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Hsu and Robbins (Proc. Nat. Acad. Sci. USA 33, 25–31, 1947) introduced the concept of complete convergence as a complement to the Kolmogorov strong law, in that they proved that ∑ n = 1 ∞ P ( | S n | > nε ) < ∞ $ {\sum }_{n=1}^{\infty } P(|S_{n}|>n\varepsilon )<\infty $ provided the mean of the summands is zero and that the variance is finite. Later, Erdős proved...
In this paper, joint limit distributions of maxima and minima on independent and non-identically distributed bivariate Gaussian triangular arrays is derived as the correlation coefficient of ith vector of given nth row is the function of i/n. Furthermore, second-order expansions of joint distributions of maxima and minima are established if the correlation function satisfies some regular conditions.
In 1975 James Pickands III showed that the excesses over a high threshold are approximatly Generalized Pareto distributed. Since then, a variety of estimators for the parameters of this cdf have been studied, but always assuming the underlying data to be independent. In this paper we consider the special case where the underlying data arises from a linear process with regularly varying (i.e. heavy-tailed)...
This paper lists the continuous limit distributions for central order statistics normalized by power transformations, and describes their domains of attraction. One may argue that power transformations are the natural normalizations to use if one wants to study the asymptotic behaviour of central order statistics. Power transformations preserve the origin, which may be assumed to be the quantile to...
Weighted laws of large numbers are established for components which are independent copies of a positive relatively stable law and the weights comprise a regularly varying sequence. The index of regular variation of the weights must be at least −1 for a weak law and be exactly −1 for a strong law. Consideration is given to the special case where the truncated moment function is proportional to the...
For a stochastic process {Xt}t∈T with identical one-dimensional margins and upper endpoint τup its tail correlation function (TCF) is defined through χ ( X ) ( s , t ) = lim τ → τ up P ( X s > τ ∣ X t > τ ) $\chi ^{(X)}(s,t) = \lim _{\tau \to \tau _{\text {up}}} P(X_{s} > \tau \,\mid \, X_{t} > \tau )$ . It is a popular bivariate summary measure...
We extend the setting of the right endpoint estimator introduced in Fraga Alves and Neves (Statist. Sinica 24, 1811–1835, 2014) to the broader class of light-tailed distributions with finite endpoint, belonging to some domain of attraction induced by the extreme value theorem. This stretch enables a general estimator for the finite endpoint, which does not require estimation of the (supposedly non-positive)...
In this paper, we mainly investigate the converse of a well-known theorem proved by Shorrock (J. Appl. Prob. 9, 316–326 1972b), which states that the regular variation of tail distribution implies a non-degenerate limit for the ratios of the record values. Specifically, the converse is proved by using Beurling extension of Wiener’s Tauberian theorem. This equivalence is extended to the Weibull and...
In environmental sciences, it is often of interest to assess whether the dependence between extreme measurements has changed during the observation period. The aim of this work is to propose a statistical test that is particularly sensitive to such changes. The resulting procedure is also extended to allow the detection of changes in the extreme-value dependence under the presence of known breaks...
Heatwaves are defined as a set of hot days and nights that cause a marked short-term increase in mortality. Obtaining accurate estimates of the probability of an event lasting many days is important. Previous studies of temporal dependence of extremes have assumed either a first-order Markov model or a particularly strong form of extremal dependence, known as asymptotic dependence. Neither of these...
To improve the forecasts of weather extremes, we propose a joint spatial model for the observations and the forecasts, based on a bivariate Brown-Resnick process. As the class of stationary bivariate Brown-Resnick processes is fully characterized by the class of pseudo cross-variograms, we contribute to the theorical understanding of pseudo cross-variograms refining the knowledge of the asymptotic...
The analysis of seasonal or annual block maxima is of interest in fields such as hydrology, climatology or meteorology. In connection with the celebrated method of block maxima, we study several tests that can be used to assess whether the available series of maxima is identically distributed. It is assumed that block maxima are independent but not necessarily generalized extreme value distributed...
Let X1, ⋯ , Xn be iid random vectors and f≥0 be a homogeneous non–negative function interpreted as a loss function. Let also k(n)=Argmaxi=1c⋯ , nf(Xi). We are interested in the asymptotic behavior of Xk(n) as n→∞. In other words, what is the distribution of the random vector leading to maximal loss. This question is motivated by a kind of inverse problem where one wants to determine the extremal behavior...
Let X = {X(p), p ∈ M} be a centered Gaussian random field, where M is a smooth Riemannian manifold. For a suitable compact subset D ⊂ M , we obtain approximations to the excursion probabilities ℙ { sup p ∈ D X ( p ) ≥ u } $\mathbb {P}\{\sup _{p\in D} X(p) \ge u \}$ , as u → ∞ , for two cases: (i) X is smooth and isotropic; (ii) X is non-smooth...
Let X ( t ) , t ∈ 𝓣 $X(t), t\in \mathcal {T}$ be a centered Gaussian random field with variance function σ2(⋅) that attains its maximum at the unique point t 0 ∈ 𝓣 , and let M ( 𝓣 ) = sup t ∈ T X ( t ) $M(\mathcal {T})=\sup _{t\in \mathcal {T}} X(t)$ . For 𝓣 a compact subset of ℝ, the current literature explains the...
In this paper, we study the asymptotic behavior of supremum distribution of some classes of iterated stochastic processes { X ( Y ( t ) ) : t ∈ [ 0 , ∞ ) } $\{X(Y(t)) : t \in [0, \infty )\}$ , where { X ( t ) : t ∈ ℝ } $\{X(t) : t \in \mathbb {R} \}$ is a centered Gaussian process and { Y ( t ) : t ∈ [ 0 , ∞ ) } $\{Y(t): t \in [0, \infty )\}$ is an independent of {X(t)} stochastic...
A common approach to modelling extreme values is to consider the excesses above a high threshold as realisations of a non-homogeneous Poisson process. While this method offers the advantage of modelling using threshold-invariant extreme value parameters, the dependence between these parameters makes estimation more difficult. We present a novel approach for Bayesian estimation of the Poisson process...
The joint distribution of the maximum loss and the maximum gain is obtained for a spectrally negative Lévy process until the passage time of a given level. Their marginal distributions up to an independent exponential time are also provided. The existing formulas for Brownian motion with drift are recovered using the particular scale functions.
Let (ξ1, η1), (ξ2, η2),… be a sequence of i.i.d. two-dimensional random vectors. In the earlier article Iksanov and Pilipenko (2014) weak convergence in the J1-topology on the Skorokhod space of n − 1 / 2 max 0 ≤ k ≤ [ n ⋅ ] ( ξ 1 + … + ξ k + η k + 1 ) $n^{-1/2}\underset {0\leq k\leq [n\cdot ]}{\max }\,(\xi _{1}+\ldots +\xi _{k}+\eta _{k+1})$ was proved under...
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