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

SLIDE 1

Stein-Malliavin Approximations for Nonlinear Functionals

f Random Eigenfunctions on Sd

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

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 1 / 27

SLIDE 2

utline

1

Introduction What are random eigenfunctions on Sd ? Why are them so important ? Aim of our work: quantitative CLT’s for nonlinear functionals of random eigenfunctions on Sd

2

Stein-Malliavin Normal approximations on Sd CLT’s via Wiener chaos decomposition Fourth moment theorems

3

Quantitative CLT’s for nonlinear functionals of random eigenfunctions

n Sd

Hermite transforms Arbitrary polynomial transforms General nonlinear functionals

Empirical measure of excursion sets

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 2 / 27

SLIDE 3

eigenfunctions on the d-dimensional sphere, d ≥ 2 Sd ⊂ Rd+1 − → unit d-dim sphere ∆Sd − → Laplacian operator on Sd

The eigenvalues of ∆Sd are Eℓ := −ℓ(ℓ + d − 1), ℓ ∈ N.
M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 3 / 27

SLIDE 4

eigenfunctions on the d-dimensional sphere, d ≥ 2 Sd ⊂ Rd+1 − → unit d-dim sphere ∆Sd − → Laplacian operator on Sd

The eigenvalues of ∆Sd are Eℓ := −ℓ(ℓ + d − 1), ℓ ∈ N.
The dimension of the eigenspace Hℓ corresponding to Eℓ is

nℓ,d = 2ℓ + d − 1 ℓ ℓ + d − 2 ℓ − 1

∼

2 (d − 1)!ℓd−1 as ℓ → +∞ (the number of l.i. homogeneous polynomials of degree ℓ in d + 1 variables).

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 3 / 27

SLIDE 5

eigenfunctions on the d-dimensional sphere, d ≥ 2 Sd ⊂ Rd+1 − → unit d-dim sphere ∆Sd − → Laplacian operator on Sd

The eigenvalues of ∆Sd are Eℓ := −ℓ(ℓ + d − 1), ℓ ∈ N.
The dimension of the eigenspace Hℓ corresponding to Eℓ is

nℓ,d = 2ℓ + d − 1 ℓ ℓ + d − 2 ℓ − 1

∼

2 (d − 1)!ℓd−1 as ℓ → +∞ (the number of l.i. homogeneous polynomials of degree ℓ in d + 1 variables).

We consider the real orthonormal basis for Hℓ given by the spherical

harmonics in d + 1-dimension (Yℓ,m;d)m , m = 1, 2, . . . , nℓ,d , ∆SdYℓ,m;d = −ℓ(ℓ + d − 1)Yℓ,m;d .

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 3 / 27

SLIDE 6

eigenfunctions on the d-dimensional sphere, d ≥ 2 Sd ⊂ Rd+1 − → unit d-dim sphere ∆Sd − → Laplacian operator on Sd

The eigenvalues of ∆Sd are Eℓ := −ℓ(ℓ + d − 1), ℓ ∈ N.
The dimension of the eigenspace Hℓ corresponding to Eℓ is

nℓ,d = 2ℓ + d − 1 ℓ ℓ + d − 2 ℓ − 1

∼

2 (d − 1)!ℓd−1 as ℓ → +∞ (the number of l.i. homogeneous polynomials of degree ℓ in d + 1 variables).

We consider the real orthonormal basis for Hℓ given by the spherical

harmonics in d + 1-dimension (Yℓ,m;d)m , m = 1, 2, . . . , nℓ,d , ∆SdYℓ,m;d = −ℓ(ℓ + d − 1)Yℓ,m;d .

Each real-valued f ∈ L2(Sd) admits the Fourier development

f =

ℓ∈N

nℓ,d

m=1

f, Yℓ,m;d Yℓ,m;d

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 3 / 27

SLIDE 7

random eigenfunctions on the d-dimensional sphere, d ≥ 2

What are random eigenfunctions on Sd ?

They are a linear combination of spherical harmonics of fixed degree with random coefficients.

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 4 / 27

SLIDE 8

random eigenfunctions on the d-dimensional sphere, d ≥ 2

What are random eigenfunctions on Sd ?

They are a linear combination of spherical harmonics of fixed degree with random coefficients.

Precisely: for each ℓ ∈ N, we construct the random eigenfunction Tℓ as

Tℓ =

nℓ,d

m=1

aℓ,m;d Yℓ,m;d ,

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 4 / 27

SLIDE 9

random eigenfunctions on the d-dimensional sphere, d ≥ 2

What are random eigenfunctions on Sd ?

They are a linear combination of spherical harmonics of fixed degree with random coefficients.

Precisely: for each ℓ ∈ N, we construct the random eigenfunction Tℓ as

Tℓ =

nℓ,d

m=1

aℓ,m;d Yℓ,m;d ,

where (aℓ,m;d)m=1,...,nℓ,d are i.i.d. zero-mean Gaussian r.v.’s with

E[(aℓ,m;d)2] = µd nℓ,d , µd = 2π

d+1 2

Γ( d+1

2 )

denoting the measure of the surface of Sd.

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 4 / 27

SLIDE 10

Main properties

Tℓ is a Gaussian and isotropic field, i.e. its law is invariant under the action
f SO(d + 1), that is ∀g ∈ SO(d + 1),

Tℓ(·)

law

= Tℓ(g·) .

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 5 / 27

SLIDE 11

Main properties

Tℓ is a Gaussian and isotropic field, i.e. its law is invariant under the action
f SO(d + 1), that is ∀g ∈ SO(d + 1),

Tℓ(·)

law

= Tℓ(g·) .

Tℓ is centered and

E[Tℓ(x)Tℓ(y)] = Gℓ;d(cos d(x, y)) , d(x, y)= spherical geodesic distance between x and y Gℓ;d = ℓ-th Gegenbauer polynomial, normalized in such a way that Gℓ;d(1) = 1.

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 5 / 27

SLIDE 12

Main properties

Tℓ is a Gaussian and isotropic field, i.e. its law is invariant under the action
f SO(d + 1), that is ∀g ∈ SO(d + 1),

Tℓ(·)

law

= Tℓ(g·) .

Tℓ is centered and

E[Tℓ(x)Tℓ(y)] = Gℓ;d(cos d(x, y)) , d(x, y)= spherical geodesic distance between x and y Gℓ;d = ℓ-th Gegenbauer polynomial, normalized in such a way that Gℓ;d(1) = 1.

Precisely

Gℓ;d = P

( d

2 −1, d 2 −1)

ℓ

ℓ+ d

2 −1

ℓ

,

where P (α,β)

ℓ

are Jacobi polynomials.

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 5 / 27

SLIDE 13

Main properties

Tℓ is a Gaussian and isotropic field, i.e. its law is invariant under the action
f SO(d + 1), that is ∀g ∈ SO(d + 1),

Tℓ(·)

law

= Tℓ(g·) .

