Tail inequalities for order statistics of log-concave vectors and applications

Transcription

1 Tail inequalities for order statistics of log-concave vectors and applications Rafał Latała Based in part on a joint work with R.Adamczak, A.E.Litvak, A.Pajor and N.Tomczak-Jaegermann Banff, May 2011

2 Basic definitions and Notation Let X = X 1,..., X n ) be a random vector in R n with full dimensional support. We say that the distribution of X is logaritmically concave, if X has density of the form e hx) with h : R n, ] convex; isotropic, if EX i = 0 and EX i X j = δ i,j. For x R n we put ni=1 ) 1/2 x = x 2 = xi 2 ) ni=1 x r = x i r 1/r, 1 r <, x = max i x i P I x - canonical projection of x onto {y R n : suppy) I}, I {1,..., n}.

3 Order Statistics For an n dimensional random vector X by X1 X 2... X n we denote the nonincreasing rearrangement of X 1,..., X n in particular X1 = max{ X 1,..., X n } and Xn = min{ X 1,..., X n }). Random variables Xk, 1 k n, are called order statistics of X. Problem Find upper bound for PX k t). If coordinates of X i are independent symmetric exponential r.v. with variance 1 then MedXk ) logen/k) for k n/2.

4 Union bound We have for isotropic logconcave vectors X, PXk ) t) = P { X i1 t,..., X ik t} i 1 <...<i k Pη 1 X i1 t,..., η k X ik t) i 1 <...<i k η 1 =±1,...,η k =±1 i 1 <...<i k η 1 =±1,...,η k =±1 ) n k 2 k exp 1 C t k 1 P k η 1 X i η k X ik ) t ) k ). Therefore PXk t) exp 1 ) C t k for t C en ) k log. k

5 Exponential concentration Random vector X in R n satisfies exponential concentration inequality with a constant α if PX A + αtb n 2) 1 exp t) if PX A) 1 2 and t > 0. Conjecture Kannan-Lovasz-Simonovits) Isotropic log-concave vectors satisfy exponential concentration with universal α Known to hold for unconditional permutationally invariant isotropic log-concave vectors Klartag 11+). The best known bound for general case is α n 5/12 Guedon-Milman 10+).

6 Order statistics under exponential concentration Proposition If X is isotropic n-dimensional and satisfies exponential concentration inequality with a constant α 1 then PXk t) exp 1 ) kt 3α Sketch of the proof. The set A := en ) for t 8α log. k { { en )} x R n : # i : x i 4α log < k }. k 2 has measure µ at least 1/2. If z = x + y A + ksb2 n then less than k/2 of x i s are bigger than 4α logen/k) and less than k/2 of y i s are bigger than 2s, so en ) P Xk 4α log + ) 2s 1 µa+ ksb n k 2) exp α 1 ks).

7 Order Statistics for isotropic log-concave vectors Kannan-Lovasz-Simonovits Conjecture is open, nevertheless one may show the estimate for order statistics. Theorem Let X be n-dimensional log-concave isotropic vector. Then PXk t) exp 1 ) kt C en ) for t C log. k Our approach is based on the suitable estimate of moments of the process N X t), where N X t) := n 1 {Xi t} t 0. i=1

8 Estimate for N X Theorem For any isotropic log-concave vector X and p 1 we have Et 2 N X t)) p Cp) 2p nt 2 ) for t C log p 2. To get estimate for order statistics we observe that X k t implies that N X t) k/2 or N X t) k/2 and vector X is also isotropic and log-concave. Estimates for N X and Chebyshev s inequality gives 2 ) p PXk t) ENX t) p + EN X t) p) Cp 2 k t k provided that t C lognt 2 /p 2 ). We take p = 1 ec t k and notice that the restriction on t follows by the assumption that t C logen/k). ) 2p

9 Paouris Theorem Proof of estimate for N X t) is based on two ideas. First is that the restriction of a log-concave vector X to a convex set is log-concave. Second is Paouris concentration of mass result. Theorem Paouris) For any isotropic log-concave vector X in R n, P X t) exp 1 ) C t for t C n, equivalently E X p ) 1/p ) C n + p for p 2.

10 Estimate for N X implies Paouris concentration Proposition Suppose that X is a random vector in R n such that E t 2 N UX t) ) l A 1 l) 2l for t A 2, l n, U On), where A 1, A 2 are finite constants. Then P X t n) exp 1 t ) n CA 1 for t max{ca 1, A 2 }. Idea of the proof. For any U 1,..., U l On), l l E N Ui X t) EN Ui X t) l) 1/l A1 l ) 2l for l n. t i=1 i=1 If U 1,..., U l are random rotations then one may show that l E X E U N Ui X t) = E X E U1 N U1 X t)) l n l C l P X 2t n) i=1 and we take l = nt/ ec A ).

11 Concentration of l r norms, 1 r < 2 Problem. What is the concentration for l r norms of X? Case 1 r 2 reduces to the Paouris result for r = 2, since by the Hölder s inequality X r n 1/r 1/2 X. Thus E X p r ) 1/p Cn 1/r + n 1/r 1/2 p) and P X r t) exp 1 ) C tn1/2 1/r for t Cn 1/r. These bounds are optimal.

