An almost sure invariance principle for the range of random walks (Q1807273)

The author investigates invariance principles for the range of a random walk moving on the four or more dimensional integer lattice and starting from the origin. The range of random walks, \(R_n\), is considered as the number of distinct lattice sites visited at least once by the random walk in the first \(n\) steps. It is proved that the centralized and linearly interpolated range of the random walk can be asymptotically equal to a Brownian motion almost surely.