Stable Approximation for Call Function Via Stein's method

TL;DR

This paper develops bounds for approximating call functions using Stein's method for sums of i.i.d. variables in the domain of attraction of stable laws, broadening applicability under weaker moment conditions.

Contribution

It introduces a novel application of Stein's method to stable law approximation for call functions without requiring higher moments, extending existing theoretical frameworks.

Findings

Provides uniform and non-uniform bounds for stable approximation

Applies to variables in the domain of attraction of stable laws

Expands the use of call functions in fields with lower moment conditions

Abstract

Let $S_{n}$ be a sum of independent identically distribution random variables with finite first moment and $h_{M}$ be a call function defined by $g_{M}(x)=\max\{x-M,0\}$ for $x\in\mathbb{R}$, $M>0$. In this paper, we assume the random variables are in the domain $\mathcal{R}_{\alpha}$ of normal attraction of a stable law of exponent $\alpha$, then for $\alpha\in(1,2)$, we use the Stein's method developed in \cite{CNX21} to give uniform and non uniform bounds on $\alpha$-stable approximation for the call function without additional moment assumptions. These results will make the approximation theory of call function applicable to the lower moment conditions, and greatly expand the scope of application of call function in many fields.