A Gaussian/Poisson alternative on configuration spaces Giovanni - PowerPoint PPT Presentation

A Gaussian/Poisson alternative on configuration spaces Giovanni Peccati (Luxembourg University) SSP 2012/Kansas University: March 24, 2012 Giovanni Peccati Gaussian/Poisson

Introduction Partially based on joint works with R. Lachièze-Rey (Paris V). This is one of the latest installments in a recent series of papers, focussing on probabilistic approximations by means of the Malliavin calculus of variations and the Stein’s and Chen-Stein method for probabilistic approximations. Strong motivations come from stochastic geometry, in particular in connection with the notion of geometric U -statistics (Reitzner and Schulte, 2011). Giovanni Peccati Gaussian/Poisson

An example Let η be a Poisson measure on R 2 , with control equal to the Lebesgue measure. Define � � 2 √ √ − 1 n , 1 W n = n , n = 1 , 2 , ..., . 2 2 Let { r n } be a non-increasing sequence of positive numbers. For every n , we consider the disk graph G n = ( V n , E n ) , where V n = W n ∩ η, E n = { ( x , y ) : 0 < | x − y | < r n } . We are interested in the asymptotic behavior of M n = M n − E [ M n ] � M n = # { edges of G n } , � Var ( M n ) . Giovanni Peccati Gaussian/Poisson

An example One has that: LAW then � (i) If nr 2 n → ∞ , M n → N ( 0 , 1 ) ; TV (ii) If nr 2 n → c ∈ ( 0 , + ∞ ) , → Poisson ; then M n L 1 then M n , � (iii) If nr 2 n → 0, M n → 0. Remark: Same array of behaviors in any dimension, and with different graphical rules (P . - Lachièze-Rey, 2011). Giovanni Peccati Gaussian/Poisson

General aim We shall develop unified analytic techniques ( = based on differential operators) allowing to: (a) Measure the quantity d W ( F , N ) , for F a regular functional of a Poisson measure and N Gaussian; (b) Measure the quantity d TV ( F , Po ( λ )) , for F a Z + -valued functional of a Poisson measure. Giovanni Peccati Gaussian/Poisson

Related research Similar results in a Gaussian framework : several papers since 2005, e.g. by I. Nourdin, D. Nualart, G.P ., ... (forthcoming book, 2012). Applications: Breuer-Major Theorems, harmonic analysis, density estimates, concentration inequalities, needlet estimation, ... Stochastic geometry : Reitzner and Schulte (2011), Schulte and Thaele (2011, 2012), Decreusefond et al. (2011). A U -statistic is an object of the type: � F = ϕ ( x 1 , ..., x d ) . { x 1 ,..., x d }∈ η d � = Very few (extremely technical) results about CLTs for non-Poissonized U -statistics: de Jong (1990), Jammalamadaka and Janson (1986), Battahkhraya and Ghosh (1992). Giovanni Peccati Gaussian/Poisson

Digression: a connection with universality Let { ξ i : i ≥ 1 } be i.i.d. N ( 0 , 1 ) and fix d ≥ 2. In Nourdin, P ., Reinert (2010), it is proved that if � a ( n ) LAW i 1 ,..., i d ξ i 1 · · · ξ i d → N ( 0 , 1 ) , then � a ( n ) i 1 ,..., i d Y i 1 · · · Y i d verifies the same CLT for every centered i.i.d. sequence { Y i : i ≥ 1 } with unit variance. This result is very much in line with recent examples of “universality of discrete Gaussian families”: see e.g. recent proofs of the circular law , or generalized Lindberg principles by Rotar’ (1979), Mossel, O’Donnel and Oleszkiewicz (2008) or Chatterjee (2007). Giovanni Peccati Gaussian/Poisson

Digression: a connection with universality Let G be a Gaussian measure with Lebesgue control. Our results show that there exist multilinear functionals Φ n s.t. Φ n ( G ) verifies a CLT, but Φ n ( � η ) is asymptotically Poisson. This shows that a naive version of the universality phenomenon for random measures fails. Giovanni Peccati Gaussian/Poisson

