On the convex infimum convolution inequality with optimal cost function
aa r X i v : . [ m a t h . P R ] F e b ON THE CONVEX INFIMUM CONVOLUTION INEQUALITYWITH OPTIMAL COST FUNCTION
MARTA STRZELECKA, MICHA L STRZELECKI, AND TOMASZ TKOCZ
Abstract.
We show that every symmetric random variable with log-concavetails satisfies the convex infimum convolution inequality with an optimal costfunction (up to scaling). As a result, we obtain nearly optimal comparisonof weak and strong moments for symmetric random vectors with independentcoordinates with log-concave tails. Introduction
Functional inequalities such as the Poincar´e, log-Sobolev, or Marton-Talagrandinequality to name a few, play a crucial role in studying concentration of measure,an important cornerstone of the local theory of Banach spaces. In this paper wefocus on another example of such inequalities, the infimum convolution inequality,introduced by Maurey in [11].Let X be a random vector with values in R n and let ϕ : R n → [0 , ∞ ] be a mea-surable function. We say that the pair ( X, ϕ ) satisfies the infimum convolutioninequality (ICI for short) if for every bounded measurable function f : R n → R ,(1.1) E e f (cid:3) ϕ ( X ) E e − f ( X ) ≤ , where f (cid:3) ϕ denotes the infimum convolution of f and ϕ defined as f (cid:3) ϕ ( x ) =inf { f ( y ) + ϕ ( x − y ) : y ∈ R n } for x ∈ R n . The function ϕ is called a cost function and f is called a test function . We also say that the pair ( X, ϕ ) satisfies the convexinfimum convolution inequality if (1.1) holds for every convex function f : R n → R bounded from below.Maurey showed that Gaussian and exponential random variables satisfy the ICIwith a quadratic and quadratic-linear cost function respectively. Thanks to the ten-sorisation property of the ICI, he recovered the Gaussian concentration inequalityas well as the so-called Talagrand two-level concentration inequality for the expo-nential product measure. Moreover, Maurey proved that bounded random variablessatisfy the convex ICI with a quadratic cost function (see also Lemma 3.2 in [14]for an improvement).Later on, Maurey’s idea was developed further by Lata la and Wojtaszczyk whostudied comprehensively the ICI in [10]. By testing with linear functions, theyobserved that the optimal cost function is given by the Legendre transform of thecumulant-generating function (here optimal means largest possible, up to a scalingconstant, because the larger the cost function is, the better (1.1) gets). They intro-duced the notion of optimal infimum convolution inequalities, established them for Date : February 23, 2017.2010
Mathematics Subject Classification.
Primary: 60E15. Secondary: 26A51, 26B25.
Key words and phrases.
Infimum convolution, log-concave tails, convex functions, weak andstrong moments.Research partially supported by the National Science Centre, Poland, grants no.2015/19/N/ST1/02661 (M. Strzelecka) and 2015/19/N/ST1/00891 (M. Strzelecki) as well as theSimons Foundation (T. Tkocz). log-concave product measures and uniform measures on ℓ p -balls, and put forwardimportant, challenging and far-reaching conjectures (see also [6]).The recent works [4] and [3] enable to view the ICI from a different perspective.In [4] the authors introduce weak transport-entropy inequalities and establish theirdual formulations. The dual formulations are exactly the convex ICIs. In [3] theauthors investigate extensively the weak transport cost inequalities on the real line,obtaining a characterisation for arbitrary cost functions which are convex and qua-dratic near zero, thus providing a tool for studying the convex ICI. Around the sametime, the convex ICI for the quadratic-linear cost function was fully understood in[2].In this paper we go along Lata la and Wojtaszczyk’s line of research and studythe optimal convex ICI. Using the aforementioned novel tools from [3], we showthat product measures with symmetric marginals having log-concave tails satisfythe optimal convex ICI, which complements Lata la and Wojtaszczyk’s result aboutlog-concave product measures. This has applications to concentration and momentcomparison of any norm of such vectors in the spirit of celebrated Paouris’ inequality(see [13] and [1]) and addresses some questions posed lately in [7]. We also offeran example showing that the assumption of log-concave tails cannot be weakenedsubstantially. 2. Main results
For a random vector X in R n we defineΛ ∗ X ( x ) := L Λ X ( x ) := sup y ∈ R n {h x, y i − ln E e h y,X i } , which is the Legendre transform of the cumulant-generating functionΛ X ( x ) := ln E e h x,X i , x ∈ R n . If X is symmetric and the pair ( X, ϕ ) satisfies the ICI, then ϕ ( x ) ≤ Λ ∗ X ( x )for every x ∈ R n (see Remark 2.12 in [10]). In other words, Λ ∗ X is the optimalcost function ϕ for which the ICI can hold. Since this conclusion is obtained bytesting (1.1) with linear functions, the same holds for the convex ICI. Following[10] we shall say that X satisfies (convex) IC( β ) if the pair ( X, Λ ∗ X ( · /β )) satisfiesthe (convex) ICI.We are ready to present our first main result. Theorem 2.1.
