Stein-Malliavin Approximations for Nonlinear Functionals of Random - PowerPoint PPT Presentation

Stein-Malliavin Approximations for Nonlinear Functionals of Random Eigenfunctions on S d Maurizia Rossi Department of Mathematics, University of Rome “Tor Vergata” (joint work with Domenico Marinucci) Berlin − Padova Y oung Researchers Meeting Stochastic Analysis and applications in Biology, Finance and Physics WIAS Berlin - October 23, 2014 Research supported by ERC Grant 277742 Pascal Stein-Malliavin approximations on S d M. Rossi (Rome Tor Vergata) WIAS Berlin - October 23, 2014 1 / 27

outline Introduction 1 What are random eigenfunctions on S d ? Why are them so important ? Aim of our work: quantitative CLT’s for nonlinear functionals of random eigenfunctions on S d Stein-Malliavin Normal approximations on S d 2 CLT’s via Wiener chaos decomposition Fourth moment theorems Quantitative CLT’s for nonlinear functionals of random eigenfunctions 3 on S d Hermite transforms Arbitrary polynomial transforms General nonlinear functionals Empirical measure of excursion sets Stein-Malliavin approximations on S d M. Rossi (Rome Tor Vergata) WIAS Berlin - October 23, 2014 2 / 27

eigenfunctions on the d -dimensional sphere, d ≥ 2 S d ⊂ R d +1 − → unit d -dim sphere → Laplacian operator on S d ∆ S d − • The eigenvalues of ∆ S d are E ℓ := − ℓ ( ℓ + d − 1) , ℓ ∈ N . Stein-Malliavin approximations on S d M. Rossi (Rome Tor Vergata) WIAS Berlin - October 23, 2014 3 / 27

eigenfunctions on the d -dimensional sphere, d ≥ 2 S d ⊂ R d +1 − → unit d -dim sphere → Laplacian operator on S d ∆ S d − • The eigenvalues of ∆ S d are E ℓ := − ℓ ( ℓ + d − 1) , ℓ ∈ N . • The dimension of the eigenspace H ℓ corresponding to E ℓ is n ℓ,d = 2 ℓ + d − 1 � ℓ + d − 2 � 2 ( d − 1)! ℓ d − 1 as ℓ → + ∞ ∼ ℓ ℓ − 1 (the number of l.i. homogeneous polynomials of degree ℓ in d + 1 variables). Stein-Malliavin approximations on S d M. Rossi (Rome Tor Vergata) WIAS Berlin - October 23, 2014 3 / 27

eigenfunctions on the d -dimensional sphere, d ≥ 2 S d ⊂ R d +1 − → unit d -dim sphere → Laplacian operator on S d ∆ S d − • The eigenvalues of ∆ S d are E ℓ := − ℓ ( ℓ + d − 1) , ℓ ∈ N . • The dimension of the eigenspace H ℓ corresponding to E ℓ is n ℓ,d = 2 ℓ + d − 1 � ℓ + d − 2 � 2 ( d − 1)! ℓ d − 1 as ℓ → + ∞ ∼ ℓ ℓ − 1 (the number of l.i. homogeneous polynomials of degree ℓ in d + 1 variables). • We consider the real orthonormal basis for H ℓ given by the spherical harmonics in d + 1 -dimension ( Y ℓ,m ; d ) m , m = 1 , 2 , . . . , n ℓ,d , ∆ S d Y ℓ,m ; d = − ℓ ( ℓ + d − 1) Y ℓ,m ; d . Stein-Malliavin approximations on S d M. Rossi (Rome Tor Vergata) WIAS Berlin - October 23, 2014 3 / 27

