## 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

### Keywords:

intersection local time; Wiener sausage
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### References:

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