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(nk/2λn) 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. © Springer-Verlag Berlin Heidelberg 2004.