Multidimensional Limit Theorems Allowing Large Deviations for Densities of Regular Variation

The sums of i.i.d. random vectors are considered. It is assumed that the underlying distribution is absolutely continuous and its density possesses the property which can be referred to as regular variation. The asymptotic expressions for the probability of large deviations are established in the case of a normal limiting law. Furthermore, the role of the maximal summand is emphasized.