Complex Fourth Moment Theorems Simon Campese RTG 2131 opening - PowerPoint PPT Presentation

Complex Fourth Moment Theorems Simon Campese RTG 2131 opening workshop, November 27, 2015

Introduction n n Generic Fourth Moment Theorem: n d X n X 2 X 3 X 4 [ ] [ ] [ ] → X ∼ µ − ⇐ ⇒ P ( E [ X n ] , E , E , E ) → 0 • Nualart-Peccati (2005): X n = I p ( f n ) , µ ∼ N (0 , σ 2 ) , P = X 4 n − 3 • Peccati-Tudor (2005): Extension to multivariate case for µ ∼ N d (0 , Σ) • Nualart-Ortiz-Latorre (2008): New proof using Malliavin calculus • Nourdin-Peccati (2009): Quantitative FMT via Malliavin-Stein for µ ∼ N (0 , σ 2 ) and µ ∼ Gamma ( ν ) . • Nourdin-Peccati-Revéillac (2010): Quantitative FMT via Malliavin-Stein for µ ∼ N d (0 , Σ)

Introduction • Ledoux (2012): new, pathbreaking proofs by spectral methods • Azmoodeh-C.-Poly (2014): FMT for chaotic eigenfunctions of generic Markov diffusion generators, µ ∼ N (0 , σ 2 ) , µ ∼ Gamma ( ν ) or µ ∼ Beta ( α, β ) • C.-Nourdin-Peccati-Poly (2015+): Multivariate extension for µ ∼ N d (0 , Σ) • In this talk: Extension to complex valued random variables and µ ∼ CN d (0 , Σ)

Complex normal distribution p j E j q j • Z ∼ CN d (0 , Σ) if its density f is given by 1 − z T Σ − 1 z ( ) f ( z ) = π d | det Σ | exp • E [ ZZ T ] = Σ and E [ ZZ T ] = 0 [∏ ] • Completely characterized by its moments E j Z j Z • For Z ∼ CN 1 (0 , 1) : { p ! if p = q [ Z p Z q ] = 0 if p ̸ = q .

Wirtinger calculus Wirtinger derivatives and variables when differentiating, for example: • ∂ z = 1 and ∂ z = 1 ( ) ( ) ∂ x − i ∂ y ∂ x + i ∂ y 2 2 • ∂ z and ∂ z satisfy product and chain rules • Heuristic: z and z can be treated as algebraically independent ∂ z z p z q = pz p − 1 z q ∂ z z p z q = qz p z q − 1

Stein’s method for the complex normal distribution Lemma Looks nice, but associated Stein equation can not be solved in general. Z ∼ CN 1 (0 , 1) if, and only if, [ ] E [ ∂ z f ( Z )] − E Z f ( Z ) = 0 for suitable f : C → C .

Stein’s method for the complex normal distribution Lemma has nice solution for suitable h . Z ∼ CN 1 (0 , 1) if, and only if, [ ] 2 E [ ∂ zz f ( Z )] − E Z ∂ z f ( Z ) − E [ Z ∂ z f ( Z )] = 0 for suitable f : C → C . For W ∼ CN 1 (0 , 1) , associated Stein equation 2 ∂ zz f ( z ) − z ∂ z f ( z ) − z ∂ z f ( z ) = h ( z ) − E [ h ( W )]

Abstract setting • Symmetric diffusion Markov generator L acting on L 2 ( E , F , µ ) • Discrete spectrum S = {· · · < − λ 2 < − λ 1 < − λ 0 = 0 } • Spectral theorem: ∞ L 2 ( E , F , µ ) = ⊕ ker ( L + λ k Id ) k =0 • Eigenspaces closed under conjugation as L F = L F

Carré du champ operator E d E • Carré du champ operator Γ : Γ( F , G ) = 1 ( L ( FG ) − F L G − G L F ) 2 ∫ ∫ L ( FG ) d µ = L (1) FG d µ = 0 • Integration by parts: ∫ ∫ Γ( F , G ) d µ = − F L G d µ • Diffusion property: ( ) ∑ Γ( ϕ ( F 1 , . . . , F d ) , G ) = ∂ z j ϕ ( F ) Γ( F j , G ) + ∂ z j ϕ ( F ) Γ( F j , G ) j =1

Pseudo inverse of the generator E E E E E E • L − 1 pseudo-inverse of generator (compact) • Bears its name as ∫ L L − 1 F = F − F d µ • In particular: ∫ ∫ Γ( F , − L − 1 G ) d µ = F L L − 1 G d µ ∫ ∫ ∫ FG d µ − = F d µ G d µ

Quantitative bound for the Wasserstein distance Theorem variable. Then it holds that E E Let Z ∼ CN 1 (0 , 1) and denote by F a centered smooth complex random √ ( 1 ∫ � 2 d µ � Γ( F , − L − 1 F � � d W ( F , Z ) ≤ 2 2 ) 1/2 ∫ ) 2 d µ Γ( F , − L − 1 F ) − 1 ( + .

Quantitative bound for the Wasserstein distance Theorem vector. Then it holds that op E E Let Z ∼ C N d (0 , Σ) and denote by F a centered smooth complex random √ ( 1 ∫ � 2 2 ∥ Σ − 1 ∥ op ∥ Σ ∥ 1/2 � Γ( F , − L − 1 F ) � � d W ( F , Z ) ≤ HS d µ 2 ) 1/2 ∫ � 2 � Γ( F , − L − 1 F ) − Σ � � + HS d µ , where Γ( F , − L − 1 F ) = Γ( F j , − L − 1 F k ) ( ) 1 ≤ j , k ≤ d and ∥ A ∥ HS = tr ( A A T ) .

