Functional convergence to stable Lévy motions for iterated random Lipschitz mappings
Résumé
It is known that, in the dependent case, partial sums processes which are elements of D([0,1]) (the space of right-continuous functions on [0,1] with left limits) do not always converge weakly in the J1-topology sense. The purpose of our paper is to study this convergence in D([0,1]) equipped with the M1-topology, which is weaker than the J1 one. We prove that if the jumps of the partial sum process are associated then a functional limit theorem holds in D([0,1]) equipped with the M1-topology, as soon as the convergence of the finite-dimensional distributions holds. We apply our result to some stochastically monotone Markov chains arising from the family of iterated Lipschitz models.