Naslov
An invariance principle for sums and record times of regularly varying stationary sequences

Autori
Basrak, Bojan ; Planinić, Hrvoje ; Soulier, Philippe

Izvornik
Probability theory and related fields (0178-8051) 172 (2018), 3-4; 869-914

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
Point process ; Regular variation ; Invariance principle ; Functional limit theorem ; Record times

Sažetak
We prove a sequence of limiting results about weakly dependent stationary and regularly varying stochastic processes in discrete time. After deducing the limiting distribution for individual clusters of extremes, we present a new type of point process convergence theorem. It is designed to preserve the entire information about the temporal ordering of observations which is typically lost in the limit after time scaling. By going beyond the existing asymptotic theory, we are able to prove a new functional limit theorem. Its assumptions are satisfied by a wide class of applied time series models, for which standard limiting theory in the space D of càdlàg functions does not apply. To describe the limit of partial sums in this more general setting, we use the space E of so-called decorated càdlàg functions. We also study the running maximum of partial sums for which a corresponding functional theorem can be still expressed in the familiar setting of space D. We further apply our method to analyze record times in a sequence of dependent stationary observations, even when their marginal distribution is not necessarily regularly varying. Under certain restrictions on dependence among the observations, we show that the record times after scaling converge to a relatively simple compound scale invariant Poisson process.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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