The Annals of Applied Probability

Approximation of stable law in Wasserstein-1 distance by Stein’s method

Lihu Xu

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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 $\mu$.

Article information

Ann. Appl. Probab., Volume 29, Number 1 (2019), 458-504.

Received: December 2017
Revised: August 2018
First available in Project Euclid: 5 December 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60E07: Infinitely divisible distributions; stable distributions 60E17 60F05: Central limit and other weak theorems 60G52: Stable processes

Stable approximation Wasserstein-1 distance ($W_{1}$ distance) Stein’s method $L^{1}$ discrepancy normal domain of attraction of stable law $\alpha$-stable processes


Xu, Lihu. Approximation of stable law in Wasserstein-1 distance by Stein’s method. Ann. Appl. Probab. 29 (2019), no. 1, 458--504. doi:10.1214/18-AAP1424.

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