Stein’s Method and Stochastic Geometry Giovanni Peccati (Luxembourg University) Firenze — 16 marzo 2018 1 / 39
I NTRODUCTION “Stein’s method” , as devised by Charles Stein at the end of the 60s, is a collection of probabilistic techniques, for measuring the distance be- tween probability distribu- tions, by means of character- ising differential operators . Stein’s motivation was to de- velop an effective alternative to Fourier methods , for deal- ing with functionals of de- pendent random variables. 2 / 39
I NTRODUCTION ⋆ Applications of Stein’s method now span an enormous amount of domains, e.g.: random matrices, statistics, biology, alge- bra, mathematical physics, finance, geometry, ... ⋆ Main features: quantitative , and “local to global” . ⋆ In these lectures: my view of Stein’s method, with focus on Gaussian random fields and random geometric graphs . 3 / 39
S OME N AMES L.H.Y. Chen A.D. Barbour E. Bolthausen F. Götze S. Chatterjee I. Nourdin L. Goldstein G. Reinert 4 / 39
C ONVENTIONS ⋆ From now on: everything is defined on a suitable triple ( Ω , F , P ) ⋆ We write N ∼ N ( 0, 1 ) for a standard Gaussian random variable: � dy B e − y 2 /2 . √ P ( N ∈ B ) = 2 π ⋆ Often: given a random element Y , we write Y 1 , Y 2 , ... to indi- cate a sequence of independent copies of Y . 5 / 39
T HE (Q UANTITATIVE ) C ENTRAL L IMIT T HEOREM Theorem (CLT & Berry-Esseen bound) Let X 1 , X 2 , ... be a sequence of independent and identically distributed r.v.’s, such that E [ X 1 ] = 0 , and Var ( X 1 ) = 1 . Write S n : = X 1 + · · · + X n . Then, as n → ∞ , � 1 � � z e − y 2 /2 ∆ n ( z ) : = P → 0, z ∈ R . √ nS n ≤ z − √ dy − 2 π − ∞ Moreover, � � | ∆ n ( z ) | ≤ C E | X 1 | 3 sup 0.4 < C optimal < 0.48 . √ n z 6 / 39
F IRST P ROOF : F OURIER (L YAPOUNOV , L ÉVY ) ⋆ Write the characteristic function f n ( z ) of n − 1/2 S n as a n - product. ⋆ Prove that → exp {− z 2 /2 } , as n → ∞ , f n ( z ) − by a direct analytical argument. 7 / 39
S ECOND P ROOF : S WAPPING (L INDEBERG , T ROTTER ) ⋆ For a smooth ϕ , with red and blue independent, write � � � � � E [ ϕ ( N )] − E [ ϕ ( n − 1/2 S n )] � � � � E [ ϕ ( n − 1/2 ( N 1 + · · · + N n ))] = � � − E [ ϕ ( n − 1/2 ( X 1 + · · · + X n )] � . ⋆ Deduce that: � � � � � E [ ϕ ( N )] − E [ ϕ ( n − 1/2 S n )] � � n � � E ϕ ( n − 1/2 ( N 1 + · · · + N i + X i + 1 + · · · + X n )) ∑ ≤ i = 1 � � − E ϕ ( n − 1/2 ( N 1 + · · · + N i − 1 + X i + · · · + X n )) � , and control each summand by a Taylor expansion. 8 / 39
Q UESTION ⋆ What happens if the summands X 1 , X 2 , ... display some form of dependence and/or the considered random element is not a linear mapping ? ⋆ Typical example: length, number of edges / triangles / con- nected components / ... in a random geometric graph : ⋆ Even more extreme: random graphs arising in combinato- rial optimisation (MST, TSP, MM, ...) 9 / 39
S ETTING ⋆ In what follows, I will mainly focus on one-dimensional normal approximations in the 1- Wasserstein distance W 1 ( • , • ) . ⋆ Recall that W 1 ( X , Y ) : = inf A ∼ X ; B ∼ Y E | A − B | sup | E [ h ( X )] − E [ h ( Y )] | , = h ∈ Lip ( 1 ) whenever E | X | , E | Y | < ∞ . 10 / 39
I NGREDIENTS In order to implement Stein’s method, one typically needs: 1. A Lemma 2. A heuristic 3. An equation 4. Uniform bounds 11 / 39
T HE L EMMA Stein’s Lemma Let Z be a real-valued random variable. Then, Z ∼ N ( 0, 1 ) if and only if E [ f ′ ( Z )] = E [ Z f ( Z )] , for every smooth f. [ Proof : ( = ⇒ ) integration by parts. ( ⇐ = ) method of moments (or unicity of Fourier transform) ] 12 / 39
T HE H EURISTIC Stein’s Heuristic Assume Z is a real random variable such that E [ f ′ ( Z )] ≈ E [ Z f ( Z )] for a large class of smooth mappings f. Then, the distribution of Z has to be close to Gaussian . 13 / 39
T HE E QUATION ⋆ For h ∈ Lip ( K ) fixed and N ∼ N ( 0, 1 ) , define the Stein’s equation f ′ ( x ) − x f ( x ) = h ( x ) − E [ h ( N )] , x ∈ R ; equivalent to d dxe − x 2 /2 f ( x ) = e − x 2 /2 ( h ( x ) − E [ h ( N )]) . ⋆ Every solution has the form � x f ( x ) = ce x 2 /2 + e x 2 /2 − ∞ ( h ( y ) − E [ h ( N )]) e − y 2 /2 dy , x ∈ R . ⋆ Set � x − ∞ ( h ( y ) − E [ h ( N )]) e − y 2 /2 dy , f h ( x ) : = x ∈ R . 14 / 39
