Matrix normalized convergence of a Lévy process to normality at zero

We give a necessary and sufficient condition for a d-dimensional Lévy process to be in the matrix normalized domain of attraction of a d-dimensional normal random vector, as t↓0. This transfers to the Lévy case classical results of Feller, Khinchin, Lévy and Hahn and Klass for random walks. A specific construction of the norming matrix is given, and it is shown that centering constants may be taken as 0. Functional and self-normalization results are also given, as is a necessary and sufficient condition for the process to be in the matrix normalized domain of partial attraction of the normal.