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| − | =Strong Mixing Conditions=
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| − | :Richard C. Bradley
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| − | :Department of Mathematics, Indiana University, Bloomington, Indiana, USA
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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 chain]]s, 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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| | | | |
| − | The field of strong mixing conditions is a vast area,
| + | {{MSC|62E}} |
| − | and a short note such as this cannot even begin to do
| + | {{TEX|done}} |
| − | 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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| | | | |
| − | '''The strong mixing ($\alpha$-mixing) condition.'''
| + | A |
| − | Suppose
| + | [[Probability distribution|probability distribution]] of a random variable $X$ which takes non-negative integer values, defined by the formula |
| − | $X := (X_k, k \in {\bf Z})$ is a sequence of
| + | \begin{equation}\label{*} |
| − | random variables on a given probability space
| + | P(X=k)=\frac{ {k+m-1 \choose k}{N-m-k \choose M-m} } { {N \choose M} } \tag{*} |
| − | $(\Omega, {\cal F}, P)$.
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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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| − | $X_k,\ j \le k \leq \ell\ (k \in {\bf Z})$.
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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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| − | \begin{equation} \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)|.
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| | \end{equation} | | \end{equation} |
| − | For the given random sequence $X$, for any positive
| + | where the parameters <math>N,M,m</math> are non-negative integers which satisfy the condition <math>m\leq M\leq N</math>. A negative hypergeometric distribution often arises in a scheme of sampling without replacement. If in the total population of size <math>N</math>, there are <math>M</math> "marked" and <math>N-M</math> "unmarked" elements, and if the sampling (without replacement) is performed until the number of "marked" elements reaches a fixed number <math>m</math>, then the random variable <math>X</math> — the number of "unmarked" elements in the sample — has a negative hypergeometric distribution \eqref{*}. The random variable <math>X+m</math> — the size of the sample — also has a negative hypergeometric distribution. The distribution \eqref{*} is called a negative hypergeometric distribution by analogy with the |
| − | integer $n$, define the dependence coefficient
| + | [[Negative binomial distribution|negative binomial distribution]], which arises in the same way for sampling with replacement. |
| − | \begin{equation}\alpha(n) = \alpha(X,n) := | + | |
| − | \sup_{j \in {\bf Z}}
| + | The mathematical expectation and variance of a negative hypergeometric distribution are, respectively, equal to |
| − | \alpha({\cal F}_{-\infty}^j, {\cal F}_{j + n}^{\infty}).
| |
| − | \end{equation}
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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", | |
| − | 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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| | | | |
| − | In the case where the given sequence $X$ is strictly
| + | \begin{equation} |
| − | stationary (i.e. its distribution is invariant under a
| + | m\frac{N-M} {M+1} |
| − | shift of the indices), eq. (2) also has the simpler form
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| − | \begin{equation}\alpha(n) = \alpha(X,n) := | |
| − | \alpha({\cal F}_{-\infty}^0, {\cal F}_n^{\infty}). | |
| | \end{equation} | | \end{equation} |
| − | For simplicity, ''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.)
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| | | | |
| − | In particular, for strictly stationary sequences,
| + | and |
| − | 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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| | | | |
| − | '''Comments on limit theory under $\alpha$-mixing.'''
| + | \begin{equation} |
| − | Under $\alpha$-mixing and other similar conditions
| + | m\frac{(N+1)(N-M)} {(M+1)(M+2)}\Big(1-\frac{m}{M+1}\Big) \, . |
| − | (including ones reviewed below), there has been a vast development of limit theory — for example, | + | \end{equation} |
| − | CLTs, weak invariance principles,
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| − | laws of the iterated logarithm, almost sure invariance
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| − | principles, and rates of convergence in the strong law of
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| − | 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 | |
| − | of its proof, see e.g. [Br, v1,
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| − | Theorems 1.19 and 10.2].)
