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We investigate partial maxima of the uniform AR(1) processes with parameter r ⩾ 2. Positively and negatively correlated processes are considered. New limit theorems for maxima in complete and incomplete samples are obtained.
The notion of maximal-spacing in several dimensions was introduced and studied by Deheuvels (Probab. Theory Related Fields 64(4), 411–424, 1983), for data uniformly distributed on the unit cube. Later on, Janson (Ann. Prob. 15, 274–280, 1987) extended the results to data uniformly distributed on any bounded set, and obtained a very fine result, namely, he derived the asymptotic distribution of different...
We study clusters of threshold exceedances caused by dependence in time series. The clusters are defined as conglomerates containing consecutive threshold exceedances of the series separated by return intervals with consecutive non-exceedances. We derive asymptotic distributions of the cluster and inter-cluster sizes for processes with the extremal index equal to zero, the asymptotic expectation of...
Let {Xt, t = 1} be a time series. A (upper) record is a value Xj such that Xj> max{X1,…,Xj-1}. Some popular models in record theory are the Yang-Nevzorov and the Linear Drift models. The stochastic behavior of records under these models has been much studied and many interesting distribution-free properties have been unearthed. However the estimation of the parameters of these models has been less...
Pickands constants play a crucial role in the asymptotic theory of Gaussian processes. They are commonly defined as the limits of a sequence of expectations involving fractional Brownian motions and, as such, their exact value is often unknown. Recently, Dieker and Yakir (Bernoulli, 20(3), 1600–1619, 2014) derived a novel representation of Pickands constant as a simple expected value that does not...
The risk of extreme environmental events is of great importance for both the authorities and the insurance industry. This paper concerns risk measures in a spatial setting, in order to introduce the spatial features of damages stemming from environmental events into the measure of the risk. We develop a new concept of spatial risk measure, based on the spatially aggregated loss over the region of...
There exist two ways of defining regular variation of a time series in a star-shaped metric space: either by the distributions of finite stretches of the series or by viewing the whole series as a single random element in a sequence space. The two definitions are shown to be equivalent. The introduction of a norm-like function, called modulus, yields a polar decomposition similar to the one in Euclidean...
We consider the asymptotic behavior of the probability of “physical extremes” of a Gaussian field which means the probability of excursions above a high level with diameters of their bases exceeding a fixed positive number. Also we deal with the path behaviour of such excursions in case they occur.
Does the human lifespan have an impenetrable biological upper limit which ultimately will stop further increase in life lengths? This question is important for understanding aging, and for society, and has led to intense controversies. Demographic data for humans has been interpreted as showing existence of a limit, or even as an indication of a decreasing limit, but also as evidence that a limit...
Let BH={BH(t):t∈ℝ} $B_{H}=\{B_{H}(t):t\in \mathbb R\}$ be a fractional Brownian motion with Hurst parameter H ∈ (0,1). For the stationary storage process QBH(t)=sup−∞<s≤t(BH(t)−BH(s)−(t−s)) $Q_{B_{H}}(t)=\sup _{-\infty <s\le t}(B_{H}(t)-B_{H}(s)-(t-s))$, t ≥ 0, we provide a tractable criterion for assessing whether, for any positive, non-decreasing function f, ℙ(QBH(t)>f(t)i.o.) $ {\mathbb...
The vanilla method in univariate extreme-value theory consists of fitting the three-parameter Generalized Extreme-Value (GEV) distribution to a sample of block maxima. Despite claims to the contrary, the asymptotic normality of the maximum likelihood estimator has never been established. In this paper, a formal proof is given using a general result on the maximum likelihood estimator for parametric...
Conditional extreme value models have been introduced by Heffernan and Resnick (Ann. Appl. Probab., 17, 537–571, 2007) to describe the asymptotic behavior of a random vector as one specific component becomes extreme. Obviously, this class of models is related to classical multivariate extreme value theory which describes the behavior of a random vector as its norm (and therefore at least one of its...
Data exhibiting heavy-tails in one or more dimensions is often studied using the framework of regular variation. In a multivariate setting this requires identifying specific forms of dependence in the data; this means identifying that the data tends to concentrate along particular directions and does not cover the full space. This is observed in various data sets from finance, insurance, network traffic,...
The purpose of this paper is to construct a new non-parametric detector of univariate outliers and to study its asymptotic properties. This detector is based on a Hill’s type statistic. It satisfies a unique asymptotic behavior for a large set of probability distributions with positive unbounded support (for instance: for the absolute value of Gaussian, Gamma, Weibull, Student or regular variations...
The asymptotic results that underlie applications of extreme random fields often assume that the variables are located on a regular discrete grid, identified with Z2 $\mathbb {Z}^{2}$, and that they satisfy stationarity and isotropy conditions. Here we extend the existing theory, concerning the asymptotic behavior of the maximum and the extremal index, to non-stationary and anisotropic random fields,...
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