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 Markov Processes Relat. Fields, 2014, Volume 20, Issue 4, Pages 633–652 (Mi mprf2)

Random walk with heavy tail and negative drift conditioned by its minimum and final values

V. Bansayea, V. Vatutinb

a Ecole Polytech, CMAP, F-91128 Palaiseau, France
b VA Steklov Math Inst, Dept Discrete Math, Moscow 119991, Russia

Abstract: We consider a random walk with finite second moment which drifts to $-\infty$ and has a heavy tail. We focus on the events when the minimum and the final value of this walk belong to some compact set. We first specify the associated probability. Then, conditionally on such an event, we finely describe the trajectory of the random walk. It yields a decomposition theorem with respect to a random time giving a big jump whose distribution can be described explicitly.

 Funding Agency Grant Number Agence Nationale de la Recherche 09-BLAN-0215 Russian Academy of Sciences - Federal Agency for Scientific Organizations 17 This work was partially funded by the project MANEGE 'Modeles Aleatoires en Ecologie, Genetique et Evolution' 09-BLAN-0215 of ANR (French national research agency), Chair Modelisation Mathematique et Biodiversite VEOLIA-Ecole Polytechnique-MNHN-F.X. and the professorial chair Jean Marjoulet. The second author was also supported by the Program of the Russian Academy of Sciences "Dynamical systems and control theory".

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Document Type: Article
Language: English