## Electronic Communications in Probability

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

#### Article information

Source
Electron. Commun. Probab., Volume 3 (1998), paper no. 3, 21-27.

Dates
Accepted: 13 April 1998
First available in Project Euclid: 2 March 2016

https://projecteuclid.org/euclid.ecp/1456935909

Digital Object Identifier
doi:10.1214/ECP.v3-989

Mathematical Reviews number (MathSciNet)
MR1625695

Zentralblatt MATH identifier
0903.60060

Rights

#### Citation

Evans, Steven; Peres, Yuval. Eventual Intersection for Sequences of Lévy Processes. Electron. Commun. Probab. 3 (1998), paper no. 3, 21--27. doi:10.1214/ECP.v3-989. https://projecteuclid.org/euclid.ecp/1456935909

#### References

• R. Arratia, Coalescing Brownian motions on the line, Ph.D. thesis, University of Wisconsin, 1979. No
• R. Arratia, Coalescing Brownian motions on R and the voter model on Z, Preprint, 1981. No
• J. Bertoin, Levy Processes, Cambridge University Press, Cambridge, 1996.
• S.N. Evans, Multiple points in the sample paths of a Levy process, Probab. Th. Rel. Fields 76 (1987), 359-367.
• S.N. Evans, Coalescing Markov labelled partitions and a continuous sites genetics model with infinitely many types, Ann. Inst. Henri Poincare B 33 (1997), 339-358. No
• S.N. Evans and K. Fleischmann, Cluster formation in a stepping-stone model with continuous, hierarchically structured sites, Ann. Probab. 24 (1996), 1926-1952. No
• P.J. Fitzsimmons and T.S. Salisbury, Capacity and energy for multiparameter Markov processes, Ann. Inst. Henri Poincare 25 (1989), 325-350.
• T.E. Harris, Coalescing and noncoalescing stochastic flows in R_1, Stochastic Process. Appl. 17 (1984), 187-210.
• J.-P. Kahane, Some Random Series of Functions, Cambridge University Press, Cambridge, 1985.