A Further Note on Runs in Independent Sequences

Given a sequence of letters generated independently froma finite alphabet, we consider the case when more than one, but notall, letters are generated with the highest probability. the length ofthe longest run of any of these letters is shown to be one greater thanthe length of the longest run in a particular state of an associatedmarkov chain. using results of foulser and karlin (1987), a conjectureof a previous paper (smythe, 2003) concerning the expectationof this length is verified.