Stationarity and mixing properties of replicating character strings

Abstract

In this article, some models for random replication of character strings are considered that involve random mutations, deletions and insertions of characters. We derive some sufficient conditions on the replication process and the ancestor chain that ensure stationarity and mixing properties of the replicated chain. We also give examples of replication processes which lead to descendant chains not having any mixing properties even if the ancestor chain is i.i.d. in nature. Stationarity and mixing properties are two properties of dependent processes that are of fundamental importance and well studied in the literature. These properties are quite useful in generalizing many asymptotic results for i.i.d. processes to dependent processes and, in many situations, they are useful in justifying statistical estimation and inference based on dependent data. The presence of random deletions and insertions makes our stochastic replication model considerably different from simpler models that involve only mutations, and it leads to some interesting theoretical problems.