Functional Convergence to Stable Lévy Motions for Iterated Random Lipschitz Mappings

Sana Louhichi (Université Joseph Fourier)
Emmanuel Rio (Université de Versailles Saint-Quentin en Y.)

Abstract


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 $J_1$-topology sense. The purpose of our paper is to study this convergence in $D([0,1])$ equipped with the $M_1$-topology, which is weaker than the $J_1$ 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 $M_1$-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.

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Pages: 2452-2480

Publication Date: November 26, 2011

DOI: 10.1214/EJP.v16-965

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