Eventual intersection for sequences of L\'evy processes

Eventual intersection for sequences of L\'evy processes

Report Number
505
Authors
Steven N. Evans and Yuval Peres
Citation
Electronic Journal of Probability</em>, Vol. 5 (2000) Paper no. 2, pages 1-18
Abstract

Consider the events $\{F_n \cap \bigcup_{k=1}^{n-1} F_k = \emptyset\}$, $n \in \bN$, where $(F_n)_{n=1}^\infty$ is an i.i.d. sequence of stationary random subsets of a compact group $\bG$. A plausible conjecture is that these events will not occur infinitely often with positive probability if $\bP\{F_i \cap F_j \ne \emptyset \, | \, F_j\} > 0$ 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 L\'evy process $X^n$ on the $d$-dimensional torus with uniformly distributed initial position and $\bP\{\exists 0 \le s, t \le T : X_s^i = X_t^j \} > 0$ 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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