Waiting Time Distribution for Pattern Occurrence in a Constrained Sequence: an Embedding Markov Chain Approach

Abstract

In this paper we consider the distribution of a pattern of interest in a binary random $(d,k)$-sequence generated by a Markov source. Such constrained sequences are frequently encountered in communication systems. Unlike previous approach based on generating function we have chosen here to use Markov chain embedding techniques. Doing so, we get both previous results (sequence constrained up to the rth occurrence) and new ones (sequence constrained up to its end). We also provide in both cases efficient algorithms that use only basic linear algebra. We then compare our numerical results to previous ones and finally propose several interesting extensions of our method which further illustrate the usefulness of our approach: higher order Markov chains, renewal occurrences rather than overlapping ones, heterogeneous models, more complex patterns of interest and multistate trial sequences.