Let X be a symmetric random variable with log-concave tails, i.e.such that the function t N ( t ) := − ln P ( | X | ≥ t ) , t ≥ , is convex. Then there exists a universal constant β ≤ e such that X satisfiesconvex IC ( β ) . The (convex) ICI tensorises and, consequently, the property (convex) IC ten-sorises: if independent random vectors X i satisfy (convex) IC( β i ), i = 1 , . . . , n , thenthe vector ( X , . . . , X n ) satisfies (convex) IC(max β i ) (see [11] and [10]). Thereforewe have the following corollary. Corollary 2.2.
Let X be a symmetric random vector with values in R n and in-dependent coordinates with log-concave tails. Then X satisfies convex IC ( β ) witha universal constant β ≤ e . Note that the class of distributions from Theorem 2.1 is wider than the class ofsymmetric log-concave product distributions considered by Lata la and Wojtaszczykin [10]. Among others, it contains measures which do not have a connected support,e.g. a symmetric Bernoulli random variable.
ONVEX INFIMUM CONVOLUTION INEQUALITY 3
In order to comment on the relevance of the assumptions of Theorem 2.1 andpresent applications to comparison of weak and strong moments, we need the fol-lowing definition. Let X be a random vector with values in R n . We say that themoments of X grow α -regularly if for every p ≥ q ≥ θ ∈ S n − we have kh X, θ ik p ≤ α pq kh X, θ ik q , where k Y k p := ( E | Y | p ) /p is the p -th integral norm of a random variable Y . Clearly,if the moments of X grow α -regularly, then α has to be at least 1 (unless X = 0a.s.). Remark . If X is a symmetric random variable with log-concave tails, then itsmoments grow 1-regularly (this classical fact follows for instance from Proposition5.5 from [5] and the proof of Proposition 3.8 from [10]).The assumption of log-concave tails in Theorem 2.1 cannot be replaced bya weaker one of α -regularity of moments: if X is a symmetric random variabledefined by(2.1) P ( | X | > t ) = 1 [0 , ( t ) + ∞ X k =1 e − k [2 k , k +1 ) ( t ) , t ≥ , then the moments of X grow α -regularly (for some α < ∞ ), but there does notexists C >
X, x max { ( Cx ) , C | x |} ) satisfies the convexICI. All the more, X cannot satisfy convex IC( β ) with any β < ∞ (see Section 5for details). Thus it seems that the assumptions of Theorem 2.1 are not far fromnecessary conditions for the convex ICI to hold with an optimal cost function (ran-dom variables with moments growing regularly are akin to random variables withlog-concave tails as the former can essentially be sandwiched between the latter,see (4.6) in [9]).Our second main result is an application of Theorem 2.1 to moment comparison.Recall that for a random vector X its p -th weak moment associated with a norm k · k is the quantity defined as σ k·k ,X ( p ) := sup k t k ∗ ≤ kh t, X ik p , where k · k ∗ is the dual norm of k · k . The following version of [10, Proposition 3.15]holds (some non-trivial modifications of the proof are necessary in order to dealwith the fact that the inequality (1.1) only holds for convex functions). Theorem 2.4.
Let X be a symmetric random vector with values in R n whichmoments grow α -regularly. Suppose moreover that X satisfies convex IC ( β ) . Thenfor every norm k · k on R n and every p ≥ we have (cid:16) E (cid:12)(cid:12) k X k − E k X k (cid:12)(cid:12) p (cid:17) /p ≤ Cαβσ k·k ,X ( p ) , where C is a universal constant (one can take C = 4 √ e < ). Immediately we obtain the following corollary in the spirit of the results from[13, 1, 7, 8]. Similar inequalities for Rademacher sums with the emphasis on exactvalues of constants have also been studied by Oleszkiewicz (see [12, Theorem 2.1]).
Corollary 2.5.
