Localization for controlled random walks and martingales
Yuval Peres (Microsoft Research)
Ofer Zeitouni (Weizmann Institute)
Abstract
We consider controlled random walks that are martingales with uniformly bounded increments and nontrivial jump probabilities and show that such walks can be constructed so that $P(S_n^u=0)$ decays at polynomial rate $n^{-\alpha}$ where $\alpha>0$ can be arbitrarily small. We also show, by means of a general delocalization lemma for martingales, which is of independent interest, that slower than polynomial decay is not possible.
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Pages: 1-8
Publication Date: April 26, 2014
DOI: 10.1214/ECP.v19-3081
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