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| <!-- {{Mergewith|Mixing (physics)|date=March 2008}} -->
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| In [[mathematics]], '''mixing''' is an abstract concept originating from [[physics]]: the attempt to describe the irreversible [[thermodynamic process]] of [[mixing (physics)|mixing]] in the everyday world: mixing paint, mixing drinks, ''etc''.
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| The concept appears in [[ergodic theory]]—the study of [[stochastic process]]es and [[measure-preserving dynamical system]]s. Several different definitions for mixing exist, including ''strong mixing'', ''weak mixing'' and ''topological mixing'', with the last not requiring a [[measure (mathematics)|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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| [[File:Ergodic mixing of putty ball after repeated Smale horseshoe map.jpg|thumb|Upright|Mixing in a ball of colored putty after consecutive iterations of "Smale horseshoe map" (i.e. squashing and folding in two)]]
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| ==Mixing in stochastic processes==
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| Let <math> \langle X_t \rangle = \{ \ldots, X_{t-1}, X_t, X_{t+1}, \ldots \}</math> be a sequence of [[random variable]]s. Such a sequence is naturally endowed with a topology, the [[product topology]]. The [[open set]]s of this topology are called [[cylinder set]]s. These cylinder sets generate a [[sigma algebra]], the [[Borel sigma algebra]]; it is the smallest (coarsest) sigma algebra that contains the topology.
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| Define a function <math>\alpha(s)</math>, called the '''strong mixing coefficient''', as
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| :<math>\alpha(s) \equiv \sup \left\{\,|P(A \cap B) - P(A)P(B)| : -\infty < t < \infty, A\in X_{-\infty}^{t}, B\in X_{t+s}^\infty \,\right\}. </math>
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| In this definition, ''P'' is the probability [[measure (mathematics)|measure]] on the sigma algebra. The symbol <math>X_a^b</math>, with <math>-\infty \leq a \leq b \leq \infty </math> denotes a subalgebra of the sigma algebra; it is the set of cylinder sets that are specified between times ''a'' and ''b''. Given specific, fixed values <math>X_a</math>, <math>X_{a+1}</math>, ''etc.'', of the random variable, at times <math>a</math>, <math>a+1</math>, ''etc.'', then it may be thought of as the [[sigma-algebra]] generated by
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| :<math>\{X_a, X_{a+1},\ldots, X_b \}.</math>
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| The process <math>\langle X_t \rangle</math> is '''strong mixing''' if <math>\alpha(s)\rightarrow 0</math> as <math>s\rightarrow \infty</math>.
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| One way to describe this is that '''strong mixing''' implies that for any two possible states of the system (realizations of the random variable), when given a sufficient amount of time between the two states, the occurrence of the states is [[Statistical independence|independent]].
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| === Types of mixing ===
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| Suppose {''X<sub>t</sub>''} is a stationary [[Markov process]], with stationary distribution ''Q''. Denote ''L''²(''Q'') the space of Borel-measurable functions that are square-integrable with respect to measure ''Q''. Also let {{nowrap|ℰ<sub>''t''</sub>''ϕ''(''x'') {{=}} E[''ϕ''(''X<sub>t</sub>'') {{!}} ''X''<sub>0</sub> {{=}} ''x'']}} denote the conditional expectation operator on ''L''²(''Q''). Finally, let {{nowrap|''Z'' {{=}} {''ϕ''∈''L''²(''Q''): ∫ ''ϕdQ'' {{=}} 0}}} denote the space of square-integrable functions with mean zero.
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| The '''''ρ''-mixing coefficients''' of the process {''x<sub>t</sub>''} are
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| : <math>
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| \rho_t = \sup_{\phi\in Z:\,\|\phi\|_2=1} \| \mathcal{E}_t\phi \|_2.