eigenfunctions on the d -dimensional sphere, d ≥ 2 S d ⊂ R d +1 − → unit d -dim sphere → Laplacian operator on S d ∆ S d − • The eigenvalues of ∆ S d are E ℓ := − ℓ ( ℓ + d − 1) , ℓ ∈ N . • The dimension of the eigenspace H ℓ corresponding to E ℓ is n ℓ,d = 2 ℓ + d − 1 � ℓ + d − 2 � 2 ( d − 1)! ℓ d − 1 as ℓ → + ∞ ∼ ℓ ℓ − 1 (the number of l.i. homogeneous polynomials of degree ℓ in d + 1 variables). • We consider the real orthonormal basis for H ℓ given by the spherical harmonics in d + 1 -dimension ( Y ℓ,m ; d ) m , m = 1 , 2 , . . . , n ℓ,d , ∆ S d Y ℓ,m ; d = − ℓ ( ℓ + d − 1) Y ℓ,m ; d . • Each real-valued f ∈ L 2 ( S d ) admits the Fourier development n ℓ,d � � f = � f, Y ℓ,m ; d � Y ℓ,m ; d ℓ ∈ N m =1 Stein-Malliavin approximations on S d M. Rossi (Rome Tor Vergata) WIAS Berlin - October 23, 2014 3 / 27

random eigenfunctions on the d -dimensional sphere, d ≥ 2 • What are random eigenfunctions on S d ? They are a linear combination of spherical harmonics of fixed degree with random coefficients. Stein-Malliavin approximations on S d M. Rossi (Rome Tor Vergata) WIAS Berlin - October 23, 2014 4 / 27

random eigenfunctions on the d -dimensional sphere, d ≥ 2 • What are random eigenfunctions on S d ? They are a linear combination of spherical harmonics of fixed degree with random coefficients. • Precisely: for each ℓ ∈ N , we construct the random eigenfunction T ℓ as n ℓ,d � T ℓ = a ℓ,m ; d Y ℓ,m ; d , m =1 Stein-Malliavin approximations on S d M. Rossi (Rome Tor Vergata) WIAS Berlin - October 23, 2014 4 / 27

random eigenfunctions on the d -dimensional sphere, d ≥ 2 • What are random eigenfunctions on S d ? They are a linear combination of spherical harmonics of fixed degree with random coefficients. • Precisely: for each ℓ ∈ N , we construct the random eigenfunction T ℓ as n ℓ,d � T ℓ = a ℓ,m ; d Y ℓ,m ; d , m =1 • where ( a ℓ,m ; d ) m =1 ,...,n ℓ,d are i.i.d. zero-mean Gaussian r.v.’s with E [( a ℓ,m ; d ) 2 ] = µ d , n ℓ,d d +1 µ d = 2 π 2 Γ( d +1 2 ) denoting the measure of the surface of S d . Stein-Malliavin approximations on S d M. Rossi (Rome Tor Vergata) WIAS Berlin - October 23, 2014 4 / 27

Main properties • T ℓ is a Gaussian and isotropic field, i.e. its law is invariant under the action of SO ( d + 1) , that is ∀ g ∈ SO ( d + 1) , law T ℓ ( · ) = T ℓ ( g · ) . Stein-Malliavin approximations on S d M. Rossi (Rome Tor Vergata) WIAS Berlin - October 23, 2014 5 / 27

Main properties • T ℓ is a Gaussian and isotropic field, i.e. its law is invariant under the action of SO ( d + 1) , that is ∀ g ∈ SO ( d + 1) , law T ℓ ( · ) = T ℓ ( g · ) . • T ℓ is centered and E [ T ℓ ( x ) T ℓ ( y )] = G ℓ ; d (cos d ( x, y )) , d ( x, y )= spherical geodesic distance between x and y G ℓ ; d = ℓ -th Gegenbauer polynomial, normalized in such a way that G ℓ ; d (1) = 1 . Stein-Malliavin approximations on S d M. Rossi (Rome Tor Vergata) WIAS Berlin - October 23, 2014 5 / 27

