1,721,145 research outputs found
RANDOM LIPSCHITZ-KILLING CURVATURES: REDUCTION PRINCIPLES, INTEGRATION BY PARTS AND WIENER CHAOS
No full textA note on the reduction principle for the nodal length of planar random waves
Full text linkInspired by Marinucci et al. (2020), we prove that the nodal length of a planar random wave BE, i.e. the length of its zero set BE−1(0), is asymptotically equivalent, in the L2-sense and in the high-frequency limit E→∞, to the integral of H4(BE(x)), H4 being the fourth Hermite polynomial. As straightforward consequences, we obtain Moderate Deviation estimates and a central limit theorem in Wasserstein distance. This complements recent findings by Nourdin et al. (2019) and Peccati and Vidotto (2020)
An Improved Second-Order Poincaré Inequality for Functionals of Gaussian Fields
Full text linkWe present an improved version of the second-order Gaussian Poincaré inequality, first introduced in Chatterjee (Probab Theory Relat Fields 143(1):1–40, 2009) and Nourdin et al. (J Funct Anal 257(2):593–609, 2009). These novel estimates are used in order to bound distributional distances between functionals of Gaussian fields and normal random variables. Several applications are developed, including quantitative central limit theorems for nonlinear functionals of stationary Gaussian fields related to the Breuer–Major theorem, improving previous findings in the literature and obtaining presumably optimal rates of convergence
Gaussian Random Measures Generated by Berry’s Nodal Sets
Full text linkWe consider vectors of random variables, obtained by restricting the length of the nodal set of Berry’s random wave model to a finite collection of (possibly overlapping) smooth compact subsets of R2. Our main result shows that, as the energy diverges to infinity and after an adequate normalisation, these random elements converge in distribution to a Gaussian vector, whose covariance structure reproduces that of a homogeneous independently scattered random measure. A by-product of our analysis is that, when restricted to rectangles, the dominant chaotic projection of the nodal length field weakly converges to a standard Wiener sheet, in the Banach space of real-valued continuous mappings over a fixed compact set. An analogous study is performed for complex-valued random waves, in which case the nodal set is a locally finite collection of random points
Fourth moment theorems on the Poisson space in any dimension
Full text linkWe extend to any dimension the quantitative fourth moment theorem on the Poisson setting, recently proved by C. Döbler and G. Peccati (2017). In particular, by adapting the exchangeable pairs couplings construction introduced by I. Nourdin and G. Zheng (2017) to the Poisson framework, we prove our results under the weakest possible assumption of finite fourth moments. This yields a Peccati-Tudor type theorem, as well as an optimal improvement in the univariate case. Finally, a transfer principle “from-Poisson-to-Gaussian” is derived, which is closely related to the universality phenomenon for homogeneous multilinear sums
Non-universal fluctuations of the empirical measure for isotropic stationary fields on S2×R
No full textIn this paper, we consider isotropic and stationary real Gaussian random fields defined on S2 × R and we investigate the asymptotic behavior, as T->+∞, of the empirical measure (excursion area) in S2 × [0,T] at any threshold, covering both cases when the field exhibits short and long memory, that is, integrable and non-integrable temporal covariance. It turns out that the limiting distribution is not universal, depending both on the memory parameters and the threshold. In particular, in the long memory case a form of Berry's cancellation phenomenon occurs at zero-level, inducing phase transitions for both variance rates and limiting laws
Non-universal moderate deviation principle for the nodal length of arithmetic Random Waves
No full textInspired by the recent work Macci et al. (2021), we prove a non-universal non-central
Moderate Deviation Principle for the nodal length of arithmetic random waves (Gaussian Laplace eigenfunctions on the standard flat torus) both on the whole manifold and on shrinking toral domains. Second order fluctuations for the latter were established by Marinucci et al. (2016) and Benatar et al. (2020) respectively, by means of chaotic expansions, number theoretical estimates and full correlation phenomena. Our proof is simple and relies on the interplay between the long memory behavior of arithmetic random waves and the chaotic expansion of the nodal length, as well as on well-known techniques in Large Deviation theory (the contraction principle and the concept
of exponential equivalence)
LASSO estimation for spherical autoregressive processes
Full text linkThe purpose of the present paper is to investigate a class of spherical functional autoregressive processes in order to introduce and study LASSO (Least Absolute Shrinkage and Selection Operator) type estimators for the corresponding autoregressive kernels, defined in the harmonic domain by means of their spectral decompositions. Some crucial properties for these estimators are proved, in particular, consistency and oracle inequalities
Modelling the hidden magnetic field of low-mass stars
No full textPL acknowledges support from a Science and Technology Facilities Council studentship. JM, AAV and RF acknowledge support from fellowships of the Alexander von Humboldt foundation, the Royal Astronomical Society and Science and Technology Facilities Council, respectively.Zeeman-Doppler imaging is a spectropolarimetric technique that is used to map the large-scale surface magnetic fields of stars. These maps in turn are used to study the structure of the stars' coronae and winds. This method, however, misses any small-scale magnetic flux whose polarization signatures cancel out. Measurements of Zeeman broadening show that a large percentage of the surface magnetic flux may be neglected in this way. In this paper we assess the impact of this 'missing flux' on the predicted coronal structure and the possible rates of spin-down due to the stellar wind. To do this we create a model for the small-scale field and add this to the Zeeman-Doppler maps of the magnetic fields of a sample of 12 M dwarfs. We extrapolate this combined field and determine the structure of a hydrostatic, isothermal corona. The addition of small-scale surface field produces a carpet of low-lying magnetic loops that covers most of the surface, including the stellar equivalent of solar 'coronal holes' where the large-scale field is opened up by the stellar wind and hence would be X-ray dark. We show that the trend of the X-ray emission measure with rotation rate (the so-called 'activity-rotation relation') is unaffected by the addition of small-scale field, when scaled with respect to the large-scale field of each star. The addition of small-scale field increases the surface flux; however, the large-scale open flux that governs the loss of mass and angular momentum in the wind remains unaffected. We conclude that spin-down times and mass-loss rates calculated from surface magnetograms are unlikely to be significantly influenced by the neglect of small-scale field.Peer reviewe
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