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Eventual Intersection for Sequences of Lévy Processes

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	Eventual Intersection for Sequences of Lévy Processes
2.	Creator	Author's name, affiliation, country	Steven N. Evans; University of California at Berkeley
2.	Creator	Author's name, affiliation, country	Yuval Peres; University of California, Berkeley
3.	Subject	Discipline(s)	Mathematics
3.	Subject	Keyword(s)	Lévy process, hitting probability, range, graph, random measure, random set, stationary
3.	Subject	Subject classification	60J30, 60G17, 60G57, 60J45
4.	Description	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 $P\{F_i \cap F_j \ne \emptyset \mid F_j\} > 0$ 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 $\{(t,X_t^n) : 0 \le t \le T\}$.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	1998-04-13
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ecp.ejpecp.org/article/view/989
10.	Identifier	Digital Object Identifier	10.1214/ECP.v3-989
11.	Source	Journal/conference title; vol., no. (year)	Electronic Communications in Probability; Vol 3
12.	Language	English=en	en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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