Ergodic Theorem

Definition

A finite IRR HMC $\{X_t{\}}_{t\in {\mathbb{N}}_0}$ holds the following for an arbitrary function $f:E\to \mathbb{R}$ ${\lim }_{T\to \infty }\frac{1}{T}{\sum }_{t=0}^{T-1}f(X_t)=E_{\mathbf{\pi }}[f(X_0)]$ where $\mathbf{\pi }$ is the unique Stationary Distribution, and $\mathbf{P}$ is a Transition Matrix of $\{X_t\}$

IRR PR HMC $\{X_t\}$ holds the following for a function $f:E\to \mathbb{R}$ that satisfies ${\sum }_{i\in E}|f(i)|{\pi }_i<\infty$ ${\lim }_{T\to \infty }\frac{1}{T}{\sum }_{t=0}^{T-1}f(X_t)=E_{\mathbf{\pi }}[f(X_0)]$ where $\mathbf{\pi }$ is the unique Stationary Distribution, and $\mathbf{P}$ is a Transition Matrix of $\{X_t\}$

Facts

Approximate Stationary Distribution Using Ergodic Theorem (Finite)

If the Markov Chain has a unique Stationary Distribution, it can be approximated using the time average by the ergodic theorem.

Let function $f$ be $f(i)=I(i=0)=\{\begin{array}{ll}1& i=0\\ 0& o.w\end{array}$, The stationary distribution can be approximated by using the ergodic theorem ${\lim }_{T\to \infty }\frac{1}{T}{\sum }_{t=0}^{T-1}I(X_t=0)=E_{\pi }[I(X_0=0)]=P_{\pi }[X_0=0]={\pi }_0$

Approximate Stationary Distribution Using Ergodic Theorem (Infinite)

IRR PR HMC $\{X_t\}$ holds $\bar{\mathbf{\pi }}\to \mathbf{\pi }$

where $\bar{\mathbf{\pi }}$ is the time average, and $\mathbf{\pi }$ is the Stationary Distribution, of $\{X_t\}$

Examples

Identity Matrix

$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

Since the IRR condition is not satisfied, there is no unique Stationary Distribution. So the Ergodic theorem cannot be used.

Exchange Matrix

$\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$

$(\frac{1}{T}{\sum }_{t=0}^{T-1}I(X_t=0),\frac{1}{T}{\sum }_{t=0}^{T-1}I(X_t=1))=({\hat{\pi }}_0,{\hat{\pi }}_1)\approx (1/2,1/2)$