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Propriétés d’intersection des marches aléatoires. I: Convergence vers le temps local d’intersection. (Properties of intersection of random walks. I: Convergence to local time intersection). (French) Zbl 0609.60078

We study intersection properties of multi-dimensional random walks. Let X and Y be two independent random walks with values in \({\mathbb{Z}}^ d\) (d\(\leq 3)\), satisfying suitable moment assumptions, and let \(I_ n\) denote the number of common points to the paths of X and Y up to time n. The sequence \((I_ n)\), suitably normalized, is shown to converge in distribution towards the ”intersection local time” of two independent Brownian motions.
Results are applied to the proof of a central limit theorem for the range of a two-dimensional recurrent random walk, thus answering a question raised by N. C. Jain and W. E. Pruitt [Proc. 6th Berkeley Sympos. math. Statist. Probab., Univ. Calif. 1970, 3, 31-50 (1972; Zbl 0276.60066)].

MSC:

60G50 Sums of independent random variables; random walks
60J55 Local time and additive functionals
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