Distributional representations and dominance of a Lévy process over its maximal jump processes

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 $t\downarrow 0$) 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 ${\sup }_{0<s\le t}|\Delta X_s|$. Thus, we obtain necessary and sufficient conditions for $X_t/{\sup }_{0<s\le t}\Delta X_s$ and $X_t/{\sup }_{0<s\le t}|\Delta X_s|$ to converge in probability to 1, or to $\infty$, as $t\downarrow 0$, 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.

Published at http://dx.doi.org/10.3150/15-BEJ731 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (2)

• Limit theorems for mixed max-sum processes with renewal stopping
• Asymptotic behavior of the generalized St. Petersburg sum conditioned on its maximum

Cited by in corpus (7)

• The limit distribution of ratios of jumps and sums of jumps of subordinators
• Convergence to stable limits for ratios of trimmed Levy processes and their jumps
• Generalised Poisson-Dirichlet Distributions and the Negative Binomial Point Process
• Convergence of Trimmed Lévy Processes to Trimmed Stable Random Variables at $0$
• Tightness and Convergence of Trimmed Lévy Processes to Normality at Small Times
• Small Time Convergence of Subordinators with Regularly or Slowly Varying Canonical Measure
• Conditions for a Lévy process to stay positive near 0, in probability