
 
Ibragimov in 1975).
 
For further limit theory under $\rho$-mixing,
 
For further limit theory under $\rho$-mixing,
see e.g. [LL] or [Br, v1, Chapter 11].
+
see e.g. {{Cite|LL}} or {{Cite|Br| v1, Chapter 11}}.
\smallskip
+
<br>
 +
 
  
 
Under (9) and an "interlaced" variant of the
 
Under (9) and an "interlaced" variant of the
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allowed to be "interlaced" instead of just "past" and
 
allowed to be "interlaced" instead of just "past" and
 
"future"), a CLT does hold.
 
"future"), a CLT does hold.
For this and related material, see e.g. [Br, v1, Sections 11.18-11.28].
+
For this and related material, see e.g. {{Cite|Br| v1, Sections 11.18-11.28}}.
\smallskip
+
<br>
 +
 
  
 
There is a vast literature on central limit theory for
 
There is a vast literature on central limit theory for
 
random fields satisfying various strong mixing conditions.
 
random fields satisfying various strong mixing conditions.
See e.g. [Ro3], [\v Zu], [Do], and [Br, v3].
+
See e.g. {{Cite|Ro3}}, {{Cite|&#381;u}}, {{Cite|Do}}, and {{Cite|Br| v3}}.
 
In the formulation of mixing conditions for random fields
 
In the formulation of mixing conditions for random fields
 
--- and also "interlaced" mixing conditions for random
 
--- and also "interlaced" mixing conditions for random
sequences --- some caution is needed; see e.g.\
+
sequences --- some caution is needed; see e.g.  
 
[Br, v1&3, Theorems 5.11, 5.13, 29.9, and 29.12].
 
[Br, v1&3, Theorems 5.11, 5.13, 29.9, and 29.12].
\hfil\break
 
  
'''Connections with specific types of models.''' \
+
 
 +
'''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
 
Now let us return briefly to a theme from the beginning of this write-up: the connection between strong mixing
 
conditions and specific structures.
 
conditions and specific structures.
\smallskip
+
<br>
  
''Markov chains.'' \ Suppose
+
 
 +
''Markov chains.'' Suppose
 
$X := (X_k, k \in {\mathbf Z})$ is a strictly stationary
 
$X := (X_k, k \in {\mathbf Z})$ is a strictly stationary
 
Markov chain.
 
Markov chain.
Line 401: Line 395:
 
For this and other information on strong mixing
 
For this and other information on strong mixing
 
conditions for Markov chains,
 
conditions for Markov chains,
see e.g. [Ro2, Chapter 7], [Do], [MT], and
+
see e.g. {{Cite|Ro2| Chapter 7}}, {{Cite|Do}}, {{Cite|MT}}, and
[Br, v1&2, Chapters 7 and 21].
+
{{Cite|Br| v1&2, Chapters 7 and 21}}.
\smallskip
+
<br>
  
''Stationary Gaussian sequences.'' \ For
+
 
 +
''Stationary Gaussian sequences.'' For
 
stationary Gaussian sequences
 
stationary Gaussian sequences
$X := (X_k, k \in {\mathbf Z})$, Ibragimov and Rozanov [IR]
+
$X := (X_k, k \in {\mathbf Z})$, Ibragimov and Rozanov {{Cite|IR}}
 
give characterizations of various strong mixing
 
give characterizations of various strong mixing
 
conditions in terms of properties of spectral density
 
conditions in terms of properties of spectral density
Line 421: Line 416:
 
For some further closely related information on
 
For some further closely related information on
 
stationary Gaussian sequences, see also
 
stationary Gaussian sequences, see also
[Br, v1&3, Chapters 9 and 27].
+
{{Cite|Br| v1&3, Chapters 9 and 27}}.
\smallskip
+
<br>
  
''Dynamical systems.'' \ Many dynamical systems
+
 
 +
''Dynamical systems.'' Many dynamical systems
 
have strong mixing properties.
 
have strong mixing properties.
 
Certain one-dimensional "Gibbs states"
 
Certain one-dimensional "Gibbs states"
Line 431: Line 427:
 
A well known standard "continued fraction" process
 
A well known standard "continued fraction" process
 
is $\psi$-mixing with at least exponentially fast
 
is $\psi$-mixing with at least exponentially fast
mixing rate (see [Io]).
+
mixing rate (see {{Cite|Io}}).
 
