/pdf/review-m-r-leadbetter-georg-lindgren-and-holger-rootzen-2flt32nd9c.pdf

Review: M. R. Leadbetter, Georg Lindgren and Holger Rootzen, Extremes and related properties of random sequences and processes

Regularly varying random fields

Let $X(t), t\in \mathcal {T}$ be a centered Gaussian random field with variance function σ2(⋅) that attains its maximum at the unique point $t_{0}\in \mathcal {T}$, and let $M(\mathcal {T})=\sup _{t\in \mathcal {T}} X(t)$. For $\mathcal {T}$ a compact subset of ℝ, the current literature explains the asymptotic tail behaviour of $M(\mathcal {T})$ under some regularity conditions including that 1 − σ(t) has a polynomial decrease to 0 as t → t0. In this contribution we consider more general case that 1 − σ(t) is regularly varying at t0. We extend our analysis to Gaussian random fields defined on some compact set $\mathcal {T}\subset \mathbb {R}^{2}$, deriving the exact tail asymptotics of $M(\mathcal {T})$ for the class of Gaussian random fields with variance and correlation functions being regularly varying at t0. A crucial novel element is the analysis of families of Gaussian random fields that do not possess locally additive dependence structures, which leads to qualitatively new types of asymptotics.

https://serval.unil.ch/resource/serval:BIB_AAB79CDAAC9F.P001/REF.pdf

Extremes of Gaussian random fields with regularly varying dependence structure

Normal comparison lemma and Slepian’s inequality are essential tools for the analysis of extremes of Gaussian processes. In this paper we show that the Normal comparison lemma for Gaussian vectors can be extended to order statistics of Gaussian arrays. Our applications include the derivation of mixed Gumbel limit laws for the order statistics of stationary Gaussian processes and the investigation of lower tail behavior of order statistics of self-similar Gaussian processes.

Comparison inequalities for order statistics of Gaussian arrays

For a given stationary max-stable random field X(t), t ∈ ℤd, the corresponding generalized Pickands constant coincideswith the classical extremal index θX ∈ [0, 1]. In this contribution,we discuss necessary and sufficient conditions for θX to be 0, positive, or equal to 1 and also show that θX 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 θX.

On Extremal Index of Max-Stable Random Fields

Let {X(s,t) : s,t > 0} be a centered homogeneous Gaussian field with a.s. continuous sample paths and correlation function r(s,t) = Cov(X(s,t),X(0,0)) such that r(s,t) = 1 − |s| �1 − |t| �2 + o(|s| �1 + |t| �2 ), s,t → 0, with α1,α2 ∈ (0,2], and r(s,t) < 1 for (s,t) 6 (0,0). In this contribution we derive an exact asymptotic expansion (as u → ∞) of

Extremes of homogeneous Gaussian random fields

In the paper we solve the limit problem for partial maxima of m-dependent stationary random fields and we extend the obtained solution to fields satisfying some local mixing conditions. New methods for describing the limitting distribution of maxima are proposed. A notion of a phantom distribution function for a random field is investigated. As an application, several original formulas for calculation of the extremal index are provided. Moving maxima and moving averages as well as Gaussian fields satisfying the Berman condition are considered.

/pdf/managing-local-dependencies-in-asymptotic-theory-for-maxima-2gjqvoxk3e.pdf

Managing local dependencies in asymptotic theory for maxima of stationary random fields

Let $\{X(s,t):s,t\geqslant 0\}$ be a centered homogeneous Gaussian field with a.s. continuous sample paths and correlation function $r(s,t)=Cov(X(s,t),X(0,0))$ such that \[r(s,t)=1-|s|^{\alpha_1}-|t|^{\alpha_2}+o(|s|^{\alpha_1}+|t|^{\alpha_2}), \quad s,t \to 0,\] with $\alpha_1,\alpha_2\in(0,2],$ and $r(s,t)<1$ for $(s,t)\neq(0,0)$. In this contribution we derive an exact asymptotic expansion (as $u\to \infty$) of $$\mathbb{P}\left(\sup_{(s n_1(u),t n_2(u))\in\left[0,x\right]\times\left[0,y\right]}X(s,t)\leqslant u\right),$$ where $n_1(u)n_2(u)=u^{2/\alpha_1+2/\alpha_2}\Psi(u)$, which holds uniformly for $(x,y) \in [ A , B ]^2$ with $ A , B $ two positive constants and $\Psi$ the survival function of an $N(0,1)$ random variable. We apply our findings to the analysis of asymptotics of extremes of homogeneous Gaussian fields over more complex parameter sets and a ball of random radius. Additionally we determine the extremal index of the discretised random field determined by $X(s,t)$.

Let be a weakly dependent stationary random field with maxima for finite and for . In a general setting we prove that for some increasing sequence of sets of size , where and . The sets are determined by a translation-invariant total order on . For a class of fields satisfying a local mixing condition, including m-dependent ones, the main theorem holds with a constant finite A replacing . The above results lead to new formulas for the extremal index for random fields. The new method for calculating limiting probabilities for maxima is compared with some known results and applied to the moving maximum field.

On maxima of stationary fields

Postoperative recovery can be impaired by many conditions, some of which are difficult to diagnose clinically. These include type 2 neurological complications such as hypoactive subtype of postoperative delirium (PD) and early postoperative cognitive dysfunction (ePOCD). Hope for their timely detection may lie with novel biomarkers. Plasma concentrations of microRNA-1-3p, microRNA-21-5p, glial fibrillary acidic protein (GFAP), neuroserpin (NSP), phosphorylated axonal neurofilament subunit H (pNfH) and visinin-like protein 1 (VILIP-1) were investigated in 30 patients undergoing elective off-pump coronary artery bypass grafting. Blood samples were collected at the start and end of a surgery as well as 24 h postoperatively. Associations between the studied biomarkers’ perioperative expression and type 2 neurological complications were analyzed. PD was associated with postoperative expression of GFAP; ePOCD was associated with postoperative expression of microRNA-21-5p and GFAP as well as intraoperative expression of NSP. The predictive accuracy of these molecules was found acceptable, with all their areas under the curve (AUC) values above 0.7. Multivariable regression indicated that microRNA-21-5p, GFAP and NSP were the only significant predictors of ePOCD. Evaluation of a multi-marker model including these three molecules revealed its outstanding predictive accuracy for ePOCD (AUC = 0.95). The use of microRNA-21-5p, GFAP and NSP for monitoring postoperative recovery warrants further research considering their potential to predict PD and ePOCD.

https://www.mdpi.com/2077-0383/10/14/3091/pdf

Natalia Soja-Kukieła

Papers

Extremes of homogeneous Gaussian random fields

Managing local dependencies in asymptotic theory for maxima of stationary random fields

Extremes of homogeneous Gaussian random fields

On maxima of stationary fields

Circulating MicroRNAs and Novel Proteins as Potential Biomarkers of Neurological Complications after Heart Bypass Surgery.