A half-normal distribution scheme for generating functions

We present an extension of a theorem by Michael Drmota and Michèle Soria [Images and Preimages in Random Mappings, 1997] which can be used to identify the limiting distribution for a class of combinatorial schemata. This is achieved by determining analytic and algebraic properties of the associated bivariate generating function. We give sufficient conditions implying a half-normal limiting distribution, extending the known conditions leading to either a Rayleigh, a Gaussian, or a convolution of the last two distributions. We conclude with three natural appearances of such a limiting distribution in the domain of lattice paths.