Linear functional equations with a catalytic variable and area limit laws for lattice paths and polygons

We study limit distributions for random variables defined in terms of coefficients of a power series which is determined by a certain linear functional equation. Our technique combines the method of moments with the kernel method of algebraic combinatorics. As limiting distributions the area distributions of the Brownian excursion and meander occur. As combinatorial applications we compute the area laws for discrete excursions and meanders with an arbitrary finite set of steps and the area distribution of column convex polygons. As a by-product of our approach we find the joint distribution of signed areas and final altitude of a Brownian motion in terms of joint moments.