  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
polyominoes. As a by-product of our approach we find the joint distribution of
area and final altitude for meanders with an arbitrary step set, and for
unconstrained Bernoulli walks (and hence for Brownian Motion) the joint
distribution of signed areas and final altitude. We give these distributions in
terms of their moments.
