Editing Mixing (mathematics) (section)

== Mixing in stochastic processes ==
Let <math>(X_t)_{-\infty < t < \infty}</math> be a [[stochastic process]] on a probability space {{tmath|1= (\Omega, \mathcal{F}, \mathbb{P}) }}. The sequence space into which the process maps can be 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|σ-algebra]], the [[Borel sigma-algebra|Borel σ-algebra]]; this is the smallest σ-algebra that contains the topology.

Define a function <math>\alpha</math>, called the '''strong mixing coefficient''', as
: <math>\alpha(s) = \sup \left\{|\mathbb{P}(A \cap B) - \mathbb{P}(A)\mathbb{P}(B)| : -\infty < t < \infty, A\in X_{-\infty}^t, B\in X_{t+s}^\infty \right\}</math>
for all {{tmath|1= -\infty < s < \infty }}. The symbol <math>X_a^b</math>, with <math>-\infty \le a \le b \le \infty </math> 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 {{tmath|1= \{X_a, X_{a+1}, \ldots, X_b \} }}.

The process <math>(X_t)_{-\infty < t < \infty}</math> is said to be '''strongly mixing''' if <math>\alpha(s)\to 0</math> as {{tmath|1= s\to \infty }}. That is to say, a strongly mixing process is such that, in a way that is uniform over all times <math>t</math> and all events, the events before time <math>t</math> and the events after time <math>t+s</math> tend towards being [[Statistical independence|independent]] as <math>s \to \infty</math>; more colloquially, the process, in a strong sense, forgets its history.

=== Mixing in Markov processes ===
Suppose <math>(X_t)</math> were a stationary [[Markov process]] with stationary distribution <math>\mathbb{Q}</math> and let <math>L^2(\mathbb{Q})</math> denote the space of Borel-measurable functions that are square-integrable with respect to the measure <math>\mathbb{Q}</math>. Also let
: <math>\mathcal{E}_t \varphi (x) = \mathbb{E}[\varphi (X_t) \mid X_0 = x] </math>
denote the conditional expectation operator on <math>L^2(\mathbb{Q}).</math> Finally, let
: <math> Z = \left \{ \varphi \in L^2(\mathbb{Q}) : \int \varphi \, d\mathbb{Q} = 0 \right \}</math>
denote the space of square-integrable functions with mean zero.

The '''''ρ''-mixing coefficients''' of the process {''x<sub>t</sub>''} are
: <math>\rho_t = \sup_{\varphi\in Z :\,\|\varphi\|_2=1} \| \mathcal{E}_t\varphi \|_2.</math>

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>

The '''''α''-mixing coefficients''' of the process {{mset|''x<sub>t</sub>''}} are
: <math>\alpha_t = \sup_{\varphi \in Z : \|\varphi\|_\infty=1} \| \mathcal{E}_t\varphi \|_1.  </math>

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>'' &lt; ''γe''<sup>−''δt''</sup>}} for some {{nowrap|''δ'' > 0}}, and it is '''''α''-mixing with a sub-exponential decay rate''' if {{nowrap|''α<sub>t</sub>'' &lt; ''ξ''(''t'')}} for some non-increasing function <math>\xi</math> satisfying
: <math>\frac{\ln \xi(t)}{t} \to 0</math>
as {{tmath|1= t \to \infty }}.<ref name=Chen_et_al/>

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.

The '''''β''-mixing coefficients''' are given by
: <math>\beta_t = \int \sup_{0 \le \varphi \le 1} \left | \mathcal{E}_t\varphi(x) - \int \varphi \,d\mathbb{Q} \right| \,d\mathbb{Q}.</math>

The process is called '''''β''-mixing''' if these coefficients converge to zero as {{nowrap|''t'' → ∞}}, it is '''''β''-mixing with an exponential decay rate''' if {{nowrap|''β<sub>t</sub>'' &lt; ''γe''<sup>−''δt''</sup>}} for some {{nowrap|''δ'' > 0}}, and it is '''''β''-mixing with a sub-exponential decay rate''' if {{nowrap|''β<sub>t</sub>ξ''(''t'') → 0}} as {{nowrap|''t'' → ∞}} for some non-increasing function <math>\xi</math> satisfying
: <math>\frac{\ln \xi(t)}{t} \to 0</math>
as <math>t \to \infty</math>.<ref name=Chen_et_al/>

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.