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| − | Content:
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| − | \magnification=\magstep1
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| − |
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| − | \def\refs{\medskip\hangindent=25pt\hangafter=1\noindent}
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| − |
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| − | \centerline {\bf Strong Mixing Conditions}
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| − | \bigskip
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| − | \noindent Richard C. Bradley \hfil\break
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| − | \noindent Department of Mathematics, Indiana University,
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| − | Bloomington, Indiana, USA
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| − | \bigskip
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| − |
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| − | There has been much research on stochastic models
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| − | that have a well defined, specific structure --- for
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| − | example, Markov chains, Gaussian processes, or
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| − | linear models, including ARMA
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| − | (autoregressive -- moving average) models.
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| − | However, it became clear in the middle of the last century
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| − | that there was a need for
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| − | a theory of statistical inference (e.g.\ central limit
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| − | theory) that could be used in the analysis of time series
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| − | that did not seem to ``fit'' any such specific structure
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| − | but which did seem to have some ``asymptotic
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| − | independence'' properties.
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| − | That motivated the development of a broad theory of
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| − | ``strong mixing conditions'' to handle such situations.
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| − | This note is a brief description of that theory.
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| − | \smallskip
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| − |
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| − | The field of strong mixing conditions is a vast area,
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| − | and a short note such as this cannot even begin to do
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| − | justice to it.
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| − | Journal articles (with one exception) will not be cited;
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| − | and many researchers who made important contributions to
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| − | this field will not be mentioned here.
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| − | All that can be done here is to give a narrow snapshot
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| − | of part of the field.
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| − | \hfil\break
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| − |
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| − |
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| − | {\bf The strong mixing ($\alpha$-mixing) condition.}\ \ Suppose
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| − | $X := (X_k, k \in {\bf Z})$ is a sequence of
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| − | random variables on a given probability space
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| − | $(\Omega, {\cal F}, P)$.
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| − | For $-\infty \leq j \leq \ell \leq \infty$, let
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| − | ${\cal F}_j^\ell$ denote the $\sigma$-field of events
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| − | generated by the random variables
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| − | $X_k,\ j \leq k \leq \ell\ (k \in {\bf Z})$.
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| − | For any two $\sigma$-fields ${\cal A}$ and
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| − | ${\cal B} \subset {\cal F}$, define the ``measure of
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| − | dependence''
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| − | $$ \alpha({\cal A}, {\cal B}) :=
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| − | \sup_{A \in {\cal A}, B \in {\cal B}}
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| − | |P(A \cap B) - P(A)P(B)|. \eqno (1) $$
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| − | For the given random sequence $X$, for any positive
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| − | integer $n$, define the dependence coefficient
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| − | $$\alpha(n) = \alpha(X,n) :=
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| − | \sup_{j \in {\bf Z}}
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| − | \alpha({\cal F}_{-\infty}^j, {\cal F}_{j + n}^{\infty}).
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| − | \eqno (2) $$
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| − | By a trivial argument, the sequence of numbers
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| − | $(\alpha(n), n \in {\bf N})$ is nonincreasing.
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| − | The random sequence $X$ is said to be ``strongly mixing'',
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| − | or ``$\alpha$-mixing'', if $\alpha(n) \to 0$ as
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| − | $n \to \infty$.
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| − | This condition was introduced in 1956 by Rosenblatt [Ro1],
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| − | and was used in that paper in the proof of a central limit
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| − | theorem.
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| − | (The phrase ``central limit theorem'' will henceforth
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| − | be abbreviated CLT.)
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| − | \smallskip
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| − |
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| − | In the case where the given sequence $X$ is strictly
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| − | stationary (i.e.\ its distribution is invariant under a
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| − | shift of the indices), eq.\ (2) also has the simpler form
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| − | $$\alpha(n) = \alpha(X,n) :=
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| − | \alpha({\cal F}_{-\infty}^0, {\cal F}_n^{\infty}).
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| − | \eqno (3) $$
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| − | For simplicity, {\it in the rest of this note,
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| − | we shall restrict to strictly stationary sequences\/}.
