  Assume one observes independent categorical variables or, equivalently, one
observes the corresponding multinomial variables. Estimating the distribution
of the observed sequence amounts to estimating the expectation of the
multinomial sequence. A new estimator for this mean is proposed that is
nonparametric, non-asymptotic and implementable even for large sequences. It is
a penalized least-squares estimator based on wavelets, with a penalization term
inspired by papers of Birg\'{e} and Massart. The estimator is proved to satisfy
an oracle inequality and to be adaptive in the minimax sense over a class of
Besov bodies. The method is embedded in a general framework which allows us to
recover also an existing method for segmentation. Beyond theoretical results, a
simulation study is reported and an application on real data is provided.
