Contraction Principles some applications

Transcription

1 Contraction Principles some applications Fabrice Gamboa (Institut de Mathématiques de Toulouse) 7th of June 2017 Christian and Patrick 59th Birthday

2 Overview Appetizer : Christian and Patrick secret lives 1 Appetizer : Christian and Patrick secret lives 2 Starter : Around large deviations 3 Main course : Sanov and Markov 4 Trou normand : Marginal problem and LDP 5 Cheese : Markov generalized moment problem and LDP 6 Dessert : Sum rules F. Gamboa (IMT Toulouse) 59th Birthday 7th of June / 35

3 Starter : Around large deviations Overview 1 Appetizer : Christian and Patrick secret lives 2 Starter : Around large deviations 3 Main course : Sanov and Markov 4 Trou normand : Marginal problem and LDP 5 Cheese : Markov generalized moment problem and LDP 6 Dessert : Sum rules F. Gamboa (IMT Toulouse) 59th Birthday 7th of June / 35

4 Starter : Around large deviations Starter I : Large deviations (P n ) on E satisfies a LDP with good rate I means that at log scale with 8A measurable set, P n (A) exp(-n inf x2a I(A)). I I > 0 is lower semi-continuous, I 8L >0, {x 2 E : I(x) 6 L} is compact. I I is unique F. Gamboa (IMT Toulouse) 59th Birthday 7th of June / 35

5 Starter : Around large deviations Starter II : Contraction principles Contraction principle f : E! F a continuous map (P n ) on E LDP with good rate I ) (P n f -1 ) LDP on F with good rate J( ) := inf I(x). {x:f(x)= } Inverse contraction principle g : F! E a continuous bijective map + exponential tightness of (P n ) (P n g -1 ) on E LDP with rate I ) (P n ) LDP on F with good rate J( ) :=I(g( )). F. Gamboa (IMT Toulouse) 59th Birthday 7th of June / 35

6 Starter : Around large deviations Starter III : Kullback Leibler Divergence I R. Leibler ( ) S. Kullback ( ) P, Q probability measures on some space E K(P, Q) = R dp dp E log dqdp if P Q and log dq 2 L1 (P) +1 otherwise F. Gamboa (IMT Toulouse) 59th Birthday 7th of June / 35

7 Starter : Around large deviations Starter III : large deviations two examples I Sanov Theorem : (X n ) i.i.d. Q on E ) F n (dx) = 1 n P n i=1 X i (dx) LDP on P(E) rate K(, Q) I Weighted measures : (Z n ) i.i.d. E(1), (x n ) 2 supp P E, 1 n P n i=1 x i (dx)! P ) n (dx) = 1 n P n i=1 Z i x i (dx) LDP on M + (E) rate K(P, ) F. Gamboa (IMT Toulouse) 59th Birthday 7th of June / 35

8 Main course : Sanov and Markov Overview 1 Appetizer : Christian and Patrick secret lives 2 Starter : Around large deviations 3 Main course : Sanov and Markov 4 Trou normand : Marginal problem and LDP 5 Cheese : Markov generalized moment problem and LDP 6 Dessert : Sum rules F. Gamboa (IMT Toulouse) 59th Birthday 7th of June / 35

9 Main course : Sanov and Markov Main course : Sanov and Markov C. Léonard P. Cattiaux I Cattiaux, P., & Léonard C. (1994). Minimization of the Kullback information of diffusion processes. Annales de l IHP probabilités et statistique, 30, I Cattiaux, P., & Léonard C. (1995). Large deviations and Nelson processes. Forum Mathematicum, 7, I Cattiaux, P., & Léonard C. Minimization of the Kullback information for some Markov processes. (1996). Séminaire de Probabilités XXX, Springer, Sanov Theorem : Q law of a Markov process on [0, T], (X n ) i.i.d. Q F n (dx) = 1 n nx X i (dx) i=1 Random law of a random process-satisfies LDP rate K(, Q) F. Gamboa (IMT Toulouse) 59th Birthday 7th of June / 35

10 Main course : Sanov and Markov Main course (bis) : Sanov and Markov C. Léonard P. Cattiaux Various computation of the contracted functional J( ) := f(p) is the marginals flow of P inf K(P, Q). {x:f(p)= } Feasibility conditions for the optimization problem (existence of solution having finite entropy) F. Gamboa (IMT Toulouse) 59th Birthday 7th of June / 35

11 Trou normand : Marginal problem and LDP Overview 1 Appetizer : Christian and Patrick secret lives 2 Starter : Around large deviations 3 Main course : Sanov and Markov 4 Trou normand : Marginal problem and LDP 5 Cheese : Markov generalized moment problem and LDP 6 Dessert : Sum rules F. Gamboa (IMT Toulouse) 59th Birthday 7th of June / 35