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| − | (Some comments below will have obvious adaptations to
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| − | nonstationary processes.) \smallskip
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| − |
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| − | In particular, for strictly stationary sequences,
| |
| − | the strong mixing ($\alpha$-mixing) condition implies Kolmogorov regularity
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| − | (a trivial ``past tail'' $\sigma$-field),
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| − | which in turn implies ``mixing'' (in the ergodic-theoretic
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| − | sense), which in turn implies ergodicity.
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| − | (None of the converse implications holds.)\ \
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| − | For further related information, see
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| − | e.g.\ [Br, v1, Chapter 2].
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| − | \hfil\break
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| − |
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| − | {\bf Comments on limit theory under
| |
| − | $\alpha$-mixing.}\ \
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| − | Under $\alpha$-mixing and other similar conditions
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| − | (including ones reviewed below), there has been a vast development of limit theory --- for example,
| |
| − | CLTs, weak invariance principles,
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| − | laws of the iterated logarithm, almost sure invariance
| |
| − | principles, and rates of convergence in the strong law of
| |
| − | large numbers.
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| − | For example, the CLT in [Ro1] evolved through
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| − | subsequent refinements by several researchers
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| − | into the following ``canonical'' form.
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| − | (For its history and a generously detailed presentation
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| − | of its proof, see e.g.\ [Br, v1,
| |
| − | Theorems 1.19 and 10.2].)
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| − | \bigskip
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| − |
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| − | {\bf Theorem 1.}\ \ {\sl Suppose $(X_k, k \in {\bf Z})$
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| − | is a strictly stationary sequence of random variables
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| − | such that
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| − | $EX_0 = 0$, $EX_0^2 < \infty$,
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| − | $\sigma_n^2 := ES_n^2 \to \infty$ as $n \to \infty$,
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| − | and $\alpha(n) \to 0$ as $n \to \infty$.
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| − | Then the following two conditions (A) and (B) are
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| − | equivalent:
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| − |
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| − | \noindent (A) The family of random variables
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| − | $(S_n^2/\sigma_n^2, n \in {\bf N})$ is uniformly
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| − | integrable.
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| − |
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| − | \noindent (B) $S_n/\sigma_n \Rightarrow N(0,1)$ as
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| − | $n \to \infty$.
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| − |
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| − | If (the hypothesis and) these two equivalent
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| − | conditions (A) and (B) hold, then
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| − | $\sigma_n^2 = n \cdot h(n)$ for some
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| − | function $h(t),\ t \in (0, \infty)$ which is slowly
| |
| − | varying as $t \to \infty$.}
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| − | \bigskip
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| − |
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| − | Here $S_n := X_1 + X_2 + \dots + X_n$; and\
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| − | $\Rightarrow$\ denotes convergence in distribution.
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| − | The assumption $ES_n^2 \to \infty$ is needed here in
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| − | order to avoid trivial $\alpha$-mixing (or even
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| − | 1-dependent) counterexamples in which a kind of ``cancellation'' prevents the partial sums $S_n$ from
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| − | ``growing'' (in probability) and becoming asymptotically
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| − | normal.
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| − | \hfil\break
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| − |
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| − | In the context of Theorem 1, if one wants to obtain asymptotic normality of the
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| − | partial sums (as in condition (B)) without an explicit
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| − | uniform integrability assumption on the partial sums
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| − | (as in condition (A)),
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| − | then as an alternative, one can impose a combination of assumptions on, say, (i) the (marginal) distribution
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| − | of $X_0$ and (ii) the rate of decay of the
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| − | numbers $\alpha(n)$ to 0 (the ``mixing rate'').
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| − | This involves a ``trade-off''; the weaker one assumption
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| − | is, the stronger the other has to be.
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| − | One such CLT of Ibragimov in 1962
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| − | involved such a ``trade-off'' in which it is assumed that
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| − | for some $\delta > 0$,
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| − | $E|X_0|^{2 + \delta} < \infty$ and
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| − | $\sum_{n=1}^\infty [\alpha(n)]^{\delta/(2 + \delta)}
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| − | < \infty$.
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| − | Counterexamples of Davydov in 1973
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| − | (with just slightly weaker properties) showed that that
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| − | result is quite sharp.
