By István Berkes, Lajos Horváth, Johannes Schauer (auth.), Paul Doukhan, Gabriel Lang, Donatas Surgailis, Gilles Teyssière (eds.)

This quantity collects contemporary works on weakly established, long-memory and multifractal procedures and introduces new dependence measures for learning complicated stochastic platforms. different themes comprise the statistical thought for bootstrap and permutation records for endless variance approaches, the dependence constitution of max-stable methods, and the statistical houses of spectral estimators of the lengthy reminiscence parameter. The asymptotic habit of Fejér graph integrals and their use for proving valuable restrict theorems for tapered estimators are investigated. New multifractal approaches are brought and their multifractal houses analyzed. Wavelet-based tools are used to check multifractal procedures with diverse multiresolution amounts, and to become aware of adjustments within the variance of random methods. Linear regression versions with long-range established error are studied, as is the problem of detecting adjustments of their parameters.

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Stoev defin the moving maxima processes: X(t) := e R f (t − u)Mα (du), t ∈ R, (13) where f ≥ 0, R f α (u)du < ∞, and where Mα is an α −Fréchet random sup–measure with the Lebesgue control measure. More generally, we defin a mixed moving maxima process or fiel as follows: X(t) ≡ X(t1 , · · · ,td ) := e Rd ×V f (t − u, v)Mα (du, dv), t = (ti )di=1 ∈ Rd , (14) where f ≥ 0, Rd ×V f α (u, v)duν (dv) < ∞ and where now the random sup–measure Mα is define on the product space R d ×V and has control measure du × ν (dv), for some measure ν (dv) on the set V .

Stoev 1 T T 0 h(X(τ + t1 ), · · · , X(τ + tk ))d τ −→ ξ , almost surely and in the L 1 −sense, where Eξ = Eh(X(t1 ), · · · , X(tk ). The limit ξ is shift–invariant, that is S τ (ξ ) = ξ , almost surely, for all τ > 0, and therefore ξ is measurable with respect to F inv . s. & L1 h(X(τ + t1 ), · · · , X(τ + tk ))d τ −→ Eh(X(t1 ), · · · , X(tk )), as T → ∞. (16) In fact, one can show that X is ergodic if and only if Relation (16) holds, for all such Borel functions h and all k ∈ N. g. [21]. Relation (16) indicates the importance of knowing whether a process X is ergodic or not.

3. Suppose that (25) holds for the maxima of independent Y k∗ ’s. We say that the time series Y has an extremal index θ , if P 1 w Mn − bn ≤ x −→ Gθ (x), as n → ∞, an (26) where the an ’s and bn ’s are as in (25). It turns out that if the time series Y = {Yk }k∈Z has an extremal index θ , then it necessarily follows that 0 ≤ θ ≤ 1. Observe that if the Yk ’s are independent and belong to the maximum domain of attraction of an extreme value distribution, then trivially, Y has extremal index θ = 1.