Some combinatorial results on Bernoulli sets and codes

A Bernoulli set is a set X of words over a finite alphabet A such that for any positive Bernoulli distribution pi in A* one has that pi(X) = 1. In the case of a two-letter alphabet A = {a,b} a characterization of finite Bernoulli sets is given in terms of the function x(i,j) counting the number of words of X having i occurrences of the letter a and j occurrences of the letter b. Moreover, we also derive a necessary and sufficient condition on the distribution x(i,j) which characterizes Bernoulli sets which are commutatively equivalent to prefix codes. (C) 2002 Elsevier Science B.V. All rights reserved.