Explicit Large-Deviation Error Bounds in the Infinite-mean $$\boldsymbol{\alpha=1}$$ Domain via Characteristic-Function Inversion

Lobachevskii Journal of Mathematics, vol. 47, pp. 1546–1561

Abstract

How quickly do sums of independent identically distributed heavy-tailed random variables variables exceed large thresholds, and with what error rate? To our knowledge, no prior results provide explicit error rates in this setting. In this note we obtain explicit uniform estimates for $${\mathbb{P}}(X_{1}+\dotsb+X_{n}>N)$$ for large $$N$$ , where $$X_{j}$$ are independent identically distributed in the domain of a non-symmetric, infinite-mean, $$\alpha=1$$ stable law. The proof uses only characteristic-function inversion, smoothing inequalities, and contour estimates.

Authors 2

1. Jonny Imbierski Aachen Mathematisch Instituut

   Leiden University

   Affiliation as printed

   Mathematisch Instituut, University of Leiden, Einsteinweg 55, 2333, CC Leiden, Netherlands

2. Dalia Terhesiu corresponding Aachen Mathematisch Instituut

   Leiden University

   Affiliation as printed

   Mathematisch Instituut, University of Leiden, Einsteinweg 55, 2333, CC Leiden, Netherlands