Distributional representations and dominance of a L\'{e}vy process over its maximal jump processes

TL;DR

This paper derives distributional identities for Lévy processes and their jump-related processes, enabling asymptotic comparisons and conditions for self-normalized convergence, revealing how jumps influence process behavior near zero.

Contribution

It introduces new distributional identities for Lévy processes and their maximal jumps, facilitating analysis of their asymptotic behavior and self-normalization properties.

Findings

Conditions for convergence of normalized Lévy processes as time approaches zero.

Insights into the dominance of Lévy processes over their maximal jumps.

Relationships between jump behavior and process stability or attraction to normality.

Abstract

Distributional identities for a L\'evy 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 $\sup_{0<s\le t}| \Delta X_s|$. 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 $t\downarrow0$, 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 L\'evy 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.