12 Trou normand : Marginal problem and LDP Trou normand : Marginal problem and LDP P. Cattiaux F.G I Cattiaux, P., & G F. (1999). Large deviations and variational theorems for marginal problems. Bernoulli, 5, P on E F, on E, µ on F données 9 Q P on E F marginals and µ+conditions? Tool : LDP weighted measures ( n ) +contraction I (Z n ) i.i.d., (x n, y n ) 2 E F, I 1 n P n i=1 (x i,y i )(dx, dy)! P n (dx, dy) = 1 n nx Z i (xi,y i )(dx, dy). i=1 F. Gamboa (IMT Toulouse) 59th Birthday 7th of June / 35

13 Trou normand : Marginal problem and LDP Trou normand (bis) : Marginal problem and LDP P. Cattiaux F.G n (dx, dy) = 1 nx Z i n (xi,y i )(dx, dy). i=1 ( n ) satisfies a LDP with good rate I Z I(Q) := E F is the Cramér transform of Z dq dp dp, Q P. F. Gamboa (IMT Toulouse) 59th Birthday 7th of June / 35

14 Trou normand : Marginal problem and LDP Trou normand (ter) : Marginal problem and LDP P. Cattiaux F.G Example : Z LB(1/2) (L>1) Contraction on the two marginals 9 Q P on E F having marginals and µ with dq dp 6 L Z E iff 8h 1 2 C(E), h 2 2 C(F), Z Z h 1 d + h 2 dµ 6 log (1 + exp (L(h 1 (x)+h 2 (y)))) P(dx, dy) F E F F. Gamboa (IMT Toulouse) 59th Birthday 7th of June / 35

15 Cheese : Markov generalized moment problem and LDP Overview 1 Appetizer : Christian and Patrick secret lives 2 Starter : Around large deviations 3 Main course : Sanov and Markov 4 Trou normand : Marginal problem and LDP 5 Cheese : Markov generalized moment problem and LDP 6 Dessert : Sum rules F. Gamboa (IMT Toulouse) 59th Birthday 7th of June / 35

16 Cheese : Markov generalized moment problem and LDP Cheese : Markov generalized moment problem and LDP D. Dacunha-Castelle F.G I Dacunha-Castelle, D., & G F. (1990). Maximum d entropie et problème des moments. Annales de l IHP probabilités et statistique, 26, P = dx on [0,1] and contraction on the -moments ( 2 C k ([0, 1]) given) The feasibility of the Markov moment problem for c 2 R k 9G 2 M + ([0, 1]) G dx on [0, 1] with Z 1 0 (x)g(dx) =c and dg dx 6 1 iff 8v 2 R k, hv, ci 6 Z 1 0 log (1 + exphv, (x)i) dx F. Gamboa (IMT Toulouse) 59th Birthday 7th of June / 35

17 Overview 1 Appetizer : Christian and Patrick secret lives 2 Starter : Around large deviations 3 Main course : Sanov and Markov 4 Trou normand : Marginal problem and LDP 5 Cheese : Markov generalized moment problem and LDP 6 Dessert : Sum rules F. Gamboa (IMT Toulouse) 59th Birthday 7th of June / 35

18 Sum rules and LDP coworks with J. Nagel (Munich and EURANDOM) A. Rouault (Versailles) F. Gamboa (IMT Toulouse) 59th Birthday 7th of June / 35

19 Orthogonal polynomial recursion on T I µ probability measure on T 1) (p n ) sequence of orthogonal polynomials associated to µ 2) p n is monic and has degree n 3) k 6= n, R T p n(z)p k (z)µ(dz) =0 I Satisfies the recursion! p n+1 (z) =zp n (z) - n p n(z) where p n(z) :=z n p n (1/ z).! n = -p n+1 (0) is the Verblunsky coefficient F. Gamboa (IMT Toulouse) 59th Birthday 7th of June / 35

20 Orthogonal polynomial recursion on T I µ probability measure on T 1) (p n ) sequence of orthogonal polynomials associated to µ 2) p n is monic and has degree n 3) k 6= n, R T p n(z)p k (z)µ(dz) =0 I Satisfies the recursion! p n+1 (z) =zp n (z) - n p n(z) where p n(z) :=z n p n (1/ z).! n = -p n+1 (0) is the Verblunsky coefficient F. Gamboa (IMT Toulouse) 59th Birthday 7th of June / 35

