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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.

##### 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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##### References:
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