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=Strong Mixing Conditions=
  
 +
:Richard C. Bradley
 +
:Department of Mathematics, Indiana University, Bloomington, Indiana, USA
  
\begin{equation}\label{ab}
+
There has been much research on stochastic models
E=mc^2
+
that have a well defined, specific structure — for
\end{equation}
+
example, Markov chains, Gaussian processes, or
By \eqref{ab}, it is possible. But see \eqref{ba} below:
+
linear models, including ARMA
\begin{equation}\label{ba}
+
(autoregressive – moving average) models.
E\ne mc^3,
+
However, it became clear in the middle of the last century
\end{equation}
+
that there was a need for
which is a pity.
+
a theory of statistical inference (e.g. central limit
 +
theory) that could be used in the analysis of time series
 +
that did not seem to "fit" any such specific structure
 +
but which did seem to have some "asymptotic
 +
independence" properties.
 +
That motivated the development of a broad theory of
 +
"strong mixing conditions" to handle such situations.
 +
This note is a brief description of that theory.
  
------------------------------
+
The field of strong mixing conditions is a vast area,
 
+
and a short note such as this cannot even begin to do
 
+
justice to it.
A function $f$, defined on some interval, satisfying the condition
+
Journal articles (with one exception) will not be cited;
 
+
and many researchers who made important contributions to
$$
+
this field will not be mentioned here.
f \left( \frac{x_1 + x_2}{2} \right) \le \frac{f(x_1) + f(x_2)}{2} \tag{1}
+
All that can be done here is to give a narrow snapshot
$$
+
of part of the field.
 
 
for  every two points $x_1$ and $x_2$ from this interval. The geometrical  meaning of this condition is that the midpoint of any chord of the graph  of the function $f$ is located either above the graph or on it. If the  inequality (1) is strict for all $x_1$ and $x_2$, then $f$ is called  strictly convex. Examples of convex functions include $x^p, p \ge 1$, $x  \ln x$ for $x > 0$, and $\left| x \right|$ for all $x$. If the sign  of inequality (1) is reversed, the function is called concave. All  measurable convex functions on open intervals are continuous. There  exist convex functions which are not continuous, but they are very  irregular: If a function $f$ is convex on the interval $(a, b)$ and is  bounded from above on some interval lying inside $(a, b)$, it is  continuous on $(a, b)$. Thus, a discontinuous convex function is  unbounded on any interior interval and is not measurable.
 
 
 
If  a function $f$ is continuous on an interval, and if each chord of its  graph contains at least one point other than the end points of the chord  and lying above the graph or on it, $f$ is convex. It follows from  condition (1) that for a continuous function the centre of gravity of  any finite number of material points lying on the graph of the function  lies either above the graph or on it: For any numbers $p_k > 0, k =  1, \ldots, n$ (where $n$ is arbitrary), the [[Jensen inequality|Jensen  inequality]]
 
 
 
$$
 
f \left( \frac{\sum_{k = 1}^{n}  p_k x_k}{\sum_{k = 1}^{n} p_k} \right) \le \frac{\sum_{k = 1}^{n} p_k  f(x_k)}{\sum_{k = 1}^{n} p_k} \tag{2}
 
$$
 
 
 
is valid.
 
 
 
If,  for some function $f$, inequality (2) is true for any two points $x_1$  and $x_2$ in some interval and any $p_1 > 0$ and $p_2 > 0$, the  function $f$ is continuous and, of course, convex on this interval. Any  chord of the graph of a continuous convex function coincides with the  corresponding part of the graph or lies entirely above the graph except  for its end points. This means that if a continuous convex function is  not linear on any interval, strict inequality is realized in (1) and (2)  for any pairwise different values of the argument, i.e. $f$ is a  strictly convex function.
 
 
 
A continuous function is  convex if and only if the set of points of the plane located above its  graph, i.e. its [[Supergraph|supergraph]], is a [[Convex set|convex  set]]. For a continuous function $f$, defined on an interval $(a, b)$,  to be convex, it is necessary and sufficient that for each point on the  graph there be at least one straight line (known as a supporting line)  situated under the graph (on the interval $(a, b)$) or partly on the  graph that passes through the point, i.e. for any point $x_0 \in (a, b)$  there exists a $k = k(x_0)$ such that
 
 
 
$$
 
f(x_0) + k(x - x_0) \le f(x) \tag{3}
 
$$
 
 
 
for all $x \in (a, b)$.
 
