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Scaling limit of the prudent walk. (English) Zbl 1201.60029

Summary: We describe the scaling limit of the nearest neighbour prudent walk on \(Z^{2}\), which performs steps uniformly in directions in which it does not see sites already visited. We show that the scaling limit is given by the process
\[ Z_u =\int_0^{3u/7}(\sigma_1 \mathbf 1_{\{W(s)\geq 0\}} \vec e_1+\sigma_2\mathbf 1_{\{W(s)< 0\}} \vec e_2)\, ds,\quad u \in [0,1], \] where \(W\) is the one-dimensional Brownian motion and \(\sigma_1,\sigma_2\) two random signs. In particular, the asymptotic speed of the walk is well-defined in the \(L^1\)-norm and equals 3/7.

MSC:

60F17 Functional limit theorems; invariance principles
60G50 Sums of independent random variables; random walks
60G52 Stable stochastic processes
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