Tℓ is centered and

E[Tℓ(x)Tℓ(y)] = Gℓ;d(cos d(x, y)) , d(x, y)= spherical geodesic distance between x and y Gℓ;d = ℓ-th Gegenbauer polynomial, normalized in such a way that Gℓ;d(1) = 1.

Precisely

Gℓ;d = P

( d

2 −1, d 2 −1)

ℓ

ℓ+ d

2 −1

ℓ

,

where P (α,β)

ℓ

are Jacobi polynomials.

Gegenbauer polynomials (Gℓ;d)ℓ are orthogonal polynomials on the interval

[−1, 1] with respect to the weight w(t) = (1 − t2)

d 2 −1.

For instance, if d = 2, then Gℓ;2 = Pℓ the Legendre polynomials.

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 5 / 27

SLIDE 14

why do we study spherical random eigenfunctions? There are many reasons.

Every Gaussian and isotropic random field T on Sd is mean-square continuous

(Marinucci and Peccati, 2013) and satisfy in L2 the spectral representation T(x) =

∞

ℓ=1

cℓTℓ(x) , E

T(x)2

=

∞

ℓ=1

c2

ℓ < ∞ ,

where the deterministic sequence (cℓ)ℓ is the power spectrum of T.

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 6 / 27

SLIDE 15

why do we study spherical random eigenfunctions? There are many reasons.

Every Gaussian and isotropic random field T on Sd is mean-square continuous

(Marinucci and Peccati, 2013) and satisfy in L2 the spectral representation T(x) =

∞

ℓ=1

cℓTℓ(x) , E

T(x)2

=

∞

ℓ=1

c2

ℓ < ∞ ,

where the deterministic sequence (cℓ)ℓ is the power spectrum of T.

⇒ (Tℓ)ℓ∈N can be viewed as the Fourier components of T.
M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 6 / 27

SLIDE 16

why do we study spherical random eigenfunctions? There are many reasons.

Every Gaussian and isotropic random field T on Sd is mean-square continuous

(Marinucci and Peccati, 2013) and satisfy in L2 the spectral representation T(x) =

∞

ℓ=1

cℓTℓ(x) , E

T(x)2

=

∞

ℓ=1

c2

ℓ < ∞ ,

where the deterministic sequence (cℓ)ℓ is the power spectrum of T.

⇒ (Tℓ)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 S2, (in connection with the analysis of Cosmic Microwave Background).

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 6 / 27

SLIDE 17

why do we study spherical random eigenfunctions? There are many reasons.

Every Gaussian and isotropic random field T on Sd is mean-square continuous

(Marinucci and Peccati, 2013) and satisfy in L2 the spectral representation T(x) =

∞

ℓ=1

cℓTℓ(x) , E

T(x)2

=

∞

ℓ=1

c2

ℓ < ∞ ,

where the deterministic sequence (cℓ)ℓ is the power spectrum of T.

⇒ (Tℓ)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 S2, (in connection with the analysis of Cosmic Microwave Background).

According to Berry’s Universality conjecture (Berry, 1977)

random Gaussian monochromatic waves (similar to e.g. random Gaussian spherical harmonics) could model deterministic eigenfunctions on a “generic” manifold with or without boundary.

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 6 / 27

SLIDE 18

ur work

⇒ growing interest for geometric functionals of spherical random eigenfunctions

more generally nonlinear functionals of random eigenfunctions on compact

manifolds (e.g. Nazarov and Sodin, 2009 - Granville and Wigman, 2011

Krishnapur, Kurlberg and Wigman, 2013 - Marinucci and Wigman,

2014).

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 7 / 27

SLIDE 19

ur work

⇒ growing interest for geometric functionals of spherical random eigenfunctions

more generally nonlinear functionals of random eigenfunctions on compact

manifolds (e.g. Nazarov and Sodin, 2009 - Granville and Wigman, 2011

Krishnapur, Kurlberg and Wigman, 2013 - Marinucci and Wigman,

2014).

AIM : investigate quantitative CLT’s for nonlinear functionals of Tℓ, as

ℓ → +∞. We consider functionals of the form Sℓ(M) =

Sd M(Tℓ(x)) dx ,

where M : R → R is any square integrable measurable function.

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 7 / 27

SLIDE 20

ur work

⇒ growing interest for geometric functionals of spherical random eigenfunctions

more generally nonlinear functionals of random eigenfunctions on compact

manifolds (e.g. Nazarov and Sodin, 2009 - Granville and Wigman, 2011

Krishnapur, Kurlberg and Wigman, 2013 - Marinucci and Wigman,

2014).

AIM : investigate quantitative CLT’s for nonlinear functionals of Tℓ, as

ℓ → +∞. We consider functionals of the form Sℓ(M) =

Sd M(Tℓ(x)) dx ,

where M : R → R is any square integrable measurable function.

MAIN RESULT : Under a mild assumption on M, the Normal approximation

Sℓ(M) − E[Sℓ(M)]

Var[Sℓ(M)]

L

− → N(0, 1) . The rate of convergence w.r.t. the Wasserstein distance = O(ℓ−1/2).

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 7 / 27

SLIDE 21

ur work

⇒ growing interest for geometric functionals of spherical random eigenfunctions

more generally nonlinear functionals of random eigenfunctions on compact

manifolds (e.g. Nazarov and Sodin, 2009 - Granville and Wigman, 2011

Krishnapur, Kurlberg and Wigman, 2013 - Marinucci and Wigman,

2014).

AIM : investigate quantitative CLT’s for nonlinear functionals of Tℓ, as

ℓ → +∞. We consider functionals of the form Sℓ(M) =

Sd M(Tℓ(x)) dx ,

where M : R → R is any square integrable measurable function.

MAIN RESULT : Under a mild assumption on M, the Normal approximation

Sℓ(M) − E[Sℓ(M)]

Var[Sℓ(M)]

L

− → N(0, 1) . The rate of convergence w.r.t. the Wasserstein distance = O(ℓ−1/2).

Important application (Excursion area) If M = 1(· ≤ z), for z ∈ R, then

Sℓ(z) =

Sd 1(Tℓ(x) ≤ z) dx
M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 7 / 27

SLIDE 22

background: wiener chaoses

Isonormal Gaussian field X on L2(Sd)

Centered Gaussian family {X(f), f ∈ L2(Sd)} with covariance structure Cov (X(f), X(f ′)) = f, f ′2

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 8 / 27

SLIDE 23

background: wiener chaoses

Isonormal Gaussian field X on L2(Sd)

Centered Gaussian family {X(f), f ∈ L2(Sd)} with covariance structure Cov (X(f), X(f ′)) = f, f ′2

For each q ≥ 0, the q-th Wiener chaos Hq of X is the closure in L2(P) of the

subspace generated by r.v.’s of the form Hq(X(f)) , f ∈ L2(Sd), f2 = 1 , where Hq is the q-th Hermite polynomial.