 
 
A  continuous function which is convex on an open interval has no strict  local maximum. If a function $f$ is continuous and convex on an interval  $(a, b)$, it has, at each one of its points $x_0$, a finite left,  $D_{-} f(x_0)$, and right, $D_{+} f(x_0)$, derivative; moreover, $D_{-}  f(x_0) \le D_{+} f(x_0)$ and, in addition, if the number $k = k(x_0)$  satisfies condition (3), the inequalities $D_{-} f(x_0) \le k(x_0) \le  D_{+} f(x_0)$< hold. The functions $D_{-} f(x)$ and $D_{+} f(x)$ do not decrease, and at all points except, possibly, a countable number of  them, $D_{-} f(x) = D_{+} f(x) = f'(x)$, so that $f$ is differentiable  at these points. On each closed interval located inside $(a, b)$ the  function $f$ satisfies a Lipschitz condition and is thus absolutely  continuous. This makes it possible to establish the following convexity  criterion: A continuous function is convex if and only if it is the  indefinite integral of a non-decreasing function.
 
 
 
If a  function $f$ is differentiable on an interval, it is (strictly) convex  on this interval if and only if its derivative does not decrease (is  increasing). At a point of the graph of a continuous convex function at  which the function is differentiable there exists a unique supporting  line — the tangent at this point. On the other hand, if, at any point of  the graph of a function which is differentiable on an interval, the  tangent to the graph at that point lies under the graph in some  neighbourhood of that point (except at the tangency point itself), the  function is strictly convex; if it lies under the graph or partly on it,  it is just a convex function.
 
 
 
If the function $f$ is  twice-differentiable on the interval, it is convex on this interval if  and only if its second derivative is non-negative on this interval (this  theorem is valid for the second symmetric derivative, as well as for  the ordinary second derivative). If the function has a positive second  derivative at each point of some interval, it is strictly convex on that  interval.
 
 
 
If the functions $f_i$ are convex on an interval $(a, b)$ and $p_i > 0, i = 1, \ldots, n$, then the function
 
 
 
$$
 
f = \sum\limits_{i = 1}^{n} p_i f_i
 
$$
 
 
 
is also convex on this interval; also, if even one of the functions $f_i$ is strictly convex, $f$ is strictly convex as well.
 
 
 
There  exist various generalizations of the concept of convexity to functions  of several variables. For instance, let a function $y = f(x^1, \ldots,  x^n)$ be defined on a convex set $M$ of the $n$-dimensional affine space  $E^n$. The function $f$ is called convex if inequality (1) is valid for  all points $x_1 \in M$ and $x_2 \in M$, where $x_1 + x_2$ denotes the  sum of the $n$-dimensional vectors $x_1$ and $x_2$. The properties of a  convex function of one variable are correspondingly generalized to  functions of several variables; for example, inequality (2) is satisfied  only for continuous convex functions. A continuous function is convex  if and only if the set of points $(x^1, \ldots, x^n, y)$ of the space  $E^{n + 1}$ lying above its graph is convex.
 
 
 
A  continuous function $f$ defined on a convex domain $G$ is convex if and only if for each point $x \in G$ there exists a linear function
 
 
 
$$
 
l(y) = a_1 y^1 + \ldots + a_n y^n + b,
 
$$
 
 
 
such that
 
 
 
$$
 
f(x) = l(x),\qquad f(y) \ge l(y),\qquad y \in G. \tag{4}
 
$$
 
 
 
The hyperplane defined by the equation $l(y) = 0$ is called a supporting hyperplane.
 
 
 
If a function $f$ is continuously differentiable in $G$, condition (4) is equivalent to the condition
 
 
 
$$
 
f(y) - f(x) - \sum\limits_{i = 1}^{n} \frac{\partial f(x)}{\partial x^i} (y^i - x^i) \ge 0,\qquad x, y \in G.
 
$$
 
 
 
If  $f$ is twice-differentiable, condition (4) is equivalent to the  condition that the second differential of the function, i.e. the  quadratic form
 
 
 
$$
 
\sum\limits_{i = 1}^{n} \sum\limits_{j = 1}^{n} \frac{\partial f(x)}{\partial x^i \partial x^j} \xi^i \xi^j
 
$$
 
 
 
is non-negative for all $x \in G$.
 
