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$d$.
Publié le : 2001-10-14
Classification:
product measure,
random shift,
Poisson process,
tagged particle,
shift coupling,
60G60,
60G55,
60K35
@article{1015345755,
author = {Holroyd, Alexander E. and Liggett, Thomas M.},
title = {How to Find an extra Head: Optimal Random Shifts of Bernoulli and
Poisson Random Fields},
journal = {Ann. Probab.},
volume = {29},
number = {1},
year = {2001},
pages = { 1405-1425},
language = {en},
url = {http://dml.mathdoc.fr/item/1015345755}
}
Holroyd, Alexander E.; Liggett, Thomas M. How to Find an extra Head: Optimal Random Shifts of Bernoulli and
Poisson Random Fields. Ann. Probab., Tome 29 (2001) no. 1, pp. 1405-1425. http://gdmltest.u-ga.fr/item/1015345755/