Self-intersections of 1-dimensional random walks.(English)Zbl 0602.60055

Suppose that $$S_ n$$, $$n\geq 0$$ is a random walk on the integers and that its steps have mean 0 and variance $$\sigma^ 2$$. It is shown that the time $$T_ 1$$ of first self-intersection of the random walk grows at rate $$\sigma^{2/3}$$ as $$\sigma\to \infty$$. In fact the following limit theorem is proved for the entire process $$(T_ i)$$ of self- intersections, where $$T_ i=\min \{n>T_{i-1}:S_ n=S_ m$$ for some $$m<n\}$$, $$i\geq 2$$. Let W(t), $$t\geq 0$$ be Brownian motion and define $$0<U_ 1<U_ 2<..$$. by stipulating that, conditioned on W, the times $$(U_ i)$$ are the points of a non-homogeneous Poisson process of rate L(t,W(t)), $$t\geq 0$$ where L(t,x) is the local time of W. It is shown that if $$S^*(t)=\sigma^{-4/3}S_{[t\sigma^{2/3}]}$$, $$t\geq 0$$ and $$T_ i^*=\sigma^{-2/3}T_ i$$, then under certain technical conditions $(S^*,T^{*}_{1},T^{*}_{2},...)\to^{{\mathcal D}}(W,U_ 1,U_ 2,...)$ as $$\sigma\to \infty$$, in the sense of weak convergence on D[0,$$\infty)\times [0,\infty)\times [0,\infty)\times..$$. It is further shown that $$\sigma^{-2/3}ET_ 1\to EU_ 1=2^{2/3}\Gamma (5/3)E(Y^{-2/3})\approx 1.43$$, where $$Y=\int^{\infty}_{-\infty}L^ 2(1,x)dx$$.
Reviewer: F.Papangelou

MSC:

 60G50 Sums of independent random variables; random walks 60F05 Central limit and other weak theorems 60J65 Brownian motion
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References:

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