## Matrix normalized sums of independent identically distributed random vectors.(English)Zbl 0602.60031

Let $$X,X_ 1,X_ 2,..$$. be a sequence of i.i.d. random vectors taking values in $${\mathbb{R}}^ d$$, $$S_ n=\sum^{n}_{i=1}X_ i$$. Assume that the distribution of X is full, i.e. X is not supported an a (d-1)- dimensional hyperplane. Let $$S^{d-1}$$ be the unit sphere in $${\mathbb{R}}^ d$$ and $$<, >$$ the usual inner product. For $$r>0$$ and $$\theta \in S^{d- 1}$$ let $G(\theta,r)=P\{| <X,\theta >| >r\},\quad K(\theta,r)=r^{-2}\int_{| <X,\theta >| \leq r}<X,\theta >^ 2dP.$ The author proves that there exist matrices $$B_ n$$ and vectors $$\gamma_ n$$ such that $$\{B_ n(S_ n-\gamma_ n)\}$$ is stochastically compact, i.e., every subsequence contains a further subsequence which converges weakly to a full limit random vector, if and only if $\liminf_{r\to \infty}\quad \inf_{\theta \in S^{d- 1}}(K(\theta,r)/G(\theta,r))>0.$ In this case, estimates of the distribution of $$S_ n$$ are given. These results are then specialized to the case where $$X_ 1$$ is in the generalized domain of attraction of an operator stable law and a local limit theorem is proved which generalizes the classical local limit theorem where the normalization is done by scalars.
Reviewer: L.Hahn

### MSC:

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