An almost sure invariance principle for the range of random walks

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Abstract

The range of random walks means the number of distinct sites visited at least once by the random walk before time n. We study an almost sure invariance principle for the range of random walks on the four or more dimensional integer lattice and obtain that the centralized and linearly interpolated range of the random walk can be asymptotically equal to a Brownian motion almost surely.

MSC

primary 60J15
secondary 60G15
60G17

Keywords

Almost sure invariance principle
Range of random walks
Skorohod’s representation theorem

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