Main properties • T ℓ is a Gaussian and isotropic field, i.e. its law is invariant under the action of SO ( d + 1) , that is ∀ g ∈ SO ( d + 1) , law T ℓ ( · ) = T ℓ ( g · ) . • T ℓ is centered and E [ T ℓ ( x ) T ℓ ( y )] = G ℓ ; d (cos d ( x, y )) , d ( x, y )= spherical geodesic distance between x and y G ℓ ; d = ℓ -th Gegenbauer polynomial, normalized in such a way that G ℓ ; d (1) = 1 . • Precisely ( d 2 − 1 , d 2 − 1) G ℓ ; d = P ℓ , � ℓ + d 2 − 1 � ℓ where P ( α,β ) are Jacobi polynomials. ℓ Stein-Malliavin approximations on S d M. Rossi (Rome Tor Vergata) WIAS Berlin - October 23, 2014 5 / 27

Main properties • T ℓ is a Gaussian and isotropic field, i.e. its law is invariant under the action of SO ( d + 1) , that is ∀ g ∈ SO ( d + 1) , law T ℓ ( · ) = T ℓ ( g · ) . • T ℓ is centered and E [ T ℓ ( x ) T ℓ ( y )] = G ℓ ; d (cos d ( x, y )) , d ( x, y )= spherical geodesic distance between x and y G ℓ ; d = ℓ -th Gegenbauer polynomial, normalized in such a way that G ℓ ; d (1) = 1 . • Precisely ( d 2 − 1 , d 2 − 1) G ℓ ; d = P ℓ , � ℓ + d 2 − 1 � ℓ where P ( α,β ) are Jacobi polynomials. ℓ • Gegenbauer polynomials ( G ℓ ; d ) ℓ are orthogonal polynomials on the interval d [ − 1 , 1] with respect to the weight w ( t ) = (1 − t 2 ) 2 − 1 . For instance, if d = 2 , then G ℓ ;2 = P ℓ the Legendre polynomials. Stein-Malliavin approximations on S d M. Rossi (Rome Tor Vergata) WIAS Berlin - October 23, 2014 5 / 27

why do we study spherical random eigenfunctions? There are many reasons. • Every Gaussian and isotropic random field T on S d is mean-square continuous ( Marinucci and Peccati, 2013 ) and satisfy in L 2 the spectral representation ∞ ∞ � � � T ( x ) 2 � c 2 T ( x ) = c ℓ T ℓ ( x ) , = ℓ < ∞ , E ℓ = 1 ℓ = 1 where the deterministic sequence ( c ℓ ) ℓ is the power spectrum of T . Stein-Malliavin approximations on S d M. Rossi (Rome Tor Vergata) WIAS Berlin - October 23, 2014 6 / 27

why do we study spherical random eigenfunctions? There are many reasons. • Every Gaussian and isotropic random field T on S d is mean-square continuous ( Marinucci and Peccati, 2013 ) and satisfy in L 2 the spectral representation ∞ ∞ � � � T ( x ) 2 � c 2 T ( x ) = c ℓ T ℓ ( x ) , = ℓ < ∞ , E ℓ = 1 ℓ = 1 where the deterministic sequence ( c ℓ ) ℓ is the power spectrum of T . • ⇒ ( T ℓ ) ℓ ∈ N can be viewed as the Fourier components of T . Stein-Malliavin approximations on S d M. Rossi (Rome Tor Vergata) WIAS Berlin - October 23, 2014 6 / 27

why do we study spherical random eigenfunctions? There are many reasons. • Every Gaussian and isotropic random field T on S d is mean-square continuous ( Marinucci and Peccati, 2013 ) and satisfy in L 2 the spectral representation ∞ ∞ � � � T ( x ) 2 � c 2 T ( x ) = c ℓ T ℓ ( x ) , = ℓ < ∞ , E ℓ = 1 ℓ = 1 where the deterministic sequence ( c ℓ ) ℓ is the power spectrum of T . • ⇒ ( T ℓ ) ℓ ∈ N can be viewed as the Fourier components of T . • Moreover they are important in physical contexts, mainly related to the analysis of isotropic spherical random fields on S 2 , (in connection with the analysis of Cosmic Microwave Background). Stein-Malliavin approximations on S d M. Rossi (Rome Tor Vergata) WIAS Berlin - October 23, 2014 6 / 27