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 8 / 27

SLIDE 24

background: wiener chaoses

Isonormal Gaussian field X on L2(Sd)

Centered Gaussian family {X(f), f ∈ L2(Sd)} with covariance structure Cov (X(f), X(f ′)) = f, f ′2

For each q ≥ 0, the q-th Wiener chaos Hq of X is the closure in L2(P) of the

subspace generated by r.v.’s of the form Hq(X(f)) , f ∈ L2(Sd), f2 = 1 , where Hq is the q-th Hermite polynomial.

Wiener-Itˆ
chaos decomposition into orthogonal subspaces

L2(P) =

q≥0 Hq

F ∈ L2(P) ⇒ F = E[F] +

q≥1 Fq

Fq being the projection of F on Hq.

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 8 / 27

SLIDE 25

background: wiener chaoses

Isonormal Gaussian field X on L2(Sd)

Centered Gaussian family {X(f), f ∈ L2(Sd)} with covariance structure Cov (X(f), X(f ′)) = f, f ′2

For each q ≥ 0, the q-th Wiener chaos Hq of X is the closure in L2(P) of the

subspace generated by r.v.’s of the form Hq(X(f)) , f ∈ L2(Sd), f2 = 1 , where Hq is the q-th Hermite polynomial.

Wiener-Itˆ
chaos decomposition into orthogonal subspaces

L2(P) =

q≥0 Hq

F ∈ L2(P) ⇒ F = E[F] +

q≥1 Fq

Fq being the projection of F on Hq.

Borel function ϕ : R → R : F = ϕ(X(f)) ∈ L2(P)

ϕ(X(f)) =

q≥0 Jq(ϕ) q!

Hq(X(f)) , Jq(ϕ) = E[ϕ(X(f))Hq(X(f))]

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 8 / 27

SLIDE 26

clt’s via chaos decomposition

Isonormal representation Tℓ(x) = X(fx) ,

fx(·) = nℓ;d

µd Gℓ(cos d(x, ·))

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 9 / 27

SLIDE 27

clt’s via chaos decomposition

Isonormal representation Tℓ(x) = X(fx) ,

fx(·) = nℓ;d

µd Gℓ(cos d(x, ·))

Sℓ(M) =
Sd M(X(fx)) dx ∈ L2(P) ⇒ Wiener-Itˆ
chaos expansion

Sℓ(M) =

∞

q=0

Jq(M) q!

Sd Hq(Tℓ(x)) dx

(2.1)

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 9 / 27

SLIDE 28

clt’s via chaos decomposition

Isonormal representation Tℓ(x) = X(fx) ,

fx(·) = nℓ;d

µd Gℓ(cos d(x, ·))

Sℓ(M) =
Sd M(X(fx)) dx ∈ L2(P) ⇒ Wiener-Itˆ
chaos expansion

Sℓ(M) =

∞

q=0

Jq(M) q!

Sd Hq(Tℓ(x)) dx

(2.1)

In order to prove a quantitative CLT for Sℓ(M), as ℓ → +∞, we would
M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 9 / 27

SLIDE 29

clt’s via chaos decomposition

Isonormal representation Tℓ(x) = X(fx) ,

fx(·) = nℓ;d

µd Gℓ(cos d(x, ·))

Sℓ(M) =
Sd M(X(fx)) dx ∈ L2(P) ⇒ Wiener-Itˆ
chaos expansion

Sℓ(M) =

∞

q=0

Jq(M) q!

Sd Hq(Tℓ(x)) dx

(2.1)

In order to prove a quantitative CLT for Sℓ(M), as ℓ → +∞, we would
1) prove a quantitative CLT for hℓ;q,d :=
Sd Hq(Tℓ(x)) dx, that are the basic

building blocks for (2.1);

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 9 / 27

SLIDE 30

clt’s via chaos decomposition

Isonormal representation Tℓ(x) = X(fx) ,

fx(·) = nℓ;d

µd Gℓ(cos d(x, ·))

Sℓ(M) =
Sd M(X(fx)) dx ∈ L2(P) ⇒ Wiener-Itˆ
chaos expansion

Sℓ(M) =

∞

q=0

Jq(M) q!

Sd Hq(Tℓ(x)) dx

(2.1)

In order to prove a quantitative CLT for Sℓ(M), as ℓ → +∞, we would
1) prove a quantitative CLT for hℓ;q,d :=
Sd Hq(Tℓ(x)) dx, that are the basic

building blocks for (2.1);

⇒ then the CLT for finite linear combinations of hℓ;q,d immediately follows,

Peccati and Taqqu, 2010 (but not the rate of convergence, which will require some work);

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 9 / 27

SLIDE 31

clt’s via chaos decomposition

Isonormal representation Tℓ(x) = X(fx) ,

fx(·) = nℓ;d

µd Gℓ(cos d(x, ·))

Sℓ(M) =
Sd M(X(fx)) dx ∈ L2(P) ⇒ Wiener-Itˆ
chaos expansion

Sℓ(M) =

∞

q=0

Jq(M) q!

Sd Hq(Tℓ(x)) dx

(2.1)

In order to prove a quantitative CLT for Sℓ(M), as ℓ → +∞, we would
1) prove a quantitative CLT for hℓ;q,d :=
Sd Hq(Tℓ(x)) dx, that are the basic

building blocks for (2.1);

⇒ then the CLT for finite linear combinations of hℓ;q,d immediately follows,

Peccati and Taqqu, 2010 (but not the rate of convergence, which will require some work);

2) show that, under a mild assumption on M, the asymptotic behaviour of

the r.v. in (2.1) is dominated by a single Wiener chaos or at most a finite linear combination of hℓ;q,d.

