A. A. Balkema

University of Amsterdam | UVA · Department of Mathematics

This paper compares the shape of the level sets for two multivariate densities. The densities are positive and continuous, and have the same dependence structure. The density f is heavy-tailed. It decreases at the same rate–up to a positive constant–along all rays. The level sets {f>c} for c↓0, have a limit shape, a bounded convex set. We transform...

Quantitative Risk Management (QRM) has become a field of research of considerable importance to numerous areas of application, including insurance, banking, energy, medicine, reliability. Mainly motivated by examples from insurance and finance, the authors develop a theory for handling multivariate extremes. The approach borrows ideas from portfoli...

A Pareto distribution has the property that any tail of the distribution has the same shape as the original distribution. The exponential distribution and the uniform distribution have the tail property too. The tail property characterizes the univariate generalized Pareto distributions. There are three classes of univariate GPDs: Pareto distributi...

With the df F of the rv X we associate the natural exponential family of df's F[lambda] wheredF[lambda](x)=e[lambda]x dF(x)/Ee[lambda]Xfor . Assume [lambda][infinity]=sup [Lambda][less-than-or-equals, slant][infinity] does not lie in [Lambda]. Let [lambda][short up arrow][lambda][infinity], then non-degenerate limit laws for the normalised distribu...

This paper describes for any given logconcave density f the set of all finite measures mi whose Laplace transforms are asymptotic to the Laplace transform of f. It is shown that the density of mi is asymptotic to f if it is logconcave. Thus logconcavity is a Tauberian condition for Laplace transforms of finite measures .

A random vector X generates a natural exponential family of vectors X , 2 , where is the set where the moment generating function (mgf) K() = Ee X is finite. Assume that is open and X non-degenerate. Suppose there exist affine transformations ff (x) = A x+a depending continuously on the parameter and a non-degenerate vector Y so that ff Gamma1 (X )...

Convergence of a sequence of deterministic functions in the Skorohod topology D([0,¥))D([0,\infty )) implies convergence of the jumps. For processes with independent additive increments the fixed discontinuities converge. In this paper it will be shown that this is not true for processes with independent max-increments. The limit in D([0,¥))D([0,...

The behavior of queues with excess capacity whose service times have very thick tails is studied.

With the df F of the rv X is associated the natural exponential family of dfs F where dF (x) = e x dF (x)=Ee X for 2 := f 2 R j Ee X ! 1g. Assume 1 = sup 1 does not lie in . The non-degenerate limit laws for the normalised distributions F (a x + b ) for " 1 are the normal and gamma distributions. Their domains of attractions are determined. These r...

For a real random variable [math] with distribution function [math] , define ¶ [math] ¶ The distribution [math] generates a natural exponential family of distribution functions [math] , where ¶ [math] ¶ We study the asymptotic behaviour of the distribution functions [math] as [math] increases to [math] . If [math] then [math] pointwise on [math] ....

The paper is organized as follows:(1) A probabilistic part. We introduce multivariate residual lifetimes and discuss different ways of investigating their asymptotic behavior. We briefly go into the relation with multivariate record sequences. We introduce the concepts of stability and strict stability. (2) An algebraic part. Our problem: to descri...

It is possible to introduce the concept of a stopping time in an algebraic way, and define for each stopping time a natural sigma-field. This yields a simple theorem about sample path regularity for processes with a complete separable metric state space.

The probability distribution of an extremal process in R with independent max-increments is completely determined by its distribution function. The df of an extremal process is similar to the cdf of a random vector. It is a monotone function on (0, ∞) × R with values in the interval [0,1]. On the other hand the probability distribution of an extrem...

Consider densities f i ( t ), for i = 1, …, d , on the real line which have thin tails in the sense that, for each i , f i ( t ) ∼ γ i ( t ) e −ψ i ( t ), where γ i behaves roughly like a constant and ψ i is convex, C 2, with ψ′ → ∞ and ψ″ > 0 and l/√ψ″ is self-neglecting. (The latter is an asymptotic variation condition.) Then the convo...

Limits in distribution of maxima of independent stochastic processes are characterized in terms of spectral functions acting on a Poisson point process.

Limits in distribution of maxima of independent stochastic processes are characterized in terms of spectral functions acting on a Poisson point process.

A uniform convergence rate is determined for maxima of i.i.d. random variables from a distribution, in the domain of attraction of the double-exponential distribution. The result is proved under a second-order condition on the underlying distribution parallelling the one given in Smith (1982) for the domain of attraction of the bounded-below and bo...

A uniform convergence rate is determined for maxima of i.i.d. random variables from a distribution in the domain of attraction of the double-exponential distribution. The result is proved under a second-order condition on the underlying distribution parallelling the one given in Smith (1982) for the domain of attraction of the bounded-below and bou...

Rootzen (1978) gives a sufficient condition for sample continuity of moving average processes with respect to stable motion with index $\alpha$ less than two. We provide a simple proof of this criterion for $\alpha < 1$ and show that the condition is then also necessary for continuity of the process. The same result holds for the moving-maximum pro...

The asymptotic behaviour of the residual life time at time $t$ is investigated (for $t \rightarrow \infty$). We derive weak limit laws and their domains of attraction and treat rates of convergence and moment convergence. The presentation exploits the close similarity with extreme value theory.

There exist well-known necessary and sufficient conditions for a distribution function to belong to the domain of attraction of the double exponential distribution $\Lambda$. For practical purposes a simple sufficient condition due to von Mises is very useful. It is shown that each distribution function $F$ in the domain of attraction of $\Lambda$...