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| − | However, it is not at the exact ``borderline''.
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| − | From a covariance inequality of Rio in 1993 and a
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| − | CLT (in fact a weak invariance principle)
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| − | of Doukhan, Massart, and Rio in 1994, it became clear that
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| − | the ``exact borderline'' CLTs of this
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| − | kind have to involve quantiles of the (marginal)
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| − | distribution of $X_0$ (rather than just moments).
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| − | For a generously detailed exposition of such CLTs,
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| − | see [Br, v1, Chapter 10]; and for further
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| − | related results, see also Rio [Ri].
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| − | \smallskip
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| − |
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| − | Under the hypothesis (first sentence) of Theorem 1
| |
| − | (with just finite second moments),
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| − | there is no mixing rate, no matter how fast
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| − | (short of $m$-dependence), that can insure that
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| − | a CLT holds.
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| − | That was shown in 1983 with two different
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| − | counterexamples, one by the author and the other by
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| − | Herrndorf.
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| − | See [Br, v1\&3, Theorem 10.25 and Chapter 31].
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| − | \hfil\break
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| − |
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| − | {\bf Several other classic strong mixing conditions.}\ \
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| − | As indicated above, the terms ``$\alpha$-mixing'' and
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| − | ``strong mixing condition'' (singular) both refer to the condition $\alpha(n) \to 0$.
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| − | (A little caution is in order;
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| − | in ergodic theory, the term ``strong mixing'' is often
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| − | used to refer to the condition of
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| − | ``mixing in the ergodic-theoretic sense'',
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| − | which is weaker than
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| − | $\alpha$-mixing as noted earlier.)\ \
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| − | The term ``strong mixing conditions'' (plural) can
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| − | reasonably be thought of as referring
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| − | to all conditions that are at least as strong
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| − | as (i.e.\ that imply) $\alpha$-mixing.
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| − | In the classical theory, five strong mixing conditions
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| − | (again, plural) have emerged as the most prominent ones:
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| − | $\alpha$-mixing itself and four others that will be
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| − | defined here.
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| − | \smallskip
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| − |
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| − | Recall our probability space $(\Omega, {\cal F}, P)$.
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| − | For any two $\sigma$-fields ${\cal A}$ and
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| − | ${\cal B} \subset {\cal F}$, define the following four ``measures of dependence'':
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| − | $$ \eqalignno{
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| − | \phi({\cal A}, {\cal B}) &:=
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| − | \sup_{A \in {\cal A}, B \in {\cal B}, P(A) > 0}
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| − | |P(B|A) - P(B)|; & (4) \cr
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| − | \psi({\cal A}, {\cal B}) &:=
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| − | \sup_{A \in {\cal A}, B \in {\cal B}, P(A) > 0, P(B) > 0}
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| − | |P(B \cap A)/[P(A)P(B)]\thinspace -\thinspace 1|; & (5) \cr
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| − | \rho({\cal A}, {\cal B}) &:=
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| − | \sup_{f \in {\cal L}^2({\cal A}),\thinspace g \in {\cal L}^2({\cal B})}
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| − | |{\rm Corr}(f,g)|; \quad {\rm and} & (6) \cr
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| − | \beta ({\cal A}, {\cal B}) &:= \sup\ (1/2)
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| − | \sum_{i=1}^I \sum_{j=1}^J |P(A_i \cap B_j) - P(A_i)P(B_j)|
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| − | & (7) \cr } $$
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| − | where the latter supremum is taken over all pairs of finite
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| − | partitions $(A_1, A_2, \dots, A_I)$ and
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| − | $(B_1, B_2, \dots, B_J)$ of $\Omega$
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| − | such that $A_i \in {\cal A}$ for
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| − | each $i$ and $B_j \in {\cal B}$ for each $j$.
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| − | In (6), for a given $\sigma$-field
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| − | ${\cal D} \subset {\cal F}$,
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| − | the notation ${\cal L}^2({\cal D})$ refers to the space of
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| − | (equivalence classes of) square-integrable,
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| − | ${\cal D}$-measurable random variables.
