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.
Citation
Ori Gurel-Gurevich. Yuval Peres. Ofer Zeitouni. "Localization for controlled random walks and martingales." Electron. Commun. Probab. 19 1 - 8, 2014. https://doi.org/10.1214/ECP.v19-3081
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