Matrix normalised stochastic compactness for a Lévy process at zero

We give necessary and sufficient conditions for a d–dimensional Lévy process (Xt)t≥0 to be in the matrix normalised Feller (stochastic compactness) classes FC and FC0 as t ↓ 0. This extends earlier results of the authors concerning convergence of a Lévy process in R to normality, as the time parameter tends to 0. It also generalises and transfers to the Lévy case classical results of Feller and Griffin concerning realand vector-valued random walks. The process (Xt) and its quadratic variation matrix together constitute a matrix-valued Lévy process, and, in a further extension, we show that the condition derived for the process itself also guarantees the stochastic compactness of the combined matrix-valued process. This opens the way to further investigations regarding self-normalised processes.