Truncations of unitary matrices and Brownian bridges Alain Rouault - PowerPoint PPT Presentation

Truncations of unitary matrices and Brownian bridges Alain Rouault (Laboratoire de Mathématiques de Versailles) joint work with C. Donati-Martin (Versailles) and V. Beffara (Grenoble) 22 may 2018 Paris 13 Seminar

Motivation Plan 1 Motivation 2 Main result 3 The 1-marginals Towards the fidi convergence and tightness 4 Combinatorics of the unitary and orthogonal groups 5 Random truncation 6 Main result 7 Subordination 8 A. Rouault (LMV) Paris 13 Seminar 22 may 2018 2 / 35

Motivation Motivation : Computational biology Aim : measure of the similarity between two genomic (long) sequences. Let S n be the set of permutations of [ n ] . If σ , τ ∈ S n , set O p ( σ , τ ) = # { i � p : σ ◦ τ − 1 ( i ) � p } , p = 1, · · · , n . and compare them with the results of a random permutation. G. Chapuy introduced the discrepancy process T ( n ) ⌊ ns ⌋ , ⌊ nt ⌋ ( σ ) = # { i � ⌊ ns ⌋ : σ ( i ) � ⌊ nt ⌋ } , s , t ∈ [ 0, 1 ] , Theorem (G. Chapuy 2007) The sequence n − 1 / 2 � � T ( n ) ⌊ ns ⌋ , ⌊ nt ⌋ ( σ ) − stn , s , t ∈ [ 0, 1 ] converges in distribution to the bivariate tied down Brownian bridge, of covariance ( s ∧ s ′ − ss ′ )( t ∧ t ′ − tt ′ ) . A. Rouault (LMV) Paris 13 Seminar 22 may 2018 3 / 35

Motivation Motivation : Computational biology Aim : measure of the similarity between two genomic (long) sequences. Let S n be the set of permutations of [ n ] . If σ , τ ∈ S n , set O p ( σ , τ ) = # { i � p : σ ◦ τ − 1 ( i ) � p } , p = 1, · · · , n . and compare them with the results of a random permutation. G. Chapuy introduced the discrepancy process T ( n ) ⌊ ns ⌋ , ⌊ nt ⌋ ( σ ) = # { i � ⌊ ns ⌋ : σ ( i ) � ⌊ nt ⌋ } , s , t ∈ [ 0, 1 ] , Theorem (G. Chapuy 2007) The sequence n − 1 / 2 � � T ( n ) ⌊ ns ⌋ , ⌊ nt ⌋ ( σ ) − stn , s , t ∈ [ 0, 1 ] converges in distribution to the bivariate tied down Brownian bridge, of covariance ( s ∧ s ′ − ss ′ )( t ∧ t ′ − tt ′ ) . A. Rouault (LMV) Paris 13 Seminar 22 may 2018 3 / 35

Motivation Motivation : Computational biology Aim : measure of the similarity between two genomic (long) sequences. Let S n be the set of permutations of [ n ] . If σ , τ ∈ S n , set O p ( σ , τ ) = # { i � p : σ ◦ τ − 1 ( i ) � p } , p = 1, · · · , n . and compare them with the results of a random permutation. G. Chapuy introduced the discrepancy process T ( n ) ⌊ ns ⌋ , ⌊ nt ⌋ ( σ ) = # { i � ⌊ ns ⌋ : σ ( i ) � ⌊ nt ⌋ } , s , t ∈ [ 0, 1 ] , Theorem (G. Chapuy 2007) The sequence n − 1 / 2 � � T ( n ) ⌊ ns ⌋ , ⌊ nt ⌋ ( σ ) − stn , s , t ∈ [ 0, 1 ] converges in distribution to the bivariate tied down Brownian bridge, of covariance ( s ∧ s ′ − ss ′ )( t ∧ t ′ − tt ′ ) . A. Rouault (LMV) Paris 13 Seminar 22 may 2018 3 / 35

