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Stability of overshoots of recurrent random walks

Le : 22/05/2017 11h00
Par : Vladislav Vysotsky (University of Sussex)
Lieu : i103
Lien web :
Résumé : Take a one-dimensional random walk with zero mean increments, and consider the sizes of its overshoots over the zero level. It turns out that this sequence, which forms a Markov chain, always has a unique stationary distribution of a simple explicit form. The question of convergence to this distribution is surprisingly hard. We were able to prove only the total variation convergence, which holds for random walks with lattice and spread out distributions (i.e., essentially, the ones with density). We also obtained the rate of this convergence under additional mild assumptions. We shall also discuss connections to related topics: local times of random walks and the number of level-crossings, stability of reflected random walks, ergodic theory, and renewal theory. This is a joint work with Alex Mijatovic (King’s College London).