Trimmed L\'evy Processes and their Extremal Components

TL;DR

This paper investigates the asymptotic behavior of a trimmed Lévy process and its largest jumps, establishing conditions under which the joint distribution converges to a normal or non-normal limit as the number of trimmed jumps increases.

Contribution

It introduces a detailed analysis of the large-trimming limit behavior of Lévy processes, including joint convergence results with both random and deterministic normalizations.

Findings

Joint convergence to a bivariate normal distribution with random normalization.

Non-normal limit distributions under deterministic normalization.

Conditions on the Lévy measure for the limit behavior.

Abstract

We analyse a trimmed stochastic process of the form ${}^{(r)}X_t= X_t - \sum_{i=1}^r \Delta_t^{(i)}$, where $(X_t)_{t \geq 0}$ is a driftless subordinator on $\mathbb{R}$ with its jumps on $[0,t]$ ordered as $ \Delta_t^{(1)}\ge \Delta_t^{(2)} \cdots$. When $r\to\infty$, both ${}^{(r)}X_t \to 0$ and $\Delta_t^{(r)} \to 0$ a.s. for each $t>0$, and it is interesting to study the weak limiting behaviour of $\bigl({}^{(r)}X_t, \Delta_t^{(r)}\bigr)$ in this case. We term this "large-trimming" behaviour. Concentrating on the case $t=1$, we study joint convergence of $\bigl({}^{(r)}X_1, \Delta_1^{(r)}\bigr)$ under linear normalization, assuming extreme value-related conditions on the L\'evy measure of $X$ which guarantee that $\Delta_1^{(r)}$ has a limit distribution with linear normalization. Allowing ${}^{(r)}X_1$ to have random centering and scaling in a natural way, we show that $\bigl({}^{(r)}X_1, \Delta_1^{(r)}\bigr)$ has a bivariate normal limiting distribution, as $r\to\infty$; but replacing the random normalizations with natural deterministic ones produces non-normal limits which we can specify.