A limit theorem for recursively defined processes in L p

In this paper we derive a limit theorem for recursively defined processes. For several instances of recursive processes like for depth first search processes in random trees with logarithmic height or for fractal processes it turns out that convergence can not be expected in the space of continuous functions or in the Skorohod space D. We therefore weaken the Skorohod topology and establish a convergence result in Lp spaces in which D is continuously imbedded. The proof of our convergence result is based on an extension of the contraction method. An application of the limit theorem is given to the FIND process. The paper extends in particular results in [HR00] on the existence and uniqueness of random fractal measures and processes. The depth first search processes of Catalan and of logarithmically growing trees do however not fit the assumptions of our limit theorem and lead to the so far unsolved problem of degenerate limits.