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Weak Limit (weak + limit)

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Mathematics and Statistics100%

Selected Abstracts

Density results relative to the Dirichlet energy of mappings into a manifold

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 12 2006
Mariano Giaquinta
Let ,, be a smooth, compact, oriented Riemannian manifold without boundary. Weak limits of graphs of smooth maps uk:Bn , ,, with an equibounded Dirichlet integral give rise to elements of the space cart2,1 (Bn × ,,). Assume that ,, is 1-connected and that its 2-homology group has no torsion. In any dimension n we prove that every element T in cart2,1 (Bn × ,,) with no singular vertical part can be approximated weakly in the sense of currents by a sequence of graphs of smooth maps uk:Bn , ,, with Dirichlet energies converging to the energy of T. © 2006 Wiley Periodicals, Inc. [source]

The Limiting Density of Unit Root Test Statistics: A Unifying Technique

JOURNAL OF TIME SERIES ANALYSIS, Issue 3 2000
Mithat Gonen
In this note we introduce a simple principle to derive a constructive expression for the density of the limiting distribution, under the null hypothesis, of unit root statistics for an AR(1)-process in a variety of situations. We consider the case of unknown mean and reconsider the well-known situation where the mean is zero. For long-range dependent errors we indicate how the principle might apply again. We also show that in principle the method also works for a near unit root case. Weak convergence and subsequent Karhunen-Loeve expansion of the weak limit of the partial sum process of the errors plays an important role, along with the evaluation of a certain normal type integral with complex mean and variance. For independent and long range dependent errors this weak limit is ordinary and fractional Brownian motion respectively. AMS 1991 subject classification. Primary 62M10; secondary 62E20. [source]

Some notes on poisson limits for empirical point processes

THE CANADIAN JOURNAL OF STATISTICS, Issue 3 2009
André Dabrowski
Abstract The authors define the scaled empirical point process. They obtain the weak limit of these point processes through a novel use of a dimension-free method based on the convergence of compensators of multiparameter martingales. The method extends previous results in several directions. They obtain limits at points where the density may be zero, but has regular variation. The joint limit of the empirical process evaluated at distinct points is given by independent Poisson processes. They provide applications both to nearest-neighbour density estimation in high dimensions, and to the asymptotic behaviour of multivariate extremes such as those arising from bivariate normal copulas. The Canadian Journal of Statistics 37: 347,360; 2009 © 2009 Statistical Society of Canada Les auteurs définissent un processus ponctuel empirique normalisé. Ils obtiennent une limite faible de ces processus ponctuels grâce à l'utilisation novatrice d'une méthode indépendante de la dimension basée sur la convergence des compensateurs de martingales à plusieurs paramètres. La méthode généralise des résultats précédents de différentes façons. Ils obtiennent des limites à des points où la densité peut être égale à 0, mais qui est à variation régulière. La limite conjointe du processus empirique évalué à des points distincts est représentée par des processus de Poisson indépendants. Les auteurs présentent deux applications, l'une sur l'estimation de densité de dimension élevée basée sur le plus proche voisin et l'autre sur le comportement asymptotique des extrêmes multidimensionnels provenant de copules normales bidimensionnelles. La revue canadienne de statistique 37: 347,360; 2009 © 2009 Société statistique du Canada [source]

Nonstationary weak limit of a stationary harmonic map sequence

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 2 2003
Weiyue Ding
Let M and N be two compact Riemannian manifolds. Let uk be a sequence of stationary harmonic maps from M to N with bounded energies. We may assume that it converges weakly to a weakly harmonic map u in H1,2 (M, N) as k , ,. In this paper, we construct an example to show that the limit map u may not be stationary. © 2002 Wiley Periodicals, Inc. [source]