T HE B OUNDS By direct inspection, one proves Stein’s “Magic Factors” and Bounds For every h ∈ Lip ( K ) , f h ∈ C 1 , and � 2 � f ′ π K . h � ∞ ≤ As a consequence, for X integrable, � � � � W 1 ( X , N ) sup = � E [ h ( X )] − E [ h ( N )] � h ∈ Lip ( 1 ) � � � � � E [ f ′ sup = h ( X ) − X f h ( X )] � h ∈ Lip ( 1 ) � � � � � E [ f ′ ( X ) − X f ( X )] sup � . ≤ f : | f ′ |≤ 1 15 / 39
A ND N OW ? ⋆ The name of the game is now to compare as sharply as possible E [ f ′ ( X )] and E [ X f ( X )] , for every smooth mapping f . ⋆ Several techniques: exchangeable pairs , dependency graphs , zero-bias transforms , size-bias transforms , ... 16 / 39
A S IMPLE E XAMPLE : B ACK TO THE CLT ⋆ For a fixed n , write Z : = n − 1/2 ( X 1 + · · · + X n ) , and Z i = Z − n − 1/2 X i . ⋆ One has, by Taylor and independence, E [ X i ( f ( Z ) − f ( Z i ))] ≈ E [ X i ( Z − Z i ) f ′ ( Z )] E [ X i f ( Z )] = n − 1/2 E [ X 2 i f ′ ( Z )] . = ⋆ It follows that E [ Z f ( Z )] = n − 1/2 ∑ E [ X i f ( Z )] ≈ E [ n − 1 ∑ X 2 i × f ′ ( Z )] . i i ⋆ By the law of large numbers, E [ Z f ( Z )] ≈ E [ f ′ ( Z )] for n large, and using Stein’s bounds one deduces the CLT. 17 / 39
S ECOND O RDER P OINCARÉ E STIMATES ⋆ Assume now g = ( g 1 , ..., g d ) ∼ N d ( 0, Id. ) , and define F = ψ ( g 1 , ..., g d ) , for some smooth ψ : R d → R s.t. E [ F ] = 0 and Var ( F ) = 1. ⋆ Remember the Poincaré inequality : Var ( F ) ≤ E [ �∇ ψ ( g ) � 2 ] . 18 / 39
S ECOND O RDER P OINCARÉ E STIMATES ⋆ It turns out that F verifies an exact integration by parts for- mula: � � f ′ ( F ) �∇ ψ ( g ) , −∇ L − 1 ψ ( g ) � , E [ F f ( F )] = E where L − 1 is the pseudo-inverse of the Ornstein-Uhlenbeck generator L = −� x , ∇� + ∆ . ⋆ Plugging this into Stein’s bound and applying once more Poincaré yields that, for N ∼ N ( 0, 1 ) , � W 1 ( F , N ) Var ( �∇ ψ ( g ) , −∇ L − 1 ψ ( g ) � ) ≤ op ] 1/4 × E [ �∇ ψ ( g ) � 4 ] 1/4 . 2 E [ � Hess ψ ( g ) � 4 ≤ ⋆ This is a second order Poincaré inequality , — see Chatter- jee (2007), Nourdin, Peccati and Reinert (2010), and Vidotto (2017). Applications in random matrix theory & analysis of fractional fields. 19 / 39
B EYOND G AUSSIAN Stein’s approach extends to much more gen- eral densities — for instance to elements of the Pearson family . See Stein’s 1986 mono- graph. In the discrete setting, the equivalent of Stein’s method is the Chen-Stein method . See the monograph by Barbour, Holst and Janson (1990). 20 / 39
T WO E XAMPLES In what follows, I will illustrate two striking applications of Stein’s method, that are relevant in a geometric setting : (1) capturing the fluctuations of chaotic random variables , and (2) quantifying second order interactions . Both are connected to (generalized) integration by parts formulae . 21 / 39
B ERRY ’ S R ANDOM W AVES (1977) ⋆ Let E > 0. The Berry’s random wave model on R 2 , with parameter E , written B E = { B E ( x ) : x ∈ R 2 } , is defined as the unique (in law) centred, isotropic Gaussian field on R 2 such that ∆ B + E · B = 0, where ∆ = ∂ 2 + ∂ 2 . ∂ x 2 ∂ x 2 1 2 √ ⋆ Equivalently, E [ B E ( x ) B E ( y )] = J 0 ( E � x − y � ) ( J 0 = Bessel function of the 1st kind). ⋆ Its high-energy local behaviour is conjectured to be a “uni- versal model” for Laplace eigenfunctions on arbitrary man- ifolds (Berry, 1977). ⋆ It is the local scaling limit of monochromatic random waves on arbitrary manifolds (Canzani & Hanin, 2016). 22 / 39
N ODAL S ETS One is interested in the length L E of the nodal set (components are the nodal lines ): B − 1 E ( { 0 } ) ∩ Q : = { x ∈ Q : B E ( x ) = 0 } , where Q is e.g. a square of size 1, as E → ∞ . Image: D. Belyaev (2016) 23 / 39
C HLADNI P LATES (1787) 24 / 39
M EAN AND V ARIANCE (B ERRY , 2002) ⋆ Berry (J. Phys. A, 2002) : semi-rigorous computations give √ E , Var ( L E ) ∼ log E , E [ L E ] ∼ although the natural guess for the order of the variance is √ E . See Wigman (2010) for the spherical case. ∼ ⋆ Such a variance reduction “... results from a cancellation whose meaning is still obscure... ” (Berry (2002), p. 3032). ⋆ Question: can one explain such a ‘cancellation phenomenon’, and characterise second-order fluctuations, involving the normalised length L E : = L E − E [ L E ] � ? � Var ( L E ) 25 / 39
Recommend
More recommend
