Weak and strong moments of l r -norms of log-concave vectors

Weak and strong moments of l r -norms of log-concave vectors Rafał Latała based on the joint work with Marta Strzelecka) University of Warsaw Minneapolis, April 14 2015

Log-concave measures/vectors A measure µ on a locally convex linear space F is called logarithmically concave log-concave in short) if for any compact nonempty sets K, L F and λ [0, 1], µλk + 1 λ)l) µk) λ µl) 1 λ. A random vector with values in F is called log-concave if its distribution is logarithmically concave. By the result of Borell an n-dimensional vector with a full dimensional support is log-concave iff it has a log-concave density, i.e. a density of the form e h, where h is a convex function with values in, ].

Examples of log-concave vectors Gaussian vectors Vectors with independent log-concave coordinates in particular vectors with product exponential distribution) Vectors uniformly distributed on convex bodies Affine images of log-concave vectors Sums of independent log-concave vectors Weak limits of log-concave vectors It may be shown that the class of log-concave distributions on R n is the smallest class that contains uniform distributions on convex bodies and is closed under affine transformations and weak limits.

Isotropic vectors Let X = X 1,..., X n ) be a random vector in R n such that E X 2 <. We say that the distribution of X is isotropic, if EX i = 0 and EX i X j = δ i,j for all 1 i, j n. If E X 2 < and X has a full dimensional support then there exists an affine transformation T such that TX is isotropic. Here and in the sequel x = x 2, where x r = x i r i n 1/r for x R d, r 1. Remark. By the result of Borell, for any log-concave vector X and any seminorm, E X p <. Moreover E X p ) 1/p C p q E X q ) 1/q for p q 1. By C we denote universal constants that may differ from line to line).

Paouris inequality One of the fundamental properties of log-concave vectors is the Paouris inequality. Theorem Paouris 06) Fo any log-concave vector X in R n, ) E X p ) 1/p C E X 2 ) 1/2 + σ X p) for p 1, where n p ) 1/p σ X p) := sup E t t 2 1 i X i. i=1 Equivalently, in terms of tails we have ) P X CtE X ) exp σx 1 te X ) for t 1,

Paouris inequality in the isotropic case For an isotropic vectorx, E X E X 2 ) 1/2 = n, and σ X 2) = 1. So if X is isotropic log-concave then σ X p) p for p 1. Hence we have the following weaker form of the Paouris inequality. Corollary Fo any isotropic log-concave vector X in R n, E X p ) 1/p C n + p ) for p 1. and P X Ct n) exp t n ) for t 1.

Example Let Y = ngu, where U has a uniform distribution on S n 1 and g is the standard normal N 0, 1) r.v., independent of U. Then it is easy to see that Y is isotropic, rotationally invariant and for any seminorm on R n E Y p ) 1/p = ne g p ) 1/p E U p ) 1/p pne U p ) 1/p for p 1. In particular this implies that for any t R n, n p ) 1/p E t i Y i C p n q ) 1/q E t q i Y i for p q 1. i=1 i=1 Therefore E Y p ) 1/p pn, E Y 2 ) 1/2 = n, σ Y p) Cp and for 1 p n, E Y p ) 1/p E Y 2 ) 1/2 + σ Y p). It would be very valuable to have a nice characterization of random vectors which satisfy the Paouris inequality.

Conjecture about weak and strong moments It is natural to ask whether the Paouris inequality may be generalized to non-euclidean norms. One may risk the following conjecture. Conjecture There exists a universal constant C such that for any log-concave vector X with values in a normed space F, ), E X p ) 1/p C E X + sup ϕ F, ϕ 1 E ϕx) p ) 1/p ) for p 1. Remark. Obviously for p 1, E X p ) 1/p E X and strong moments dominate weak moments, i.e. E X p ) 1/p sup E ϕx) p ) 1/p. ϕ F, ϕ 1

Main result Theorem Let X be a log-concave vector with values in a normed space F, ) which may be isometrically embedded in l r for some r [2, ). Then for p 1, E X p ) 1/p Cr E X + sup ϕ F, ϕ 1 E ϕx) p ) 1/p ) Remark. Let X and F be as above. Then by Chebyshev s inequality we obtain large deviation estimate for X : ) P X CrtE X ) exp σx,f 1 te X ) for t 1, where σ X,F p) := denotes the weak p-th moment of X. sup EϕX) p ) 1/p for p 1 ϕ F, ϕ 1.

