The statistics of words on rings

• Published in: Communications on pure and applied mathematics Vol. 59; no. 1; pp. 145 - 160
• Main Authors: Percus, Jerome K., Percus, Ora E.
• Format: Journal Article
• Language: English
• Published: Hoboken Wiley Subscription Services, Inc., A Wiley Company 01.01.2006; Wiley
• Subjects: Exact sciences and technology; Markov processes; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical methods in probability and statistics; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Markov chain; Statistical method; Word; Generating function; Letter sequence; Ring; Distribution function; Probability distribution; Poisson distribution; Statistics
• ISSN: 0010-3640; 1097-0312
• DOI: 10.1002/cpa.20096

We analyze sequences of letters on a ring. Our objective is to determine the statistics of the occurrences of a set of r‐letter words when the sequence is chosen as a periodic Markov chain of order ≤ r − 1. We first obtain a generating function for the associated probability distribution and then display its Poisson limit. For an i.i.d. letter sequence, correction terms to the Poisson limit are given. Finally, we indicate how a hidden Markov chain fits into this scheme. © 2005 Wiley Periodicals, Inc.