Markovian embeddings of general random strings

Abstract

Let A be a finite set and X a sequence of A-valued random variables. We do not assume any particular correlation structure between these random variables; in particular, X may be a non-Markovian sequence. An adapted embedding of X is a sequence of the form R(X_1), R(X_1,X_2), R(X_1,X_2,X_3), etc where R is a transformation defined over finite length sequences. In this extended abstract we characterize a wide class of adapted embeddings of X that result in a first-order homogeneous Markov chain. We show that any transformation R has a unique coarsest refinement R' in this class such that R'(X_1), R'(X_1,X_2), R'(X_1,X_2,X_3), etc is Markovian. (By refinement we mean that R'(u)=R'(v) implies R(u)=R(v), and by coarsest refinement we mean that R' is a deterministic function of any other refinement of R in our class of transformations.) We propose a specific embedding that we denote as R^X which is particularly amenable for analyzing the occurrence of patterns described by regular expressions in X. A toy example of a non-Markovian sequence of 0's and 1's is analyzed thoroughly: discrete asymptotic distributions are established for the number of occurrences of a certain regular pattern in X_1,...,X_n, as n tends to infinity, whereas a Gaussian asymptotic distribution is shown to apply for another regular pattern.