1. Introduction

A class of problems — important for their applications to computer science and computational biology as well as for their inherent mathematical interest — is the statistical analysis of a string of random symbols. The symbols, called letters, are assumed to belong to an alphabet A of fixed size k. The set of all such strings (or words) of length N, W(A,N), forms the sample space in the statistical analysis of these strings. A natural measure on W is to assign each letter equal probability, namely 1/ k, and to define the probability measure on words by the product measure. Thus each letter in a word occurs independently and with equal probability. We call such random word models homogeneous. Of course, for some applications, each letter in the alphabet does not occur with the same frequency and it is therefore natural to assign to each letter i a probability pi. If we again use the product measure for the words (letters in a word occur independently), then the resulting random word models are called inhomogeneous. Fixing an ordering of the alphabet A, a weakly increasing subsequence of a Word.