Some characterizations of the spherical harmonics coefficients for isotropic random fields

In this paper we provide some simple characterizations for the spherical harmonics coefficients of an isotropic random field on the sphere. The main result is a characterization of isotropic gaussian fields through independence of the coefficients of their development in spherical harmonics.Comment: 9 pages. Submitted June 200

On the characterization of isotropic Gaussian fields on homogeneous spaces of compact groups

Let T be a random field invariant under the action of a compact group G We give conditions ensuring that independence of the random Fourier coefficients is equivalent to Gaussianity. As a consequence, in general it is not possible to simulate a non-Gaussian invariant random field through its Fourier expansion using independent coefficients

Subsampling needlet coefficients on the sphere

In a recent paper, we analyzed the properties of a new kind of spherical wavelets (called needlets) for statistical inference procedures on spherical random fields; the investigation was mainly motivated by applications to cosmological data. In the present work, we exploit the asymptotic uncorrelation of random needlet coefficients at fixed angular distances to construct subsampling statistics evaluated on Voronoi cells on the sphere. We illustrate how such statistics can be used for isotropy tests and for bootstrap estimation of nuisance parameters, even when a single realization of the spherical random field is observed. The asymptotic theory is developed in detail in the high resolution sense.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ164 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Asymptotics for spherical needlets

We investigate invariant random fields on the sphere using a new type of spherical wavelets, called needlets. These are compactly supported in frequency and enjoy excellent localization properties in real space, with quasi-exponentially decaying tails. We show that, for random fields on the sphere, the needlet coefficients are asymptotically uncorrelated for any fixed angular distance. This property is used to derive CLT and functional CLT convergence results for polynomial functionals of the needlet coefficients: here the asymptotic theory is considered in the high-frequency sense. Our proposals emerge from strong empirical motivations, especially in connection with the analysis of cosmological data sets.Comment: Published in at http://dx.doi.org/10.1214/08-AOS601 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Spherical Needlets for CMB Data Analysis

We discuss Spherical Needlets and their properties. Needlets are a form of spherical wavelets which do not rely on any kind of tangent plane approximation and enjoy good localization properties in both pixel and harmonic space; moreover needlets coefficients are asymptotically uncorrelated at any fixed angular distance, which makes their use in statistical procedures very promising. In view of these properties, we believe needlets may turn out to be especially useful in the analysis of Cosmic Microwave Background (CMB) data on the incomplete sky, as well as of other cosmological observations. As a final advantage, we stress that the implementation of needlets is computationally very convenient and may rely completely on standard data analysis packages such as HEALPix.Comment: 7 pages, 7 figure

Explicit computation of second-order moments of importance sampling estimators for fractional Brownian motion

We study a family of importance sampling estimators of the probability of level crossing when the crossing level is large or the intensity of the noise is small. We develop a method which gives, explicitly the asymptotics of the second-order moment. Some of the results apply to fractional Brownian motion, some are more general. The main tools are refined versions of classical large-deviations results

A Chandra archival study of the temperature and metal abundance profiles in hot Galaxy Clusters at 0.1 < z < 0.3

We present the analysis of the temperature and metallicity profiles of 12 galaxy clusters in the redshift range 0.1--0.3 selected from the Chandra archive with at least ~20,000 net ACIS counts and kT>6 keV. We divide the sample between 7 Cooling-Core (CC) and 5 Non-Cooling-Core (NCC) clusters according to their central cooling time. We find that single power-laws can describe properly both the temperature and metallicity profiles at radii larger than 0.1 r_180 in both CC and NCC systems, showing the NCC objects steeper profiles outwards. A significant deviation is only present in the inner 0.1 r_180. We perform a comparison of our sample with the De Grandi & Molendi BeppoSAX sample of local CC and NCC clusters, finding a complete agreement in the CC cluster profile and a marginally higher value (at ~1sigma) in the inner regions of the NCC clusters. The slope of the power-law describing kT(r) within 0.1 r_180 correlates strongly with the ratio between the cooling time and the age of the Universe at the cluster redshift, being the slope >0 and tau_c/tau_age<=0.6 in CC systems.Comment: 12 pages, 6 figures, Accepted for publication by the Astrophysical Journa

The evolution of the spatially-resolved metal abundance in galaxy clusters up to z=1.4

We present the combined analysis of the metal content of 83 objects in the redshift range 0.09-1.39, and spatially-resolved in the 3 bins (0-0.15, 0.15-0.4, >0.4) R500, as obtained with similar analysis using XMM-Newton data in Leccardi & Molendi (2008) and Baldi et al. (2012). We use the pseudo-entropy ratio to separate the Cool-Core (CC) cluster population, where the central gas density tends to be relatively higher, cooler and more metal rich, from the Non-Cool-Core systems. The average, redshift-independent, metal abundance measured in the 3 radial bins decrease moving outwards, with a mean metallicity in the core that is even 3 (two) times higher than the value of 0.16 times the solar abundance in Anders & Grevesse (1989) estimated at r>0.4 R500 in CC (NCC) objects. We find that the values of the emission-weighted metallicity are well-fitted by the relation $Z(z) = Z_0 (1+z)^{-\gamma}$ at given radius. A significant scatter, intrinsic to the observed distribution and of the order of 0.05-0.15, is observed below 0.4 R500. The nominal best-fit value of $\gamma$ is significantly different from zero in the inner cluster regions ($\gamma = 1.6 \pm 0.2$) and in CC clusters only. These results are confirmed also with a bootstrap analysis, which provides a still significant negative evolution in the core of CC systems (P>99.9 per cent). No redshift-evolution is observed when regions above the core (r > 0.15 R500) are considered. A reasonable good fit of both the radial and redshift dependence is provided from the functional form $Z(r,z)=Z_0 (1+(r/0.15 R500)^2)^{-\beta} (1+z)^{-\gamma}$, with $(Z_0, \beta, \gamma) = (0.83 \pm 0.13, 0.55 \pm 0.07, 1.7 \pm 0.6)$ in CC clusters and $(0.39 \pm 0.04, 0.37 \pm 0.15, 0.5 \pm 0.5)$ for NCC systems. Our results represent the most extensive study of the spatially-resolved metal distribution in the cluster plasma as function of redshift.Comment: 5 pages. Research Note accepted for publication in A&

Depth-bounded Belief functions

This paper introduces and investigates Depth-bounded Belief functions, a logic-based representation of quantified uncertainty. Depth-bounded Belief functions are based on the framework of Depth-bounded Boolean logics [4], which provide a hierarchy of approximations to classical logic. Similarly, Depth-bounded Belief functions give rise to a hierarchy of increasingly tighter lower and upper bounds over classical measures of uncertainty. This has the rather welcome consequence that \u201chigher logical abilities\u201d lead to sharper uncertainty quantification. In particular, our main results identify the conditions under which Dempster-Shafer Belief functions and probability functions can be represented as a limit of a suitable sequence of Depth-bounded Belief functions