Illinois Journal of Mathematics

On high-frequency limits of $U$-statistics in Besov spaces over compact manifolds

Solesne Bourguin and Claudio Durastanti

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In this paper, quantitative bounds in high-frequency central limit theorems are derived for Poisson based $U$-statistics of arbitrary degree built by means of wavelet coefficients over compact Riemannian manifolds. The wavelets considered here are the so-called needlets, characterized by strong concentration properties and by an exact reconstruction formula. Furthermore, we consider Poisson point processes over the manifold such that the density function associated to its control measure lives in a Besov space. The main findings of this paper include new rates of convergence that depend strongly on the degree of regularity of the control measure of the underlying Poisson point process, providing a refined understanding of the connection between regularity and speed of convergence in this framework.

Article information

Illinois J. Math., Volume 61, Number 1-2 (2017), 97-125.

Received: 17 March 2017
Revised: 15 November 2017
First available in Project Euclid: 3 March 2018

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Zentralblatt MATH identifier

Primary: 60B05: Probability measures on topological spaces 60F05: Central limit and other weak theorems 60G57: Random measures 62E20: Asymptotic distribution theory


Bourguin, Solesne; Durastanti, Claudio. On high-frequency limits of $U$-statistics in Besov spaces over compact manifolds. Illinois J. Math. 61 (2017), no. 1-2, 97--125. doi:10.1215/ijm/1520046211.

