Matrix normalised stochastic compactness for a Levy process at zero

Abstract

We give necessary and sufficient conditions for a d-dimensional Levy process (X-t)(t >= 0) to be in the matrix normalised Feller (stochastic compactness) classes FC and FC0 as t down arrow 0. This extends earlier results of the authors concerning convergence of a Levy process in R-d to normality, as the time parameter tends to 0. It also generalises and transfers to the Levy case classical results of Feller and Griffin concerning realand vector-valued random walks. The process (X-t) and its quadratic variation matrix together constitute a matrix-valued Levy 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. Show more