Mixing (mathematics)
In mathematics, mixing is an abstract concept originating from physics: the attempt to describe the irreversible thermodynamic process of mixing in the everyday world: mixing paint, mixing drinks, etc.
The concept appears in ergodic theory—the study of stochastic processes and measure-preserving dynamical systems. Several different definitions for mixing exist, including strong mixing, weak mixing and topological mixing, with the last not requiring a measure to be defined. Some of the different definitions of mixing can be arranged in a hierarchical order; thus, strong mixing implies weak mixing. Furthermore, weak mixing (and thus also strong mixing) implies ergodicity: that is, every system that is weakly mixing is also ergodic (and so one says that mixing is a "stronger" notion than ergodicity).
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Mixing in stochastic processes[edit]
Let be a stochastic process on a probability space . The sequence space into which the process maps can be endowed with a topology, the product topology. The open sets of this topology are called cylinder sets. These cylinder sets generate a σ-algebra, the Borel σ-algebra; this is the smallest σ-algebra that contains the topology.
Define a function , called the strong mixing coefficient, as
for all . The symbol , with denotes a sub-σ-algebra of the σ-algebra; it is the set of cylinder sets that are specified between times a and b, i.e. the σ-algebra generated by .
The process is said to be strongly mixing if as . That is to say, a strongly mixing process is such that, in a way that is uniform over all times and all events, the events before time and the events after time tend towards being independent as ; more colloquially, the process, in a strong sense, forgets its history.
Types of mixing[edit]
Suppose were a stationary Markov process with stationary distribution and let denote the space of Borel-measurable functions that are square-integrable with respect to the measure . Also let
denote the conditional expectation operator on Finally, let
denote the space of square-integrable functions with mean zero.
The ρ-mixing coefficients of the process {x_{t}} are
The process is called ρ-mixing if these coefficients converge to zero as t → ∞, and “ρ-mixing with exponential decay rate” if ρ_{t} < e^{−δt} for some δ > 0. For a stationary Markov process, the coefficients ρ_{t} may either decay at an exponential rate, or be always equal to one.^{[1]}
The α-mixing coefficients of the process {x_{t}} are
The process is called α-mixing if these coefficients converge to zero as t → ∞, it is “α-mixing with exponential decay rate” if α_{t} < γe^{−δt} for some δ > 0, and it is α-mixing with a sub-exponential decay rate if α_{t} < ξ(t) for some non-increasing function satisfying
as .^{[1]}
The α-mixing coefficients are always smaller than the ρ-mixing ones: α_{t} ≤ ρ_{t}, therefore if the process is ρ-mixing, it will necessarily be α-mixing too. However, when ρ_{t} = 1, the process may still be α-mixing, with sub-exponential decay rate.
The β-mixing coefficients are given by
The process is called β-mixing if these coefficients converge to zero as t → ∞, it is β-mixing with an exponential decay rate if β_{t} < γe^{−δt} for some δ > 0, and it is β-mixing with a sub-exponential decay rate if β_{t}ξ(t) → 0 as t → ∞ for some non-increasing function satisfying
as .^{[1]}
A strictly stationary Markov process is β-mixing if and only if it is an aperiodic recurrent Harris chain. The β-mixing coefficients are always bigger than the α-mixing ones, so if a process is β-mixing it will also be α-mixing. There is no direct relationship between β-mixing and ρ-mixing: neither of them implies the other.
Mixing in dynamical systems[edit]
A similar definition can be given using the vocabulary of measure-preserving dynamical systems. Let be a dynamical system, with T being the time-evolution or shift operator. The system is said to be strong mixing if, for any , one has
For shifts parametrized by a continuous variable instead of a discrete integer n, the same definition applies, with replaced by with g being the continuous-time parameter.
To understand the above definition physically, consider a shaker full of an incompressible liquid, which consists of 20% wine and 80% water. If is the region originally occupied by the wine, then, for any region within the shaker, the percentage of wine in after repetitions of the act of stirring is
In such a situation, one would expect that after the liquid is sufficiently stirred (), every region of the shaker will contain approximately 20% wine. This leads to
where , because measure-preserving dynamical systems are defined on probability spaces, and hence the final expression implies the above definition of strong mixing.
