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

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.