Matrix normalized sums of independent identically distributed random vectors (Q1081958)

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.