Motivation Motivation : Computational biology Aim : measure of the similarity between two genomic (long) sequences. Let S n be the set of permutations of [ n ] . If σ , τ ∈ S n , set O p ( σ , τ ) = # { i � p : σ ◦ τ − 1 ( i ) � p } , p = 1, · · · , n . and compare them with the results of a random permutation. G. Chapuy introduced the discrepancy process T ( n ) ⌊ ns ⌋ , ⌊ nt ⌋ ( σ ) = # { i � ⌊ ns ⌋ : σ ( i ) � ⌊ nt ⌋ } , s , t ∈ [ 0, 1 ] , Theorem (G. Chapuy 2007) The sequence n − 1 / 2 � � T ( n ) ⌊ ns ⌋ , ⌊ nt ⌋ ( σ ) − stn , s , t ∈ [ 0, 1 ] converges in distribution to the bivariate tied down Brownian bridge, of covariance ( s ∧ s ′ − ss ′ )( t ∧ t ′ − tt ′ ) . A. Rouault (LMV) Paris 13 Seminar 22 may 2018 3 / 35

Motivation Motivation : Computational biology Aim : measure of the similarity between two genomic (long) sequences. Let S n be the set of permutations of [ n ] . If σ , τ ∈ S n , set O p ( σ , τ ) = # { i � p : σ ◦ τ − 1 ( i ) � p } , p = 1, · · · , n . and compare them with the results of a random permutation. G. Chapuy introduced the discrepancy process T ( n ) ⌊ ns ⌋ , ⌊ nt ⌋ ( σ ) = # { i � ⌊ ns ⌋ : σ ( i ) � ⌊ nt ⌋ } , s , t ∈ [ 0, 1 ] , Theorem (G. Chapuy 2007) The sequence n − 1 / 2 � � T ( n ) ⌊ ns ⌋ , ⌊ nt ⌋ ( σ ) − stn , s , t ∈ [ 0, 1 ] converges in distribution to the bivariate tied down Brownian bridge, of covariance ( s ∧ s ′ − ss ′ )( t ∧ t ′ − tt ′ ) . A. Rouault (LMV) Paris 13 Seminar 22 may 2018 3 / 35

Motivation Matrix representation If σ is represented by the matrix U ( σ ) , the integer T n p , q ( σ ) is the sum of all elements of the upper-left p × q submatrix of U ( σ ) , i.e. p , q ( σ ) = Tr [ D p U ( σ ) D q U ( σ ) ∗ ] T n where D k = diag ( 1, · · · , 1, 0, · · · , 0 ) ( k times 1) . Instead of picking randomly σ in the group S n , we propose to pick a random element U in the group U ( n ) (resp. O ( n ) ) and to study The main statistic � | U ij | 2 . p , q = Tr ( D 1 UD 2 U ∗ ) = T n i � p , j � q A. Rouault (LMV) Paris 13 Seminar 22 may 2018 4 / 35

Motivation Matrix representation If σ is represented by the matrix U ( σ ) , the integer T n p , q ( σ ) is the sum of all elements of the upper-left p × q submatrix of U ( σ ) , i.e. p , q ( σ ) = Tr [ D p U ( σ ) D q U ( σ ) ∗ ] T n where D k = diag ( 1, · · · , 1, 0, · · · , 0 ) ( k times 1) . Instead of picking randomly σ in the group S n , we propose to pick a random element U in the group U ( n ) (resp. O ( n ) ) and to study The main statistic � | U ij | 2 . p , q = Tr ( D 1 UD 2 U ∗ ) = T n i � p , j � q A. Rouault (LMV) Paris 13 Seminar 22 may 2018 4 / 35

Main result Plan 1 Motivation 2 Main result 3 The 1-marginals Towards the fidi convergence and tightness 4 Combinatorics of the unitary and orthogonal groups 5 Random truncation 6 Main result 7 Subordination 8 A. Rouault (LMV) Paris 13 Seminar 22 may 2018 5 / 35

Main result Main result Theorem (CDM+AR, RMTA 2011) The process � � W ( n ) = T ( n ) ⌊ ns ⌋ , ⌊ nt ⌋ − E T ( n ) ⌊ ns ⌋ , ⌊ nt ⌋ , s , t ∈ [ 0, 1 ] converges in distribution in the Skorokhod space D ([ 0, 1 ] 2 ) to the bivariate � β W ( ∞ ) where W ( ∞ ) is a centered 2 tied-down Brownian bridge continuous Gaussian process on [ 0, 1 ] 2 of covariance E [ W ( ∞ ) ( s , t ) W ( ∞ ) ( s ′ , t ′ )] = ( s ∧ s ′ − ss ′ )( t ∧ t ′ − tt ′ ) , β = 2 in the unitary case and β = 1 in the orthogonal case. A. Rouault (LMV) Paris 13 Seminar 22 may 2018 6 / 35

