Theoretical Statistics and Mathematics Unit, ISI Delhi

On the coverage of space by random sets

by Siva Athreya, Rahul Roy and Anish Sarkar

Let $\xi_1 , \xi_2 , . . .$ be a Poisson point process of density $\lambda$ on $(0, \infty)^d$ , $d \ge 1$ and let
$\rho, \rho_1 , \rho_2 , . . .$ be i.i.d. positive random variables independent of the point process. Let
$C := \cup_{i \ge 1} \{ \xi_i + [0, \rho_i ]^d \}$. If, for some $t > 0$, $(t, \infty)^d \subseteq C$, then we say that $(0, \infty)^d$ is
eventually covered. We show that the eventual coverage of $(0, \infty)^d$ depends on the behaviour of $xP (\rho > x)$ as $x \Rightarrow \infty$ as well as on whether $d = 1$ or $d \ge 2$. These results
are quite dissimilar to those known for complete coverage of $\mathbb{R}^d$ by such Poisson Boolean
models (Hall [3]).
In addition, we consider the region $C := \cup_{\{i\ge 1: X_i =1\}}$ $[i, i + \rho_i ]$, where $X_1 , X_2 , . . .$ is
a $\{0, 1\}$ valued Markov chain and $\rho, \rho_1 , \rho_2 , . . .$ are i.i.d. positive integer valued random
variables independent of the Markov chain. We study the eventual coverage properties of
this random set $C$.

isid/ms/2003/01 [fulltext]

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