12 Concentration of l r norms, r > 2 Example It is not hard to see that if X 1,..., X n are independent symmetric exponential r.v. s with variance one then E X p r ) 1/p 1 C rn1/r + p) for p 2, r 2, n C r. Theorem For any δ > 0 there exist constants C 1 δ), C 2 δ) Cδ 1/2 such that for any isotropic logconcave vector X and r 2 + δ, Equivalently ) E X p r ) 1/p C 2 δ) rn 1/r + p for p 2. P X r t) exp 1 ) C 1 δ) t for t C 1 δ)rn 1/r.

13 Concentration of l r norms, r > 2 - idea of the proof We have n ) X r = X i r 1/r n ) = Xi r 1/r s 1 2 ) 2 k X 2 r 1/r, k i=1 i=1 where s = log 2 n. It is easy to check that k=0 k=0 s 2 k log r en2 k ) Cn j r 2 j Cr) r n. Thus for t 1,..., t k 0 we get P X r C rn 1/r + s ) 1/r )) t k j=1 k=0 s P 2 k X 2 r C r k 3 log r en2 k ) ) k=0 s k=0 exp 1 ) C 2 k 2 k 1/r r t k. s ) t k k=0

14 Uniform Paouris-type estimate Theorem For any m n and any isotropic log-concave vector X in R n we have for t 1, P sup I =m P I X Ct en )) m log exp t m en )) log. m logem) m Idea of the proof. We have sup P I X = I {1,...,N} I =m where s = log 2 m. m Xk 2) 1/2 s i X2 i 2) 1/2, k=1 i=0

15 Weak parameter For a vector X in R n we define Examples σ X p) := sup E t, X p ) 1/p p 2. t S n 1 For isotropic log-concave vectors X, σ X p) p/ 2. For subgaussian vectors X, σ X p) C p. We say that an isotropic vector X is ψ α if σ X p) Cp 1/α uniform distributions on suitable normalized Br n balls are ψ α with α = minr, 2))

16 Paouris theorem with weak parameter Theorem Paoouris) For any log-concave random vector X, ) E X p ) 1/p C E X 2 ) 1/2 + σ X p) for p 2, P X t) exp t )) σx 1 C for t CE X 2 ) 1/2. Corollary For any log-concave vector X in R n, any Euclidean norm on R n and p 1 we have E X p ) 1/p C E X 2 ) 1/2 + sup t 1 where R n, ) is a dual space to R n, ). E t, X p ) 1/p), 1) It is an open problem whether 1) holds for arbitrary norms

17 Bounds with use of weak parameter Theorem For any n-dimensional log-concave isotropic vector X, PX l t) exp 1 σx l)) 1 C t for t C log As before the proof is based on a suitable estimate of N X : Theorem Let X be an isotropic log-concave vector in R n. Then en ). l Et 2 N X t)) p Cσ X p)) 2p nt 2 ) for p 2, t C log σx 2 p).

18 Uniform bound for projections with weak parameter Theorem Let X be an isotropic log-concave vector in R n. Then for any t 1, P sup P I X Ct en m log I =m m where m 0 = m 0 X, t) = sup ) ) exp { en ) k m : k log k σ 1 X ) t m log en m logem/m0 )) ), σx 1 t en ))} m log. m

19 Weak parameter for convolution of log-concave measures Proposition Let X 1),..., X d) be independent isotropic log-concave vectors and Y = d i=1 x i X i). Then σ Y p) C p x + p x ). for p 2. Sketch of the proof. Fix t S n 1. Let E i be independent symmetric exponential random variables with variance 1. The result of Borell gives E t, X i) p C p E E i p for p 1. Hence E t, Y p ) 1/p d = E x i t, X i) p ) 1/p d p) 1/p C E x i E i i=1 C p x + p x ), where the last inequality follows by the Gluskin and Kwapień bound. i=1

20 Order statistics of convolutions Corollary Let X 1),..., X m) be independent isotropic log-concave vectors and Y = m i=1 x i X i). Then PY l t) exp 1 { t 2 C min l x 2, t l }) x for t x log en ). l

21 Uniform bound for projections of convolutions Theorem Let Y = d i=1 x i X i), where X 1),..., X d) are independent isotropic n-dimensional log-concave vectors. Assume that x 1 and x b 1. i) If b 1 m, then for any t 1, P sup P I Y Ct en ) ) t ) m log en ) m m log exp. I {1,...,n} m b loge 2 b 2 m) I =m ii) If b 1 m then for any t 1, P sup I {1,...,n} I =m P I Y Ct en ) ) m log m exp min {t 2 m log 2 en ), t en )}) m log. m b m

22 Bibliography R. Adamczak, R. Latała, A.E. Litvak, A. Pajor and N. Tomczak-Jaegermann, Geometry of log-concave Ensembles of random matrices and approximate reconstruction, R. Adamczak, R. Latała, A.E. Litvak, A. Pajor and N. Tomczak-Jaegermann, Tail estimates for convolutions of log-concave measures and applications, in preparation. R. Latała, Order statistics and concentration of l r norms for log-concave vectors, J. Funct. Anal ), G. Paouris, Concentration of mass on convex bodies, Geom. Funct. Anal ),