Analytic proof of multivariate stable local large deviations and application to deterministic dynamical systems

TL;DR

This paper provides an analytic proof of local large deviations for multivariate stable laws, applicable to both lattice and nonlattice distributions, and extends the results to deterministic dynamical systems.

Contribution

It introduces a unified analytic approach to local large deviations for multivariate stable laws, simplifying previous methods and covering a broader class of distributions.

Findings

Proof applies to lattice and nonlattice distributions

Extends results to deterministic dynamical systems

Bypasses aperiodicity considerations

Abstract

We give a short analytic proof of local large deviations for i.i.d. random variables in the domain of a multivariate $\alpha$-stable law, $\alpha\in(0,1)\cup(1,2]$. Our method simultaneously covers lattice and nonlattice distributions (and mixtures thereof), bypassing aperiodicity considerations. The proof applies also to the dynamical setting.