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Matrix normalised stochastic compactness for a Lévy process at zero. (English) Zbl 1395.60053

Summary: We give necessary and sufficient conditions for a \(d\)-dimensional Lévy process \((\mathbf{X}_t)_{t\geq 0}\) to be in the matrix normalised Feller (stochastic compactness) classes \(FC\) and \(FC_0\) as \(t\downarrow 0\). This extends earlier results of the authors concerning convergence of a Lévy process in \(\mathbb{R}^d\) 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 real- and vector-valued random walks. The process \((\mathbf{X}_t)\) 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.

MSC:

60G51 Processes with independent increments; Lévy processes
60F05 Central limit and other weak theorems
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References:

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