Analysis of the Rate Functions of Large Deviations for Symbol Statistics

We study the properties of large deviations for symbol statistics in primitive rational models. These models are defined by generalized automata over a binary alphabet for which the overall matrix of the transition weights is primitive. The corresponding symbol statistics enjoy a property of large deviations, and here we study the associated rate functions, which are always defined in an open interval (U, V ) ⊆ (0, 1). In particular, we prove some properties of symmetry of such functions, and show that their limits at the endpoints U and V are always finite. Moreover, under a rather mild condition, we prove that the same values U and V are rational numbers of the form i/j such that i, j ∈ {0, 1, . . . , d}, where d is the number of states of the generalized automaton defining the model. Under the same hypotheses we also yield a precise value for the corresponding limits depending on the characteristic polynomial of the matrix of the transition weights.