A half-normal distribution scheme for generating functions. (June 2020) - Wallner, Michael
Abstract: We present a general theorem on the structure of bivariate generating functions which gives sufficient conditions such that the limiting probability distribution is a half-normal distribution. If X is a normally distributed random variable with zero mean, then | X | obeys a half-normal distribution. In the second part, we apply our result to prove three natural appearances in the domain of lattice paths: the number of returns to zero, the height, and the sign changes are under zero drift distributed according to a half-normal distribution. This extends known results to a general step set. Finally, our result also gives a new proof of Banach's matchbox problem.