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 9 / 27

SLIDE 32

fourth moment theorems

Z, N r.v.’s : Kolmogorov dK, TotalVariation dT V and Wasserstein dW

distances dK(Z, N) = sup

z∈R

|P(Z ≤ z) − P(N ≤ z)| , dT V (Z, N) = sup

A∈B(R)

|P(Z ∈ A) − P(N ∈ A)| , dW (Z, N) = sup

h∈Lip(1)

|E[h(Z)] − E[h(N)]| ,

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 10 / 27

SLIDE 33

fourth moment theorems

Z, N r.v.’s : Kolmogorov dK, TotalVariation dT V and Wasserstein dW

distances dK(Z, N) = sup

z∈R

|P(Z ≤ z) − P(N ≤ z)| , dT V (Z, N) = sup

A∈B(R)

|P(Z ∈ A) − P(N ∈ A)| , dW (Z, N) = sup

h∈Lip(1)

|E[h(Z)] − E[h(N)]| ,

to prove a quantitative CLT for hℓ;q,d =
Sd Hq(Tℓ(x)) dx ⇒ fourth moment

theorem by Nourdin and Peccati: for dD = dT V , dW , dK dD

hℓ;q,d
V ar[hℓ;q,d]

, N(0, 1)

≤
q − 1

3q cum4[hℓ;q,d] (Var[hℓ;q,d])2 (2.2)

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 10 / 27

SLIDE 34

fourth moment theorems

Z, N r.v.’s : Kolmogorov dK, TotalVariation dT V and Wasserstein dW

distances dK(Z, N) = sup

z∈R

|P(Z ≤ z) − P(N ≤ z)| , dT V (Z, N) = sup

A∈B(R)

|P(Z ∈ A) − P(N ∈ A)| , dW (Z, N) = sup

h∈Lip(1)

|E[h(Z)] − E[h(N)]| ,

to prove a quantitative CLT for hℓ;q,d =
Sd Hq(Tℓ(x)) dx ⇒ fourth moment

theorem by Nourdin and Peccati: for dD = dT V , dW , dK dD

hℓ;q,d
V ar[hℓ;q,d]

, N(0, 1)

≤
q − 1

3q cum4[hℓ;q,d] (Var[hℓ;q,d])2 (2.2)

Therefore we have to study the variance of hℓ;q,d and the fourth cumulant of

hℓ;q,d and show that the r.h.s. in (2.2) goes to 0, as ℓ → +∞

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 10 / 27

SLIDE 35

the variance of hℓ;q,d

Var[hℓ;q,d] =E
Sd Hq(Tℓ(x)) dx

2 = q!

(Sd)2 Gℓ;d(cos d(x, y))q dxdy =

= q! µdµd−1 π

0 Gℓ;d(cos ϑ)q(sin ϑ)d−1 dϑ .

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 11 / 27

SLIDE 36

the variance of hℓ;q,d

Var[hℓ;q,d] =E
Sd Hq(Tℓ(x)) dx

2 = q!

(Sd)2 Gℓ;d(cos d(x, y))q dxdy =

= q! µdµd−1 π

0 Gℓ;d(cos ϑ)q(sin ϑ)d−1 dϑ .

Gℓ;d(t) = (−1)ℓGℓ;d(−t), so hℓ;q,d = 0 a.s. when both ℓ and q are odd,
therwise

Var[hℓ;q,d] = 2q!µdµd−1 π

2

0 Gℓ;d(cos ϑ)q(sin ϑ)d−1 dϑ .

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 11 / 27

SLIDE 37

the variance of hℓ;q,d

Var[hℓ;q,d] =E
Sd Hq(Tℓ(x)) dx

2 = q!

(Sd)2 Gℓ;d(cos d(x, y))q dxdy =

= q! µdµd−1 π

0 Gℓ;d(cos ϑ)q(sin ϑ)d−1 dϑ .

Gℓ;d(t) = (−1)ℓGℓ;d(−t), so hℓ;q,d = 0 a.s. when both ℓ and q are odd,
therwise

Var[hℓ;q,d] = 2q!µdµd−1 π

2

0 Gℓ;d(cos ϑ)q(sin ϑ)d−1 dϑ .

Our first result: upper bound for these variances, asymptotic as ℓ → +∞

(for ”notational” simplicity: just even ℓ).

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 11 / 27

SLIDE 38

the variance of hℓ;q,d

Var[hℓ;q,d] =E
Sd Hq(Tℓ(x)) dx

2 = q!

(Sd)2 Gℓ;d(cos d(x, y))q dxdy =

= q! µdµd−1 π

0 Gℓ;d(cos ϑ)q(sin ϑ)d−1 dϑ .

Gℓ;d(t) = (−1)ℓGℓ;d(−t), so hℓ;q,d = 0 a.s. when both ℓ and q are odd,
therwise

Var[hℓ;q,d] = 2q!µdµd−1 π

2

0 Gℓ;d(cos ϑ)q(sin ϑ)d−1 dϑ .

Our first result: upper bound for these variances, asymptotic as ℓ → +∞

(for ”notational” simplicity: just even ℓ).

Method: investigate the asymptotic behaviour as ℓ → +∞ of all order

moments of Gegenbauer polynomials

π

2

Gℓ;d(cos ϑ)q(sin ϑ)d−1 dϑ .

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 11 / 27

SLIDE 39

the variance of hℓ;q,d

Var[hℓ;q,d] =E
Sd Hq(Tℓ(x)) dx

2 = q!

(Sd)2 Gℓ;d(cos d(x, y))q dxdy =

= q! µdµd−1 π

0 Gℓ;d(cos ϑ)q(sin ϑ)d−1 dϑ .

Gℓ;d(t) = (−1)ℓGℓ;d(−t), so hℓ;q,d = 0 a.s. when both ℓ and q are odd,
therwise

Var[hℓ;q,d] = 2q!µdµd−1 π

2

0 Gℓ;d(cos ϑ)q(sin ϑ)d−1 dϑ .

Our first result: upper bound for these variances, asymptotic as ℓ → +∞

(for ”notational” simplicity: just even ℓ).

Method: investigate the asymptotic behaviour as ℓ → +∞ of all order

moments of Gegenbauer polynomials

π

2

Gℓ;d(cos ϑ)q(sin ϑ)d−1 dϑ .

We are inspired by the proof for moments of Legendre polynomials (d = 2) in

Marinucci and Wigman, 2011.

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 11 / 27

SLIDE 40

main tool for investigating these variances Recall the definition of the Bessel functions Jν(ψ) =

∞

k=0

(−1)k ψ2k+ν 22k+νk!Γ(k + ν + 1) , ν ≥ 0 , ψ ≥ 0 ;

Hilb’s asymptotic formula for Jacobi polynomials (Szego, 1975):

(sin ϑ)

d 2 −1Gℓ;d(cos ϑ) =

2

d 2 −1

ℓ+ d

2 −1

ℓ



      Γ(ℓ + d

2)

(ℓ + d−1

2 ) d 2 −1ℓ!

∼1 as ℓ→+∞
ϑ

sin ϑJ d

2 −1

(ℓ + d − 1

2 )ϑ

+ δ(ϑ)

       The remainder is δ(ϑ) =    √ ϑ ℓ− 3

2

if ℓ−1 < ϑ < π

2 ,

ϑ

d 2 −1

+2 ℓ

d 2 −1

if 0 < ϑ < ℓ−1 .

