  Invariance principles are obtained for a Markov process on a half-line with
continuous paths on the interior. The domains of attraction of the two
different types of self-similar processes are investigated. Our approach is to
establish convergence of excursion point processes, which is based on It\^{o}'s
excursion theory and a recent result on convergence of excursion measures by
Fitzsimmons and the present author.