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| − | \smallskip
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| − |
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| − | Now suppose $X := (X_k, k \in {\bf Z})$ is a strictly
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| − | stationary sequence of random variables on
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| − | $(\Omega, {\cal F}, P)$.
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| − | For any positive integer $n$, analogously to (3), define
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| − | the dependence coefficient
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| − | $$\phi(n) = \phi(X,n) :=
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| − | \phi({\cal F}_{-\infty}^0, {\cal F}_n^{\infty}),
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| − | \eqno (8) $$
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| − | and define analogously the dependence coefficients
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| − | $\psi(n)$, $\rho(n)$, and $\beta(n)$.
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| − | Each of these four sequences of dependence
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| − | coefficients is trivially nonincreasing.
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| − | The (strictly stationary) sequence $X$ is said to be
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| − | \hfil\break
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| − | ``$\phi$-mixing'' if $\phi(n) \to 0$ as $n \to \infty$;
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| − | \hfil\break
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| − | ``$\psi$-mixing'' if $\psi(n) \to 0$ as $n \to \infty$;
| |
| − | \hfil\break
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| − | ``$\rho$-mixing'' if $\rho(n) \to 0$ as $n \to \infty$;
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| − | and
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| − | \hfil\break
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| − | ``absolutely regular'', or ``$\beta$-mixing'', if $\beta(n) \to 0$ as $n \to \infty$.
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| − | \smallskip
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| − |
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| − | The $\phi$-mixing condition was introduced by
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| − | Ibragimov in 1959 and was also studied by Cogburn in 1960 .
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| − | The $\psi$-mixing condition evolved through papers of Blum,
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| − | Hanson, and Koopmans in 1963 and Philipp in 1969; and
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| − | (see e.g.\ [Io]) it was also implicitly present
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| − | in earlier work of Doeblin in 1940 involving the metric
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| − | theory of continued fractions.
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| − | The $\rho$-mixing condition was introduced by
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| − | Kolmogorov and Rozanov 1960.
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| − | (The ``maximal correlation coefficient''
| |
| − | $\rho({\cal A}, {\cal B})$ itself was first studied by
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| − | 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
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| − | is also called the ``weak Bernoulli'' condition.
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| − | \smallskip
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| − |
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| − | For the five measures of dependence in (1) and (4)--(7),
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| − | one has the following well known inequalities:
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| − | $$ \eqalignno{
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| − | 2\alpha({\cal A}, {\cal B}) \thinspace & \leq \thinspace
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| − | \beta({\cal A}, {\cal B}) \thinspace \leq \thinspace
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| − | \phi({\cal A}, {\cal B}) \thinspace \leq \thinspace
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| − | (1/2) \psi({\cal A}, {\cal B}); \cr
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| − | 4 \alpha({\cal A}, {\cal B})\thinspace &\leq \thinspace
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| − | \rho({\cal A}, {\cal B}) \thinspace \leq \thinspace
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| − | \psi({\cal A}, {\cal B}); \quad {\rm and} \cr
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| − | \rho({\cal A}, {\cal B}) \thinspace &\leq \thinspace
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| − | 2 [\phi({\cal A}, {\cal B})]^{1/2}
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| − | [\phi({\cal B}, {\cal A})]^{1/2} \thinspace \leq
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| − | \thinspace
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| − | 2 [\phi({\cal A}, {\cal B})]^{1/2}. \cr
| |
| − | } $$
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| − | For a history and proof of these inequalities, see e.g.\
| |
| − | [Br, v1, Theorem 3.11].
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| − | As a consequence of these inequalities and some
| |
| − | well known examples, one has the following ``hierarchy''
| |
| − | of the five strong mixing conditions here: \hfil\break
| |
| − | \indent (i) $\psi$-mixing implies $\phi$-mixing. \hfil\break
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| − | \indent (ii) $\phi$-mixing implies both $\rho$-mixing and
| |
| − | $\beta$-mixing (absolute regularity). \hfil\break
| |
| − | \indent (iii) $\rho$-mixing and $\beta$-mixing each imply
| |
| − | $\alpha$-mixing (strong mixing). \hfil\break
| |
| − | \indent (iv) Aside from ``transitivity'', there are in
| |
| − | general
| |
| − | no other implications between these five mixing conditions.