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 12 / 27

SLIDE 41

sketch of the proof (d, q ≥ 3) By Hilb’s asymptotic

π

2

Gℓ;d(cos ϑ)q(sin ϑ)d−1 dϑ = =   2

d 2 −1

ℓ+ d

2 −1

ℓ





q

π 2

(sin ϑ)−q( d

2 −1)

ϑ sin ϑ q

2 J d 2 −1

(ℓ + d − 1

2 )ϑ q (sin ϑ)d−1 dϑ + error

error = o

1

Ld

! By the change of variable ψ = (ℓ + d−1

2 )ϑ := Lϑ in the integral

=   2

d 2 −1

ℓ+ d

2 −1

ℓ





q

1 L L π

2

(sin ψ/L)−q( d

2 −1)

ψ/L sin ψ/L q

2 Jq d 2 −1(ψ)(sin ψ/L)d−1 dψ

and since ψ/L = o(1), ∼ 1 Ld  (2

d 2 −1)L( d 2 −1)

ℓ+ d

2 −1

ℓ





q L π

2

ψ/L

sin ψ/L q( d

2 − 1 2 )−d+1

Jq

d 2 −1(ψ)ψ−q( d 2 −1)+d−1 dψ

−

→ cq

, cq :=

2

d 2 −1

d 2 − 1

!

q +∞ J d

2 −1(ψ)qψ

−q d 2 −1

+d−1 dψ .
M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 13 / 27

SLIDE 42

summary on the variance of hℓ;q,d for d ≥ 2

Our result: as ℓ → ∞, for d, q ≥ 3,

Var[hℓ;q,d] = 2q!µdµd−1cq 1 ℓd (1 + o(1)) .

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 14 / 27

SLIDE 43

summary on the variance of hℓ;q,d for d ≥ 2

Our result: as ℓ → ∞, for d, q ≥ 3,

Var[hℓ;q,d] = 2q!µdµd−1cq 1 ℓd (1 + o(1)) .

Second moment: for d ≥ 2, there exists an exact formula when q = 2 :

π Gℓ;d(cos ϑ)2(sin ϑ)d−1 dϑ = µd µd−1 nℓ,d ∼ c2 1 ℓd−1

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 14 / 27

SLIDE 44

summary on the variance of hℓ;q,d for d ≥ 2

Our result: as ℓ → ∞, for d, q ≥ 3,

Var[hℓ;q,d] = 2q!µdµd−1cq 1 ℓd (1 + o(1)) .

Second moment: for d ≥ 2, there exists an exact formula when q = 2 :

π Gℓ;d(cos ϑ)2(sin ϑ)d−1 dϑ = µd µd−1 nℓ,d ∼ c2 1 ℓd−1

Case d = 2, q ≥ 3: in Marinucci and Wigman, 2011.
q = 3, 5, . . .

Var[hℓ;q,2] = (4π)2q!cq 1 ℓ2 (1 + o(1)) , cq = +∞ J0(ψ)qψ dψ ,

q = 4, as ℓ → +∞,

Var[hℓ;4,2] ∼ 242 logℓ ℓ2 .

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 14 / 27

SLIDE 45

the fourth cumulant of hℓ;q,d

circulant diagrams → give the major contribution for cum4[hℓ;q,d] (Nourdin

and Peccati) ↓ corresponds to multiple integrals of Gegenbauer polynomials as follows:

Kℓ(q; r) :=

(Sd)4 Gq−r

ℓ;d (cos d(x1, x2))Gr ℓ;d(cos d(x2, x3))×

×Gq−r

ℓ;d (cos d(x3, x4))Gr ℓ;d(cos d(x4, x1)) dx1dx2dx3dx4 ,

for q = 2, 3, 4, ..., r = 1, 2, ...[ q

2].

How do we estimate them ? : We bound |Gr

ℓ;d(cos d(x4, x1))| in |Kℓ(q; r)| by 1 and

we use previous results for all order moments of Gℓ;d. Example: q = 5 and d = 2 − → Gℓ;2 ≡ Pℓ Legendre polynomials |Kℓ(5; 1)| ≤

(S2)4 |Pℓ(x1, x2)|4 |Pℓ(x2, x3)| |Pℓ(x3, x4)|4 |Pℓ(x4, x1)|
≤1

dx1dx2dx3dx4 = O

(log ℓ)ℓ−2

×

(S2)2 |Pℓ(x1, x2)|4
S2 |Pℓ(x2, x3)| dx3
dx1dx2

= O

(log ℓ)ℓ−2

× O

ℓ− 1

2

×
(S2)2 |Pℓ(x1, x2)|4 dx1dx2 = O
(log2 ℓ)ℓ−9/2

.

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 15 / 27

SLIDE 46

results for cum4[hℓ;q,2] (d = 2)

For r = 1, 2, ..., [ q

2] we have

|Kℓ(q; r)| = O

ℓ−5

for q = 3 , (3.1) |Kℓ(q; r)| = O

ℓ−4

for q = 4 , (3.2) |Kℓ(q; r)| = O

(log ℓ)ℓ−9/2

for q = 5, 6 , (3.3) |Kℓ(q; r)| = O

ℓ−9/2

for q ≥ 7 . (3.4)

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 16 / 27

SLIDE 47

results for cum4[hℓ;q,2] (d = 2)

For r = 1, 2, ..., [ q

2] we have

|Kℓ(q; r)| = O

ℓ−5

for q = 3 , (3.1) |Kℓ(q; r)| = O

ℓ−4

for q = 4 , (3.2) |Kℓ(q; r)| = O

(log ℓ)ℓ−9/2

for q = 5, 6 , (3.3) |Kℓ(q; r)| = O

ℓ−9/2

for q ≥ 7 . (3.4)

The bounds (3.1), (3.2) are known and indeed the corresponding integrals can

be evaluated explicitly in terms of Wigner’s 3j and 6j coefficients (Marinucci, 2008, Marinucci and Peccati, 2011, Marinucci and Wigman, 2014).

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 16 / 27

SLIDE 48

results for cum4[hℓ;q,2] (d = 2)

For r = 1, 2, ..., [ q

2] we have

|Kℓ(q; r)| = O

ℓ−5

for q = 3 , (3.1) |Kℓ(q; r)| = O

ℓ−4

for q = 4 , (3.2) |Kℓ(q; r)| = O

(log ℓ)ℓ−9/2

for q = 5, 6 , (3.3) |Kℓ(q; r)| = O

ℓ−9/2

for q ≥ 7 . (3.4)

The bounds (3.1), (3.2) are known and indeed the corresponding integrals can

be evaluated explicitly in terms of Wigner’s 3j and 6j coefficients (Marinucci, 2008, Marinucci and Peccati, 2011, Marinucci and Wigman, 2014).

The bounds in (3.3),(3.4) improve the existing bounds in Marinucci and

Wigman, 2014, where they use a different method to bound cum4[hℓ;q,2].

