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M. Wallner:
"A half-normal distribution scheme for generating function";
European Journal of Combinatorics, 87 (2020), 103138.

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.

http://dx.doi.org/10.1016/j.ejc.2020.103138