Markovian Embeddings of General Random Strings

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(X1), R(X1,X2), R(X1,X2,X3), 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'(X1), R'(X1,X2), R'(X1,X2,X3), 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 RX 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 X1, ..., Xn as n → ∞ whereas a Gaussian asymptotic distribution is shown to apply for another regular pattern.