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| | | | |
| − | '''Theorem 1.'''
| + | When <math>N, M, N-M \to \infty</math> such that <math>M/N\to p</math>, the negative hypergeometric distribution tends to the |
| − | ''Suppose'' $(X_k, k \in {\bf Z})$
| + | [[negative binomial distribution]] with parameters <math>m</math> and <math>p</math>. |
| − | ''is a strictly stationary sequence of random variables 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 equivalent:''
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| | | | |
| − | (A) ''The family of random variables''
| + | The distribution function <math>F(n)</math> of the negative hypergeometric function with parameters <math>N,M,m</math> is related to the |
| − | $(S_n^2/\sigma_n^2, n \in {\bf N})$ ''is uniformly integrable.''
| + | [[Hypergeometric distribution|hypergeometric distribution]] <math>G(m)</math> with parameters <math>N,M,n</math> by the relation |
| − | | + | \begin{equation} |
| − | (B) $S_n/\sigma_n \Rightarrow N(0,1)$ ''as''
| + | F(n) = 1-G(m-1) \, . |
| − | $n \to \infty$.
| + | \end{equation} |
| − | | + | This means that in solving problems in mathematical statistics related to negative hypergeometric distributions, tables of hypergeometric distributions can be used. The negative hypergeometric distribution is used, for example, in |
| − | ''If (the hypothesis and) these two equivalent conditions'' (A) ''and'' (B) ''hold, then''
| + | [[Statistical quality control|statistical quality control]]. |
| − | $\sigma_n^2 = n \cdot h(n)$ ''for some function'' $h(t),\ t \in (0, \infty)$ ''which is slowly varying as'' $t \to \infty$.
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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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| − | | |
| − | 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). | |
| − | 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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| − | | |
| − | Under the hypothesis (first sentence) of Theorem 1
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| − | (with just finite second moments), | |
| − | 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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| − | | |
| − | '''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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| − | | |
| − | 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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| − | \begin{eqnarray}
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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)|; \\
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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|; \\
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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} \\
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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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| − | \end{eqnarray} | |
| − | 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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| − | | |
| − | ==References==
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| − | | |
| − | | |
| − | [Br] R.C. Bradley.
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| − | ''Introduction to Strong Mixing Conditions,''
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| − | Vols. 1, 2, and 3.
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| − | Kendrick Press, Heber City (Utah), 2007.
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| − | | |
| − | [DDLLLP] J. Dedecker, P. Doukhan, G. Lang,
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| − | J.R. León, S. Louhichi, and C. Prieur.
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| − | ''Weak Dependence: Models, Theory, and Applications.''
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| − | Lecture Notes in Statistics 190. Springer-Verlag,
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| − | New York, 2007.
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| − | | |
| − | [DMS] H. Dehling, T. Mikosch, and M. Sørensen,
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| − | eds.
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| − | "Empirical Process Techniques for Dependent Data."
| |
| − | Birkhäuser, Boston, 2002.
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| − | | |
| − | [De] M. Denker. The central limit theorem for
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| − | dynamical systems.
| |
| − | In: ''Dynamical Systems and Ergodic Theory,''
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| − | (K. Krzyzewski, ed.), pp. 33-62.
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| − | Banach Center Publications, Polish Scientific Publishers,
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| − | Warsaw, 1989.
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| − | | |
| − | [Do] P. Doukhan.
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| − | ''Mixing: Properties and Examples.''
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| − | Springer-Verlag, New York, 1995.
| |
| − | | |
| − | ---------------------------------------------------
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| − | | |
| − | \noindent [HH] P.\ Hall and C.C.\ Heyde.
| |
| − | {\it Martingale Limit Theory and its Application\/}.
| |
| − | Academic Press, San Diego, 1980.
| |
| − | | |
| − | \noindent [IR] I.A.\ Ibragimov and Yu.A.\ Rozanov.
| |
| − | {\it Gaussian Random Processes\/}.
| |
| − | Springer-Verlag, New York, 1978.
| |
| − | | |
| − | \noindent [Io] M.\ Iosifescu.
| |
| − | Doeblin and the metric theory of continued fractions: a
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| − | functional theoretic solution to Gauss' 1812 problem.
| |
| − | In: {\it Doeblin and Modern Probability\/},
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| − | (H.\ Cohn, ed.), pp.\ 97-110.
