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\(J_1\) convergence of partial sum processes with a reduced number of jumps. (English) Zbl 1325.60039

Summary: Various functional limit theorems for partial sum processes of strictly stationary sequences of regularly varying random variables in the space of cádlág functions \(D[0,1]\) with one of the Skorokhod topologies have already been obtained. The mostly used Skorokhod \(J_1\) 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, the Skorokhod \(M_1\) 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 allows 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 \(\mathrm{GARCH}(1,1)\) process and its squares, the stochastic volatility models and \(m\)-dependent sequences.

MSC:

60F17 Functional limit theorems; invariance principles
60G52 Stable stochastic processes
60G70 Extreme value theory; extremal stochastic processes
60G51 Processes with independent increments; Lévy processes
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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