For certain stationary finite-state stochastic processes
 
For certain stationary finite-state stochastic processes
 
built on piecewise expanding mappings of the
 
built on piecewise expanding mappings of the
Line 438: Line 434:
 
with at least exponentially fast mixing rate.
 
with at least exponentially fast mixing rate.
 
For more detains on the mixing properties of these and
 
For more detains on the mixing properties of these and
other dynamical systems, see e.g. Denker [De].
+
other dynamical systems, see e.g. Denker {{Cite|De}}.
\smallskip
+
<br>
  
''Linear and related processes.'' \ There is
+
 
 +
''Linear and related processes.'' There is
 
a large literature on strong mixing properties of
 
a large literature on strong mixing properties of
 
strictly stationary linear processes (including strictly
 
strictly stationary linear processes (including strictly
Line 448: Line 445:
 
and linear random fields) and also of some other related processes such as bilinear, ARCH, or GARCH models.
 
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,
 
For details on strong mixing properties of these and other related processes,
see e.g. Doukhan [Do, Chapter 2].
+
see e.g. Doukhan {{Cite|Do| Chapter 2}}.
\smallskip
+
<br>
 +
 
  
 
However, many strictly stationary linear
 
However, many strictly stationary linear
processes ''fail\/'' to be $\alpha$-mixing.
+
processes ''fail'' to be $\alpha$-mixing.
 
A well known classic example is the
 
A well known classic example is the
 
strictly stationary AR(1) process
 
strictly stationary AR(1) process
Line 463: Line 461:
 
It has long been well known that this random sequence $X$
 
It has long been well known that this random sequence $X$
 
is not $\alpha$-mixing.
 
is not $\alpha$-mixing.
For more on this example, see e.g.\
+
For more on this example, see e.g.  
[Br, v1, Example 2.15] or [Do, Section 2.3.1].
+
{{Cite|Br| v1, Example 2.15}} or {{Cite|Do| Section 2.3.1}}.
\hfil\break
+
 
  
'''Further related developments.''' \ The AR(1)
+
'''Further related developments.''' The AR(1)
 
example spelled out above, together with many other
 
example spelled out above, together with many other
 
examples that are not $\alpha$-mixing but seem to
 
examples that are not $\alpha$-mixing but seem to
Line 478: Line 476:
 
for strictly stationary sequences under weak dependence assumptions explicitly involving characteristic functions
 
for strictly stationary sequences under weak dependence assumptions explicitly involving characteristic functions
 
in connection with "block sums"; much of that theory
 
in connection with "block sums"; much of that theory
is codified in [Ja].
+
is codified in {{Cite|Ja}}.
 
There is a substantial development of limit theory of
 
There is a substantial development of limit theory of
 
various kinds under weak dependence assumptions that involve
 
various kinds under weak dependence assumptions that involve
Line 484: Line 482:
 
(in the spirit of, but much less restrictive than, say,
 
(in the spirit of, but much less restrictive than, say,
 
the dependence coefficient $\rho(n)$ defined analogously
 
the dependence coefficient $\rho(n)$ defined analogously
to (3) and (8)); see e.g. [DDLLLP].
+
to (3) and (8)); see e.g. {{Cite|DDLLLP}}.
 
There is a substantial development of limit theory under
 
There is a substantial development of limit theory under
 
weak dependence assumptions that involve dependence
 
weak dependence assumptions that involve dependence
Line 492: Line 490:
 
$\{X_k > c\}$ for appropriate indices $k$ and
 
$\{X_k > c\}$ for appropriate indices $k$ and
 
appropriate real numbers $c$; for the use of such
 
appropriate real numbers $c$; for the use of such
conditions in extreme value theory, see e.g. [LLR].
+
conditions in extreme value theory, see e.g. {{Cite|LLR}}.
 
In recent years, there has been a considerable
 
In recent years, there has been a considerable
 
development of central limit theory under "projective"
 
development of central limit theory under "projective"
 
criteria related to martingale theory (motivated
 
criteria related to martingale theory (motivated
 
by Gordin's martingale-approximation
 
by Gordin's martingale-approximation
technique --- see [HH]); for details,
+
technique --- see {{Cite|HH}}); for details,
see e.g. [Pe].
+
see e.g. {{Cite|Pe}}.
 
