Convergence to stable limits for ratios of trimmed Levy processes and their jumps

TL;DR

This paper establishes the convergence of ratios of trimmed Levy processes and their jumps to stable limits, providing new identities and distributions in the context of processes in the domain of attraction of stable laws.

Contribution

It introduces characteristic function identities for conditioned trimmed Levy processes and proves their convergence to stable process-based limits, extending previous subordinator results.

Findings

Joint convergence of trimmed process ratios to stable process quantities

New discrete distributions on the infinite simplex in the limit

Generalization of 1D subordinator results to Levy processes

Abstract

We derive characteristic function identities for conditional distributions of an r-trimmed Levy process given its r largest jumps up to a designated time t. Assuming the underlying Levy process is in the domain of attraction of a stable process as t goes to 0, these identities are applied to show joint convergence of the trimmed process divided by its large jumps to corresponding quantities constructed from a stable limiting process. This generalises related results in the 1-dimensional subordinator case developed in Kevei & Mason (2014) and produces new discrete distributions on the infinite simplex in the limit.