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  • M. Axelsson, H. T. Ihle, S. Scodeller and F. K. Hansen, Testing for foreground residuals in the Planck foreground cleaned maps: A new method for designing confidence masks, A&A 5788, (2015).
  • P. Baldi, G. Kerkyacharian, D. Marinucci and D. Picard, Adaptive density estimation for directional data using needlets, Ann. Statist. 37 (2009), no. 6A, 3362–3395.
  • P. Baldi, G. Kerkyacharian, D. Marinucci and D. Picard, Asymptotics for spherical needlets, Ann. Statist. 37 (2009), 1150–1171.
  • J. Bobin, J.-L. Starck and S. Basak, Sparse component separation for accurate cmb map estimation, A&A 550, (2013).
  • S. Bourguin and C. Durastanti, On normal approximations for the two-sample problem on multidimensional tori, to appear in J. Statist. Plann. Inference (2018).
  • S. Bourguin, C. Durastanti, D. Marinucci and G. Peccati, Gaussian approximations of nonlinear statistics on the sphere, J. Math. Anal. Appl. 436 (2016), 1121–1148.
  • S. Bourguin and G. Peccati, A portmanteau inequality on the Poisson space, Electron. J. Probab. 19 (2014), 1–42.
  • P. Cabella and D. Marinucci, Statistical challenges in the analysis of cosmic microwave background radiation, Ann. Appl. Stat. 2 (2009), 61–95.
  • V. Cammarota and D. Marinucci, On the limiting behaviour of needlets polyspectra, Ann. Inst. Henri Poincaré Probab. Stat. 51 (2015), 1159–1189.
  • S. Dodelson, Modern cosmology, Academic Press, San Diego, 2003.
  • C. Durastanti, Adaptive global thresholding on the sphere, J. Multivariate Anal. 151 (2016), 110–132.
  • C. Durastanti, Quantitative central limit theorems for Mexican needlet coefficients on circular Poisson fields, Stat. Methods Appl. 25 (2016), no. 4, 651–673.
  • C. Durastanti, X. Lan and D. Marinucci, Gaussian semiparametric estimates on the unit sphere, Bernoulli 20 (2014), 28–77.
  • C. Durastanti, D. Marinucci and G. Peccati, Normal approximations for wavelet coefficients on spherical Poisson fields, J. Math. Anal. Appl. 409 (2014), 212–227.
  • D. Geller and D. Marinucci, Spin wavelets on the sphere, J. Fourier Anal. Appl. 16 (2010), 840–884.
  • D. Geller and D. Marinucci, Mixed needlets, J. Math. Anal. Appl. 375 (2011), 610–630.
  • D. Geller and A. Mayeli, Besov spaces and frames on compact manifolds, Indiana Univ. Math. J. 58 (2009), 2003–2042.
  • D. Geller and A. Mayeli, Continuous wavelets on manifolds, Math. Z. 262 (2009), 895–927.
  • D. Geller and A. Mayeli, Nearly tight frames and space-frequency analysis on compact manifolds, Math. Z. 263 (2009), 235–264.
  • D. Geller and I. Pesenson, Band-limited localized Parseval frames and Besov spaces on compact homogeneous manifolds, J. Geom. Anal. 21 (2011), 334–371.
  • T. Ghosh, J. Delabrouille, M. Remazeilles, J.-F. Cardoso and T. Souradeep, Foreground maps in wmap frequency bands, Mon. Not. R. Astron. Soc. 412 (2011), 883–899.
  • W. Hardle, G. Kerkyacharian, D. Picard and A. Tsybakov, Wavelets, approximations and statistical applications, Springer, Berlin, 1997.
  • V. Hoeffding, A class of statistics with asymptotically normal distribution, Ann. Math. Stat. 19 (1948), 293–325.
  • G. Kerkyacharian, R. Nickl and D. Picard, Concentration inequalities and confidence bands for needlet density estimators on compact homogeneous manifolds, Probab. Theory Related Fields 153 (2012), 363–404.
  • R. Lachieze-Rey and G. Peccati, Fine Gaussian fluctuations on the Poisson space, I: Contractions, cumulants and geometric random graphs, Electron. J. Probab. 18 (2013), 1–32.
  • M. Ledoux, Chaos of a Markov operator and the fourth moment condition, Ann. Probab. 40 (2012), 2439–2459.
  • A. J. Lee, U-statistics: Theory and practice, Mathematics and Its Applications., vol. 465, Marcel Dekker, Inc., New York, 1990.
  • D. Marinucci and G. Peccati, Random fields on the sphere: Representations, limit theorems and cosmological applications, Cambridge University Press, Cambridge, 2011.
  • F. J. Narcowich, P. Petrushev and J. D. Ward, Decomposition of Besov and Triebel–Lizorkin spaces on the sphere, J. Funct. Anal. 238 (2006), 530–564.
  • F. J. Narcowich, P. Petrushev and J. D. Ward, Localized tight frames on spheres, SIAM J. Math. Anal. 38 (2006), 574–594.
  • I. Nourdin and G. Peccati, Stein's method on Wiener chaos, Probab. Theory Related Fields 145 (2009), 75–118.
  • G. Peccati, J.-L. Solé, M. S. Taqqu and F. Utzet, Stein's method and normal approximation of Poisson functionals, Ann. Probab. 38 (2010), 443–478.
  • G. Peccati and M. S. Taqqu, Wiener chaos: Moments, cumulants and diagrams: A survey with computer implementation, Bocconi & Springer Series, vol. 1, Springer, Milan, 2011.
  • G. Peccati and C. Zheng, Multi-dimensional Gaussian fluctuations on the Poisson space, Electron. J. Probab. 15 (2010), 1487–1527.
  • I. Z. Pesenson, Multiresolution analysis on compact Riemannian manifolds, Multiscale analysis and nonlinear dynamics, Rev. Nonlinear Dyn. Complex, Wiley-VCH, Weinheim, 2013, pp. 65–82.
  • D. Pietrobon, P. Cabella, A. Balbi, G. De Gasperis and N. Vittorio, Constraints on primordial non-gaussianity from a needlet analysis of the wmap-5 data, Mon. Not. R. Astron. Soc. 396 (2009), 1682–1688.
  • N. Privault, Stochastic analysis in discrete and continuous settings with normal martingales, Lecture Notes in Mathematics, vol. 1982, Springer-Verlag, Berlin, 2009.
  • M. Reitzner and M. Schulte, Central limit theorems for $u$-statistics of Poisson point processes, Ann. Probab. 41 (2013), 3879–3909.
  • S. Scodeller, O. Rudjord, F. K. Hansen, D. Marinucci, D. Geller and A. Mayeli, Introducing Mexican needlets for cmb analysis: Issues for practical applications and comparison with standard needlets, Astrophys. J. 733, (2011).
  • A. W. van der Vaart, Asymptotic statistics, Cambridge, 1998.