There are far too many other types of weak dependence
 
There are far too many other types of weak dependence
 
conditions, of the general spirit of strong mixing
 
conditions, of the general spirit of strong mixing
 
conditions but less restrictive, to describe here;
 
conditions but less restrictive, to describe here;
 
for more details, see
 
for more details, see
e.g. [DDLLLP] or [Br, v1, Chapter 13].
+
e.g. {{Cite|DDLLLP}} or {{Cite|Br| v1, Chapter 13}}.
 +
 
  
\hfil\break
 
  
\centerline
 
 
====References====
 
====References====
 
{|
 
{|
 
|-
 
|-
|valign="top"|{{Ref|1}}||valign="top"| [Br] R.C. Bradley.  ''Introduction to Strong Mixing Conditions\/'', Vols. 1, 2, and 3. Kendrick Press, Heber City (Utah), 2007.  
+
|valign="top"|{{Ref|Br}}||valign="top"|   R.C. Bradley.  ''Introduction to Strong Mixing Conditions'', Vols. 1, 2, and 3. Kendrick Press, Heber City (Utah), 2007.  
 
|-
 
|-
|valign="top"|{{Ref|2}}||valign="top"| [DDLLLP] J. Dedecker, P. Doukhan, G. Lang, J.R. Le&oacute;n, S. Louhichi, and C. Prieur.  ''Weak Dependence: Models, Theory, and Applications\/''. Lecture Notes in Statistics 190. Springer-Verlag, New York, 2007.  
+
|valign="top"|{{Ref|DDLLLP}}||valign="top"|   J. Dedecker, P. Doukhan, G. Lang, J.R. Le&oacute;n, S. Louhichi, and C. Prieur.  ''Weak Dependence: Models, Theory, and Applications''. Lecture Notes in Statistics 190. Springer-Verlag, New York, 2007.  
 
|-
 
|-
|valign="top"|{{Ref|3}}||valign="top"| [DMS] H. Dehling, T. Mikosch, and M. S\o rensen, eds.  ''Empirical Process Techniques for Dependent Data\/''. Birkh&auml;user, Boston, 2002.  
+
|valign="top"|{{Ref|DMS}}||valign="top"|   H. Dehling, T. Mikosch, and M. S&oslash;rensen, eds.  ''Empirical Process Techniques for Dependent Data''. Birkh&auml;user, Boston, 2002.  
 
|-
 
|-
|valign="top"|{{Ref|4}}||valign="top"| [De] M. Denker. The central limit theorem for dynamical systems. In:  ''Dynamical Systems and Ergodic Theory\/'', (K. Krzyzewski, ed.), pp. 33-62. Banach Center Publications, Polish Scientific Publishers, Warsaw, 1989.  
+
|valign="top"|{{Ref|De}}||valign="top"|   M. Denker. The central limit theorem for dynamical systems. In:  ''Dynamical Systems and Ergodic Theory'', (K. Krzyzewski, ed.), pp. 33-62. Banach Center Publications, Polish Scientific Publishers, Warsaw, 1989.  
 
|-
 
|-
|valign="top"|{{Ref|5}}||valign="top"| [Do] P. Doukhan.  ''Mixing: Properties and Examples\/''. Springer-Verlag, New York, 1995.  
+
|valign="top"|{{Ref|Do}}||valign="top"|   P. Doukhan.  ''Mixing: Properties and Examples''. Springer-Verlag, New York, 1995.  
 
|-
 
|-
|valign="top"|{{Ref|6}}||valign="top"| [HH] P. Hall and C.C. Heyde.  ''Martingale Limit Theory and its Application\/''. Academic Press, San Diego, 1980.  
+
|valign="top"|{{Ref|HH}}||valign="top"|   P. Hall and C.C. Heyde.  ''Martingale Limit Theory and its Application''. Academic Press, San Diego, 1980.  
 
