Approximation of stable law by Stein's method

Lihu Xu

Published 2017-09-04Version 1

Let $n \in \mathbb N$, let $\zeta_{n,1},...,\zeta_{n,n}$ 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 $S_{n}=\zeta_{n,1}+...+\zeta_{n,n}$ and its distribution by $\mathcal L(S_n)$, we bound the Wasserstein distance of $\mathcal L(S_{n})$ and $\mu$ essentially by an $L^{1}$ discrepancy between two kernels, this bound can be interpreted as a generalization of the Stein discrepancy (in $L^{2}$ sense) introduced by Ledoux, Nourdin and Peccati. More precisely, we prove the following inequality: \begin{equation} \begin{split} d_W\left(\mathcal L (S_n), \mu\right) \ \le C \left[\sum_{i=1}^n\int_{-N}^N \left|\frac{\mathcal K_\alpha(t,N)}n -\frac{ K_i(t,N)}{\alpha}\right| d t \ +\ \mathcal R_{N,n}\right], \end{split} \end{equation} where $d_{W}$ is the Wasserstein distance of probability measures, $\mathcal K_\alpha(t,N)$ is the kernel of a decomposition of the fractional Laplacian $\Delta^{\frac \alpha2}$, $ K_i(t,N)$ is a kernel introduced by Chen, Goldstein and Shao with a truncation which can be interpreted as an $L^1$ Stein kernel, and $\mathcal R_{N,n}$ is a small remainder. The integral term $$\sum_{i=1}^n\int_{-N}^N \left|\frac{\mathcal K_\alpha(t,N)}n -\frac{ K_i(t,N)}{\alpha}\right| d t$$ can be interpreted as an $L^{1}$ Stein 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 $\mu$. We also study four examples with comparing our convergence rates and those known for these examples, among which the distribution in the second example is not in the normal domain of attraction of $\mu$.