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2018 Matrix normalised stochastic compactness for a Lévy process at zero
Ross A. Maller, David M. Mason
Electron. J. Probab. 23: 1-37 (2018). DOI: 10.1214/18-EJP193

Abstract

We give necessary and sufficient conditions for a $d$–dimensional Lévy process $({\bf X}_t)_{t\ge 0}$ 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 $\Bbb{R} ^d$ 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 $({\bf X}_t)$ 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.

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Ross A. Maller. David M. Mason. "Matrix normalised stochastic compactness for a Lévy process at zero." Electron. J. Probab. 23 1 - 37, 2018. https://doi.org/10.1214/18-EJP193

Information

Received: 30 August 2017; Accepted: 3 July 2018; Published: 2018
First available in Project Euclid: 26 July 2018

zbMATH: 1395.60053
MathSciNet: MR3835475
Digital Object Identifier: 10.1214/18-EJP193

Subjects:
Primary: 62B15 , 62E17 , 62G05

Keywords: domain of attraction , matrix normalisation , matrix-valued Lévy process , Quadratic Variation , small time convergence , stochastic compactness , vector-valued Lévy Process

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