So I was going over some Markov chain problems and came across a question where you have some standard Markov chain, and you find the stationary distribution $\pi = [\pi_{1}, \pi_{2}, \pi_{3}]$. Then the question asks you to find $$\lim_{n \rightarrow \infty} E[X_{n} | X_{0} = 0]$$ But this notation isn't in my notes anywhere, and I can't consider it intuitively. It seems like it would just be the element in my stationary distribution with the largest value, but that seems strange..
1 Answer
$\begingroup$
$\endgroup$
In Markov chain some one use this notation:
$$E[I_A(X_n)|X_0=0]=P(X_n\in A|X_0=0).$$
But in classical probability theory this notation rappresent the conditional expectation of X given an event H (which may be the event Y=y for a random variable Y) is the average of X over all outcomes in H, that is
$$E(X|H)=\frac{\sum_{w\in H}X(w)}{|H|}$$
so I think that in your exercise the question is to find
$$\lim_{n\rightarrow \infty}\frac{\sum_{w\in\{X_0=0\}}X_n(w)}{|\{X_0=0\}|}.$$
