Error estimates for the robust $\alpha$-stable central limit theorem under sublinear expectation by discrete approximation method

TL;DR

This paper introduces a numerical method to estimate errors in the robust $ ext{alpha}$-stable central limit theorem under sublinear expectation, using discrete approximation of nonlinear integro-differential equations.

Contribution

It develops a discrete approximation scheme for nonlinear PIDE and establishes error bounds for the $ ext{alpha}$-stable CLT under sublinear expectation, covering both integrable and non-integrable cases.

Findings

Established error bounds for the approximation scheme.

Derived convergence rates for the CLT under sublinear expectation.

Validated results with concrete examples.

Abstract

In this work, we develop a numerical method to study the error estimates of the $\alpha$-stable central limit theorem under sublinear expectation with $\alpha \in(0,2)$, whose limit distribution can be characterized by a fully nonlinear integro-differential equation (PIDE). Based on the sequence of independent random variables, we propose a discrete approximation scheme for the fully nonlinear PIDE. With the help of the nonlinear stochastic analysis techniques and numerical analysis tools, we establish the error bounds for the discrete approximation scheme, which in turn provides a general error bound for the robust $\alpha$-stable central limit theorem, including the integrable case $\alpha \in(1,2)$ as well as the non-integrable case $\alpha \in(0,1]$. Finally, we provide some concrete examples to illustrate our main results and derive the precise convergence rates.