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 Electronic Communications in Probability > Vol. 3 (1998) > Paper 3 open journal systems 


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 ${(t,X_t^n) : 0 le t le T}$.


Full text: PDF

Pages: 21-27

Published on: April 13, 1998


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Electronic Communications in Probability. ISSN: 1083-589X