首页/论文/1908.06840

arXiv 1908.068402019-08-19math.PR

Implicit max-stable extremal integrals

Dustin Kremer

英文摘要：

Recently, the notion of implicit extreme value distributions has been established, which is based on a given loss function $f \ge 0$. From an application point of view, one is rather interested in extreme loss events that occur relative to $f$ than in the corresponding extreme values itself. In this context, so-called $f$-implicit $α$-Fréchet max-stable distributions arise and have been used to construct independently scattered sup-measures that possess such margins. In this paper we solve an open problem in [7] by developing a stochastic integral of a deterministic function $g\ge 0$ with respect to implicit max-stable sup-measures. The resulting theory covers the construction of max-stable extremal integrals (see [14]) and, at the same time, reveals striking parallels.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

On Extremal Index of Max-Stable Random Fields

For a given stationary max-stable random field $X(t),t\in Z^d$ the corresponding generalised Pickands constant coincides with the classical extremal index $θ$ which always exists. In this contribution we discuss necessary and sufficient conditions for $θ$ to be 0, positive or equal to 1 and also show that $θ$ is equal to the so-called block extremal index. Further, we consider some general functional indices of $X$ and prove that for a large class of functionals they coincide with $θ$. Our study of max-stable and stationary random fields is important since the formulas are valid with obvious m…

2020-10-06math.PR

Extremal shot noises, heavy tails and max-stable random fields

We consider the extremal shot noise defined by $$M(y)=\sup\{mh(y-x);(x,m)\inΦ\},$$ where $Φ$ is a Poisson point process on $\bbR^d\times (0,+\infty)$ with intensity $λdxG(dm)$ and $h:\bbR^d\to [0,+\infty]$ is a measurable function. Extremal shot noises naturally appear in extreme value theory as a model for spatial extremes and serve as basic models for annual maxima of rainfall or for coverage field in telecommunications. In this work, we examine their properties such as boundedness, regularity and ergodicity. Connections with max-stable random fields are established: we prove a limit theorem…

2010-05-31math.PR

On Extremal Index of max-stable stationary processes

In this contribution we discuss the relation between Pickands-type constants defined for certain Brown-Resnick stationary process $W(t),t\in R$ as $$\mathcal{H}_W^δ= \lim_{T\to\infty} T^{-1} E{ \left(\sup_{t\in δZ \cap [0,T]} e^{W(t)}\right) },\ δ\ge 0$$ (set $0 Z=R$ if $δ=0$) and the extremal index of the associated max-stable stationary process $ξ_W$. We derive several new formulas and obtain lower bounds for $\mathcal{H}_W^δ$ if $W$ is a Gaussian or a Lévy process. As a by-product we show an interesting relation between Pickands constants and lower tail probabilities for fractional Brownian…

2017-04-05math.PR

An ergodic theorem for the extremal process of branching Brownian motion

In a previous paper, the authors proved a conjecture of Lalley and Sellke that the empirical (time-averaged) distribution function of the maximum of branching Brownian motion converges almost surely to a Gumbel distribution. The result is extended here to the entire system of particles that are extremal, i.e. close to the maximum. Namely, it is proved that the distribution of extremal particles under time-average converges to a Poisson cluster process.

2012-09-26math.PR

Extremal linkage networks

We demonstrate how sophisticated graph properties, such as small distances and scale-free degree distributions, arise naturally from a reinforcement mechanism on layered graphs. Every node is assigned an a-priori i.i.d. fitness with max-stable distribution. The fitness determines the node attractiveness w.r.t. incoming edges as well as the spatial range for outgoing edges. For max-stable fitness distributions, we thus obtain complex spatial network, which we coin extremal linkage network.

