Random walks in Euclidean space. (English) Zbl 1315.60034

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.


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