A half-normal distribution scheme for generating functions

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.

Long version of "A half-normal distribution scheme for generating functions and the unexpected behavior of Motzkin paths" appeared in the Proceedings of 27th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms, Krakow, Poland, 4-8 July 2016, see arXiv:1605.03046. To appear in the European Journal of Combinatorics

References in corpus (6)

• Prediction by Random-Walk Perturbation
• Lattice paths with catastrophes
• Maxima of Two Random Walks: Universal Statistics of Lead Changes
• Local time for lattice paths and the associated limit laws
• Some reflections on directed lattice paths
• A half-normal distribution scheme for generating functions and the unexpected behavior of Motzkin paths

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