  We study in limit law the complexity of some anticipated rejection random
sampling algorithms. We express this complexity in terms of a probabilistic
process, the threshold sum process. We show that, under the right conditions,
the complexity is linear and admits as a limit law a so-called
Darling-Mandelbrot distribution, studied by Darling (Trans Am Math Soc
73:95-107, 1952) and Lew (Constr Approx 10(1):15-30, 1994). We also give an
explicit form to the density of the Darling-Mandelbrot distribution and derive
some of its analytic properties.