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| </math>
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| The process is called '''''ρ''-mixing''' if these coefficients converge to zero as {{nowrap|''t'' → ∞}}, and “ρ-mixing with exponential decay rate” if {{nowrap|''ρ<sub>t</sub>'' < ''e''<sup>−''δt''</sup>}} for some {{nowrap|''δ'' > 0}}. For a stationary Markov process, the coefficients ''ρ<sub>t</sub>'' may either decay at an exponential rate, or be always equal to one.<ref name=Chen_et_al>{{harvtxt|Chen|Hansen|Carrasco|2010}}</ref>
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| The '''''α''-mixing coefficients''' of the process {''x<sub>t</sub>''} are
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| : <math>
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| \alpha_t = \sup_{\phi\in Z:\,\|\phi\|_\infty=1} \| \mathcal{E}_t\phi \|_1.
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| </math>
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| The process is called '''''α''-mixing''' if these coefficients converge to zero as {{nowrap|''t'' → ∞}}, it is “α-mixing with exponential decay rate” if {{nowrap|''α<sub>t</sub>'' < ''γe''<sup>−''δt''</sup>}} for some {{nowrap|''δ'' > 0}}, and it is “α-mixing with sub-exponential decay rate” if {{nowrap|''α<sub>t</sub>'' < ''ξ''(''t'')}} for some non-increasing function ''ξ''(''t'') satisfying {{nowrap|''t''<sup>−1</sup>ln ''ξ''(''t'') → 0}} as {{nowrap|''t'' → ∞}}.<ref name=Chen_et_al/>
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| The ''α''-mixing coefficients are always smaller than the ''ρ''-mixing ones: {{nowrap|''α<sub>t</sub>'' ≤ ''ρ<sub>t</sub>''}}, therefore if the process is ''ρ''-mixing, it will necessarily be ''α''-mixing too. However when {{nowrap|''ρ<sub>t</sub>'' {{=}} 1}}, the process may still be ''α''-mixing, with sub-exponential decay rate.
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| The '''''β''-mixing coefficients''' are given by
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| : <math>
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| \beta_t = \int \sup_{0\leq\phi\leq1} \Big| \mathcal{E}_t\phi(x) - \int \phi dQ \Big| dQ.
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| </math>
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| The process is called '''''β''-mixing''' if these coefficients converge to zero as {{nowrap|''t'' → ∞}}, it is “β-mixing with exponential decay rate” if {{nowrap|''β<sub>t</sub>'' < ''γe''<sup>−''δt''</sup>}} for some {{nowrap|''δ'' > 0}}, and it is “β-mixing with sub-exponential decay rate” if {{nowrap|''β<sub>t</sub>ξ''(''t'') → 0}} as {{nowrap|''t'' → ∞}} for some non-increasing function ''ξ''(''t'') satisfying {{nowrap|''t''<sup>−1</sup>ln ''ξ''(''t'') → 0}} as {{nowrap|''t'' → ∞}}.<ref name=Chen_et_al/>
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| 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.
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| ==Mixing in dynamical systems==
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| A similar definition can be given using the vocabulary of [[measure-preserving dynamical system]]s. Let <math>(X, \mathcal{A}, \mu, T)</math> be a dynamical system, with ''T'' being the time-evolution or [[shift operator]]. The system is said to be '''strong mixing''' if, for any <math>A,B \in \mathcal{A}</math>, one has
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| :<math>\lim_{n\to\infty} \mu (A \cap T^{-n}B) = \mu(A)\mu(B)</math>.
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| For shifts parametrized by a continuous variable instead of a discrete integer ''n'', the same definition applies, with <math>T^{-n}</math> replaced by <math>T_g</math> with ''g'' being the continuous-time parameter.
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| To understand the above definition physically, consider a shaker <math>M</math> full of an incompressible liquid, which consists of 20% wine and 80% water. If <math>A</math> is the region originally occupied by the wine, then, for any part <math>B</math> of the shaker, the percentage of wine in <math>B</math> after n repetitions of the act of stirring is
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| :<math>\frac{\mu\left( T^n A \cap B\right)}{\mu\left( B\right)}.</math>
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| In such a situation, one would expect that after the liquid is sufficiently stirred (<math>n \rightarrow \infty</math>), every part <math>B</math> of the shaker will contain approximately 20% wine. This leads to
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| :<math> \lim_{n \rightarrow \infty} \frac{\mu\left( T^n A \cap B\right)}{\mu\left( B\right)} = \mu\left( A\right)</math>
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| which implies the above definition of strong mixing.
