Random walks and intersection local time. (English) Zbl 0717.60057

Rosen, Jay

Ann. Probab. 18, No. 3, 959-977 (1990).

Let S(1,\(\cdot),S(2,\cdot),...,S(k,\cdot)\) be k independent random walks in \({\mathbb{Z}}^ 2\), each in the domain of attraction of a nondegenerate strictly stable process X(t) of order \(\beta >1\). Denote by \[ I(n)=n^{-1}\sum^{n}_{i_ 1,...,i_ k}\delta (S(1,i_ 1),S(2,i_ 2))...\delta (S(k-1,i_{k-1}),S(k,i_ k)) \] the functional related to the number of times \((i_ 1,...,i_ k)\) for which \(S(1,i_ 1)=S(2,i_ 2)=...=S(k,i_ k)\). In the case of a single random walk S(n) in \({\mathbb{Z}}^ 2\) assume also the functional R(n) related to the number of steps which S(n) spends in sites occupied at least k times. It is proved that if \(\beta >2-2/(2k-1)\), then suitably normalized R(n) converges as \(n\to \infty\) to the intersectional local time of the limiting process X(t); if \(\beta >2-2/k\), an analogous result is proved for I(n). To make the proofs more simple the author discusses separately the cases of attraction to Gaussian process and general stable process.

Reviewer: N.M.Zinchenko

MSC:

60G60	Random fields
60J55	Local time and additive functionals
60J65	Brownian motion

Keywords:

random walk; domain of attraction; intersectional local time; strictly stable process