## The Annals of Probability

### How to Find an extra Head: Optimal Random Shifts of Bernoulli and Poisson Random Fields

#### Abstract

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$d$.

#### Article information

Source
Ann. Probab., Volume 29, Number 4 (2001), 1405-1425.

Dates
First available in Project Euclid: 5 March 2002

https://projecteuclid.org/euclid.aop/1015345755

Digital Object Identifier
doi:10.1214/aop/1015345755

Mathematical Reviews number (MathSciNet)
MR1880225

Zentralblatt MATH identifier
1019.60048

#### Citation

Holroyd, Alexander E.; Liggett, Thomas M. How to Find an extra Head: Optimal Random Shifts of Bernoulli and Poisson Random Fields. Ann. Probab. 29 (2001), no. 4, 1405--1425. doi:10.1214/aop/1015345755. https://projecteuclid.org/euclid.aop/1015345755

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