Random walks in Euclidean space. (English) Zbl 1315.60034

Varjú, Péter Pál

Ann. Math. (2) 181, No. 1, 243-301 (2015).

Let \(X_1, X_2,\dots\) be independent identically distributed random isometries of the Euclidean space \(R^d\). Let \(x_0\in R^d\) be any point, and consider the sequence of points \[ Y=x_0, \dots, Y_e(X_{l-1}(\dots (x_0))), \dots. \]
The author’s main objective is to obtain a local limit theorem in the following sense. Under some conditions on \(Y_0\) and \(X\) there are \(v_0\in \mathbb{R}^d\) and \(c>0\) depending only on the distribution of \(X_j\) such that \[ \lim_{l\to0}l^{d/2}E[f(Y_0-lv_0)]=c\int f(r)\,dx, \] where \(f\) is any continuous and compactly supported function.

Reviewer: N. G. Gamkrelidze (Moskva)

MSC:

60F05	Central limit and other weak theorems
60G50	Sums of independent random variables; random walks

Keywords:

random walks; Euclidean isometries; local limit theorem

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