Main result Normalizations ◮ No normalization here ! ◮ If σ is Haar distributed in S n , then U ij is Bernoulli of parameter 1 /n and Var ( | U ij | 2 ) ∼ n − 1 ◮ If U is Haar distributed in U ( n ) , then the column vector ( U i , j ) n i = 1 is uniform on the (complex) sphere of dim n , and | U ij | 2 is Beta distributed with parameters ( 1, n − 1 ) and Var ( | U ij | 2 ) ∼ n − 2 . ◮ If O is Haar distributed in O ( n ) , then | O ij | 2 is Beta distributed with parameters ( 1 / 2, ( n − 1 ) / 2 ) and Var ( | O ij | 2 ) ∼ 2 n − 2 . A. Rouault (LMV) Paris 13 Seminar 22 may 2018 7 / 35

Main result Normalizations ◮ No normalization here ! ◮ If σ is Haar distributed in S n , then U ij is Bernoulli of parameter 1 /n and Var ( | U ij | 2 ) ∼ n − 1 ◮ If U is Haar distributed in U ( n ) , then the column vector ( U i , j ) n i = 1 is uniform on the (complex) sphere of dim n , and | U ij | 2 is Beta distributed with parameters ( 1, n − 1 ) and Var ( | U ij | 2 ) ∼ n − 2 . ◮ If O is Haar distributed in O ( n ) , then | O ij | 2 is Beta distributed with parameters ( 1 / 2, ( n − 1 ) / 2 ) and Var ( | O ij | 2 ) ∼ 2 n − 2 . A. Rouault (LMV) Paris 13 Seminar 22 may 2018 7 / 35

Main result Normalizations ◮ No normalization here ! ◮ If σ is Haar distributed in S n , then U ij is Bernoulli of parameter 1 /n and Var ( | U ij | 2 ) ∼ n − 1 ◮ If U is Haar distributed in U ( n ) , then the column vector ( U i , j ) n i = 1 is uniform on the (complex) sphere of dim n , and | U ij | 2 is Beta distributed with parameters ( 1, n − 1 ) and Var ( | U ij | 2 ) ∼ n − 2 . ◮ If O is Haar distributed in O ( n ) , then | O ij | 2 is Beta distributed with parameters ( 1 / 2, ( n − 1 ) / 2 ) and Var ( | O ij | 2 ) ∼ 2 n − 2 . A. Rouault (LMV) Paris 13 Seminar 22 may 2018 7 / 35

Main result Normalizations ◮ No normalization here ! ◮ If σ is Haar distributed in S n , then U ij is Bernoulli of parameter 1 /n and Var ( | U ij | 2 ) ∼ n − 1 ◮ If U is Haar distributed in U ( n ) , then the column vector ( U i , j ) n i = 1 is uniform on the (complex) sphere of dim n , and | U ij | 2 is Beta distributed with parameters ( 1, n − 1 ) and Var ( | U ij | 2 ) ∼ n − 2 . ◮ If O is Haar distributed in O ( n ) , then | O ij | 2 is Beta distributed with parameters ( 1 / 2, ( n − 1 ) / 2 ) and Var ( | O ij | 2 ) ∼ 2 n − 2 . A. Rouault (LMV) Paris 13 Seminar 22 may 2018 7 / 35

Main result Previous related results ◮ If q is fixed, Silverstein (1981) proved that the process   ⌊ ns ⌋ � | U iq | 2 − s n 1 / 2   , s ∈ [ 0, 1 ] i = 1 converges in distribution to the (univariate) Brownian bridge, continuous gaussian process of covariance s ( 1 − s ) . ◮ In multivariate (real) analysis of variance, T p , q is known as the Bartlett-Nanda-Pillai statistics, used to test equalities of covariances matrices from Gaussian populations. A. Rouault (LMV) Paris 13 Seminar 22 may 2018 8 / 35