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The inner boundary of random walk range. (English) Zbl 1359.60061
For a delayed random walk in $$\mathbb{Z}^d$$ with $$d\in\mathbb{N}$$ having independent and identically distributed steps, denote by $$L_n$$ the number of the inner boundary points of the random walk range in the $$n$$ steps, that is, the number of points in the random walk range within the first $$n$$ steps for which there exists at least one neighbor point (with respect to $$\mathbb{Z}^d$$) outside the range. Further, for $$p\in\mathbb{N}$$, let $$J_n^{(p)}$$ and $$J_n^p$$ denote the number of the inner boundary points of the random walk range which are visited exactly $$p$$ times and at least $$p$$ times, respectively, within the first $$n$$ steps. For any delayed random walk in $$\mathbb{Z}^d$$ with $$d\in\mathbb{N}$$, the author proves the strong laws of large numbers for $$L_n$$, $$J_n^{(p)}$$ and $$J_n^p$$ as $$n\to\infty$$. For any delayed random walk in $$\mathbb{Z}^d$$ with $$d\geq 2$$ satisfying a standard aperiodic condition, it is shown that the limit $$\psi(x):=\lim_{n\to\infty}n^{-1}\log\mathbb{P}\{L_n>nx\}$$ exists for all $$x\in\mathbb{R}$$, and some properties of the rate function $$\psi$$ are discussed. For a delayed random walk in $$\mathbb{Z}^2$$ with the steps taking values $$(\pm 1,0)$$ and $$(0,\pm 1)$$ with probabilities $$1/4$$, it is proved that the limits $$\lim_{n\to\infty}n^{-1}(\log n)^2\mathbb{E}L_n$$, $$\lim_{n\to\infty}n^{-1}(\log n)^2\mathbb{E}J_n^{(p)}$$ and $$\lim_{n\to\infty}n^{-1}(\log n)^2\mathbb{E}J_n^p$$ exist. Since the explicit values of the limits are unknown, the author finds upper and lower bounds for these values.

MSC:
 60G50 Sums of independent random variables; random walks 60F15 Strong limit theorems 60F10 Large deviations 60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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