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 16 / 27

SLIDE 49

results for cum4[hℓ;q,d] when d ≥ 3

For all r = 1, 2, . . . , [ q

2] we have

|Kℓ(q; r)| = O

1

ℓ2d+ d−5

2

for q = 3 ,

|Kℓ(q; r)| = O

1

ℓ2d+ d−3

2

for q = 4 ,

|Kℓ(q; r)| = O

1

ℓ2d+ d−1

2

for q ≥ 5 .
M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 17 / 27

SLIDE 50

results for cum4[hℓ;q,d] when d ≥ 3

For all r = 1, 2, . . . , [ q

2] we have

|Kℓ(q; r)| = O

1

ℓ2d+ d−5

2

for q = 3 ,

|Kℓ(q; r)| = O

1

ℓ2d+ d−3

2

for q = 4 ,

|Kℓ(q; r)| = O

1

ℓ2d+ d−1

2

for q ≥ 5 .
Recall that for q ≥ 3, Var[hℓ;q,d] ∼

1 ℓd , therefore we obtain a useful upper

bound for cum4[hℓ;q,d], i.e. cum4[hℓ;q,d] Var[hℓ;q,d]2 − →0 , ℓ → +∞ but for the cases (1) q = 3 and d = 3, 4, 5; (2) q = 4 and d = 3. We leave these cases for future research as we need estimates on Clebsch Gordan coefficients for SO(d + 1), d = 3, 4, 5.

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 17 / 27

SLIDE 51

quantitative clt’s for hℓ;q,d ∀q s.t. (d, q) = (3, 4), (3, 3), (4, 3), (5, 3) and cq > 0, we have hℓ;q,d

V ar[hℓ;q,d]

law

− → N(0, 1) and the rate of convergence R(ℓ; q, d) for the metrics dD = dT V , dW , dK is

d = 2 R(ℓ; q, 2) =          ℓ− 1

2

if q = 2, 3 , (log ℓ)−1 if q = 4 , (log ℓ)ℓ− 1

4

if q = 5, 6 , ℓ− 1

4

if q = 7, 8, . . . ; d ≥ 3 R(ℓ; q, d) =        ℓ−( d−1

2 )

if q = 2, 3 , ℓ−( d−3

4 )

if q = 4 , ℓ−( d−1

4 )

if q = 5, 6, . . . .

The results for d = 2 improve the quantitative CLT in Marinucci and Wigman, 2014,

actually they show that the total variation rate satisfies (up to logarithmic terms) dT V = O(ℓ−δq), where δ4 =

1 10, δ5 = 1 7, and δq = q−6 4q−6 < 1 4 for q = 7, 8, . . . .

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 18 / 27

SLIDE 52

arbitrary polynomial transforms

The transforms of Tℓ by a finite order polynomial p are of the form

Zℓ =

Sd p(Tℓ(x)) dx =

Q

q=2

βqhℓ;q,d , Q ∈ N, βq ∈ R .

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 19 / 27

SLIDE 53

arbitrary polynomial transforms

The transforms of Tℓ by a finite order polynomial p are of the form

Zℓ =

Sd p(Tℓ(x)) dx =

Q

q=2

βqhℓ;q,d , Q ∈ N, βq ∈ R .

Assume that for d = 3, β3 = 0, β4 = 0 and for d = 4, 5, β3 = 0.
M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 19 / 27

SLIDE 54

arbitrary polynomial transforms

The transforms of Tℓ by a finite order polynomial p are of the form

Zℓ =

Sd p(Tℓ(x)) dx =

Q

q=2

βqhℓ;q,d , Q ∈ N, βq ∈ R .

Assume that for d = 3, β3 = 0, β4 = 0 and for d = 4, 5, β3 = 0.
CLT’s for hℓ;q,d ⇒ CLT for Zℓ (Peccati and Tudor, 2010).
M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 19 / 27

SLIDE 55

arbitrary polynomial transforms

The transforms of Tℓ by a finite order polynomial p are of the form

Zℓ =

Sd p(Tℓ(x)) dx =

Q

q=2

βqhℓ;q,d , Q ∈ N, βq ∈ R .

Assume that for d = 3, β3 = 0, β4 = 0 and for d = 4, 5, β3 = 0.
CLT’s for hℓ;q,d ⇒ CLT for Zℓ (Peccati and Tudor, 2010).
Our result: The rate of convergence for Zℓ is the same as what we obtained

for the Hermite polynomial case. If there exists q such that βq = 0 and cq > 0 dD

Zℓ − E[Zℓ]
Var[Zℓ]

, N(0, 1)

= O(R(Zℓ; d)) ,

where dD = dT V , dW , dK and for d ≥ 2 R(Zℓ; d) =

ℓ− d−1

2

if β2 = 0 , maxq=3,...,Q : βq=0 R(ℓ; q, d) if β2 = 0 .

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 19 / 27

SLIDE 56

why ? (sketch)

Following Nourdin and Peccati, the CLT rate for random variables with

finite chaotic expansion is dD

Zℓ − E[Zℓ]
Var[Zℓ]

, N(0, 1)

≤ (A) + (B) ,

where in (A) and (B) there are quantities of two different form as follows:

(A)

(Sd)4 Gq−r

ℓ;d (cos d(x1, x2))Gr ℓ;d(cos d(x2, x3))×

×Gq−r

ℓ;d (cos d(x3, x4))Gr ℓ;d(cos d(x4, x1)) dx1dx2dx3dx4 ,

(B)

(Sd)4 Gq1

ℓ;d(cos d(x1, x2))Gq2−q1 ℓ;d

(cos d(x2, x3))Gq1

ℓ;d(cos d(x3, x4)) dx1dx2dx3dx4 .

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 20 / 27

SLIDE 57

why ? (sketch)

Following Nourdin and Peccati, the CLT rate for random variables with

finite chaotic expansion is dD

Zℓ − E[Zℓ]
Var[Zℓ]

, N(0, 1)

≤ (A) + (B) ,

where in (A) and (B) there are quantities of two different form as follows:

(A)

(Sd)4 Gq−r

ℓ;d (cos d(x1, x2))Gr ℓ;d(cos d(x2, x3))×

×Gq−r

ℓ;d (cos d(x3, x4))Gr ℓ;d(cos d(x4, x1)) dx1dx2dx3dx4 ,

(B)

(Sd)4 Gq1

ℓ;d(cos d(x1, x2))Gq2−q1 ℓ;d

(cos d(x2, x3))Gq1

ℓ;d(cos d(x3, x4)) dx1dx2dx3dx4 .

