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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 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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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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'''The strong mixing ($\alpha$-mixing) condition.'''
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{{MSC|62E}}
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Suppose
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{{TEX|done}}
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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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A
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$(\Omega, {\cal F}, P)$.
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[[Probability distribution|probability distribution]] of a random variable $X$ which takes non-negative integer values, defined by the formula
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For $-\infty \leq j \leq \ell \leq \infty$, let
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\begin{equation}\label{*}
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${\cal F}_j^\ell$ denote the $\sigma$-field of events
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P(X=k)=\frac{ {k+m-1 \choose k}{N-m-k \choose M-m} } { {N \choose M} } \tag{*}
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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)|.
 
 
\end{equation}
 
\end{equation}
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For the given random sequence $X$, for any positive
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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
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integer $n$, define the dependence coefficient
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[[Negative binomial distribution|negative binomial distribution]], which arises in the same way for sampling with replacement.
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\begin{equation}\alpha(n) = \alpha(X,n) :=
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\sup_{j \in {\bf Z}}
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The mathematical expectation and variance of a negative hypergeometric distribution are, respectively, equal to
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\alpha({\cal F}_{-\infty}^j, {\cal F}_{j + n}^{\infty}).
 
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\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",
 
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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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In the case where the given sequence $X$ is strictly
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\begin{equation}
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stationary (i.e. its distribution is invariant under a
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m\frac{N-M} {M+1}
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shift of the indices), eq. (2) also has the simpler form
 
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\begin{equation}\alpha(n) = \alpha(X,n) :=
 
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\alpha({\cal F}_{-\infty}^0, {\cal F}_n^{\infty}).
 
 
\end{equation}
 
\end{equation}
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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,
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and
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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.'''
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\begin{equation}
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Under $\alpha$-mixing and other similar conditions
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m\frac{(N+1)(N-M)} {(M+1)(M+2)}\Big(1-\frac{m}{M+1}\Big) \, .
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(including ones reviewed below), there has been a vast development of limit theory — for example,
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\end{equation}
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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
 
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of its proof, see e.g. [Br, v1,
 
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Theorems 1.19 and 10.2].)
 
  
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'''Theorem 1.'''
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When <math>N, M, N-M \to \infty</math> such that <math>M/N\to p</math>, the negative hypergeometric distribution tends to the
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''Suppose'' $(X_k, k \in {\bf Z})$
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[[negative binomial distribution]] with parameters <math>m</math> and <math>p</math>.
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''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''
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The distribution function <math>F(n)</math> of the negative hypergeometric function with parameters <math>N,M,m</math> is related to the
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$(S_n^2/\sigma_n^2, n \in {\bf N})$ ''is uniformly integrable.''
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[[Hypergeometric distribution|hypergeometric distribution]] <math>G(m)</math> with parameters <math>N,M,n</math> by the relation
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\begin{equation}
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(B) $S_n/\sigma_n \Rightarrow N(0,1)$ ''as''
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F(n) = 1-G(m-1) \, .
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$n \to \infty$.
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\end{equation}
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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
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[[Statistical quality control|statistical quality control]].
  
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''If (the hypothesis and) these two equivalent conditions'' (A) ''and'' (B) ''hold, then''
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====References====
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$\sigma_n^2 = n \cdot h(n)$ ''for some function'' $h(t),\ t \in (0, \infty)$ ''which is slowly varying as'' $t \to \infty$.
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{|
 +
|-
 +
|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}}
 +
|-
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|valign="top"|{{Ref|PaJo}}||valign="top"|  G.P. Patil,   S.W. Joshi,  "A dictionary and bibliography of discrete distributions", Hafner  (1968) {{MR|0282770}}
 +
|-
 +
|}

Latest revision as of 14:46, 5 June 2017



2020 Mathematics Subject Classification: Primary: 62E [MSN][ZBL]


A probability distribution of a random variable $X$ which takes non-negative integer values, defined by the formula \begin{equation}\label{*} P(X=k)=\frac{ {k+m-1 \choose k}{N-m-k \choose M-m} } { {N \choose M} } \tag{*} \end{equation} where the parameters \(N,M,m\) are non-negative integers which satisfy the condition \(m\leq M\leq N\). A negative hypergeometric distribution often arises in a scheme of sampling without replacement. If in the total population of size \(N\), there are \(M\) "marked" and \(N-M\) "unmarked" elements, and if the sampling (without replacement) is performed until the number of "marked" elements reaches a fixed number \(m\), then the random variable \(X\) — the number of "unmarked" elements in the sample — has a negative hypergeometric distribution \eqref{*}. The random variable \(X+m\) — the size of the sample — also has a negative hypergeometric distribution. The distribution \eqref{*} is called a negative hypergeometric distribution by analogy with the negative binomial distribution, which arises in the same way for sampling with replacement.

The mathematical expectation and variance of a negative hypergeometric distribution are, respectively, equal to

\begin{equation} m\frac{N-M} {M+1} \end{equation}

and

\begin{equation} m\frac{(N+1)(N-M)} {(M+1)(M+2)}\Big(1-\frac{m}{M+1}\Big) \, . \end{equation}

When \(N, M, N-M \to \infty\) such that \(M/N\to p\), the negative hypergeometric distribution tends to the negative binomial distribution with parameters \(m\) and \(p\).

The distribution function \(F(n)\) of the negative hypergeometric function with parameters \(N,M,m\) is related to the hypergeometric distribution \(G(m)\) with parameters \(N,M,n\) by the relation \begin{equation} F(n) = 1-G(m-1) \, . \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 statistical quality control.

References

[Be] Y.K. Belyaev, "Probability methods of sampling control", Moscow (1975) (In Russian) MR0428663
[BoSm] 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) MR0243650 Zbl 0529.62099
[JoKo] N.L. Johnson, S. Kotz, "Distributions in statistics, discrete distributions", Wiley (1969) MR0268996 Zbl 0292.62009
[PaJo] G.P. Patil, S.W. Joshi, "A dictionary and bibliography of discrete distributions", Hafner (1968) MR0282770
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