On the convergence of the distribution for the parameter of a shift for a composition of random motions of the Euclidean space to a multidimensional stable law. (Russian) Zbl 0532.60021

Khokhlov, Yu. S.

Teor. Veroyatn. Mat. Stat. 28, 133-138 (1983).

The paper proves the following result: Let \(\{g_ n,n\geq 1\}\) be a sequence of independent identically distributed random elements taking values in M(d) - a group of motions of \({\mathbb{R}}^ d\), \(g_ i=(U_ i,Y_ i)\), \(U_ i\)- a rotation, \(Y_ i\)- a shift. Suppose that (i) \(U(n)=U_ 1U_ 2...U_ n\) converges weakly to a Haar measure, (ii) \(Y_ i\) belong to the domain of attraction of a multidimensional distribution with \(B_ n=n^{1/\alpha}\), \(0<\alpha<2\), \(\alpha\neq 1\). Then \(n^{-1/\alpha}Y(n)\), where \(Y(n)=Y_ 1+U_ 1Y_ 2+...+U_ 1...U_{n-1}Y_ n\), converges weakly to a multidimensional stable law with characteristic function \(\phi(t)=\exp \{-c| t|^{\alpha}\}\), \(t\in {\mathbb{R}}^ d\), \(c>0\).

Reviewer: D.Szynal

MSC:

60F05	Central limit and other weak theorems
60E07	Infinitely divisible distributions; stable distributions
60D05	Geometric probability and stochastic geometry

Keywords:

Haar measure; domain of attraction; stable law; characteristic function