Poisson measures ( Z , Z ) is a Polish space. Given a σ -finite non atomic measure µ , we denote by η a Poisson measure with control µ , and its compensated counterpart is � η = η ( · ) − µ ( · ) . Recall: for every A , B such that A ∩ B = ∅ and µ ( A ) , µ ( B ) < ∞ , η ( A ) and η ( B ) are two independent Poisson r.v.’s of parameters µ ( A ) , µ ( B ) . Giovanni Peccati Gaussian/Poisson

Integral and chaos For every symmetric square-integrable function f in q variables, we define the multiple Wiener-Itô integral � � I q ( f ) = · · · f ( x 1 , ..., x q ) 1 { no diagonals } ˆ η ( dx 1 ) · · · ˆ η ( dx q ) . Z Z Recall that every F ∈ L 2 ( σ ( η )) can be written as: F = E ( F ) + � ∞ q = 1 I q ( f q ) . Giovanni Peccati Gaussian/Poisson

A tiny bit of Malliavin calculus The derivative operator is: D z F = � q qI q − 1 ( f q ( z , · )) . Nualart and Vives (1990): D z F ( η ) = F ( η + δ z ) − F ( η ) (add-one cost). The O-U generator : LF = − � q ≥ 1 qI q ( f q ) . Pseudo-inverse of the O-U generator : L − 1 F = − � q ≥ 1 q − 1 I q ( f q ) . Integration by parts : for every X derivable and F centered, E [ XF ] = E [ � DX , − DL − 1 F � µ ] . Giovanni Peccati Gaussian/Poisson

Another look at U -statistics U -statistics are typically “smooth functionals”. For instance, using η = ˆ η + µ : � F = ϕ ( x 1 , x 2 ) { x 1 , x 2 }∈ η d � = � � = ϕ ( x , y ) 1 { x � = y } η ( dx ) η ( dy ) = E [ F ] + 2 I 1 ( f ) + I 2 ( ϕ ) , � ϕ ( x , y ) µ ( dy ) . where f ( x ) = Giovanni Peccati Gaussian/Poisson

Stein’s Lemma and equations (Stein’s Lemma) A random variable Z has the N ( 0 , 1 ) distribution if and only if E [ Zf ( Z ) − f ′ ( Z )] = 0 for every smooth function f . (Stein’s equations) Let Z ∼ N ( 0 , 1 ) . For every h ∈ Lip ( 1 ) , the equation f ′ ( x ) − xf ( x ) = h ( x ) − E [ h ( Z )] admits a solution f h such that � f ′ h � ∞ ≤ 1 and � f ′′ h � ∞ ≤ 2. Giovanni Peccati Gaussian/Poisson

Applying integration by parts Now consider F centered and differentiable: for every f smooth E [ Ff ( F )] = E [ � Df ( F ) , − DL − 1 F � µ ] = E [ f ′ ( F ) � DF , − DL − 1 F � µ ]+ R f , � where | R f | ≤ 1 2 � f ′′ � ∞ E Z ( D z F ) 2 | D z L − 1 F | µ ( dz ) . Then, for Y ∼ N ( 0 , 1 ) , � � � E [ f ′ ( F )] − E [ Ff ( F )] � d W ( F , Y ) ≤ sup | f ′ |≤ 1 , | f ′′ |≤ 2 � � � � � E [ f ′ ( F )( 1 − � DF , − DL − 1 F � µ ] + R f ≤ sup � | f ′ |≤ 1 , | f ′′ |≤ 2 � ≤ E | 1 − � DF , − DL − 1 F � µ | + E ( D z F ) 2 | D z L − 1 F | µ ( dz ) . Z Giovanni Peccati Gaussian/Poisson

