Translator Disclaimer
February 2019 Approximation of stable law in Wasserstein-1 distance by Stein’s method
Lihu Xu
Ann. Appl. Probab. 29(1): 458-504 (February 2019). DOI: 10.1214/18-AAP1424

Abstract

Let $n\in\mathbb{N}$, let $\zeta_{n,1},\ldots,\zeta_{n,n}$ be a sequence of independent random variables with $\mathbb{E}\zeta_{n,i}=0$ and $\mathbb{E}|\zeta_{n,i}|<\infty$ for each $i$, and let $\mu$ be an $\alpha$-stable distribution having characteristic function $e^{-|\lambda|^{\alpha}}$ with $\alpha\in(1,2)$. Denote $S_{n}=\zeta_{n,1}+\cdots+\zeta_{n,n}$ and its distribution by $\mathcal{L}(S_{n})$, we bound the Wasserstein-1 distance of $\mathcal{L} (S_{n})$ and $\mu$ essentially by an $L^{1}$ discrepancy between two kernels. More precisely, we prove the following inequality: \[d_{W}\big(\mathcal{L}(S_{n}),\mu\big)\le C\Bigg[\sum_{i=1}^{n}\int_{-N}^{N}\bigg\vert \frac{\mathcal{K}_{\alpha}(t,N)}{n}-\frac{K_{i}(t,N)}{\alpha}\bigg\vert \,\mathrm{d}t+\mathcal{R}_{N,n}\Bigg],\] where $d_{W}$ is the Wasserstein-1 distance of probability measures, $\mathcal{K}_{\alpha}(t,N)$ is the kernel of a decomposition of the fractional Laplacian $\Delta^{\frac{\alpha}{2}}$, $K_{i}(t,N)$ is a $K$ function (Normal Approximation by Stein’s Method (2011) Springer) with a truncation and $\mathcal{R}_{N,n}$ is a small remainder. The integral term \[\sum_{i=1}^{n}\int_{-N}^{N}\bigg\vert \frac{\mathcal{K}_{\alpha}(t,N)}{n}-\frac{K_{i}(t,N)}{\alpha}\bigg\vert \,\mathrm{d}t\] can be interpreted as an $L^{1}$ discrepancy.

As an application, we prove a general theorem of stable law convergence rate when $\zeta_{n,i}$ are i.i.d. and the distribution falls in the normal domain of attraction of $\mu$. To test our results, we compare our convergence rates with those known in the literature for four given examples, among which the distribution in the fourth example is not in the normal domain of attraction of $\mu$.

Citation

Download Citation

Lihu Xu. "Approximation of stable law in Wasserstein-1 distance by Stein’s method." Ann. Appl. Probab. 29 (1) 458 - 504, February 2019. https://doi.org/10.1214/18-AAP1424

Information

Received: 1 December 2017; Revised: 1 August 2018; Published: February 2019
First available in Project Euclid: 5 December 2018

zbMATH: 07039130
MathSciNet: MR3910009
Digital Object Identifier: 10.1214/18-AAP1424

Subjects:
Primary: 60E07, 60E17, 60F05, 60G52

Rights: Copyright © 2019 Institute of Mathematical Statistics

JOURNAL ARTICLE
47 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.29 • No. 1 • February 2019
Back to Top