On the maximum coefficients of the 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 ∈ R, 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/2 λn ) 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