21 Orthogonal polynomial recursion on T I µ probability measure on T 1) (p n ) sequence of orthogonal polynomials associated to µ 2) p n is monic and has degree n 3) k 6= n, R T p n(z)p k (z)µ(dz) =0 I Satisfies the recursion! p n+1 (z) =zp n (z) - n p n(z) where p n(z) :=z n p n (1/ z).! n = -p n+1 (0) is the Verblunsky coefficient F. Gamboa (IMT Toulouse) 59th Birthday 7th of June / 35

22 Verblunsky version of Szegö Theorem G. Szegö ( ) Verblunsky-Szegö Theorem Lebesgue measure on T. 1X K(, µ) =- log(1 - n 2 ) n=0 S. Verblunsky ( ) F. Gamboa (IMT Toulouse) 59th Birthday 7th of June / 35

23 Second ingredient : the three parametrisations I µ probability measure on R! Assume that µ has all its moments finite! (p n ) normalized othogonal polynomials in L 2 (µ)! Three terms recursion xp n (x) =a n p n-1 (x)+b n+1 p n (x)+a n+1 p n+1 (x), a n > 0, b n+1 2 R I Assume moreover that µ is supported on [0, +1[ b n = z 2n-2 + z 2n-1, and a 2 n = z 2n-1 z 2n. I If µ is supported on [0, 1], µ may be seen as the pushforward of a measure on T invariant by 2 - by sin 2 ( /2). b k+1 = 1/4[2 q - (1-2k-1 ) 2k - (1 + 2k-1 ) 2k-2 ] and a k+1 = 1/4 (1-2k-1 )(1-2 2k )(1 + 2k+1) F. Gamboa (IMT Toulouse) 59th Birthday 7th of June / 35

24 Second ingredient : the three parametrisations I µ probability measure on R! Assume that µ has all its moments finite! (p n ) normalized othogonal polynomials in L 2 (µ)! Three terms recursion xp n (x) =a n p n-1 (x)+b n+1 p n (x)+a n+1 p n+1 (x), a n > 0, b n+1 2 R I Assume moreover that µ is supported on [0, +1[ b n = z 2n-2 + z 2n-1, and a 2 n = z 2n-1 z 2n. I If µ is supported on [0, 1], µ may be seen as the pushforward of a measure on T invariant by 2 - by sin 2 ( /2). b k+1 = 1/4[2 q - (1-2k-1 ) 2k - (1 + 2k-1 ) 2k-2 ] and a k+1 = 1/4 (1-2k-1 )(1-2 2k )(1 + 2k+1) F. Gamboa (IMT Toulouse) 59th Birthday 7th of June / 35

25 Semicircular distribution R = 2 SC(dx) = 1 p 4 - x 2 2 Limit of the eigenvalues distribution of the GUE [-2,2] (x)dx a k = 1, b k = 0 for all k > 1. F. Gamboa (IMT Toulouse) 59th Birthday 7th of June / 35

26 Pastur-Marchenko Distribution L. Pastur V. Marchenko p ( MP (dx) = - x)(x - - ) 2 x ( -, + )(x)dx, ± =(1 ± p ) 2 Limit of the squared singular values distribution of rectangular Gaussian matrices. 2]0, 1] the asymptotic ratio nb col/ nb line a k = p (k > 1), b 1 = 1, b k = 1 + (k > 2) and correspond to z 2n-1 = 1 and z 2n = for all n > 1. F. Gamboa (IMT Toulouse) 59th Birthday 7th of June / 35

27 Kesten Mac Kay Distribution H. Kesten B. Mc Kay KMK apple1,apple 2 (dx) = (2 + apple p 1 + apple 2 ) (u + - x)(x - u - ) 2 x(1 - x) (u -,u + )(x)dx u ± := apple2 1 - apple2 2 ± 4p (1 + apple 1 )(1 + apple 2 )(1 + apple 1 + apple 2 ) 2(2 + apple 1 + apple 2 ) 2, apple 1, apple 2 > 0. Asymptotic distribution of the eigenvalues in the Jacobi-ensemble F. Gamboa (IMT Toulouse) 59th Birthday 7th of June / 35

28 The associated Verblunsky coefficients for k > 0, Then 2k = apple 1 - apple apple 1 + apple 2, 2k+1 = - apple 1 + apple apple 1 + apple 2. a 1 = p (1 + apple1 )(1 + apple 2 ) (2 + apple 1 + apple 2 ) 3/2, b 1 = 1 + apple apple 1 + apple 2, and for k > 2 p (1 + apple1 + apple 2 )(1 + apple 1 )(1 + apple 2 ) a k = (2 + apple 1 + apple 2 ) 2, b k = 1 apple apple apple2 2 2 (2 + apple 1 + apple 2 ) 2. F. Gamboa (IMT Toulouse) 59th Birthday 7th of June / 35

