On the $J_{1}$ convergence for partial sum processes with a reduced number of jumps

TL;DR

This paper introduces a modified partial sum process that enables $J_{1}$ convergence in the presence of clustered extremes, applicable to various time series models like GARCH and stochastic volatility.

Contribution

It proposes a new approach to achieve $J_{1}$ convergence by shrinking clusters of extremes, addressing limitations of existing topologies in heavy-tailed time series.

Findings

The modified process achieves $J_{1}$ convergence in clustered extreme scenarios.

Application to GARCH(1,1) and stochastic volatility models demonstrates practical relevance.

The method extends the scope of functional limit theorems for dependent heavy-tailed data.

Abstract

Various functional limit theorems for partial sum processes of strictly stationary sequences of regularly varying random variables in the space of cadlag functions $D[0,1]$ with one of the Skorohod topologies have already been obtained. The mostly used Skorohod $J_{1}$ topology is inappropriate when clustering of large values of the partial sum processes occurs. When all extremes within each cluster of high-threshold excesses do not have the same sign, Skorohod $M_{1}$ topology also becomes inappropriate. In this paper we alter the definition of the partial sum process in order to shrink all extremes within each cluster to a single one, which allow us to obtain the functional $J_{1}$ convergence. We also show that this result can be applied to some standard time series models, including the GARCH(1,1) process and its squares, the stochastic volatility models and $m$-dependent sequences.