|-
 
|-
|valign="top"|{{Ref|7}}||valign="top"| [IR] I.A. Ibragimov and Yu.A. Rozanov.  ''Gaussian Random Processes\/''. Springer-Verlag, New York, 1978.  
+
|valign="top"|{{Ref|IR}}||valign="top"|   I.A. Ibragimov and Yu.A. Rozanov.  ''Gaussian Random Processes''. Springer-Verlag, New York, 1978.  
 
|-
 
|-
|valign="top"|{{Ref|8}}||valign="top"| [Io] M. Iosifescu. Doeblin and the metric theory of continued fractions: a functional theoretic solution to Gauss' 1812 problem. In:  ''Doeblin and Modern Probability\/'', (H. Cohn, ed.), pp. 97-110. Contemporary Mathematics 149, American Mathematical Society, Providence, 1993.  
+
|valign="top"|{{Ref|Io}}||valign="top"|   M. Iosifescu. Doeblin and the metric theory of continued fractions: a functional theoretic solution to Gauss' 1812 problem. In:  ''Doeblin and Modern Probability'', (H. Cohn, ed.), pp. 97-110. Contemporary Mathematics 149, American Mathematical Society, Providence, 1993.  
 
|-
 
|-
|valign="top"|{{Ref|9}}||valign="top"| [Ja] A. Jakubowski.  ''Asymptotic Independent Representations for Sums and Order Statistics of Stationary Sequences\/''. Uniwersytet Miko\l aja Kopernika, Toru\'n, Poland, 1991.  
+
|valign="top"|{{Ref|Ja}}||valign="top"|   A. Jakubowski.  ''Asymptotic Independent Representations for Sums and Order Statistics of Stationary Sequences''. Uniwersytet Miko&#322;aja Kopernika, Toru&#324;, Poland, 1991.  
 
|-
 
|-
|valign="top"|{{Ref|10}}||valign="top"| [LL] Z. Lin and C. Lu.  ''Limit Theory for Mixing Dependent Random Variables\/''. Kluwer Academic Publishers, Boston, 1996.  
+
|valign="top"|{{Ref|LL}}||valign="top"|   Z. Lin and C. Lu.  ''Limit Theory for Mixing Dependent Random Variables''. Kluwer Academic Publishers, Boston, 1996.  
 
|-
 
|-
|valign="top"|{{Ref|11}}||valign="top"| [LLR] M.R. Leadbetter, G. Lindgren, and H. Rootz&eacute;n.  ''Extremes and Related Properties of Random Sequences and Processes\/''. Springer-Verlag, New York, 1983.  
+
|valign="top"|{{Ref|LLR}}||valign="top"|   M.R. Leadbetter, G. Lindgren, and H. Rootz&eacute;n.  ''Extremes and Related Properties of Random Sequences and Processes''. Springer-Verlag, New York, 1983.  
 
|-
 
|-
|valign="top"|{{Ref|12}}||valign="top"| [MT] S.P. Meyn and R.L. Tweedie.  ''Markov Chains and Stochastic Stability\/'' (3rd printing). Springer-Verlag, New York, 1996.  
+
|valign="top"|{{Ref|MT}}||valign="top"|   S.P. Meyn and R.L. Tweedie.  ''Markov Chains and Stochastic Stability'' (3rd printing). Springer-Verlag, New York, 1996.  
 
|-
 
|-
|valign="top"|{{Ref|13}}||valign="top"| [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.  
+
|valign="top"|{{Ref|Pe}}||valign="top"|   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.  
 
|-
 
|-
|valign="top"|{{Ref|14}}||valign="top"| [Ri] E. Rio.  ''Th&eacute;orie Asymptotique des Processus Al&eacute;atoires Faiblement D&eacute;pendants\/''. \break Math&eacute;matiques & Applications 31. Springer, Paris, 2000.  
+
|valign="top"|{{Ref|Ri}}||valign="top"|   E. Rio.  ''Th&eacute;orie Asymptotique des Processus Al&eacute;atoires Faiblement D&eacute;pendants''.   Math&eacute;matiques & Applications 31. Springer, Paris, 2000.  
 
|-
 
|-
|valign="top"|{{Ref|15}}||valign="top"| [Ro1] M. Rosenblatt. A central limit theorem and a strong mixing condition.  ''Proc. Natl. Acad. Sci. USA\/'' 42 (1956) 43-47.  
+
|valign="top"|{{Ref|Ro1}}||valign="top"|   M. Rosenblatt. A central limit theorem and a strong mixing condition.  ''Proc. Natl. Acad. Sci. USA'' 42 (1956) 43-47.  
 