(A) directly follows from the analysis of cum4[hℓ;q,d].
M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 20 / 27

SLIDE 58

why ? (sketch)

Following Nourdin and Peccati, the CLT rate for random variables with

finite chaotic expansion is dD

Zℓ − E[Zℓ]
Var[Zℓ]

, N(0, 1)

≤ (A) + (B) ,

where in (A) and (B) there are quantities of two different form as follows:

(A)

(Sd)4 Gq−r

ℓ;d (cos d(x1, x2))Gr ℓ;d(cos d(x2, x3))×

×Gq−r

ℓ;d (cos d(x3, x4))Gr ℓ;d(cos d(x4, x1)) dx1dx2dx3dx4 ,

(B)

(Sd)4 Gq1

ℓ;d(cos d(x1, x2))Gq2−q1 ℓ;d

(cos d(x2, x3))Gq1

ℓ;d(cos d(x3, x4)) dx1dx2dx3dx4 .

(A) directly follows from the analysis of cum4[hℓ;q,d].
(B) corresponds to diagrams with no proper loop. Usually Cauchy-Schwartz

inequality is used to estimate |(B)|.

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 20 / 27

SLIDE 59

why ? (sketch)

Following Nourdin and Peccati, the CLT rate for random variables with

finite chaotic expansion is dD

Zℓ − E[Zℓ]
Var[Zℓ]

, N(0, 1)

≤ (A) + (B) ,

where in (A) and (B) there are quantities of two different form as follows:

(A)

(Sd)4 Gq−r

ℓ;d (cos d(x1, x2))Gr ℓ;d(cos d(x2, x3))×

×Gq−r

ℓ;d (cos d(x3, x4))Gr ℓ;d(cos d(x4, x1)) dx1dx2dx3dx4 ,

(B)

(Sd)4 Gq1

ℓ;d(cos d(x1, x2))Gq2−q1 ℓ;d

(cos d(x2, x3))Gq1

ℓ;d(cos d(x3, x4)) dx1dx2dx3dx4 .

(A) directly follows from the analysis of cum4[hℓ;q,d].
(B) corresponds to diagrams with no proper loop. Usually Cauchy-Schwartz

inequality is used to estimate |(B)|.

We estimate (B) in a different way.
M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 20 / 27

SLIDE 60

estimates for (b)

(B)

(Sd)4 Gq1

ℓ;d(cos d(x1, x2))Gq2−q1 ℓ;d

(cos d(x2, x3))Gq1

ℓ;d(cos d(x3, x4)) dx1dx2dx3dx4 =

(Sd)2

       Gq2−q1

ℓ;d

(cos d(x2, x3))

Sd Gq1

ℓ;d(cos d(x1, x2))dx1

Sd Gq1

ℓ;d(cos d(x3, x4))dx4

=
1

q1! Var[hℓ;q1,d]

2

       dx2dx = 1 q1!Var[hℓ;q1,d] 2

(Sd)2 Gq2−q1 ℓ;d

(cos d(x2, x3))dx2dx3 = =    if q2 − q1 = 1 ; ≤

1

q1!Var[hℓ;q1,d]

2

(Sd)2 G2 ℓ;d(cos d(x2, x3)) dx2dx3

if q2 − q1 ≥ 2 . Therefore (B) = O 1 q1!Var[hℓ;q1,d] 2 × ℓ−(d−1)

.
M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 21 / 27

SLIDE 61

It is easy to analyze the variance of Zℓ, indeed

d = 2 Var[Zℓ] =

Q

q=2

β2

qVar[hℓ;q,d] ∼

   ℓ−1, for β2 = 0 ℓ−2 log ℓ, for β2 = 0 , β4 = 0 ℓ−2,

therwise.

d ≥ 3 Var[Zℓ] =

Q

q=2

β2

qVar[hℓ;q,d] ∼

ℓ−d+1, for β2 = 0 ℓ−d , otherwise.

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 22 / 27

SLIDE 62

It is easy to analyze the variance of Zℓ, indeed

d = 2 Var[Zℓ] =

Q

q=2

β2

qVar[hℓ;q,d] ∼

   ℓ−1, for β2 = 0 ℓ−2 log ℓ, for β2 = 0 , β4 = 0 ℓ−2,

therwise.

d ≥ 3 Var[Zℓ] =

Q

q=2

β2

qVar[hℓ;q,d] ∼

ℓ−d+1, for β2 = 0 ℓ−d , otherwise.

In the special case of polynomials of Hermite rank 2, i.e. β2 = 0,

the asymptotic behaviour of Zℓ is dominated by the term hℓ;2,d, that is the projection on the 2nd Wiener chaos. The variance of hℓ;2,d is of order ℓ−(d−1), rather than O(ℓ−d) as for the other

terms. Therefore we find the simple behaviour

dD

Zℓ − E[Zℓ]
Var[Zℓ]

, N(0, 1)

∼ dD
hℓ;2,d
Var[hℓ;2,d]

, N(0, 1)

= O
ℓ−( d−1

2 )

.

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 22 / 27

SLIDE 63

general nonlinear tranforms

This simple behaviour holds also for the general case (under a mild

assumption). Precisely, consider Sℓ(M) =

Sd M(Tℓ(x))dx ;
M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 23 / 27

SLIDE 64

general nonlinear tranforms

This simple behaviour holds also for the general case (under a mild

assumption). Precisely, consider Sℓ(M) =

Sd M(Tℓ(x))dx ;
M : R → R is a measurable function such that
M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 23 / 27

SLIDE 65

general nonlinear tranforms

This simple behaviour holds also for the general case (under a mild

assumption). Precisely, consider Sℓ(M) =

Sd M(Tℓ(x))dx ;
M : R → R is a measurable function such that
E[M(Tℓ)2] < ∞
M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 23 / 27

SLIDE 66

general nonlinear tranforms

This simple behaviour holds also for the general case (under a mild

assumption). Precisely, consider Sℓ(M) =

Sd M(Tℓ(x))dx ;
M : R → R is a measurable function such that
E[M(Tℓ)2] < ∞
J2(M) = 0, where

Jq(M) = E[M(Tℓ)Hq(Tℓ)] , that is, the projection on the 2nd Wiener chaos is nonzero.

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 23 / 27

SLIDE 67

general nonlinear tranforms

This simple behaviour holds also for the general case (under a mild

assumption). Precisely, consider Sℓ(M) =

Sd M(Tℓ(x))dx ;
M : R → R is a measurable function such that
E[M(Tℓ)2] < ∞
J2(M) = 0, where

Jq(M) = E[M(Tℓ)Hq(Tℓ)] , that is, the projection on the 2nd Wiener chaos is nonzero.

w.l.o.g, the first two coefficients of Sℓ(M), i.e. J0(M), J1(M) can always be

taken = 0 in the present framework. → J0(M) = E[M(Tℓ)] = 0, assuming we work with centred variables. → hℓ;1,d =

Sd Tℓ(x) dx = 0, whence the value of J1(M) can be taken = 0.

