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Convergence of randomly oscillating point patterns to the Poisson point process

Jan Rataj, Ivan Saxl, Karol Pelikán (1993)

Applications of Mathematics

Oscillating point patterns are point processes derived from a locally finite set in a finite dimensional space by i.i.d. random oscillation of individual points. An upper and lower bound for the variation distance of the oscillating point pattern from the limit stationary Poisson process is established. As a consequence, the true order of the convergence rate in variation norm for the special case of isotropic Gaussian oscillations applied to the regular cubic net is found. To illustrate these theoretical...

Convergence of simple random walks on random discrete trees to brownian motion on the continuum random tree

David Croydon (2008)

Annales de l'I.H.P. Probabilités et statistiques

In this article it is shown that the brownian motion on the continuum random tree is the scaling limit of the simple random walks on any family of discrete n-vertex ordered graph trees whose search-depth functions converge to the brownian excursion as n→∞. We prove both a quenched version (for typical realisations of the trees) and an annealed version (averaged over all realisations of the trees) of our main result. The assumptions of the article cover the important example of simple random walks...

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