CONVERGENCE OF 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 cądląg functions
with one of the Skorokhod topologies have already been obtained. The
mostly used Skorokhod 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, Skorokhod 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
allows us to obtain the functional convergence. We also show that this result
can be applied to some standard time series models, including the GARCH
process and its squares, the stochastic volatility models and -dependent sequences.

2000 AMS Mathematics Subject Classification: Primary: 60F17; Secondary: 60G52,
60G55.

Keywords and phrases: Functional limit theorem, partial sum process, regular variation,
Skorokhod topology, Lé vy process, weak dependence, mixing.