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November 2016 Distributional representations and dominance of a Lévy process over its maximal jump processes
Boris Buchmann, Yuguang Fan, Ross A. Maller
Bernoulli 22(4): 2325-2371 (November 2016). DOI: 10.3150/15-BEJ731

Abstract

Distributional identities for a Lévy process $X_{t}$, its quadratic variation process $V_{t}$ and its maximal jump processes, are derived, and used to make “small time” (as $t\downarrow0$) asymptotic comparisons between them. The representations are constructed using properties of the underlying Poisson point process of the jumps of $X$. Apart from providing insight into the connections between $X$, $V$, and their maximal jump processes, they enable investigation of a great variety of limiting behaviours. As an application, we study “self-normalised” versions of $X_{t}$, that is, $X_{t}$ after division by $\sup_{0<s\le t}\Delta X_{s}$, or by $\sup_{0<s\le t}\vert \Delta X_{s}\vert $. Thus, we obtain necessary and sufficient conditions for $X_{t}/\sup_{0<s\le t}\Delta X_{s}$ and $X_{t}/\sup_{0<s\le t}\vert \Delta X_{s}\vert $ to converge in probability to 1, or to $\infty$, as $t\downarrow0$, so that $X$ is either comparable to, or dominates, its largest jump. The former situation tends to occur when the singularity at 0 of the Lévy measure of $X$ is fairly mild (its tail is slowly varying at 0), while the latter situation is related to the relative stability or attraction to normality of $X$ at 0 (a steeper singularity at 0). An important component in the analyses is the way the largest positive and negative jumps interact with each other. Analogous “large time” (as $t\to\infty$) versions of the results can also be obtained.

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Boris Buchmann. Yuguang Fan. Ross A. Maller. "Distributional representations and dominance of a Lévy process over its maximal jump processes." Bernoulli 22 (4) 2325 - 2371, November 2016. https://doi.org/10.3150/15-BEJ731

Information

Received: 1 October 2014; Revised: 1 March 2015; Published: November 2016
First available in Project Euclid: 3 May 2016

zbMATH: 1352.60065
MathSciNet: MR3498031
Digital Object Identifier: 10.3150/15-BEJ731

Keywords: distributional representation , domain of attraction to normality , dominance , Lévy process , maximal jump process , Relative stability

Rights: Copyright © 2016 Bernoulli Society for Mathematical Statistics and Probability

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Vol.22 • No. 4 • November 2016
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