Countable alphabet stationary processes with at least one memory word and intermittent estimation with universal rates

We present here a number of results that provide universal rates of convergence for certain non parametric estimation problems. For example consider the class C of all finite order Markov chains on a countable alphabet and the problem of estimating the conditional distribution of X_n+1 given the first n outputs of the process. We will give a sequence of stopping times with density one and estimators at those times such that almost surely our estimators will eventually differ from the true conditional distribution by no more than a certain fixed sequence tending to zero. Similar results are given for estimating the conditional expectation of X_n+1 given the first n outputs, but here some additional moment conditions are required. An example shows that this is not possible in general.