|-
 
|-
|valign="top"|{{Ref|16}}||valign="top"| [Ro2] M. Rosenblatt.  ''Markov Processes, Structure and Asymptotic Behavior\/''. Springer-Verlag, New York, 1971.  
+
|valign="top"|{{Ref|Ro2}}||valign="top"|   M. Rosenblatt.  ''Markov Processes, Structure and Asymptotic Behavior''. Springer-Verlag, New York, 1971.  
 
|-
 
|-
|valign="top"|{{Ref|17}}||valign="top"| [Ro3] M. Rosenblatt.  ''Stationary Sequences and Random Fields\/''. Birkh&auml;user, Boston, 1985.  
+
|valign="top"|{{Ref|Ro3}}||valign="top"|   M. Rosenblatt.  ''Stationary Sequences and Random Fields''. Birkh&auml;user, Boston, 1985.  
 
|-
 
|-
|valign="top"|{{Ref|18}}||valign="top"| [\v Zu] I.G. \v Zurbenko.  ''The Spectral Analysis of Time Series\/''. North-Holland, Amsterdam, 1986.
+
|valign="top"|{{Ref|&#381;u}}||valign="top"|   I.G. &#381;urbenko.  ''The Spectral Analysis of Time Series''. North-Holland, Amsterdam, 1986.
 
|-
 
|-
 
|}
 
|}

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2020 Mathematics Subject Classification: Primary: 60G10 Secondary: 60G99 [MSN][ZBL]


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 $X := (X_k, k \in {\mathbf Z})$ 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

[Br] R.C. Bradley. Introduction to Strong Mixing Conditions, Vols. 1, 2, and 3. Kendrick Press, Heber City (Utah), 2007.
[DDLLLP] J. Dedecker, P. Doukhan, G. Lang, J.R. León, S. Louhichi, and C. Prieur. Weak Dependence: Models, Theory, and Applications. Lecture Notes in Statistics 190. Springer-Verlag, New York, 2007.
[DMS] H. Dehling, T. Mikosch, and M. Sørensen, eds. Empirical Process Techniques for Dependent Data. Birkhäuser, Boston, 2002.
[De] M. Denker. The central limit theorem for dynamical systems. In: Dynamical Systems and Ergodic Theory, (K. Krzyzewski, ed.), pp. 33-62. Banach Center Publications, Polish Scientific Publishers, Warsaw, 1989.
[Do] P. Doukhan. Mixing: Properties and Examples. Springer-Verlag, New York, 1995.
[HH] P. Hall and C.C. Heyde. Martingale Limit Theory and its Application. Academic Press, San Diego, 1980.
[IR] I.A. Ibragimov and Yu.A. Rozanov. Gaussian Random Processes. Springer-Verlag, New York, 1978.
[Io] M. Iosifescu. Doeblin and the metric theory of continued fractions: a functional theoretic solution to Gauss' 1812 problem. In: Doeblin and Modern Probability, (H. Cohn, ed.), pp. 97-110. Contemporary Mathematics 149, American Mathematical Society, Providence, 1993.
[Ja] A. Jakubowski. Asymptotic Independent Representations for Sums and Order Statistics of Stationary Sequences. Uniwersytet Mikołaja Kopernika, Toruń, Poland, 1991.
[LL] Z. Lin and C. Lu. Limit Theory for Mixing Dependent Random Variables. Kluwer Academic Publishers, Boston, 1996.
[LLR] M.R. Leadbetter, G. Lindgren, and H. Rootzén. Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New York, 1983.
[MT] S.P. Meyn and R.L. Tweedie. Markov Chains and Stochastic Stability (3rd printing). Springer-Verlag, New York, 1996.
[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.
[Ro3] M. Rosenblatt. Stationary Sequences and Random Fields. Birkhäuser, Boston, 1985.
[Žu] I.G. Žurbenko. The Spectral Analysis of Time Series. North-Holland, Amsterdam, 1986.



How to Cite This Entry:
Strong mixing conditions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Strong_mixing_conditions&oldid=38548