  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.
