= Summable-memory coupling
{title2=$\|q(\cdot\mid\eta)-q(\cdot\mid\eta')\|_{\rm TV}\leq b_r,\quad\sum_rb_r<\infty$}
Suppose a finite-alphabet transition kernel is uniformly positive and histories agreeing in their most recent $r$ symbols have <total variation distance> at most $b_r$. Choose decreasing bounds with $b_r<1$ and summable tail. A <maximal coupling> can be dominated by a match-length chain that increments with probability $1-b_r$ and resets with probability $b_r$. Its chance of never resetting is $\prod_{r\geq0}(1-b_r)>0$. Successive failed attempts therefore end almost surely, leaving a final infinite run of matches. The dominating chain does not depend on the histories, so future-block laws forget the remote past uniformly. This proves uniqueness of a two-sided law with the prescribed transition kernel without assuming stationarity first.
Back to article page
