Eventual Intersection for Sequences of Lévy Processes

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 $P\{F_i \cap F_j \ne \emptyset \mid F_j\} \gt 0$ 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 $P\{\exists 0 \le s, t \le T : X_s^i = X_t^j \} \gt 0$ for $i \ne j$. We also establish an analogous result for the sequence of graphs $\{(t,X_t^n) : 0 \le t \le T\}$.