116 research outputs found
Tauberian Results for Densities with Gaussian Tails
We study a class of probability densities with very thin upper tails. These densities generate exponential families which are asymptotically normal. Furthermore the class is closed under convolution. In this paper we shall be concerned with Abelian and strong Tauberian theorems for moment generating functions and Laplace transforms with respect to these densities. We obtain a duality relation between this class of densities and the associated class of moment generating functions which is closely related to the duality relation for convex function
Densities with Gaussian Tails
Consider densities fi(t), for i = 1, ..., d, on the real line which have thin tails in the sense that, for each i, fi(t) ∼ γi(t)e−ψi(t), where γi behaves roughly like a constant and ψi is convex, C2, with ψ′ → ∞ and ψ″ > 0 and l/√ψ″ is self-neglecting. (The latter is an asymptotic variation condition.) Then the convolution is of the same form ft * ... *fd(t) ∼ γ(t)e − ψ(t) Formulae for γ, ψ are given in terms of the factor densities and involve the conjugate transform and infimal convolution of convexity theory. The derivations require embedding densities in exponential families and showing that the assumed form of the densities implies asymptotic normality of the exponential familie
Heavy-tailed max-linear structural equation models in networks with hidden nodes
Recursive max-linear vectors provide models for the causal dependence between
large values of observed random variables as they are supported on directed
acyclic graphs (DAGs). But the standard assumption that all nodes of such a DAG
are observed is often unrealistic. We provide necessary and sufficient
conditions that allow for a partially observed vector from a regularly varying
model to be represented as a recursive max-linear (sub-)model. Our method
relies on regular variation and the minimal representation of a recursive
max-linear vector. Here the max-weighted paths of a DAG play an essential role.
Results are based on a scaling technique and causal dependence relations
between pairs of nodes. In certain cases our method can also detect the
presence of hidden confounders. Under a two-step thresholding procedure, we
show consistency and asymptotic normality of the estimators. Finally, we study
our method by simulation, and apply it to nutrition intake data
Smoothing of Transport Plans with Fixed Marginals and Rigorous Semiclassical Limit of the Hohenberg–Kohn Functional
We prove rigorously that the exact N-electron Hohenberg–Kohn density functional converges in the strongly interacting limit to the strictly correlated electrons (SCE) functional, and that the absolute value squared of the associated constrained search wavefunction tends weakly in the sense of probability measures to a minimizer of the multi-marginal optimal transport problem with Coulomb cost associated to the SCE functional. This extends our previous work for N = 2 (Cotar etal. in Commun Pure Appl Math 66:548–599, 2013). The correct limit problem has been derived in the physics literature by Seidl (Phys Rev A 60 4387–4395, 1999) and Seidl, Gorigiorgi and Savin (Phys Rev A 75:042511 1-12, 2007); in these papers the lack of a rigorous proofwas pointed out.We also give amathematical counterexample to this type of result, by replacing the constraint of given one-body density—an infinite dimensional quadratic expression in the wavefunction—by an infinite-dimensional quadratic expression in the wavefunction and its gradient. Connections with the Lawrentiev phenomenon in the calculus of variations are indicated
Convergence to stable laws for multidimensional stochastic recursions: the case of regular matrices
Given a sequence of i.i.d.\ random variables with
generic copy , we consider the random
difference equation (RDE) , and assume
the existence of such that \lim_{n \to \infty}(\E{\norm{M_1 ...
M_n}^\kappa})^{\frac{1}{n}} = 1 . We prove, under suitable assumptions, that
the sequence , appropriately normalized, converges in
law to a multidimensional stable distribution with index . As a
by-product, we show that the unique stationary solution of the RDE is
regularly varying with index , and give a precise description of its
tail measure. This extends the prior work http://arxiv.org/abs/1009.1728v3 .Comment: 15 page
Tail probabilities of St. Petersburg sums, trimmed sums, and their limit
We provide exact asymptotics for the tail probabilities as , for fix , where is the -trimmed
partial sum of i.i.d. St. Petersburg random variables. In particular, we prove
that although the St. Petersburg distribution is only O-subexponential, the
subexponential property almost holds. We also determine the exact tail behavior
of the -trimmed limits.Comment: 24 pages, 2 figure
Alternative approach to the optimality of the threshold strategy for spectrally negative Levy processes
Consider the optimal dividend problem for an insurance company whose
uncontrolled surplus precess evolves as a spectrally negative Levy process. We
assume that dividends are paid to the shareholders according to admissible
strategies whose dividend rate is bounded by a constant. The objective is to
find a dividend policy so as to maximize the expected discounted value of
dividends which are paid to the shareholders until the company is ruined.
Kyprianou, Loeffen and Perez [28] have shown that a refraction strategy (also
called threshold strategy) forms an optimal strategy under the condition that
the Levy measure has a completely monotone density. In this paper, we propose
an alternative approach to this optimal problem.Comment: 16 page
A HILL TYPE ESTIMATOR OF THE WEIBULL TAIL-COEFFICIENT
International audienceWe present a new estimator of the Weibull tail-coefficient. The Weibull tail-coefficient is defined as the regular variation coefficient of the inverse cumulative hazard function. Our estimator is based on the log-spacings of the upper order statistics. Therefore, it is very similar to the Hill estimator for the extreme value index. We prove the weak consistency and the asymptotic normality of our estimator. Its asymptotic as well as its finite sample performances are compared to classical ones
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