The detection of words and an ordering for Markov chains

Abstract

This paper considers the occurrence of patterns in sequences of independent trials from a finite alphabet; Gani and Irle (1999) have described a finite state automaton which identifies exactly those sequences of symbols containing the specific pattern, which may be thought of as the word of interest. Each word generates a particular Markov chain. Motivated by a result of Guibas and Odlyzko (1981) on stochastic monotonicity for the random times when a particular word is completed for the first time, a new level-crossing ordering is introduced for stochastic processes. A process {Yn : n = 0, 1, …} is slower in level-crossing than a process {Zn}, if it takes {Yn} stochastically longer than {Zn} to exceed any given level. This relation is shown to be useful for the comparison of stochastic automata, and is used to investigate this ordering for Markov chains in discrete time.