Let X be a symmetric random vector with values in R n and withindependent coordinates which have log-concave tails. Then for every norm k · k on R n and every p ≥ we have (2.2) (cid:0) E k X k p (cid:1) /p ≤ E k X k + Dσ k·k ,X ( p ) , where D is a universal constant (one can take D = 6720 √ e < ). M. STRZELECKA, M. STRZELECKI, AND T. TKOCZ
Note that each of the terms on the right-hand side of (2.2) is, up to a constant,dominated by the left-hand side of (2.2), so (2.2) yields the comparison of weakand strong moments of the norms of X .Note also that the constant standing at E k X k is equal to 1. If we only assumethat the coordinates of X are independent and their moments grow α -regularly,then (2.2) does not always hold (the example here is a vector with independentcoordinates distributed like in (2.1); see Section 5 for details), although by [7,Theorem 1.1] it holds if we allow the constant at E k X k to be greater than 1 and todepend on α . Hence Corollary 2.5 and example (2.1) partially answer the followingquestion raised in [7]: “For which vectors does the comparison of weak and strongmoments hold with constant 1 at the first strong moment?”The organization of the paper is the following. In Section 3 and 4 we presentthe proofs of Theorem 2.1 and Proposition 2.4 respectively. In Section 5 we discussexample (2.1) in details. 3. Proof of Theorem 2.1
Our approach is based on a characterization – provided by Gozlan, Roberto,Samson, Shu, and Tetali in [3] – of measures on the real line which satisfy a weaktransport-entropy inequality. We emphasize that our optimal cost functions neednot be quadratic near the origin, therefore we cannot apply their characterizationas is, but have to first fine-tune the cost functions a bit. We shall also need thefollowing simple lemma.
Lemma 3.1. If X is a symmetric random variable and E X = β − , then Λ ∗ X ( x/β ) ≤ x for | x | ≤ . Proof.
Since X is symmetric, we have E e tX = 1 + ∞ X k =1 k X k k k t k (2 k )! ≥ ∞ X k =1 k X k k t k (2 k )! = 1 + ∞ X k =1 β − k t k (2 k )! = cosh( β − | t | ) . Moreover, L (cid:0) ln cosh( · ) (cid:1) ( | u | ) ≤ | u | for | u | ≤ ∗ X ( x/β ) = L (Λ X ( β · ))( x ) ≤ L (ln cosh( · ))( x ) ≤ x for | x | ≤ . (cid:3) Throughout the proof g − stands for the generalized inverse of a function g defined as g − ( y ) := inf { x : g ( x ) ≥ y } . Proof of Theorem 2.1.
Note that N (0) = 0 and the function N is non-decreasing.First we tweak the assumptions and change the assertion to a more straightforwardone. Step 1 (first reduction).
We claim that it suffices to prove the assertion forrandom variables for which the function N is strictly increasing on the set whereit is finite (or, in other words, N ( t ) = 0 only for t = 0). Indeed, suppose wehave done this and let now X be any random variable satisfying the assumptionsof the theorem. Let X ε be a symmetric random variable such that P ( | X ε | ≥ t ) =exp( − N ε ( t )), where N ε ( t ) = N ( t ) ∨ εt . If X and X ε are represented in the standardway by the inverses of their CDFs on the probability space (0 , | X ε | ≤ | X | a.s. (and also X ε → X a.s. as ε → + ). Hence Λ X ε ≤ Λ X and therefore alsoΛ ∗ X ε ≥ Λ ∗ X .The theorem applied to the random variable X ε and the above inequality implythat the pair ( X ε , Λ ∗ X ( · /β )) satisfies the convex ICI. Taking ε → + we get theassertion for X (in the second integral we just use the fact that the test function ONVEX INFIMUM CONVOLUTION INEQUALITY 5 f is bounded from below and thus e − f is bounded from above; for the first in-tegral it suffices to prove the convergence of integrals on any interval [ − M, M ],and on such an interval we have f (cid:3) Λ ∗ X ( x/β ) ≤ f ( x ) + Λ ∗ X (0) = f ( x ), and thusexp(max [ − M,M ] f ) is a good majorant). Step 2 (second reduction).
We claim that it suffices to prove the assertion forrandom variables such that Λ X < ∞ . Indeed, suppose we have done this andlet X be any random variable satisfying the assumptions of the theorem. Let N ε ( t ) = N ( t ) ∨ ε t and let X ε be a symmetric random variable such that P ( | X ε | ≥ t ) = exp( − N ε ( t )). Then, similarly as in Step 1., Λ X ε ≤ Λ Y < ∞ , where Y issymmetric and P ( | Y | ≥ t ) = exp( − ε t ). Thus we can apply the proposition to X ε and we continue as in Step 1. Step 3 (scaling).
Due to the scaling properties of the Legendre transform, wecan assume that E X = β − , where β := 2 e (the case where X ≡ e − N (1 / = P ( | X | ≥ ) ≤ E X = e − , so(3.1) N (1 / ≥ . Step 4 (reformulation).