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| − | Contemporary Mathematics 149,
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| − | American Mathematical Society, Providence, 1993.
| |
| − | | |
| − | \noindent [Ja] A.\ Jakubowski.
| |
| − | {\it Asymptotic Independent Representations for Sums and
| |
| − | Order Statistics of Stationary Sequences\/}.
| |
| − | Uniwersytet Miko\l aja Kopernika, Toru\'n, Poland, 1991.
| |
| − | | |
| − | \noindent [LL] Z.\ Lin and C.\ Lu.
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| − | {\it Limit Theory for Mixing Dependent Random Variables\/}.
| |
| − | Kluwer Academic Publishers, Boston, 1996.
| |
| − | | |
| − | \noindent [LLR] M.R.\ Leadbetter, G.\ Lindgren, and
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| − | H.\ Rootz\'en.
| |
| − | {\it Extremes and Related Properties of Random Sequences
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| − | and Processes\/}.
| |
| − | Springer-Verlag, New York, 1983.
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| − | | |
| − | \noindent [MT] S.P.\ Meyn and R.L.\ Tweedie.
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| − | {\it Markov Chains and Stochastic Stability\/} (3rd
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| − | printing). Springer-Verlag, New York, 1996.
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| − | | |
| − | \noindent [Pe] M.\ Peligrad.
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| − | Conditional central limit theorem via martingale
| |
| − | approximation.
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| − | In: {\it Dependence in Probability, Analysis and Number
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| − | Theory\/}, (I.\ Berkes, R.C.\ Bradley, H.\ Dehling,
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| − | M.\ Peligrad, and R.\ Tichy, eds.), pp.\ 295-309.
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| − | Kendrick Press, Heber City (Utah), 2010.
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| − | | |
| − | \noindent [Ri] E.\ Rio.
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| − | {\it Th\'eorie Asymptotique des Processus Al\'eatoires Faiblement D\'ependants\/}. \break
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| − | Math\'ematiques \& Applications 31.
| |
| − | Springer, Paris, 2000.
| |
| − | | |
| − | \noindent [Ro1] M.\ Rosenblatt. A central limit theorem and
| |
| − | a strong mixing condition.
| |
| − | {\it Proc.\ Natl.\ Acad.\ Sci.\ USA\/} 42 (1956) 43-47.
| |
| − | | |
| − | \noindent [Ro2] M.\ Rosenblatt.
| |
| − | {\it Markov Processes, Structure and Asymptotic Behavior\/}.
| |
| − | Springer-Verlag, New York, 1971.
| |
| − | | |
| − | \noindent [Ro3] M.\ Rosenblatt.
| |
| − | {\it Stationary Sequences and Random Fields\/}.
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| − | Birkh\"auser, Boston, 1985.
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| | | | |
| − | \noindent [\v Zu] I.G.\ \v Zurbenko.
| + | ====References==== |
| − | {\it The Spectral Analysis of Time Series\/}. | + | {| |
| − | North-Holland, Amsterdam, 1986.
| + | |- |
| | + | |valign="top"|{{Ref|Be}}||valign="top"| Y.K. Belyaev, "Probability methods of sampling control", Moscow (1975) (In Russian) {{MR|0428663}} |
| | + | |- |
| | + | |valign="top"|{{Ref|BoSm}}||valign="top"| L.N. Bol'shev, N.V. Smirnov, "Tables of mathematical statistics", ''Libr. math. tables'', '''46''', Nauka (1983) (In Russian) (Processed by L.S. Bark and E.S. Kedrova) {{MR|0243650}} {{ZBL|0529.62099}} |
| | + | |- |
| | + | |valign="top"|{{Ref|JoKo}}||valign="top"| N.L. Johnson, S. Kotz, "Distributions in statistics, discrete distributions", Wiley (1969) {{MR|0268996}} {{ZBL|0292.62009}} |
| | + | |- |
| | + | |valign="top"|{{Ref|PaJo}}||valign="top"| G.P. Patil, S.W. Joshi, "A dictionary and bibliography of discrete distributions", Hafner (1968) {{MR|0282770}} |
| | + | |- |
| | + | |} |