 
 
Another  important generalization of the concept of a convex function for  functions of several variables is the concept of a [[Subharmonic  function|subharmonic function]]. The concept of a convex function can be extended in a natural manner to include functions defined on  corresponding subsets of infinite-dimensional linear spaces; cf.  [[Convex functional|Convex functional]].
 
 
 
====References====
 
 
 
<table><TR><TD  valign="top">[1]</TD> <TD valign="top"> N. Bourbaki,  "Elements of mathematics. Functions of a real variable" , Addison-Wesley  (1976) pp. Chapt. 1 Sect. 4 (Translated from  French)</TD></TR><TR><TD  valign="top">[2]</TD> <TD valign="top"> A. Zygmund,  "Trigonometric series" , '''1–2''' , Cambridge Univ. Press (1988) pp.  Chapt. 1</TD></TR><TR><TD  valign="top">[3]</TD> <TD valign="top"> L.D. Kudryavtsev,  "Mathematical analysis" , '''1''' , Moscow (1973) pp. Chapt. 1 (In  Russian)</TD></TR><TR><TD  valign="top">[4]</TD> <TD valign="top"> I.P. Natanson,  "Theorie der Funktionen einer reellen Veränderlichen" , Deutsch. Verlag  Wissenschaft. , Frankfurt a.M. (1961) pp. Chapt. 10 (Translated from  Russian)</TD></TR><TR><TD  valign="top">[5]</TD> <TD valign="top"> S.M. Nikol'skii,  "A course of mathematical analysis" , '''1–2''' , MIR (1977) pp. Chapt. 5  (Translated from Russian)</TD></TR><TR><TD  valign="top">[6]</TD> <TD valign="top"> G.H. Hardy, J.E.  Littlewood, G. Pólya, "Inequalities" , Cambridge Univ. Press (1934) pp.  Chapt. 3</TD></TR></table>
 
 
 
====Comments====
 
 
 
Convexity of a real-valued function $f$ on an interval $I$ is often defined by the condition that
 
 
 
$$
 
f((1 - \alpha) x + \alpha y) \le (1 - \alpha) f(x) + \alpha f(y) \tag{c1}
 
$$
 
 
 
whenever  $x, y \in I$ and $0 \le \alpha \le 1$. This implies that $f$ is  continuous on the interior of $I$. For measurable $f$, (1) and (c1) are  equivalent. A function $f$ satisfying (1) is also called midpoint  convex.
 
 
 
====References====
 
 
 
<table><TR><TD  valign="top">[a1]</TD> <TD valign="top"> K.R. Stromberg,  "Introduction to classical real analysis" , Wadsworth (1981) pp.  199–206; 334</TD></TR><TR><TD  valign="top">[a2]</TD> <TD valign="top"> V. Barbu, Th.  Precupanu, "Convexity and optimization in Banach spaces" , Reidel (1986)  pp. Chapt. 2</TD></TR></table>
 

Revision as of 06:21, 2 August 2013

Strong Mixing Conditions

Richard C. Bradley
Department of Mathematics, Indiana University, Bloomington, Indiana, USA

There has been much research on stochastic models that have a well defined, specific structure — for example, Markov chains, Gaussian processes, or linear models, including ARMA (autoregressive – moving average) models. However, it became clear in the middle of the last century that there was a need for a theory of statistical inference (e.g. central limit theory) that could be used in the analysis of time series that did not seem to "fit" any such specific structure but which did seem to have some "asymptotic independence" properties. That motivated the development of a broad theory of "strong mixing conditions" to handle such situations. This note is a brief description of that theory.

The field of strong mixing conditions is a vast area, and a short note such as this cannot even begin to do justice to it. Journal articles (with one exception) will not be cited; and many researchers who made important contributions to this field will not be mentioned here. All that can be done here is to give a narrow snapshot of part of the field.

How to Cite This Entry:
Boris Tsirelson/sandbox. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boris_Tsirelson/sandbox&oldid=30032