Probabilistic models for pattern statistics

In this work we study some probabilistic models for the random generation of words over a given alphabet used in the literature in connection with pattern statistics. Our goal is to compare models based on Markovian processes (where the occurrence of a symbol in a given position only depends on a finite number of previous occurrences) and the stochastic models that can generate a word of given length from a regular language under uniform distribution. We present some results that show the differences between these two stochastic models and their relationship with the rational probabilistic measures

On the Maximum Coefficients of Rational Formal Series in Commuting Variables

Abstract. We study the maximum function of any R+-rational formal series S in two commuting variables, which assigns to every integer n ∈ N, the maximum coefficient of the monomials of degree n. We show that if S is a power of any primitive rational formal series, then its maximum function is of the order Θ(n k/2 λ n ) for some integer k ≥ −1 and some positive real λ. Our analysis is related to the study of limit distributions in pattern statistics. In particular, we prove a general criterion for establishing Gaussian local limit laws for sequences of discrete positive random variables

Pattern occurrences in multicomponent models

In this paper we determine some limit distributions of pattern statistics in rational stochastic models, defined by means of nondeterministic weighted finite automata. We present a general approach to analyze these statistics in rational models having an arbitrary number of connected components. We explicitly establish the limit distributions in the most significant cases; these ones are characterized by a family of unimodal density functions defined by polynomials over adjacent intervals