29 Killip Simon Theorem Dessert : Sum rules R. Killip B. Simon K(SC µ)+ N + X n=1 F + H ( + n)+ N - X n=1 F - H ( - n)=x k>1 1 2 b2 k + a2 k log(a2 k ) 8 < F H + (x) := : Z x 2 p t 2-4dt = x 2 1 otherwise, p x log p x+ x if x > 2 F - H (x) :=F+ H (-x) F. Gamboa (IMT Toulouse) 59th Birthday 7th of June / 35

30 Our sum rules I K(MP µ)+ N + X n=1 F + L ( + n)+ N - X n=1 F - L ( - n)= 1X -1 G(z 2k-1 )+G( -1 z 2k ) k=1 8 >< F L + (x) = >: 8 >< FL - (x) = >: Z x p (t - - )(t - + ) dt if x > +, + t 1 otherwise, Z - p ( - - t)( + - t) dt if x 6 -, x t 1 otherwise. G(x) =x log x, (x >0). F. Gamboa (IMT Toulouse) 59th Birthday 7th of June / 35

31 Our sum rules II K(KMK apple1,apple 2 µ)+ N + X n=1 F + J ( + n)+ N - X n=1 F - J ( - n)= 1X H 1 ( 2k+1 )+H 2 ( 2k ) k=0 8 Z x p >< (t - u + )(t - u - ) F J + (x) = dt if u + 6 x 6 1 u >: + t(1 - t) 1 otherwise. 8 Z u - p >< (u - - t)(u + - t) FJ - (x) = dt if 0 6 x 6 u - x t(1 - t) >: 1 otherwise. F. Gamboa (IMT Toulouse) 59th Birthday 7th of June / 35

32 For -1 6 x 6 1 apple 2 + apple1 + apple 2 H 1 (x) =-(1 + apple 1 + apple 2 ) log (1 - x) 2(1 + apple 1 + apple 2 ) apple 2 + apple1 + apple 2 - log (1 + x) 2 apple (2 + apple1 + apple 2 ) H 2 (x) =-(1 + apple 1 ) log (1 + x) 2(1 + apple 1 ) apple (2 + apple1 + apple 2 ) -(1 + apple 2 ) log (1 - x). 2(1 + apple 1 ) Particular case apple 1 = apple 2 = 0 ) u - = 0, u + = 1, KMK apple1,apple 2 arsine law. We recover the Szegö Theorem pushed on [0, 1] by sin 2 ( /2). F. Gamboa (IMT Toulouse) 59th Birthday 7th of June / 35

33 How it works? Main idea Large deviations +inverse contraction principle Random measures Spectral random measures associated to classical ensembles of random matrices (A n ) ( n ) defined by its moments Z x k n (dx) :=he, A k nei, (k 2 N), kek = 1, e cyclic F. Gamboa (IMT Toulouse) 59th Birthday 7th of June / 35

34 Main tool I Theorem I n is supported by the eigenvalues of A n. I For the classical ensembles the weights and the supporting points are independents I Furthermore, in this last case, the vector of weights is Dirichlet distributed. F. Gamboa (IMT Toulouse) 59th Birthday 7th of June / 35

35 Main tool II Theorem 1) In the GUE, (a n k ) and (bn k ) are independent sequences of independent variables (normal and 2 ) 2) In the Wishart model, (z n k ) is a sequence of independent variables ( 2 ). 3) In the Jacobi model, ( n k ) is a sequence of independent variables (Beta) F. Gamboa (IMT Toulouse) 59th Birthday 7th of June / 35

36 Some references around the dessert I Verblunsky, S. (1936). On positive harmonic functions. Proceedings of the London Mathematical Society, 2(1), I Killip, R., & Simon, B. (2003). Sum rules for Jacobi matrices and their applications to spectral theory. Annals of mathematics, I Simon, B. (2010). Szegö s theorem and its descendants : spectral theory for L2 perturbations of orthogonal polynomials. Princeton University Press. I GF., Nagel, J., & Rouault, A. (2016). Sum rules via large deviations. Journal of Functional Analysis, 270(2), I GF., Nagel, J., & Rouault, A. (2016). Sum rules and large deviations for spectral matrix measures. arxiv preprint arxiv : I GF., Nagel, J., & Rouault, A. (2017). Sum rules and large deviations for spectral measures on the unit circle. Random Matrices : Theory and Applications. I Breuer, J., Simon, B., & Zeitouni, O. (2017). Large Deviations and Sum Rules for Spectral Theory-A Pedagogical Approach. To appear in Jounal of Spectral Theory. I Breuer, J., Simon, B., & Zeitouni, O. (2017). Large deviations and the Lukic conjecture. arxiv preprint arxiv : F. Gamboa (IMT Toulouse) 59th Birthday 7th of June / 35