⇒ M has Hermite rank = 2 ⇒ its asymptotic behaviour is dominated by the projection on the 2nd Wiener chaos

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 23 / 27

SLIDE 68

quantitative clt’s for general nonlinear functionals

Under previous hypothesis, the 2nd Wiener chaos dominates and

Var[Sℓ(M)] ∼ 1 ℓd−1

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 24 / 27

SLIDE 69

quantitative clt’s for general nonlinear functionals

Under previous hypothesis, the 2nd Wiener chaos dominates and

Var[Sℓ(M)] ∼ 1 ℓd−1

As ℓ → ∞, we have the CLT and the rate

dW

Sℓ(M)
Var(Sℓ(M))

, N(0, 1)

= O

1 √ ℓ

,

where dW is the Wasserstein distance.

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 24 / 27

SLIDE 70

quantitative clt’s for general nonlinear functionals

Under previous hypothesis, the 2nd Wiener chaos dominates and

Var[Sℓ(M)] ∼ 1 ℓd−1

As ℓ → ∞, we have the CLT and the rate

dW

Sℓ(M)
Var(Sℓ(M))

, N(0, 1)

= O

1 √ ℓ

,

where dW is the Wasserstein distance.

Our hypothesis J2(M) = 0 covers the case of Hermite rank = 2 but also

Hermite rank = 1 for M.

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 24 / 27

SLIDE 71

quantitative clt’s for general nonlinear functionals

Under previous hypothesis, the 2nd Wiener chaos dominates and

Var[Sℓ(M)] ∼ 1 ℓd−1

As ℓ → ∞, we have the CLT and the rate

dW

Sℓ(M)
Var(Sℓ(M))

, N(0, 1)

= O

1 √ ℓ

,

where dW is the Wasserstein distance.

Our hypothesis J2(M) = 0 covers the case of Hermite rank = 2 but also

Hermite rank = 1 for M.

M does not need to be smooth in any meaningful sense ⇒ analysis of the

asymptotic behaviour of functionals of geometric interest, e.g. empirical measure of excursion sets.

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 24 / 27

SLIDE 72

empirical measure of excursion sets

excursion set: fix z ∈ R,

{x ∈ Sd : Tℓ(x) ≤ z} empirical measure: nonlinear functional of Tℓ Sℓ(z) := Sℓ(1(· ≤ z)) =

Sd 1(Tℓ(x) ≤ z) dx ,

where 1(· ≤ z) is the indicator function of the interval (−∞, z].

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 25 / 27

SLIDE 73

empirical measure of excursion sets

excursion set: fix z ∈ R,

{x ∈ Sd : Tℓ(x) ≤ z} empirical measure: nonlinear functional of Tℓ Sℓ(z) := Sℓ(1(· ≤ z)) =

Sd 1(Tℓ(x) ≤ z) dx ,

where 1(· ≤ z) is the indicator function of the interval (−∞, z].

J2(1(· ≤ z)) = zφ(z), φ being the Gaussian density ⇒
M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 25 / 27

SLIDE 74

empirical measure of excursion sets

excursion set: fix z ∈ R,

{x ∈ Sd : Tℓ(x) ≤ z} empirical measure: nonlinear functional of Tℓ Sℓ(z) := Sℓ(1(· ≤ z)) =

Sd 1(Tℓ(x) ≤ z) dx ,

where 1(· ≤ z) is the indicator function of the interval (−∞, z].

J2(1(· ≤ z)) = zφ(z), φ being the Gaussian density ⇒
As ℓ → ∞, for z = 0, we have that

dW

Sℓ(z) − µdΦ(z)
Var[Sℓ(z)]

, N(0, 1)

= O

1 √ ℓ

,

where Φ is the Gaussian cdf.

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 25 / 27

SLIDE 75

Thank you for your attention!

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 26 / 27

SLIDE 76

Berry, M. V. (1977) Regular and irregular semiclassical wavefunctions. Journal of Physics A: Mathematical and Theoretical, 10, no. 12, 2083-2091 Granville, A.; Wigman, I. (2011) The distribution of the zeros of random trigonometric polynomials. American Journal of Mathematics, 133, no. 2, 295-357. Krishnapur, M., Kurlberg, P., and Wigman, I. (2013) Nodal length fluctuations for arithmetic random waves. Ann. of Math. (2) 177, no. 2, 699–737. Marinucci, D.; Peccati, G. (2011) Random Fields on the Sphere: Representations, Limit Theorems and Cosmological Applications, London Mathematical Society Lecture Notes, Cambridge University Press Marinucci, D.; Peccati G. (2013) Mean-square continuity on homogeneous spaces, Electronic Communications in Probability, Vol.18, 37. Marinucci, D.; Rossi M. (2014) Stein-Malliavin approximations for nonlinear functionals of random eigenfunctions on Sd, Preprint, arXiv 1405.3449 Marinucci, D.; Wigman, I. (2011) On the excursion sets of spherical Gaussian eigenfunctions, Journal of Mathematical Physics, 52, 093301, arXiv 1009.4367 Marinucci, D.; Wigman, I. (2011) The defect variance of random spherical harmonics, Journal of Physics A: Mathematical and Theoretical, 44, 355206, arXiv:1103.0232 Marinucci, D.; Wigman, I. (2014) On nonlinear functionals of spherical Gaussian eigenfunctions, Communications in Mathematical Physics, in press. Nazarov, F.; Sodin, M. (2009) On the number of nodal domains of random spherical harmonics. American Journal of Mathematics 131, no. 5, 1337-1357 Nourdin, I.; Peccati, G. (2009) Stein’s method on Wiener chaos, Probability Theory and Related Fields, 145, no. 1-2, 75–118. Nourdin, I.; Peccati, G. (2010) Cumulants on the Wiener space. Journal of Functional Analysis 258, no. 11, 3775–3791. Nourdin, I.; Peccati, G.; Podoslkij, M. (2011) Quantitative Breuer-Major theorems, Stochastic Processes and their Applications, vol.121, 793-812 Nourdin, I.; Peccati, G. (2012) Normal Approximations Using Malliavin Calculus: from Stein’s Method to Universality, Cambridge University Press Peccati, G.; Taqqu, M.S. (2011) Wiener Chaos: Moments, Cumulants and Diagrams, Springer-Verlag Peccati, G.; Tudor, C.A. (2005) Gaussian limits for vector-valued multiple stochastic integrals. S´ eminaire de Probabilit´ es XXXVIII, Springer Wigman, I. (2009) On the distribution of the nodal sets of random spherical harmonics. Journal of Mathematical Physics, 50, no. 1, 013521, 44 pp. Wigman, I. (2010) Fluctuation of the nodal length of random spherical harmonics, Communications in Mathematical Physics , 298, no. 3, 787-831

M. Rossi (Rome Tor Vergata)

Stein-Malliavin approximations on Sd WIAS Berlin - October 23, 2014 27 / 27