1,721,123 research outputs found
Quantitative limit theorems for local functionals of arithmetic random waves
Full text linkWe consider Gaussian Laplace eigenfunctions on the two-dimensional flat torus (arithmetic random waves), and provide explicit Berry-Esseen bounds in the 1-Wasserstein distance for the normal and non-normal high-energy approximation of the associated Leray measures and total nodal lengths, respectively. Our results provide substantial extensions (as well as alternative proofs) of findings by Oravecz et al. (Ann Inst Fourier (Grenoble) 58(1):299–335, 2008), Krishnapur et al. (Ann Math 177(2):699–737, 2013), and Marinucci et al. (Geom Funct Anal 26(3):926–960, 2016). Our techniques involve Wiener-Itô chaos expansions, integration by parts, as well as some novel estimates on residual terms arising in the chaotic decomposition of geometric quantities that can implicitly be expressed in terms of the coarea formula
Functional Convergence of Berry's Nodal Lengths: Approximate Tightness and Total Disorder
Full text linkWe consider Berry's random planar wave model (1977), and prove spatial
functional limit theorems - in the high-energy limit - for discretized and
truncated versions of the random field obtained by restricting its nodal length
to rectangular domains. Our analysis is crucially based on a detailed study of
the projection of nodal lengths onto the so-called second Wiener chaos, whose
high-energy fluctuations are given by a Gaussian total disorder field indexed
by polygonal curves. Such an exact characterization is then combined with
moment estimates for suprema of stationary Gaussian random fields, and with a
tightness criterion by Davydov and Zikitis (2005).Comment: 38 pages, 1 figur
U-Statistics on the spherical poisson space
No full textWe review a recent stream of research on normal approximations for linear functionals and more general U-statistics of wavelets/needlets coefficients evaluated on a homogeneous spherical Poisson field. We show how, by exploiting results from Peccati and Zheng (Electron J Probab 15(48):1487-1527, 2010) based on Malliavin calculus and Stein's method, it is possible to assess the rate of convergence to Gaussianity for a triangular array of statistics with growing dimensions. These results can be exploited in a number of statistical applications, such as spherical density estimations, searching for point sources, estimation of variance, and the spherical two-sample problem
Variational Analysis of Poisson Processes
No full text© 2016 Springer International Publishing Switzerland.The expected value of a functional F(η) of a Poisson process η can be considered as a function of its intensity measure μ. The paper surveys several results concerning differentiability properties of this functional on the space of signed measures with finite total variation. Then, necessary conditions for μ being a local minima of the considered functional are elaborated taking into account possible constraints on μ, most importantly the case of μ with given total mass a. These necessary conditions can be phrased by requiring that the gradient of the functional (being the expected first difference) is constant on the support of μ. In many important cases, the gradient depends only on the local structure of μ in a neighbourhood of x and so it is possible to work out the asymptotics of the minimising measure with the total mass a growing to infinity. Examples include the optimal approximation of convex functions, clustering problem and optimal search. In non-asymptotic cases, it is in general possible to find the optimal measure using steepest descent algorithms which are based on the obtained explicit form of the gradient
Poisson point process convergence and extreme values in stochastic geometry
No full textLet η t be a Poisson point process with intensity measure tμ , t>0 , over a Borel space X , where μ is a fixed measure. Another point process ξ t on the real line is constructed by applying a symmetric function f to every k -tuple of distinct points of η t . It is shown that ξ t behaves after appropriate rescaling like a Poisson point process, as t→∞ , under suitable conditions on η t and f . This also implies Weibull limit theorems for related extreme values. The result is then applied to investigate problems arising in stochastic geometry, including small cells in Voronoi tessellations, random simplices generated by non-stationary hyperplane processes, triangular counts with angular constraints and non-intersecting k -flats. Similar results are derived if the underlying Poisson point process is replaced by a binomial point process
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
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