On Approximations of Subordinators in $L^p$ and the Simulation of Tempered Stable Distributions

TL;DR

This paper develops methods to approximate subordinators, especially tempered stable distributions, using scaled Poisson mixtures, providing convergence rates and a simulation approach for these infinitely divisible distributions.

Contribution

It introduces a novel approximation technique for subordinators via scaled Poisson mixtures with explicit convergence rates in $L^p$, applicable to tempered stable distributions.

Findings

Established $L^p$ convergence rates for Poisson mixture approximations.

Developed a practical method for simulating tempered stable subordinators.

Extended results to mixtures represented as differences of stopped Lévy processes.

Abstract

Subordinators are infinitely divisible distributions on the positive half-line. They are often used as mixing distributions in Poisson mixtures. We show that appropriately scaled Poisson mixtures can approximate the mixing subordinator and we derive a rate of convergence in $L^p$ for each $p\in[1,\infty]$. This includes the Kolmogorov and Wasserstein metrics as special cases. As an application, we develop an approach for approximate simulation of the underlying subordinator. In the interest of generality, we present our results in the context of more general mixtures, specifically those that can be represented as differences of randomly stopped L\'evy processes. Particular focus is given to the case where the subordinator belongs to the class of tempered stable distributions.