Eigenvalue Distributions and Geometry of Banach Spaces

We consider the sums Z⁽ⁿ⁾ of i.i.d. random vectors taking values in a d-dimensional Euclidean space. It is assumed that the so-called CRAMER condition holds in a neighbourhood of the origin. We establish lower bounds for the large deviation probabilities P(Z⁽ⁿ⁾ ? A) with A belonging to a large class of sets.