Eventual Intersection for Sequences of Lévy Processes
Steven N. Evans, University of California at Berkeley
Yuval Peres, University of California, Berkeley
Abstract
Consider the events
${F_n cap bigcup_{k=1}^{n-1} F_k = emptyset}$, $n in N$,
where $(F_n)_{n=1}^infty$ is an i.i.d. sequence of
stationary random subsets of a compact group $G$.
A plausible conjecture is that these events will not occur infinitely often
with positive probability if
$P{F_i cap F_j ne emptyset , | , F_j} > 0$ a.s. for $i ne j$.
We present a counterexample to show that this
condition is not sufficient, and give one that is.
The sufficient condition always holds when
$F_n = {X_t^n : 0 le t le T}$ is the range of
a Lévy process $X^n$ on the $d$-dimensional torus
with uniformly distributed initial position
and $P{exists 0 le s, t le T : X_s^i = X_t^j } > 0$ for $i ne j$.
We also establish an analogous result for the sequence of
graphs ${(t,X_t^n) : 0 le t le T}$.
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