Editing Mixing (mathematics) (section)

=== 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.