  We study the redundancy of universally compressing strings $X_1,\dots, X_n$
generated by a binary Markov source $p$ without any bound on the memory. To
better understand the connection between compression and estimation in the
Markov regime, we consider a class of Markov sources restricted by a continuity
condition. In the absence of an upper bound on memory, the continuity condition
implies that $p(X_0|X^{-1}_{-m})$ gets closer to the true probability
$p(X_0|X_{-\infty}^{-1})$ as $m$ increases, rather than vary around
arbitrarily. For such sources, we prove asymptotically matching upper and lower
bounds on the redundancy. In the process, we identify what sources in the class
matter the most from a redundancy perspective.
