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==Sandbox==
 
==Sandbox==
 
Content:
 
\magnification=\magstep1
 
 
\def\refs{\medskip\hangindent=25pt\hangafter=1\noindent}
 
 
\centerline {\bf Strong Mixing Conditions}
 
\bigskip
 
\noindent Richard C. Bradley \hfil\break
 
\noindent Department of Mathematics, Indiana University,
 
Bloomington, Indiana, USA
 
\bigskip
 
 
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.
 
\smallskip
 
 
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.
 
\hfil\break
 
 
 
{\bf The strong mixing ($\alpha$-mixing) condition.}\ \ Suppose
 
$X := (X_k, k \in {\bf Z})$ 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
 
$X_k,\ j \leq k \leq \ell\ (k \in {\bf Z})$.
 
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)|. \eqno (1) $$
 
For the given random sequence $X$, for any positive
 
integer $n$, define the dependence coefficient
 
$$\alpha(n) = \alpha(X,n) :=
 
\sup_{j \in {\bf Z}}
 
\alpha({\cal F}_{-\infty}^j, {\cal F}_{j + n}^{\infty}).
 
\eqno (2) $$
 
By a trivial argument, the sequence of numbers
 
$(\alpha(n), n \in {\bf N})$ 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.)
 
\smallskip
 
 
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}).
 
\eqno (3) $$
 
For simplicity, {\it in the rest of this note,
 
we shall restrict to strictly stationary sequences\/}.
 
(Some comments below will have obvious adaptations to
 
nonstationary processes.) \smallskip
 
 
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].
 
\hfil\break
 
 
{\bf 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].)
 
\bigskip
 
 
{\bf Theorem 1.}\ \ {\sl Suppose $(X_k, k \in {\bf Z})$
 
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:
 
 
\noindent (A) The family of random variables
 
$(S_n^2/\sigma_n^2, n \in {\bf N})$ is uniformly
 
integrable.
 
 
\noindent (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 $t \to \infty$.}
 
\bigskip
 
 
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.
 
\hfil\break
 
 
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].
 
\smallskip
 
 
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].
 
\hfil\break
 
 
{\bf 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.
 
\smallskip
 
 
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.
 
\smallskip
 
 
Now suppose $X := (X_k, k \in {\bf Z})$ 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}),
 
\eqno (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
 
\hfil\break
 
``$\phi$-mixing'' if $\phi(n) \to 0$ as $n \to \infty$;
 
\hfil\break
 
``$\psi$-mixing'' if $\psi(n) \to 0$ as $n \to \infty$;
 
\hfil\break
 
``$\rho$-mixing'' if $\rho(n) \to 0$ as $n \to \infty$;
 
and
 
\hfil\break
 
``absolutely regular'', or ``$\beta$-mixing'', if $\beta(n) \to 0$ as $n \to \infty$.
 
\smallskip
 
 
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.
 
\smallskip
 
 
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: \hfil\break
 
\indent (i) $\psi$-mixing implies $\phi$-mixing. \hfil\break
 
\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.
 
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, 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].
 
\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
 

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  2. Some Administrator rights should be granted to Editorial-Board members as well
    1. List of rights: Mark pages as patrolled.
  3. Only encyclopedic articles (only the items that will eventually be in the Main name space) should appear in the “Pages A-Z” list. This is closely related with the next item:
  4. Reorganization of Namespace:
    1. Main should only contain encyclopedic articles
    2. Rename Encyclopedia of Mathematics namespace to EOM
    3. Place all non-encyclopedic pages in the EOM namespace
    4. Place the project talk page at “EOM:Talk:This Project”
  5. Modify CSS in order to wrap article text to match viewport width.
  6. Grant users the right to edit pages within Talk namesubspaces.
  7. Administrators should have the ability to change the editing restrictions on a particular page independent of namespace rules.
  8. History pages should display time stamps in UTC not CET (only if this is the wikipedia standard).

Requested Wiki Extensions

  • Extension:HarvardReferences
  • Extension:ConfirmEdit#MathCaptcha

Requests for Software Extensions

Comment: “It requires to have /usr/bin/asy installed. If you have asy at some other place please adjust the line

$this->asymptoteCommand = "/usr/bin/asy"; in the php file "Asymptote.php".”

Requests for Information

  • More detail about how to use the rights listed under administrator on [Special:ListGroupRights]

Notes relating to user intro pages

(Taken from an email by Ulf sent on 12/4/2011)

“Here is my intro page on en.wikipedia.org: http://en.wikipedia.org/wiki/User:Urehmann

It allows me to say something about me. I have almost left it as it was, except that I modified a bit the part "About me".

Here is the user page of some colleague, which is much more elaborated: http://en.wikipedia.org/wiki/User:Tsirel

What I think of is that we draft a template page (which is edible by me) a copy of which appears as a template for each new user, allowing to give a little info. For example, there should be a field which informs about the special research fields of the user, in keywords as well as in Math Subject Classification.

Also, there could be a field in which the names of pages could be included for which the user (e.g., an editor) could list the pages for which the user feels responsible.

So: As a template, the page should be edible by me, and the local copy should be edited by the user.”

Sandbox

How to Cite This Entry:
Nbrothers. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nbrothers&oldid=21673