Bottleneck. Suppose a $($non$-$stationary$)$ Markov chain starts in one of

n states, necks down to A $<$ n states, and then fans back to m $>$ k

states. Thus X$1-*$X$2-+$X$,,$ X$,$~$\{1,2,...,$ n$\},$ X$,$E$\{1,2,...,$ k$\},$

x$3$ E $\{1,2,...,$ m$\}.$

$($a$)$ Show that the dependence of X$1$ and X$3$ is limited by the bottleneck

by proving that $1($X$,$; X$3)5$ log k$.$

$($b$)$ Evaluate $1($X$1$; X$,)$ for k $=1,$ and conclude that no dependence can

survive such a bottleneck.