On the length of the longest run in a multi-state Markov chain

Let {Xa}a[set membership, variant]Z be an irreducible and aperiodic Markov chain on a finite state space S={0,1,...,r}, r[greater-or-equal, slanted]1. Denote by Ln the length of the longest run of consecutive i's, for i=1,...,r, that occurs in the sequence X1,...,Xn. In this work, we extend a result of Goncharov (Amer. Math. Soc. Transl. 19 (1943) 1) which concerned a limit law for Ln in sequences of 0-1 i.i.d. trials. Moreover, it is shown that Ln has approximately an extreme value distribution along a certain subsequence. Finally, a weak version of an Erdös-Rényi type law for Ln is proved.