First bound Theorem 1 (P ., Solé Taqqu and Utzet, 2010) Let Z ∼ N ( 0 , 1 ) . For every differentiable centered F ∈ L 2 ( σ ( η )) E | 1 − � DF , − DL − 1 F � µ | d W ( F , Z ) ≤ � ( D z F ) 2 | D z L − 1 F | µ ( dz ) . + E Z If F = I q ( f ) , � � � � 1 − 1 � � + 1 q µ ( | DF | 2 ) q E µ ( | DF | 3 ) . � � d W ( F , Z ) ≤ E Giovanni Peccati Gaussian/Poisson

Chen-Stein’s Lemma and equations (Chen-Stein Lemma) A random variable Y in Z + has the Po ( λ ) distribution if and only if E [ Yg ( Y ) − λ g ( Y + 1 )] = 0 for every bounded function g . (Chen-Stein equations) Let Y ∼ Po ( λ ) . For every h = 1 A , the equation λ g ( k + 1 ) − kg ( k ) = h ( k ) − E [ h ( Y )] , k = 0 , 1 , 2 , ..., admits a unique bounded solution g A such that g A ( 0 ) = 0, � ∆ g A � ∞ ≤ 1 − e − λ and � ∆ 2 g A � ∞ ≤ 2 λ − 1 � ∆ g A � ∞ . λ Giovanni Peccati Gaussian/Poisson

Applying integration by parts Recall that, for g : Z + → R bounded, | g ( a ) − g ( k ) − ∆ g ( k )( a − k ) | ≤ 1 2 � ∆ 2 g � ∞ | ( a − k )( a − k − 1 ) | . Now consider F with values in Z + , differentiable and such that E [ F ] = λ : for every g bounded E [ Fg ( F ) − λ g ( F + 1 )] = E [( F − λ ) g ( F ) − λ ∆ g ( F )] = E [ � Dg ( F ) , − DL − 1 F � µ − λ ∆ g ( F )] = E [∆ g ( F )( � DF , − DL − 1 F � µ − λ )] + R g , � where | R g | ≤ 1 2 � ∆ 2 g � ∞ E Z | D z F ( D z F − 1 ) D z L − 1 F | µ ( dz ) . Giovanni Peccati Gaussian/Poisson

Applying integration by parts Then, for Y ∼ Po ( λ ) , and writing B λ = 1 − e − λ , C λ = 1 − e − λ . λ λ 2 | E [ Fg A ( F )] − λ E [ g A ( F + 1 )] | d TV ( F , Y ) = sup A � � � � � E [∆ g A ( F )( � DF , − DL − 1 F � µ − λ )] + R g A ≤ sup � A ≤ B λ E | λ − � DF , − DL − 1 F � µ | � | ( D z F )( D z F − 1 ) D z L − 1 F | µ ( dz ) . + C λ E Z Giovanni Peccati Gaussian/Poisson

Second bound Theorem 2 (P ., 2011) Let Y ∼ Po ( λ ) . For every differentiable F ∈ L 2 ( σ ( η )) with values in Z + and such that E [ F ] = λ , B λ E | λ − � DF , − DL − 1 F � µ | ≤ d TV ( F , Y ) � | ( D z F )( D z F − 1 ) D z L − 1 F | µ ( dz ) . + C λ E Z If F = λ + I q ( f ) , � � � � � λ − 1 � + C λ � q µ ( | DF | 2 ) � q E µ ( | ( DF ) 2 ( DF − 1 ) | ) . d TV ( F , Y ) ≤ B λ E Remark: E [ � DF , − DL − 1 F � µ ] = Var ( F ) . Giovanni Peccati Gaussian/Poisson

To resume For a smooth centered random variable F , E | 1 − � DF , − DL − 1 F � µ | d W ( F , N ) ≤ � ( D z F ) 2 | D z L − 1 F | µ ( dz ) . + E Z For a Z + -valued random variable F of mean λ , B λ E | λ − � DF , − DL − 1 F � µ | d TV ( F , Po ( λ )) ≤ � | ( D z F )( D z F − 1 ) D z L − 1 F | µ ( dz ) . + C λ E Z Remark . On a Gaussian space (Nourdin, P . 2009): d TV ( F , N ) ≤ 2 E | 1 − � DF , − DL − 1 F � µ | . Giovanni Peccati Gaussian/Poisson