Extremes of randomly scaled Gumbel risks

We investigate the product Y1Y2 of two independent positive risks Y1 and Y2. If Y1 has distribution in the Gumbel max-domain of attraction with some auxiliary function which is regularly varying at infinity and Y2 is bounded, then we show that Y1Y2 has also distribution in the Gumbel max-domain of attraction. If both Y1,Y2 have log-Weibullian or Weibullian tail behaviour, we prove that Y1Y2 has log-Weibullian or Weibullian asymptotic tail behaviour, respectively. We present here three theoretical applications concerned with a) the limit of point-wise maxima of randomly scaled Gaussian processes, b) extremes of Gaussian processes over random intervals, and c) the tail of supremum of iterated processes

Extremes of threshold-dependent Gaussian processes

In this contribution we are concerned with the asymptotic behaviour, as u→∞, of P{supt∈[0,T]Xu(t)>u}, where Xu(t),t∈[0,T],u>0 is a family of centered Gaussian processes with continuous trajectories. A key application of our findings concerns P{supt∈[0,T](X(t)+g(t))>u}, as u→∞, for X a centered Gaussian process and g some measurable trend function. Further applications include the approximation of both the ruin time and the ruin probability of the Brownian motion risk model with constant force of interest

Extremes of multidimensional Gaussian processes

This paper considers extreme values attained by a centered, multidimensional Gaussian process t) = (X_1(t), ..., X_n(t)) minus drift d(t) = (d_1(t), ..., d_n(t)), on an arbitrary set T. Under mild regularity conditions, we establish the asymptotics of the logarithm of the probability that for some t in T, we have that (for all i = 1, ..., n) X_i(t) - d_i(t) > q_i u, for positive thresholds q_i > 0 and u large. Our findings generalize and extend previously known results for the single-dimensional and two-dimensional case. A number of examples illustrate the theory

Extremes of order statistics of stationary processes

Let be independent copies of a stationary process . For given positive constants , define the set of th conjunctions with the th largest order statistics of . In numerous applications such as brain mapping and digital communication systems, of interest is the approximation of the probability that the set of conjunctions is not empty. Imposing the Albin's conditions on , in this paper we obtain an exact asymptotic expansion of this probability as tends to infinity. Furthermore, we establish the tail asymptotics of the supremum of the order statistics processes of skew-Gaussian processes and a Gumbel limit theorem for the minimum order statistics of stationary Gaussian processes

The planar triangular S = 3/2 magnet AgCrSe2 : magnetic frustration, short range correlations, and field tuned anisotropic cycloidal magnetic order

Funding: Deutsche Forschungsgemeinschaft (DFG) through the SFB 1143 and the Wurzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter–ct.qmat (EXC 2147, Project No. 390858490), as well as the support of the HLD at HZDR, a member of the European Magnetic Field Laboratory (EMFL). We gratefully acknowledge support from the European Research Council (through the QUESTDO project, 714193), the Leverhulme Trust, and the Royal Society. We thank the Elettra synchrotron for access to the APE-HE beamline under proposal number 20195300. The research leading to this result has been supported by the project CALIPSOplus under Grant Agreement 730872 from the EU Framework Programme for Research and Innovation HORIZON 2020. Part of this work has been performed in the framework of the Nanoscience Foundry and Fine Analysis (NFFA-MUR Italy Progetti Internazionali) project (www.trieste.NFFA.eu).Our studies evidence an anisotropic magnetic order below TN = 32~K. Susceptibility data in small fields of about 1~T reveal an antiferromagnetic (AFM) order for H ⊥ c, whereas for H || c the data are reminiscent of a field-induced ferromagnetic (FM) structure. At low temperatures and for H ⊥ c, the field-dependent magnetization and AC susceptibility data evidence a metamagnetic transition at H+ = 5~T, which is absent for H || c. We assign this to a transition from a planar cycloidal spin structure at low fields to a planar fan-like arrangement above H+. A fully FM polarized state is obtained above the saturation field of H⊥S = 23.7~T at 2~K with a magnetization of Ms = 2.8~μB/Cr. For H || c, M(H) monotonously increases and saturates at the same Ms value at HIIS = 25.1~T at 4.2~K. Above TN, the magnetic susceptibility and specific heat indicate signatures of two dimensional (2D) frustration related to the presence of planar ferromagnetic and antiferromagnetic exchange interactions. We found a pronounced nearly isotropic maximum in both properties at about T* = 45~K, which is a clear fingerprint of short-range correlations and emergent spin fluctuations. Calculations based on a planar 2D Heisenberg model support our experimental findings and suggest a predominant FM exchange among nearest and AFM exchange among third-nearest neighbors. Only a minor contribution might be assigned to the antisymmetric Dzyaloshinskii-Moriya interaction possible related to the non-centrosymmetric polar space group R3m. Due to these competing interactions, the magnetism in AgCrSe2, in contrast to the oxygen based delafossites, can be tuned by relatively small, experimentally accessible, magnetic fields, allowing us to establish the complete anisotropic magnetic H-T phase diagram in detail.PostprintPeer reviewe

Extremes of vector-valued Gaussian processes with Trend

Let $X(t)=(X_1(t), \dots, X_n(t)), t\in \mathcal{T}\subset \mathbb{R}$ be a centered vector-valued Gaussian process with independent components and continuous trajectories, and $h(t)=(h_1(t),\dots, h_n(t)), t\in \mathcal{T}$ be a vector-valued continuous function. We investigate the asymptotics of $$\mathbb{P}\left(\sup_{t\in \mathcal{T} } \min_{1\leq i\leq n}(X_i(t)+h_i(t))>u\right)$$ as $u\to\infty$. As an illustration to the derived results we analyze two important classes of $X(t)$: with locally-stationary structure and with varying variances of the coordinates, and calculate exact asymptotics of simultaneous ruin probability and ruin time in a Gaussian risk model.Comment: 24 page

Lévy-driven GPS queues with heavy-tailed input

In this paper, we derive exact large buffer asymptotics for a two-class generalized processor sharing (GPS) model, under the assumption that the input traffic streams generated by both classes correspond to heavy-tailed L,vy processes. Four scenarios need to be distinguished, which differ in terms of (i) the level of heavy-tailedness of the driving L,vy processes as well as (ii) the values of the corresponding mean rates relative to the GPS weights. The derived results are illustrated by two important special cases, in which the queues' inputs are modeled by heavy-tailed compound Poisson processes and by -stable L,vy motions