Localization for controlled random walks and martingales

Ori Gurel-Gurevich; Yuval Peres; Ofer Zeitouni

doi:10.1214/ECP.v19-3081

Localization for controlled random walks and martingales

Ori Gurel-Gurevich, Yuval Peres, Ofer Zeitouni

The Hebrew University of Jerusalem, Weizmann Institute of Science

Research output: Contribution to journal › Article › peer-review

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^u_n = 0) decays at polynomial rate n^-α where α > 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.

Original language	English
Pages (from-to)	1-8
Number of pages	8
Journal	Electronic Communications in Probability
Volume	19
DOIs	https://doi.org/10.1214/ECP.v19-3081
State	Published - 18 Apr 2014

Keywords

Martingale
Random walk
Stochastic control

All Science Journal Classification (ASJC) codes

Statistics and Probability
Statistics, Probability and Uncertainty

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keywords = "Martingale, Random walk, Stochastic control",

author = "Ori Gurel-Gurevich and Yuval Peres and Ofer Zeitouni",

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AU - Peres, Yuval

AU - Zeitouni, Ofer

N1 - NSF [DMS-1106627]; ISF grantNSF grant DMS-1106627 and ISF grant.

PY - 2014/4/18

Y1 - 2014/4/18

N2 - 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(Sun = 0) decays at polynomial rate n-α where α > 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.

AB - 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(Sun = 0) decays at polynomial rate n-α where α > 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.

KW - Martingale

KW - Random walk

KW - Stochastic control

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DO - 10.1214/ECP.v19-3081

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SP - 1

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JO - Electronic Communications in Probability

JF - Electronic Communications in Probability

ER -

Localization for controlled random walks and martingales

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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