2020-06-01math.PR

Characterizing extremal coefficient functions and extremal correlation functions

We focus on two dependency quantities of a max-stable random field $X$ on some space $T$: the extremal coefficient function $θ$ which we define on finite sets of $T$ and the extremal correlation function $χ(s,t)=\lim_{x \uparrow \infty} \PP(X_s \geq x \mid X_t \geq x)$. We fully characterize extremal coefficient functions $θ$ by a property called complete alternation and construct a corresponding max-stable random field. Simple properties and consequences concerning the convex geometry of extremal coefficients are derived. We study how the continuity of $X$, $θ$ and $χ$ are linked to each othe…

2012-05-07math.PR

更多相关论文收起更多论文

Multivariate maxima of moving multivariate maxima

We define a class of multivariate maxima of moving multivariate maxima, generalising the M4 processes. For these stationary multivariate time series we characterise the joint distribution of extremes and compute the multivariate extremal index. We derive the bivariate upper tail dependence coefficients and the extremal coefficient of the new limiting multivariate extreme value distributions.

2012-04-06math.PR

极值负相关性与强Rayleigh性质

Extremal negative dependence and the strongly Rayleigh property

本文用凸多面体几何刻画极值负相关性，证明其反链与极小性条件，并在该类中通过最大熵找到满足强Rayleigh性质的分布，进而构造到独立性的超模序链。

2025-05-07math.PR

Fluctuations of extremal Markov chains driven by the Kendall convolution

The paper deals with fluctuations of Kendall random walks, which are extremal Markov chains and iterated integral transforms with the Williamson kernel $Ψ(t) = \left(1-|t|^α\right)_+$, $α>0$. We obtain the joint distribution of the first ascending ladder epoch and height over any level $a \geq 0$ and distribution of maximum and minimum for these extremal Markovian sequences solving recursive integral equations. We show that distribution of the first crossing time of level $a \geq0$ is a mixture of geometric and negative binomial distributions. The Williamson transform is the main tool for cons…

2022-05-17math.PR

离散随机系统中的极值独立性

Extremal independence in discrete random systems

本文在弱依赖条件下证明离散随机系统中随机向量最大分量分布与独立副本最大值分布渐近等价，通过新概率上下界建立极值独立性，应用于随机离散结构极值特征分布并推广高斯向量的Berman型条件。

2025-04-19math.COmath.PR

Domination of Sample Maxima and Related Extremal Dependence Measures

For a given $d$-dimensional distribution function (df) $H$ we introduce the class of dependence measures $ μ(H,Q) = - \mathbb{E}\{ \ln H(Z_1, \ldots, Z_d)\},$ where the random vector $(Z_1, \ldots, Z_d)$ has df $Q$ which has the same marginal df's as $H$. If both $H$ and $Q$ are max-stable df's, we show that for a df $F$ in the max-domain of attraction of $H$, this dependence measure explains the extremal dependence exhibited by $F$. Moreover we prove that $μ(H,Q)$ is the limit of the probability that the maxima of a random sample from $F$ is marginally dominated by some random vector with df …

2018-04-11math.PRstat.ME

A Max-AR(1) Model with Max-Semistable Marginals

The structure of stationary first order max-autoregressive schemes with max-semi-stable marginals is studied. A connection between semi-selfsimilar extremal processes and this max-autoregressive scheme is discussed resulting in their characterizations. Corresponding cases of max-stable and selfsimilar extremal processes are also discussed.

2007-08-09math.PRmath.STstat.TH

Exact extremal statistics in the classical $1d$ Coulomb gas

We consider a one-dimensional classical Coulomb gas of $N$ like-charges in a harmonic potential -- also known as the one-dimensional one-component plasma (1dOCP). We compute analytically the probability distribution of the position $x_{\max}$ of the rightmost charge in the limit of large $N$. We show that the typical fluctuations of $x_{\max}$ around its mean are described by a non-trivial scaling function, with asymmetric tails. This distribution is different from the Tracy-Widom distribution of $x_{\max}$ for the Dyson's log-gas. We also compute the large deviation functions of $x_{\max}$ ex…

2017-04-28cond-mat.stat-mechmath-phmath.MP