From Wikipedia:
Suppose $\{X_t\}$ is a stationary Markov process, with stationary distribution $Q$. Denote $L²(Q)$ the space of Borel-measurable functions that are square-integrable with respect to measure $Q$. Also let $ℰ_tϕ(x) = E[ϕ(X_t) | X_0 = x]$ denote the conditional expectation operator on $L²(Q)$. Finally, let $Z = \{ϕ∈L²(Q): ∫ ϕdQ = 0\}$ denote the space of square-integrable functions with mean zero.
The $ρ$-mixing coefficients of the process $\{x_t\}$ are $ \rho_t = \sup_{\phi\in Z:\,\|\phi\|_2=1} \| \mathcal{E}_t\phi \|_2. $ The process is called $ρ$-mixing if these coefficients converge to zero as $t → ∞$.
I was wondering
How $ \rho_t$ can represent the "difference" between the stationary distribution $Q$ and the conditional distribution of $X_t$ given $X_0 = x$?
why restricting focus on $Z = \{ϕ∈L²(Q): ∫ ϕdQ = 0\}$?
How is $\rho$-mixing related to strong mixing defined in an earlier part of the same Wikipedia article? $\rho$-mixing seems to mean convergence of measure of $X_t$ to the limiting distribution, while strong mixing seems to mean $X_t$ become more and more independent from $X_0$, again from Wikipedia:
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 independent.
What kind of mixing is for Markov chain mixing time?
- Whenever talking about the stationary distribution, is the underlying Markov process discrete-time? If not, how is the stationary distribution for a continuous time Markov process defined?
Thanks!