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.
Citation
Sana Louhichi. Emmanuel Rio. "Functional Convergence to Stable Lévy Motions for Iterated Random Lipschitz Mappings." Electron. J. Probab. 16 2452 - 2480, 2011. https://doi.org/10.1214/EJP.v16-965
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