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| A dynamical system is said to be '''weak mixing''' if one has
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| :<math>\lim_{n\to\infty} \frac {1}{n} \sum_{k=0}^n
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| |\mu (A \cap T^{-k}B) - \mu(A)\mu(B)| = 0.</math>
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| In other words, <math>T</math> is strong mixing if <math>\mu (A \cap T^{-n}B) - \mu(A)\mu(B)</math> converges towards <math>0</math> in the usual sense, weak mixing if <math>|\mu (A \cap T^{-n}B) - \mu(A)\mu(B)|</math> converges towards <math>0</math> in the [[Cesàro mean|Cesàro]] sense, and ergodic if <math>\mu (A \cap T^{-n}B) - \mu(A)\mu(B)</math> converges towards <math>0</math> 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.
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| For a system that is weak mixing, the [[shift operator]] ''T'' will have no (non-constant) [[square-integrable]] [[eigenfunction]]s with associated eigenvalue of one.{{Citation needed|date=October 2008}} In general, a shift operator will have a [[Decomposition of spectrum (functional analysis)|continuous spectrum]], and thus will always have eigenfunctions that are [[generalized function]]s. However, for the system to be (at least) weak mixing, none of the eigenfunctions with associated eigenvalue of one can be square integrable.
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| === <math>L^2</math> formulation ===
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| 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 <math>(X, \mathcal{A}, \mu, T)</math> is equivalent to the property that, for any function <math>f \in L^2 (X, \mu)</math>, the sequence <math>(f \circ T^n)_{n \ge 0}</math> converges strongly and in the sense of Cesàro to <math> \int_X f d \mu</math>, i.e.,
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| :<math> \lim_{N \to \infty} \| {1 \over N} \sum_{n=0}^{N-1} f \circ T^n - \int_X f d \mu \|_{L^2 (X, \mu)}= 0.</math>
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| A dynamical system <math>(X, \mathcal{A}, \mu, T)</math> is weakly mixing if, for any functions <math>f</math> and <math>g \in L^2 (X, \mu)</math>,
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| :<math> \lim_{N \to \infty} {1 \over N} \sum_{n=0}^{N-1} | \int_X f \circ T^n \cdot g d \mu- \int_X f d \mu \cdot \int_X g d \mu|= 0.</math>
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| A dynamical system <math>(X, \mathcal{A}, \mu, T)</math> is strongly mixing if, for any function <math>f \in L^2 (X, \mu)</math>, the sequence <math>(f \circ T^n)_{n \ge 0}</math> converges weakly to <math> \int_X f d \mu</math>, i.e., for any function <math>g \in L^2 (X, \mu)</math>,
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| :<math> \lim_{n \to \infty} \int_X f \circ T^n \cdot g d \mu = \int_X f d \mu \cdot \int_X g d \mu.</math>
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| Since the system is assumed to be measure preserving, this last line is equivalent to saying that <math>\lim_{n \to \infty} Cov (f \circ T^n, g) = 0</math>, so that the random variables <math>f \circ T^n</math> and <math>g</math> become orthogonal as <math>n</math> grows. Actually, since this works for any function <math>g</math>, one can informally see mixing as the property that the random variables <math>f \circ T^n</math> and <math>g</math> become independent as <math>n</math> grows.