Isomorhic embeddings Remark. If i : F l r is an isomorphic embedding and λ = i F lr i 1 if ) F, then we may define another norm on F by x := ix) / i F lr. Obviously F, ) isometrically embeds in l r, moreover x x λ x for x F. Hence the previous theorem gives E X p ) 1/p λe X ) p ) 1/p Crλ E X + Crλ E X + sup ϕ F, ϕ 1 sup ϕ F, ϕ 1 E ϕx) p ) 1/p E ϕx) p ) 1/p ).

Reduction to finite dimension Since log-concavity is preserved under linear transformations and, by Hahn-Banach theorem, any linear functional on a subspace of l r is a restriction of a functional on the whole l r with the same norm, it is enough to prove Theorem 2 for F = l r. An easy approximation argument shows that we may consider finite dimensional spaces lr n. To simplify the notation for an n-dimensional vector X and p 1 we write σ r,x p) := sup t r 1 n p ) 1/p E t i X i, i=1 where r denotes the Hölder s dual of r, i.e. r = Theorem r r 1. Let X be a log-concave vector in R n and r [2, ). Then E X p r ) 1/p Cr E X r + σ r,x p)) for p 1.

Modified bound for l r -norms Proof of the main result is based on the following estimate. Theorem Suppose that r [2, ) and X is a log-concave n-dimensional random vector. Let d i := EX 2 i ) 1/2, d := Then for p r and t Cr log d σ r,x p) n ), ) 1/r di r. 1) i=1 n p/r E X i r 1 { Xi td i }) Crσ r,x p)) p. i=1

Modified bound implies comparison of weak and strong moments in l n r. Since E X p r ) 1/p CpE X r, we may assume that p r. Let d i and d be definde by 1). Then d 2 = EX 2 i ) r/2 E X 2 i ) r/2 = E X 2 r CE X r ) 2. Set p := inf{q p : σ r,x q) d}. Modified bound applied with p instead of p and t = 0 yields E X p r ) 1/p E X p r ) 1/ p Crσ r,x p) = Cr max{d, σ r,x p)} CrE X r + σ r,x p)).

Idea of the proof of the modified bound Random vector X is also log-concave, has the same values of d i and σ r, X = σ r,x. Hence it is enough to show that n ) p/r E Xi r 1 {Xi td i } Crσ r,x p)) p i=1 for t Cr logd/σ r,x p)). It is easy to reduce to the case when t Cr and l = p/r is a positive integer. For l = 1, 2,... we have n l n ) l E Xi r 1 {Xi td i }) E 2 k+1)r td i ) r 1 {Xi 2 k td i } i=1 where = 2t) rl n i=1 k=0 i 1,...,i l =1 k 1,...,k l =0 2 k 1+...+k l )r d r i 1... d r i l PB i1,k 1...,i l,k l ), B i1,k 1...,i l,k l := {X i1 2 k 1 td i1,..., X il 2 k l td il }.

Idea of the proof of modified bound II So we are to show that ml) := n k 1,...,k l =0 i 1,...,i l =1 ) rl Crσr,X rl) t { for t Cr max 1, log d σ r,x rl) 2 k 1+...+k l )r d r i 1... d r i l PB i1,k 1,...,i l,k l ). )}. This is done by dividing terms in ml) into a number of groups and estimating each of them.

Crucial technical estimate Proposition Let X, r, d i and d be as before and A := {X K}, where K is a convex set in R n satisfying 0 < PA) 1/e. Then for every t r, n i=1 ) E X i r 1 A {Xi td i } C r PA) r r σr,x r logpa))) + dt) r e t/c. The proof is based on the fact that vector Y distributed as X conditioned on the set A = {X K}, i.e. PY B) = PX B) PX B K), is again log-concave if K is convex. To see that EYi 2 cannot be large for too many i s we use the Paouris inequality for X.

Case r = Recall the general conjecture about comparison of weak and strong moments ) E X p ) 1/p C E X + sup E ϕx) p ) 1/p for p 1. ϕ F, ϕ 1 2) Since every separable Banach space embeds in l it is enough to prove 2) in l n. It is known, but under the additional assumption that X is isotropic. Theorem Let X be an isotropic log-concave vector in R n. Then for any a 1,..., a n and p 1, E max a i X i p ) 1/p C i E max i ) a i X i + maxe X i p ) 1/p i The proof is completely different than in the case of l r -norms. It uses exponential concentration of log-concave vectors.

Exponential concentration Let µ be a measure on R n. We say that µ satisfies the exponential concentration with constant α if for any Borel set A, µa) 1 2 µa + αtbn 2 ) 1 e t for t 0. The fundamental open problem Kannan-Lovász-Simonovits conjecture) states that every isotropic log-concave measure satisfies exponential concentration with universal α. Klartag proved it with α Cn 1/2 ε with ε 1/30. This was improved by Eldan with the use of the result of Guédon-E.Milman) to α Cn 1/3 log 1/2 n + 1).

