A half-normal distribution scheme for generating functions. (June 2020)

• Record Type: Journal Article
• Authors: Wallner, Michael
• Is Part Of: European journal of combinatorics. Volume 87(2020)
• Publication Date: 2020-06
• Subjects: Combinatorial analysis -- Periodicals; Analyse combinatoire -- Périodiques; Combinatorial analysis; Periodicals; Electronic journals; 511.6
• DOI: 10.1016/j.ejc.2020.103138
• Languages: English
• ISSNs: 0195-6698

Abstract

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.