Announcements

Sorry for the spam... Here is an explanation about the "strange" fact mentioned in the subject, that I owe you from last week.

Consider the chain with transition matrix $P=\begin{pmatrix} 0 & 1 \\ 1/2 & 1/2 \end{pmatrix}$. The corresponding stationary distribution is $\pi=(1/3,2/3)$.

Consider now $X_0=1$, $Y_0 \sim \pi$ and let T>=0 be the coupling time of X and Y.

Then it holds that ${\mathbb P}(X_T=1)=1$, as the chains X and Y can only meet in state 1 (including if T=0). This is saying that at the coupling time, the distribution of the chain X is everything but the stationary distribution $\pi$.