  Sequential probability assignment and universal compression go hand in hand.
We propose sequential probability assignment for non-binary (and large
alphabet) sequences with empirical distributions whose parameters are known to
be bounded within a limited interval. Sequential probability assignment
algorithms are essential in many applications that require fast and accurate
estimation of the maximizing sequence probability. These applications include
learning, regression, channel estimation and decoding, prediction, and
universal compression. On the other hand, constrained distributions introduce
interesting theoretical twists that must be overcome in order to present
efficient sequential algorithms. Here, we focus on universal compression for
memoryless sources, and present precise analysis for the maximal minimax and
the average minimax for constrained distributions. We show that our sequential
algorithm based on modified Krichevsky-Trofimov (KT) estimator is
asymptotically optimal up to $O(1)$ for both maximal and average redundancies.
This paper follows and addresses the challenge presented in \cite{stw08} that
suggested "results for the binary case lay the foundation to studying larger
alphabets".