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| − | In particular, neither of the conditions $\rho$-mixing
| |
| − | and $\beta$-mixing implies the other. \smallskip
| |
| − |
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| − | For all of these mixing conditions, the
| |
| − | ``mixing rates'' can be essentially arbitrary, and in particular, arbitrarily slow.
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| − | 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].
| |
| − | \smallskip
| |
| − |
| |
| − | The various strong mixing conditions above have been
| |
| − | used extensively in statistical inference for weakly
| |
| − | dependent data.
| |
| − | See e.g.\ [DDLLLP], [DMS], [Ro3], or [\v Zu].
| |
| − | \hfil\break
| |
| − |
| |
| − |
| |
| − | {\bf Ibragimov's conjecture and related material.}\ \
| |
| − | Suppose (as in Theorem 1) $X := (X_k, k \in {\bf Z})$
| |
| − | is a strictly stationary
| |
| − | sequence of random variables such that
| |
| − | $$ EX_0 = 0,\ \ EX_0^2 < \infty,\ \ {\rm and}\ \
| |
| − | ES_n^2 \to \infty\ {\rm as}\ n \to \infty. \eqno (9) $$
| |
| − |
| |
| − | 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].)
| |
| − | \smallskip
| |
| − |
| |
| − | 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].
| |
| − | \smallskip
| |
| − |
| |
| − | 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].
| |
| − | \smallskip
| |
| − |
| |
| − | There is a vast literature on central limit theory for
| |
| − | random fields satisfying various strong mixing conditions.
| |
| − | See e.g.\ [Ro3], [\v Zu], [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].
| |
| − | \hfil\break
| |
| − |
| |
| − |
| |
| − | {\bf 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.
| |
| − | \smallskip
| |
| − |
| |
| − | {\it Markov chains.}\ \ Suppose
| |
| − | $X := (X_k, k \in {\bf Z})$ 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].
| |
| − | \smallskip
| |
| − |
| |
| − | {\it Stationary Gaussian sequences.}\ \ For
| |
| − | stationary Gaussian sequences
| |
| − | $X := (X_k, k \in {\bf Z})$, 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].
| |
| − | \smallskip
| |
| − |
| |
| − | {\it 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].
| |
| − | \smallskip
| |
| − |
| |
| − | {\it 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].
| |
| − | \smallskip
| |
| − |
| |
| − | However, many strictly stationary linear
| |
| − | processes {\it fail\/} to be $\alpha$-mixing.
| |
| − | A well known classic example is the
| |
| − | strictly stationary AR(1) process
| |
| − | (autoregressive process of order 1)
| |
| − | $X := (X_k, k \in {\bf Z})$ of the form
| |
| − | $X_k = (1/2)X_{k-1} + \xi_k$ where
| |
| − | $(\xi_k, k \in {\bf Z})$ 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].
| |
| − | \hfil\break
| |
| − |
| |
| − | {\bf 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].
| |
| − |
| |
| − | \hfil\break
| |
| − |
| |
| − | \centerline {\bf References}
| |
| − | \bigskip
| |
| − |
| |
| − | \refs [Br] R.C.\ Bradley.
| |
| − | {\it Introduction to Strong Mixing Conditions\/},
| |
| − | Vols.\ 1, 2, and 3.
| |
| − | Kendrick Press, Heber City (Utah), 2007.
| |
| − |
| |
| − | \refs [DDLLLP] J.\ Dedecker, P.\ Doukhan, G.\ Lang,
| |
| − | J.R.\ Le\'on, S.\ Louhichi, and C.\ Prieur.
| |
| − | {\it Weak Dependence: Models, Theory, and Applications\/}.
| |
| − | Lecture Notes in Statistics 190. Springer-Verlag,
| |
| − | New York, 2007.
| |
| − |
| |
| − | \refs [DMS] H.\ Dehling, T.\ Mikosch, and M.\ S\o rensen,
| |
| − | eds.
| |
| − | {\it Empirical Process Techniques for Dependent Data\/}.
| |
| − | Birkh\"auser, Boston, 2002.
| |
| − |
| |
| − | \refs [De] M.\ Denker. The central limit theorem for
| |
| − | dynamical systems.
