Stable limits for sums of dependent infinite variance random variables

TL;DR

This paper establishes conditions under which sums of dependent infinite variance stationary processes converge to stable distributions, explicitly determining the parameters based on tail characteristics, with applications to time series models.

Contribution

It provides explicit formulas for the parameters of the limiting stable distribution based on tail properties, extending previous qualitative results.

Findings

Derived explicit stable distribution parameters from tail characteristics.

Applied results to GARCH, stochastic volatility, and stochastic recurrence models.

Confirmed convergence conditions for various standard time series models.

Abstract

The aim of this paper is to provide conditions which ensure that the affinely transformed partial sums of a strictly stationary process converge in distribution to an infinite variance stable distribution. Conditions for this convergence to hold are known in the literature. However, most of these results are qualitative in the sense that the parameters of the limit distribution are expressed in terms of some limiting point process. In this paper we will be able to determine the parameters of the limiting stable distribution in terms of some tail characteristics of the underlying stationary sequence. We will apply our results to some standard time series models, including the GARCH(1, 1) process and its squares, the stochastic volatility models and solutions to stochastic recurrence equations.