Self-intersections of 1-dimensional random walks (Q1081966)

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 \(Y=\int^{\infty}_{-\infty}L^ 2(1,x)dx\).