Palm theory for extremes of stationary regularly varying time series and random fields

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.

Shift-invariant homogeneous classes of random fields

“…Tail and spectral tail rf's are initially introduced in [1,2] in the study of regular variation of stationary time series. As shown recently in [3], Y (t), t ∈ Z l is a tail rf if and only if (iff) it is an exceedance stationary process.…”

“…In the investigation of shift-invariant K + α [Z]'s several maps including anchoring and shift-involutions play a crucial role, which is not surprising in view of [2,3,15]. We shall show that also of particular importance are certain maps belonging to some class U of universal maps.…”

“…Anchoring maps introduced in [2] play a crucial role in the investigation of tail rf's. See also [2,4,13,17,22,23] for various results concerning anchoring maps. Our definiiton of anchoring maps is adopt from [15], where L is some lattice on T and R = R ∪ {−∞, ∞}.…”

“…for all compact sets K ⊂ R l , with I(f ) = I T (f ), f ∈ D the infargsup map. (ii) Related results for the case of stochastically continuous Z and L = T , τ = 0 are given in [20], recall Remark 4.5.The discrete setup T = Z l and • a norm on R l is considered in several papers, see [2,3,5,15,29] and the references therein.…”

“…ii) Tail and spectral tail rf's are defined for general rf's in Ω T dropping the càdlàg sample path assumption in [6,10] and the restriction that T is countable imposed in [2,5,13,14,17]. Moreover, we do not assume that z = 0 iff z = 0 ∈ R l as in [14] or • is a norm as in the aforementioned papers.…”