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An almost sure invariance principle for the range of planar random walks. (English) Zbl 1085.60018

Let \(S_n=X_1+\dots+X_n\) be a random walk in \(Z^2\), where \(X_1,X_2,\ldots\) are symmetric i.i.d. vectors in \(Z^2\). We assume that \(X_i\) have \(2+\delta\) moments for some \(\delta>0\) and covariance matrix equal to the identity. We assume further that the random walk \(S_n\) is strongly aperiodic in the sense of Spitzer. The range \(\mathcal R(n)\) of the random walk \(S_n\) is the set of sites visited by the walk up to step \(n\): \( \mathcal R(n)=\{S_0,\ldots,S_{n-1}\}. \) As usual, \(| \mathcal R(n)| \) denotes the cardinality of the range up to step \(n\). The authors show that for each \(k\geq1\) \[ (\log n)^k\bigg[\frac1n| \mathcal R(n)| +\sum_{j=1}^k(-1)^j\bigg(\frac1{2\pi}\log n+c_X\bigg)^{-j}\gamma_{j,n}\bigg]\to0\qquad\text{a.s.}, \] where \(W_t\) is a Brownian motion, \(W_t^{(n)}=W_{nt}/\sqrt{n}\), \(\gamma_{j,n}\) is the renormalized intersection local time at time 1 for \(W^{(n)}\) and \(c_X\) is a constant depending on the distribution of the random walk.

MSC:

60F15 Strong limit theorems
60G50 Sums of independent random variables; random walks
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