Electronic Journal of Probability

Matrix normalised stochastic compactness for a Lévy process at zero

Ross A. Maller and David M. Mason

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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.

Article information

Electron. J. Probab., Volume 23 (2018), paper no. 69, 37 pp.

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

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Zentralblatt MATH identifier

Primary: 62E17: Approximations to distributions (nonasymptotic) 62B15: Theory of statistical experiments 62G05: Estimation

vector-valued Lévy Process matrix-valued Lévy process small time convergence matrix normalisation stochastic compactness domain of attraction quadratic variation

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Maller, Ross A.; Mason, David M. Matrix normalised stochastic compactness for a Lévy process at zero. Electron. J. Probab. 23 (2018), paper no. 69, 37 pp. doi:10.1214/18-EJP193. https://projecteuclid.org/euclid.ejp/1532570597

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