The tail measure of a regularly varying stationary time series has been recently introduced. It is used in this contribution to reconsider certain properties of the tail process and establish new ones. A new formulation of the time change formula is used to establish identities, some of which were indirectly known and some of which are new.

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.

We propose a new approach to vague convergence of measures based on the general theory of boundedness due to Hu (1966). The article explains how this connects and unifies several frequently used types of vague convergence from the literature. Such an approach allows one to translate already developed results from one type of vague convergence to another. We further analyze the corresponding notion of vague topology and give a new and useful characterization of convergence in distribution of random measures in this topology.

No abstract

The tail process Y = (Y i ) i∈Z d of a stationary regularly varying time series or random field X = (X i ) i∈Z d represents the asymptotic local distribution of X as seen from its typical exceedance over a threshold u as u → ∞. Motivated by the standard Palm theory, we show that every tail process satisfies an invariance property called exceedance-stationarity and that this property, together with the polar decomposition of the tail process, characterizes the class of all tail processes. We then restrict to the case when Y i → 0 as |i| → ∞ and establish a couple of Palm-like dualities between the tail process and the so-called anchored tail process which, under suitable conditions, represents the asymptotic distribution of a typical cluster of extremes of X. The main message is that the distribution of the tail process is biased towards clusters with more exceedances. Finally, we use these results to determine the distribution of the typical cluster of extremes for moving average processes with random coefficients and heavy-tailed innovations.