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 thatwith α1, α2 ∈ (0, 2], and r(s, t) < 1 for (s, t) = (0, 0). In this contribution we derive an exact asymptotic expansion (as u → ∞) ofwhere n1(u)n2(u) = u 2/α 1 +2/α 2 Ψ(u), which holds uniformly for (x, y) ∈ [A, B] 2 with A, B two positive constants and Ψ 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).

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.

Let {Xn : n ∈ Z d } be a weakly dependent stationary random field with maxima MA := sup{X i : i ∈ A} for finite A ⊂ Z d and Mn := sup{X i : 1 ≤ i ≤ n} for n ∈ N d . In a general setting we prove that P(M (N 1 (n),N 2 (n),...,N d (n)) ≤ vn) = exp(−n d P(X0 > vn, MA n ≤ vn)) + o(1) for some increasing sequence of sets An of size o(n d ), where (N1(n), N2(n), . . . , N d (n)) → (∞, ∞, . . . , ∞) and N1(n)N2(n) · · · N d (n) ∼ n d . The sets An are determined by a translation invariant total order on Z d . For a class of fields satisfying a local mixing condition, including m-dependent ones, the main theorem holds with a constant finite A replacing An. The above results lead to new formulas for the extremal index for random fields. The new method of calculating limiting probabilities for maxima is compared with some known results and applied to the moving maximum field.

No abstract

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.