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

Michael Grabchak, Sina Saba
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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.
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论$L^p$中的近似下位数与节制稳定分布的模拟
副基数是正半线上无限可分的分布。它们经常被用作泊松混合物中的混合分布。我们发现,适当缩放的泊松混合物可以近似于混合副集,并且我们推导出了每个$p/in[1,\infty]$的$L^p$收敛率。这包括科尔莫哥洛夫度量和瓦瑟斯坦度量这两种特殊情况。作为应用,我们开发了一种近似模拟底层从属器的方法。为了保持一般性,我们将在更一般的混合物背景下展示我们的结果,特别是那些可以表示为随机停止的 L\'evy 过程的差异的混合物。
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