Statistics & Probability Letters 75 (2005) 230–236 On some stochastic models for replication of character strings Probal Chaudhuri, Amites Dasgupta à Theoretical Statistics & Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India Received 9 March 2003; received in revised form 12 May 2005 Available online 11 July 2005 Abstract In this note, some probabilistic models for replication of character strings are considered. These replication processes involve random mutations, deletions and insertions of characters. We investigate invariance of certain probabilistic properties of replicating character strings under the proposed stochastic models for the replication process. It is shown that some well-known types of hidden Markov models with finite state spaces arise as special cases of our stochastic replication models. We also introduce the notion of a hidden mixed Markov model for a character string that arises in a situation where the replication process satisfies exchangeability conditions. r 2005 Elsevier B.V. All rights reserved. Keywords: Exchangeability; Hidden Markov process; Markov process; Mixed Markov process 1. Introduction Suppose that we have observed a random character string fY 1 ; Y 2 ; ...g, where Y i 2 A ¼ a finite alphabet of symbols (¼fa 1 ; a 2 ; ... ; a k g, say). We assume that this observed sequence is generated by a random replication process operating on an (possibly unobserved) ancestor string fX 1 ; X 2 ; ...g of characters from the same alphabet A. Such replication of character strings arise in molecular evolution of DNA, RNA and protein sequences. Several stochastic models for biological sequences (i.e., DNA, RNA and protein sequences) have been considered in the literature, and their biological significance has been investigated by several authors (see e.g., ARTICLE IN PRESS www.elsevier.com/locate/stapro 0167-7152/$-see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2005.06.002 à Corresponding author. E-mail addresses: probal@isical.ac.in (P. Chaudhuri), amites@isical.ac.in (A. Dasgupta).