Optimal concentration inequalities For a probability measure µ on R n define Λ µ y) = log e y,z dµz), Λ µx) = sup y, x Λ µ y)) y and B µ t) = {x R n : Λ µ x) t}. We say that µ satifies the optimal concentration inequality with constant α if for any Borel set A, µa) 1 2 µa + αb µt)) 1 e t for t 1. Proposition If the law of an n-dimensional random vectors X satisfies the optimal concentration inequality with constant α then E X p ) 1/p E X + Cα sup ϕ F, ϕ 1 E ϕx) p ) 1/p for p 1.

Example - Talagrand s two level concentration In the case when µ = ν n is the product exponential measure i.e. the measure with the density 2 n exp i n x i ) it is easy to check that for t 1, B ν nt) tb n 1 + tb n 2. The optimal concentration inequality in this case is the Talagrand two-level concentration ν n A) 1 2 νn A + tb n 1 + tb n 2 ) 1 e t/c for t 1. Optimal concentration inequalities are strictly related to infimum convolution inequalities.

Optimal concentration inequalities - examples Examples of vectors that satisfy the optimal concentration inequality with universal constant Gaussian vectors vectors with independent log-concave coordinates rotationally invariant log-concave vectors uniform distributions on B n r -balls log-concave vectors with densities of the form exp g x r )), 1 r <, g : [0, ), ] convex increasing. Corollary For all random vectors listed above E X p ) 1/p E X + C sup E ϕx) p ) 1/p for p 1. ϕ F, ϕ 1

Unconditional vectors We say that a random vector X = X 1,..., X n ) has unconditional distribution if the distribution of η 1 X 1,..., η n X n ) is the same as X for any choice of signs η 1,..., η n. Theorem Let X be an n-dimensional isotropic, unconditional, log-concave vector and Y = Y 1,..., Y n ), where Y i are independent symmetric exponential r.v s with variance 1 i.e. with the density 2 1/2 exp 2 x )). Then for any norm on R n and p 1, E X p ) 1/p C E Y + sup ϕ 1 E ϕx) p ) 1/p). Proof is based on the Talagrand two-sided estimate of E Y and the Bobkov-Nazarov bound for the joint d.f. of X, which implies E ϕx) p ) 1/p CE ϕy ) p ) 1/p for p 1.

Unconditional vectors ctd Using the easy estimate E Y C log n E X we get. Corollary For any n-dimensional unconditional, log-concave vector X, any norm on R n and p 1 one has E X p ) 1/p C log n E X + sup ϕ 1 E ϕx) p ) 1/p). The Maurey-Pisier result implies E Y CE X in spaces with nontrivial cotype. Corollary Let X be as above, 2 q < and F = R n, ) has a q-cotype constant bounded by β <. E X p ) 1/p Cq, β) E X + sup ϕ 1 E ϕx) p ) 1/p).

Questions The conjecture E X p ) 1/p C E X + sup ϕ F, ϕ 1 E ϕx) p ) 1/p ) for p 1. seems to be rather hard in full generality. One may try to considerfirst some simpler open cases: l r -norms with 1 r 2 Orlicz norms or more general unconditional norms Unconditional log-concave vectors It is also not clear if one may improve the Paoris inequality to E X p ) 1/p E X + Cσ X p) for p 1.

References R. Eldan, Thin shell implies spectral gap up to polylog via a stochastic localization scheme, Geom. Funct. Anal. 23 2013), 532 569. B. Klartag, Power-law estimates for the central limit theorem for convex sets, J. Funct. Anal. 245 2007), 284 310. R. Latała, Weak and strong moments of random vectors, in: Marcinkiewicz centenary volume, 115 121, Banach Center Publ. 95, Polish Acad. Sci. Inst. Math., Warsaw, 2011. R. Latała, M. Strzelecka, Weak and strong moments of l r -norms of log-concave vectors, arxiv:1501.01649. R. Latała, J. O. Wojtaszczyk, On the infimum convolution inequality, Studia Math. 189 2008), 147-187. G. Paouris, Concentration of mass on convex bodies, Geom. Funct. Anal. 16 2006), 1021 1049. M. Talagarand, A new isoperimetric inequality and the concentration of measure phenomenon, in Israel Seminar GAFA), Lecture Notes in Math. 1469, 94 124, Springer, Berlin 1991

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