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| ===Products of dynamical systems===
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| Given two measured dynamical system <math>(X, \mu, T)</math> and <math>(Y, \mu, S)</math>, one can construct a dynamical system <math>(X \times Y, \mu \otimes \nu, T \times S)</math> on the Cartesian product by defining <math>(T \times S) (x,y) = (T(x), S(y))</math>. We then have the following characterizations of weak mixing:
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| '''Proposition''' : A dynamical system <math>(X, \mu, T)</math> is weakly mixing if and only if, for any ergodic dynamical system <math>(Y, \mu, S)</math>, the system <math>(X \times Y, \mu \otimes \nu, T \times S)</math> is also ergodic.
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| '''Proposition''' : A dynamical system <math>(X, \mu, T)</math> is weakly mixing if and only if <math>(X^2, \mu \otimes \mu, T \times T)</math> is also ergodic. If this is the case, then <math>(X^2, \mu \otimes \mu, T \times T)</math> is also weakly mixing.
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| ===Generalizations===
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| 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
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| :<math>\lim_{m,n\to\infty} \mu (A \cap T^{-m}B \cap T^{-m-n}C) = \mu(A)\mu(B)\mu(C)</math>
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| 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'''.
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| It is unknown whether strong 2-mixing implies strong 3-mixing. It is known that strong ''m''-mixing implies [[ergodicity]].
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| ===Examples===
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| [[Irrational rotation]]s 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.
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| Many map considered as chaotic are strongly mixing for some well-chosen invariant measure, including: the [[dyadic map]], [[Arnold's cat map]], [[horseshoe map]]s, [[Kolmogorov automorphism]]s, the geodesic flow on the unit tangent bundle of compact surfaces of negative curvature...
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| ==Topological mixing==
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| <!-- linked from redirect [[Topological mixing]] -->
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| A form of mixing may be defined without appeal to a [[measure (mathematics)|measure]], only using the [[topology]] of the system. A [[continuous map]] <math>f:X\to X</math> is said to be '''topologically transitive''' if, for every pair of non-empty [[open set]]s <math>A,B\subset X</math>, there exists an integer ''n'' such that
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| :<math>f^n(A) \cap B \ne \varnothing</math>
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| where <math>f^n</math> is the [[iterated function|''n''th 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]].
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| '''Lemma:''' If ''X'' is a [[complete metric space|complete]] [[metric space]] with no [[isolated point]], then ''f'' is topologically transitive if and only if there exists a [[hypercyclic vector|hypercyclic point]] <math>x\in X</math>, that is, a point ''x'' such that its orbit <math>\{f^n(x): n\in \mathbb{N}\}</math> is [[dense set|dense]] in ''X''.
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| A system is said to be '''topologically mixing''' if, given open sets <math>A</math> and <math>B</math>, there exists an integer ''N'', such that, for all <math>n>N</math>, one has
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| :<math>f^n(A) \cap B \neq \varnothing</math>.
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| For a continuous-time system, <math>f^n</math> is replaced by the [[flow (mathematics)|flow]] <math>\phi_g</math>, with ''g'' being the continuous parameter, with the requirement that a non-empty intersection hold for all <math>\Vert g \Vert > N</math>.
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| A '''weak topological mixing''' is one that has no non-constant [[continuous (topology)|continuous]] (with respect to the topology) eigenfunctions of the shift operator.
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| 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.
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| ==References==
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| {{refbegin}}
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| * {{cite journal
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| | last1 = Chen | first1 = Xiaohong
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| | last2 = Hansen | first2 = Lars Peter
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| | last3 = Carrasco | first3 = Marine
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| | title = Nonlinearity and temporal dependence
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| | year = 2010
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| | journal = Journal of Econometrics
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| | volume = 155 | issue = 2 | pages = 155–169
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| | doi = 10.1016/j.jeconom.2009.10.001
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| | ref = harv
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| }}
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| * Achim Klenke, ''Probability Theory'', (2006) Springer ISBN 978-1-84800-047-6
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| * V. I. Arnold and A. Avez, ''Ergodic Problems of Classical Mechanics'', (1968) W. A. Benjamin, Inc.
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| {{refend}}
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| {{reflist}}
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| [[Category:Stochastic processes]]
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| [[Category:Ergodic theory]]
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