The inner boundary of random walk range. (English) Zbl 1359.60061

Okada, Izumi

J. Math. Soc. Japan 68, No. 3, 939-959 (2016).

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.

Reviewer: Aleksander Iksanov (Kiev)

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)

Keywords:

random walk range; inner boundary; strong laws of large numbers; large deviations