On the maximum coefficients of rational formal series in commuting variables

We study the maximum function of any R+-rational formal series S in two commuting variables, which assigns to every integer n is an element of 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 Theta(n(k/2)gimel(n)) for some integer k greater than or equal to -1 and some positive real lambda. 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.