The P.O.T. method (Peaks Over Threshold) consists in using the generalized Pareto distribution (GPD) as an approximation for the distribution of excesses over a high threshold. In this work, we use a refinement of this approximation in order to estimate second order parameters of the model using the method of probability-weighted moments (PWM): in particular, this leads to the introduction of a new estimator for the second order parameter ρ, which will be compared to other recent estimators through...

Pareto distributions are most popular for modeling heavy tailed data. Here, we obtain weak limits of a sequence of extremal and a sequence of additive processes constructed by a series of Bernoulli point processes with bivariate Pareto space components. For the limiting processes we derive the one dimensional distributions in explicit forms. Some of the main properties of these distributions are also proved.

Maximum autoregressive processes like MARMA (Davis and Resnick, [5] 1989) or power MARMA (Ferreira and Canto e Castro, [12] 2008) have singular joint distributions, an unrealistic feature in most applications. To overcome this pitfall, absolute continuous versions were presented in Alpuim and Athayde [2] (1990) and Ferreira and Canto e Castro [14] (2010b), respectively. We consider an extended version of absolute continuous maximum autoregressive processes that accommodates both asymptotic tail...

In this paper we study some properties of the distribution function of the random variable C(X,Y) when the copula of the random pair (X,Y) is M (respectively, W) – the copula for which each of X and Y is almost surely an increasing (respectively, decreasing) function of the other –, and C is any copula. We also study the distribution functions of M(X,Y) and W(X,Y) given that the joint distribution function of the random variables X and Y is any copula.

This paper contains the results concerning the weak convergence of d-dimensional extreme order statistics in a Gaussian, equally correlated array. Three types of limit distributions are found and sufficient conditions for the existence of these distributions are given.

We present a method for estimating the edge of a two-dimensional bounded set, given a finite random set of points drawn from the interior. The estimator is based both on a Parzen-Rosenblatt kernel and extreme values of point processes. We give conditions for various kinds of convergence and asymptotic normality. We propose a method of reducing the negative bias and edge effects, illustrated by some simulations.

We present a method for estimating the edge of a two-dimensional bounded set, given a finite random set of points drawn from the interior. The estimator is based both on a Parzen-Rosenblatt kernel and extreme values of point processes. We give conditions for various kinds of convergence and asymptotic normality. We propose a method of reducing the negative bias and edge effects, illustrated by some simulations.

This paper deals with a weak convergence of maximum vectors built on the base of stationary and normal sequences of relatively strongly dependent random vectors. The discussion concentrates on the normality of limits and extends some results of McCormick and Mittal [4] to the multivariate case.

We define a family of local mixing conditions that enable the computation of the extremal index of periodic sequences from the joint distributions of k consecutive variables of the sequence. By applying results, under local and global mixing conditions, to the (2m - 1)-dependent periodic sequence Xn(m) = Σj=-mm-1 cjZn-j, n ≥ 1, we compute the extremal index of the periodic moving average sequence Xn = Σj=-∞∞ cjZn-j, n ≥ 1, of random variables with regularly varying tail probabilities.This paper...

The extremal shape factor of spheroidal particles is studied. Three dimensional particles are considered to be observed via their two dimensional profiles and the problem is to predict the extremal shape factor in a given size class. We proof the stability of the domain of attraction of the spheroid’s and its profile shape factor under a tail equivalence condition. We show namely that the Farlie–Gumbel–Morgenstern bivariate distributions gives the tail uniformity. We provide a way how to find normalising...

Motivated by applications in queueing fluid models and ruin theory, we analyze the asymptotics of $\mathbb{P}(su{p}_{t\in [0,T]}({\sum }_{i=1}^n{\lambda }_i{B}_{H_i}\left(t\right)-ct)>u)$, where $B_{H_i}\left(t\right):t\ge 0$, i = 1,...,n, are independent fractional Brownian motions with Hurst parameters $H_i\in (0,1]$ and λ₁,...,λₙ > 0. The asymptotics takes one of three different qualitative forms, depending on the value of $mi{n}_{i=1,...,n}{H}_i$.

The object of this research in the queueing theory is a theorem about the Strong-Law-of-Large-Numbers (SLLN) under the conditions of heavy traffic in a multiserver open queueing network. SLLN is known as a fluid limit or fluid approximation. In this work, we prove that the long-term average rate of growth of the queue length process of a multiserver open queueing network under heavy traffic strongly converges to a particular vector of rates. SLLN is proved for the values of an important probabilistic...

A linear Boltzmann equation is interpreted as the forward equation for the probability density of a Markov process $\left(K\right(t),i(t),Y(t\left)\right)$ on $({𝕋}^2\times \{1,2\}\times {ℝ}^2)$, where ${𝕋}^2$ is the two-dimensional torus. Here $\left(K\right(t),i(t\left)\right)$ is an autonomous reversible jump process, with waiting times between two jumps with finite expectation value but infinite variance. $Y\left(t\right)$ is an additive functional of $K$, defined as ${\int }_0^{t}v\left(K\left(s\right)\right)\mathrm{d}s$, where $\left|v\right|\sim 1$ for small $k$. We prove that the rescaled process ${(NlnN)}^{-1/2}Y\left(Nt\right)$ converges in distribution to a two-dimensional Brownian motion. As a consequence, the appropriately...

Let f be a measurable, real function defined in a neighbourhood of infinity. The function f is said to be of generalised regular variation if there exist functions h ≢ 0 and g > 0 such that f(xt) - f(t) = h(x)g(t) + o(g(t)) as t → ∞ for all x ∈ (0,∞). Zooming in on the remainder term o(g(t)) eventually leads to the relation f(xt) - f(t) = h₁(x)g₁(t) + ⋯ + hₙ(x)gₙ(t) + o(gₙ(t)), each $g_i$ being of smaller order than its predecessor $g_{i-1}$. The function f is said to be generalised regularly varying of...

The Multivariate Extreme Value distributions have shown their usefulness in environmental studies, financial and insurance mathematics. The Logistic or Gumbel-Hougaard distribution is one of the oldest multivariate extreme value models and it has been extended to asymmetric models. In this paper we introduce generalized logistic multivariate distributions. Our tools are mixtures of copulas and stable mixing variables, extending approaches in Tawn [14], Joe and Hu [6] and Fougères et al. [3]. The...

Spatial environmental processes often exhibit dependence in their large values. In order to model such processes their dependence properties must be characterized and quantified. In this paper we introduce a measure that evaluates the dependence among extreme observations located in two disjoint sets of locations of ${ℝ}^2$. We compute the range of this new dependence measure, which extends the existing $\lambda$-madogram concept, and compare it with extremal coefficients, finding generalizations of the known...