This question is about the advantages of ensuring a limiting distribution in the Metropolis algorithm.
According to Wikipedia ,
uniqueness of stationary distribution: the stationary distribution $\pi(x)$ must be unique. This is guaranteed by ergodicity of the
Markov process, which requires that every state must (1) be
aperiodic—the system does not return to the same state at fixed
intervals; and (2) be positive recurrent—the expected number of steps
for returning to the same state is finite.
In the article is stated that ergodic implies aperiodic. It seems that there is not agreenment about this, like in this SE question is shown.
According to this answer an unique stationary distribution exists if all states of an irreducible Markov chain are positive recurrent. In the same reference it is said that the limiting probabilities cannot converge.
I think that the requirement of aperiodicity is done to ensure that the chain is ergodic acording to this definition, which ensures a limiting distribution.
Question: In the application of the Metropolis algorithm one take the average value of some property of each state in the realization of the Markov chain. Why convergence to a limiting distribution would be an advantage? It seems to me that ensuring that the MC is irreducible and positive recurrent should be enough.
Any clarification of incorrect statements in this question is very welcome.
Best Answer
Suppose $\{Z(t)\}_{t=0}^{\infty}$ is a discrete time Markov Chain (DTMC) with a finite or countably infinite state space $S$ and with transition probability matrix $P=(P_{ij})$.
We say that $Z(t)$ is irreducible if there is a finite path of nonzero probability from every state $i \in S$ to every other state $j \in S$.
A probability mass function (PMF) on the state space $S$ is a vector $(\pi_i)_{i \in S}$ such that $\pi_i \geq 0$ for all $i \in S$ and $\sum_{i\in S} \pi_i=1$.
Theorem: Suppose $\{Z(t)\}_{t=0}^{\infty}$ is an irreducible DTMC. If we can find a PMF $(\pi_i)_{i\in S}$ that satisfies the following stationary equations: $$ \pi_j = \sum_{i \in S} \pi_i P_{ij} \quad \forall j \in S $$ then $(\pi_i)_{i\in S}$ is the unique PMF that solves the above stationary equations, $\pi_i>0$ for all $i \in S$, and regardless of the initial condition $Z(0)$ we have for all $i \in S$: \begin{align} \lim_{T\rightarrow\infty} \frac{1}{T}\sum_{t=0}^{T-1} 1_{\{Z(t)=i\}} &= \pi_i \quad \mbox{(with prob 1)} \\ \lim_{T\rightarrow\infty} \frac{1}{T}\sum_{t=0}^{T-1}P[Z(t)=i] &= \pi_i \end{align} If there is no PMF that satisfies the stationary equations, then for all states $i \in S$ and regardless of the initial state $Z(0)$ we have: \begin{align} \lim_{T\rightarrow\infty} \frac{1}{T}\sum_{t=0}^{T-1} 1_{\{Z(t)=i\}} &= 0 \quad \mbox{(with prob 1)} \\ \lim_{T\rightarrow\infty} \frac{1}{T}\sum_{t=0}^{T-1}P[Z(t)=i] &= 0 \end{align}