[Paper Review] Stein's method for asymmetric $α$-stable distributions, with application to the stable CLT

Peng Chen, Ivan Nourdin+1 more|arXiv (Cornell University)|Aug 7, 2018

Random Matrices and Applications7 references3 citations

TL;DR

This paper develops a novel Stein's method for asymmetric $α$-stable distributions in one dimension, introducing a fractional-type generator operator that generalizes the fractional Laplacian. It establishes a Wasserstein distance bound for approximation by $α$-stable laws and applies it to derive explicit error rates in the stable central limit theorem, even for distributions outside the classical domain of normal attraction, such as Pareto-like variables with slowly varying tails.

ABSTRACT

This paper is concerned with the Stein's method associated with a (possibly) asymmetric $α$-stable distribution $Z$, in dimension one. More precisely, its goal is twofold. In the first part, we exhibit a genuine bound for the Wasserstein distance between $Z$ and any integrable random variable $X$, in terms of an operator that reduces to the classical fractional Laplacian in the symmetric case. Then, in the second part we apply the aforementioned bound to compute error rates in the stable central limit theorem, when the entries are in the domain $\mathcal{D}_α$ of normal attraction of a stable law of exponent $α$. To conclude, we study the specific case where the entries are Pareto like multiplied by a slowly varying function, which provides an example of random variables that do not belong to $\mathcal{D}_α$, but for which our approach continues to apply.

Motivation & Objective

• To extend Stein's method to asymmetric $α$-stable distributions for $α \in (1,2)$, where prior work was limited to symmetric cases.
• To develop a new generator operator $\mathcal{L}^{\alpha,\beta}$ that reduces to the fractional Laplacian when $\beta = 0$, enabling approximation bounds.
• To establish a quantitative Wasserstein distance bound between any integrable random variable $X$ and a target $Z \sim S_{\alpha}(1,\beta)$, using a leave-one-out approach.
• To apply the bound to the stable central limit theorem (stable CLT), computing explicit error rates when summands are in the domain of normal attraction $\mathcal{D}_{\alpha}$.
• To extend the method beyond $\mathcal{D}_{\alpha}$ by analyzing Pareto-like distributions with slowly varying tails, showing applicability even when classical conditions fail.

Proposed method

• Introduce a novel generator operator $\mathcal{L}^{\alpha,\beta}$ defined via a singular integral involving asymmetry parameter $\beta$, which generalizes the fractional Laplacian.
• Derive a Wasserstein distance bound via the inequality $d_W(\mathcal{L}(X), \mathcal{L}(Z)) \leq \sup \left| \mathbb{E}[(\mathcal{L}^{\alpha,\beta}\phi)(X)] - \frac{1}{\alpha}\mathbb{E}[X\phi'(X)] \right|$, over $1$-Lipschitz functions with bounded second derivatives.
• Apply a leave-one-out technique to control the error in the stable CLT by comparing the sum $S_n$ to a conditional version $\widetilde{S}_{n,i}$, removing one term at a time.
• Use a truncation and logarithmic scaling scheme with $\gamma_n = \log n$ to handle heavy-tailed increments, enabling control of tail contributions.
• Establish error bounds by decomposing the difference between $\mathbb{E}[X_i \phi'(\widetilde{S}_{n,i} + aX_i)]$ and the generator term, using integral representations and $L^\infty$ norms of $\phi'$ and $\phi''$.
• Combine estimates on the generator approximation and the leave-one-out correction to derive the final rate $O((\log n)^{-1})$ for the Wasserstein distance.

Experimental results

Research questions

• RQ1 Can Stein's method be extended to asymmetric $\alpha$-stable distributions for $\alpha \in (1,2)$, beyond the symmetric case?
• RQ2 What is a suitable generator operator $\mathcal{L}^{\alpha,\beta}$ that characterizes asymmetric $\alpha$-stable laws and reduces to the fractional Laplacian when $\beta = 0$?
• RQ3 Can explicit error rates in the stable central limit theorem be derived using this new generator-based Stein method?
• RQ4 Does the method remain effective for distributions not in the classical domain of normal attraction $\mathcal{D}_{\alpha}$, such as those with slowly varying tails?
• RQ5 What is the optimal rate of convergence in the Wasserstein distance for sums of i.i.d. heavy-tailed random variables under this framework?

Key findings

• A new generator operator $\mathcal{L}^{\alpha,\beta}$ is constructed for asymmetric $\alpha$-stable laws, which reduces to the fractional Laplacian when $\beta = 0$.
• A Wasserstein distance bound is established: $d_W(\mathcal{L}(X), \mathcal{L}(Z)) \leq \sup \left| \mathbb{E}[(\mathcal{L}^{\alpha,\beta}\phi)(X)] - \frac{1}{\alpha}\mathbb{E}[X\phi'(X)] \right|$, valid for all integrable $X$.
• For sums $S_n$ of i.i.d. random variables in $\mathcal{D}_{\alpha}$, the method yields an error rate of $d_W(\widetilde{S}_n, S\alpha S(1)) = O((\log n)^{-1})$.
• The method applies beyond $\mathcal{D}_{\alpha}$: for Pareto-like variables with slowly varying tails, the same $O((\log n)^{-1})$ rate is achieved.
• The error bound is derived via a leave-one-out approach, with key estimates relying on $L^\infty$ norms of $\phi'$ and $\phi''$, and logarithmic scaling of truncation thresholds.
• The final rate $O((\log n)^{-1})$ is shown to be sharp under the given assumptions, improving upon classical Fourier-analytic bounds in this setting.