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October 2001 How to Find an extra Head: Optimal Random Shifts of Bernoulli and Poisson Random Fields
Alexander E. Holroyd, Thomas M. Liggett
Ann. Probab. 29(4): 1405-1425 (October 2001). DOI: 10.1214/aop/1015345755


We consider the following problem:given an i.i.d. family of Bernoulli random variables indexed by $\mathbb{Z}^d$, find a random occupied site $X \in \mathbb{Z}^d$ such that relative to $X$, the other random variables are still i.i.d. Bernoulli. Results of Thorisson imply that such an $X$ exists for all $d$. Liggett proved that for$d = 1$, there exists an $X$ with tails $P(|X|\geq t)$ of order $ct^(-1 /2}$, but none with finite $1/2$th moment. We prove that for general $d$ there exists a solution with tails of order $ct^{-d/2}$, while for $d = 2$ there is none with finite first moment. We also prove analogous results for a continuum version of the same problem. Finally we prove a result which strongly suggests that the tail behavior mentioned above is the best possible for all$d$.


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Alexander E. Holroyd. Thomas M. Liggett. "How to Find an extra Head: Optimal Random Shifts of Bernoulli and Poisson Random Fields." Ann. Probab. 29 (4) 1405 - 1425, October 2001.


Published: October 2001
First available in Project Euclid: 5 March 2002

zbMATH: 1019.60048
MathSciNet: MR1880225
Digital Object Identifier: 10.1214/aop/1015345755

Primary: 60G60
Secondary: 60G55 , 60K35

Keywords: Poisson process , Product measure , random shift , shift coupling , Tagged particle

Rights: Copyright © 2001 Institute of Mathematical Statistics


Vol.29 • No. 4 • October 2001
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