A dynamical system is said to be weak mixing if one has
In other words, is strong mixing if in the usual sense, weak mixing if
in the Cesàro sense, and ergodic if in the Cesàro sense. Hence, strong mixing implies weak mixing, which implies ergodicity. However, the converse is not true: there exist ergodic dynamical systems which are not weakly mixing, and weakly mixing dynamical systems which are not strongly mixing.
For a system that is weak mixing, the shift operator T will have no (non-constant) square-integrable eigenfunctions with associated eigenvalue of one.^{[citation needed]} In general, a shift operator will have a continuous spectrum, and thus will always have eigenfunctions that are generalized functions. However, for the system to be (at least) weak mixing, none of the eigenfunctions with associated eigenvalue of one can be square integrable.
formulation[edit]
The properties of ergodicity, weak mixing and strong mixing of a measure-preserving dynamical system can also be characterized by the average of observables. By von Neumann's ergodic theorem, ergodicity of a dynamical system is equivalent to the property that, for any function , the sequence converges strongly and in the sense of Cesàro to , i.e.,
A dynamical system is weakly mixing if, for any functions and
A dynamical system is strongly mixing if, for any function the sequence converges weakly to i.e., for any function
Since the system is assumed to be measure preserving, this last line is equivalent to saying that so that the random variables and become orthogonal as grows. Actually, since this works for any function one can informally see mixing as the property that the random variables and become independent as grows.
Products of dynamical systems[edit]
Given two measured dynamical systems and one can construct a dynamical system on the Cartesian product by defining We then have the following characterizations of weak mixing:
- Proposition. A dynamical system is weakly mixing if and only if, for any ergodic dynamical system , the system is also ergodic.
- Proposition. A dynamical system is weakly mixing if and only if is also ergodic. If this is the case, then is also weakly mixing.
Generalizations[edit]
The definition given above is sometimes called strong 2-mixing, to distinguish it from higher orders of mixing. A strong 3-mixing system may be defined as a system for which
holds for all measurable sets A, B, C. We can define strong k-mixing similarly. A system which is strong k-mixing for all k = 2,3,4,... is called mixing of all orders.
It is unknown whether strong 2-mixing implies strong 3-mixing. It is known that strong m-mixing implies ergodicity.
Examples[edit]
Irrational rotations of the circle, and more generally irreducible translations on a torus, are ergodic but neither strongly nor weakly mixing with respect to the Lebesgue measure.
Many maps considered as chaotic are strongly mixing for some well-chosen invariant measure, including: the dyadic map, Arnold's cat map, horseshoe maps, Kolmogorov automorphisms, and the geodesic flow on the unit tangent bundle of compact surfaces of negative curvature.
Topological mixing[edit]
A form of mixing may be defined without appeal to a measure, using only the topology of the system. A continuous map is said to be topologically transitive if, for every pair of non-empty open sets , there exists an integer n such that
where is the nth iterate of f. In the operator theory, a topologically transitive bounded linear operator (a continuous linear map on a topological vector space) is usually called hypercyclic operator. A related idea is expressed by the wandering set.
Lemma: If X is a complete metric space with no isolated point, then f is topologically transitive if and only if there exists a hypercyclic point , that is, a point x such that its orbit is dense in X.
A system is said to be topologically mixing if, given open sets and , there exists an integer N, such that, for all , one has
For a continuous-time system, is replaced by the flow , with g being the continuous parameter, with the requirement that a non-empty intersection hold for all .
A weak topological mixing is one that has no non-constant continuous (with respect to the topology) eigenfunctions of the shift operator.
Topological mixing neither implies, nor is implied by either weak or strong mixing: there are examples of systems that are weak mixing but not topologically mixing, and examples that are topologically mixing but not strong mixing.
References[edit]
- Chen, Xiaohong; Hansen, Lars Peter; Carrasco, Marine (2010). "Nonlinearity and temporal dependence". Journal of Econometrics. 155 (2): 155–169. CiteSeerX 10.1.1.597.8777. doi:10.1016/j.jeconom.2009.10.001.
- Achim Klenke, Probability Theory, (2006) Springer ISBN 978-1-84800-047-6
- V. I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics, (1968) W. A. Benjamin, Inc.