A functional limit theorem for dependent sequences with infinite variance stable limits

TL;DR

This paper proves a functional limit theorem showing that dependent, regularly varying sequences with clustering behavior converge to a stable Lévy process, extending classical results to more complex dependent structures.

Contribution

It introduces a new limit theorem for dependent sequences with clustering, demonstrating convergence to a stable Lévy process in the Skorohod M1 topology.

Findings

Dependent sequences with clustering converge to stable Lévy processes.

The limit process's Lévy triple can differ from the independent case.

Application to moving average, GARCH(1,1), and stochastic volatility models.

Abstract

Under an appropriate regular variation condition, the affinely normalized partial sums of a sequence of independent and identically distributed random variables converges weakly to a non-Gaussian stable random variable. A functional version of this is known to be true as well, the limit process being a stable L\'{e}vy process. The main result in the paper is that for a stationary, regularly varying sequence for which clusters of high-threshold excesses can be broken down into asymptotically independent blocks, the properly centered partial sum process still converges to a stable L\'{e}vy process. Due to clustering, the L\'{e}vy triple of the limit process can be different from the one in the independent case. The convergence takes place in the space of c\`{a}dl\`{a}g functions endowed with Skorohod's $M_1$ topology, the more usual $J_1$ topology being inappropriate as the partial sum processes may exhibit rapid successions of jumps within temporal clusters of large values, collapsing in the limit to a single jump. The result rests on a new limit theorem for point processes which is of independent interest. The theory is applied to moving average processes, squared GARCH(1,1) processes and stochastic volatility models.