| |
| − | In: {\it Dynamical Systems and Ergodic Theory\/},
| |
| − | (K.\ Krzyzewski, ed.), pp.\ 33-62.
| |
| − | Banach Center Publications, Polish Scientific Publishers,
| |
| − | Warsaw, 1989.
| |
| − |
| |
| − | \refs [Do] P.\ Doukhan.
| |
| − | {\it Mixing: Properties and Examples\/}.
| |
| − | Springer-Verlag, New York, 1995.
| |
| − |
| |
| − | \refs [HH] P.\ Hall and C.C.\ Heyde.
| |
| − | {\it Martingale Limit Theory and its Application\/}.
| |
| − | Academic Press, San Diego, 1980.
| |
| − |
| |
| − | \refs [IR] I.A.\ Ibragimov and Yu.A.\ Rozanov.
| |
| − | {\it Gaussian Random Processes\/}.
| |
| − | Springer-Verlag, New York, 1978.
| |
| − |
| |
| − | \refs [Io] M.\ Iosifescu.
| |
| − | Doeblin and the metric theory of continued fractions: a
| |
| − | functional theoretic solution to Gauss' 1812 problem.
| |
| − | In: {\it Doeblin and Modern Probability\/},
| |
| − | (H.\ Cohn, ed.), pp.\ 97-110.
| |
| − | Contemporary Mathematics 149,
| |
| − | American Mathematical Society, Providence, 1993.
| |
| − |
| |
| − | \refs [Ja] A.\ Jakubowski.
| |
| − | {\it Asymptotic Independent Representations for Sums and
| |
| − | Order Statistics of Stationary Sequences\/}.
| |
| − | Uniwersytet Miko\l aja Kopernika, Toru\'n, Poland, 1991.
| |
| − |
| |
| − | \refs [LL] Z.\ Lin and C.\ Lu.
| |
| − | {\it Limit Theory for Mixing Dependent Random Variables\/}.
| |
| − | Kluwer Academic Publishers, Boston, 1996.
| |
| − |
| |
| − | \refs [LLR] M.R.\ Leadbetter, G.\ Lindgren, and
| |
| − | H.\ Rootz\'en.
| |
| − | {\it Extremes and Related Properties of Random Sequences
| |
| − | and Processes\/}.
| |
| − | Springer-Verlag, New York, 1983.
| |
| − |
| |
| − | \refs [MT] S.P.\ Meyn and R.L.\ Tweedie.
| |
| − | {\it Markov Chains and Stochastic Stability\/} (3rd
| |
| − | printing). Springer-Verlag, New York, 1996.
| |
| − |
| |
| − | \refs [Pe] M.\ Peligrad.
| |
| − | Conditional central limit theorem via martingale
| |
| − | approximation.
| |
| − | In: {\it 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.
| |
| − |
| |
| − | \refs [Ri] E.\ Rio.
| |
| − | {\it Th\'eorie Asymptotique des Processus Al\'eatoires Faiblement D\'ependants\/}. \break
| |
| − | Math\'ematiques \& Applications 31.
| |
| − | Springer, Paris, 2000.
| |
| − |
| |
| − | \refs [Ro1] M.\ Rosenblatt. A central limit theorem and
| |
| − | a strong mixing condition.
| |
| − | {\it Proc.\ Natl.\ Acad.\ Sci.\ USA\/} 42 (1956) 43-47.
| |
| − |
| |
| − | \refs [Ro2] M.\ Rosenblatt.
| |
| − | {\it Markov Processes, Structure and Asymptotic Behavior\/}.
| |
| − | Springer-Verlag, New York, 1971.
| |
| − |
| |
| − | \refs [Ro3] M.\ Rosenblatt.
| |
| − | {\it Stationary Sequences and Random Fields\/}.
| |
| − | Birkh\"auser, Boston, 1985.
| |
| − |
| |
| − | \refs [\v Zu] I.G.\ \v Zurbenko.
| |
| − | {\it The Spectral Analysis of Time Series\/}.
| |
| − | North-Holland, Amsterdam, 1986.
| |
| − |
| |
| − |
| |
| − | \bye
| |