Abstract Markov chaos are jointly chaotic, if ker and Definition ker jointly chaotic. ( ) ( ) • F ∈ ker and G ∈ ker L + λ p Id L + λ q Id p + q p + q ⊕ ⊕ ( ) ( ) FG ∈ L + λ j Id FG ∈ L + λ j Id . j =0 j =0 ( ) • F ∈ ker L + λ p Id is chaotic, if F is jointly chaotic with itself. • A vector of eigenfunctions is chaotic, if any two components are

Key lemma Lemma For chaotic eigenfunctions F,G it holds that E E ∫ ∫ � 2 d µ ≤ � Γ( F , − L − 1 G ) FG Γ( F , − L − 1 G ) d µ � � Consequence of general principle from Azmoodeh-C.-Poly (2014).

Quantitative Fourth Moment Theorem Theorem E For Z ∼ CN 1 (0 , 1) and chaotic eigenfunction F, it holds that √∫ ( 1 ) 2 | F | 4 − 2 | F | 2 + 1 d W ( F , Z ) ≤ d µ Similar bound for Z ∼ CN d (0 , Σ) and chaotic vector F involving E | F j F k | 2 d µ . ∫ ∫ E F j F k d µ and

Quantitative Fourth Moment Theorem Corollary the following two assertions are equivalent: (i) F n d (ii) For Z ∼ CN 1 (0 , 1) and normalized sequence F n of chaotic eigenfunctions, − → Z E | F n | 4 d µ → 2 ∫

Proof of moment bound E E E E By key lemma, diffusion property and integration by parts: E E Key lemma also implies that E E ∫ ∫ Γ( F , − L − 1 F ) 2 d µ ≤ FF Γ( F , − L − 1 F ) d µ = 1 (∫ ∫ ) F 2 Γ( F , − L − 1 F ) d µ Γ( F 2 F , − L − 1 F ) d µ − 2 = 1 ∫ | F | 4 d µ − 1 ∫ F 2 Γ( F , − L − 1 F ) d µ 2 2 ∫ ∫ F 2 Γ( F , − L − 1 F ) d µ. � 2 d µ ≤ � � Γ( F , − L − 1 F ) �

Proof of moment bound Therefore, E E E ∫ ( 1 � 2 + ) 2 ) � Γ( F , − L − 1 F ) Γ( F , − L − 1 F ) − 1 � � ( d µ 2 ( 1 ) ∫ � 2 + Γ( F , − L − 1 F ) 2 − 2 | F | 2 + 1 � Γ( F , − L − 1 F ) � � = d µ 2 ( 1 ) ∫ 2 | F | 4 − 2 | F | 2 + 1 ≤ d µ

Complex Peccati-Tudor Theorem F n F n d (ii) F n d (i) F n underlying generator, the following two assertions are equivalent: Theorem Proof: Adaptation of real version in C.-Nourdin-Peccati-Poly (2015+) n E Let Z ∼ CN d (0 , Σ) and ( F n ) be sequence of chaotic vectors satisfying F 2 [ ] [ ] → 0 and E → Σ . Under some technical conditions on − → Z jointly − → Z componentwise

Complex Ornstein-Uhlenbeck generator eigenfunctions of the real OU-generator. • S = − N 0 , Γ( F , G ) = ⟨ DF , DG ⟩ H • Real and imaginary parts of any eigenfunction are themselves • However, eigenspaces have much richer algebraic structure: ⊕ ker ( L + k Id ) = H p , q p , q ∈ N 0 p + q = k with H p , q = H q , p .

Complex Hermite Polynomials (Itô, 1952) First few: z z j j H p , q ( z ) = ( − 1) p + q e | z | 2 ( ∂ z ) p ( ∂ z ) q e −| z | 2 p ∧ q ( p )( q ) ∑ j ! ( − 1) j z p − j z q − j = j =0 1 | z | 2 − 1 z 2 z 2 z 3 z 2 z − 2 z z 2 z − 2 z z 3

Orthonormal basis for H p , q • Let { Z ( h ): h ∈ H } be complex isonormal Gaussian process and ( e j ) orthonormal basis of H . • Orthonormal basis of H p , q is given by   ∞   √ ∏ m p ! m q ! H m p ( j ) , m q ( j ) ( Z ( e j )): ( m p , m q ) ∈ M p × M q   j =1 • In particular: Z ( e j ) p ∈ H p , 0 • Thus, H p , 0 is sub-algebra of Dirichlet domain induced by Γ

Concluding remarks Peccati-Tudor Theorem have been proven by Chen-Liu (2014+) and Chen (2014+), respectively, by separating real and imaginary parts complexified Gamma and Beta distributions are not interesting as these are real valued) Applications: Marinucci and M. Rossi) product inequality; advances for complex unlinking conjecture (forthcoming paper with G. Poly) • For OU generator and d = 1 , a (non-quantitative) FMT and • Our method can also yield FMT for other target laws (usual • Quantitative CLT for spin random fields (joint project with D. • New proof and generalization of de Reyna’s complex Gaussian