For x ∈ R let ϕ ( x ) := (cid:0) x {| x | < } + (2 | x | − {| x |≥ } (cid:1) ∨ Λ ∗ X ( x/ (2 β )) . We claim that there exists a universal constant e b ≤ / X, ϕ (˜ b · )) satisfies the convex infimum convolution inequality. Of course the asser-tion follows immediately from that.Note that ϕ is convex, increasing on [0 , ∞ ) (because Λ ∗ X ( · / (2 β )) is convex andsymmetric and thus non-decreasing on [0 , ∞ )). Crucially, ϕ ( x ) = x for x ∈ [0 , ϕ is quadratic near zero. Moreover, by Lemma3.1, ϕ − (3) = 2.Let U = F − ◦ F ν , where F , F ν are the distribution functions of X and thesymmetric exponential measure ν on R , respectively. By [3, Theorem 1.1] we knowthat if there exists b > x, y ∈ R we have(3.2) (cid:12)(cid:12) U ( x ) − U ( y ) (cid:12)(cid:12) ≤ b ϕ − (cid:0) | x − y | (cid:1) , then the pair ( X, ϕ ( e b · )), where e b = b ϕ − (2+1 ) = b , satisfies the convex ICI. Wewill show that (3.2) holds with b = 1. Step 5 (further reformulation).
Let a = inf { t > N ( t ) = ∞} . We have threepossibilities (recall that N is left-continuous): • a = ∞ . Then N is continuous, increasing, and transforms [0 , ∞ ] onto[0 , ∞ ]. Also, F is increasing and therefore F − is the usual inverse of F . • a < ∞ and N ( a ) < ∞ . Then X has an atom at a . Moreover, N ( a ) =lim t → a − N ( t ). • a < ∞ and N ( a ) = ∞ = lim t → a − N ( t ).Of course, in the first case one can extend N by putting N ( a ) = ∞ , so that allformulas below make sense.Note that F ( t ) = ( exp( − N ( | t | )) if t < , − exp( − N + ( t )) if t ≥ , where N + ( t ) denotes the right-sided limit of N at t (which is different from N ( t ) onlyif t = a and X has an atom at a ). Hence, F is continuous on the interval ( − a, a ),the image of ( − a, a ) under F is the interval (cid:0) exp( − N ( a )) , − exp( − N ( a )) (cid:1) ,and we have F ( − a ) = exp( − N ( a )) and F ( a ) = 1. Since the image of R under U is equal to the image of (0 ,
1) under F − , we conclude that U ( R ) = ( − a, a ) if N ( a ) = ∞ and U ( R ) = [ − a, a ] if N ( a ) < ∞ . Denote A := U ( R ). M. STRZELECKA, M. STRZELECKI, AND T. TKOCZ
When N ( a ) < ∞ , it suffices to check condition (3.2) for x, y ∈ [ − N ( a ) , N ( a )](otherwise one can change x , y and decrease the right-hand side while not changingthe value of the left-hand side of (3.2)). For x ∈ [ − N ( a ) , N ( a )] we can write U − ( x ) = N ( | x | ) sgn( x ) and U − ( x ) ∈ R . When N ( a ) = ∞ , U is a bijection (onits image), so we can obviously write again U − ( x ) = N ( | x | ) sgn( x ) for any x ∈ R .Therefore, in order to verify (3.2) we need to check that(3.3) | x − y | ≤ ϕ − (cid:0) (cid:12)(cid:12) N ( | x | ) sgn( x ) − N ( | y | ) sgn( y ) (cid:12)(cid:12)(cid:1) for x, y ∈ A. Since we consider the case when Λ X ( t ) is finite for every t ∈ R , the Chernoffinequality applies, so for t ≥ E X = 0 we have12 e − N ( t ) = P ( X ≥ t ) ≤ e − Λ ∗ X ( t ) , so(3.4) N ( t ) ≥ Λ ∗ X ( t ) − ln 2 . Note that ϕ ( | x − y | ) < ∞ for x, y ∈ A , since ϕ ( | x − y | ) = ∞ would implyΛ ∗ X ( | x − y | / (2 β )) = ∞ , and hence Λ ∗ X ( | x − y | /
2) = ∞ , and – by (3.4) – also N ( | x − y | /
2) = ∞ , but for x, y ∈ A we have | x − y | / ∈ [0 , a ) when N ( a ) = ∞ or | x − y | / ∈ [0 , a ] when N ( a ) < ∞ and in either case N ( | x − y | /
2) is finite.Therefore for every x, y ∈ A we have ϕ ( | x − y | ) < ∞ . Since ϕ − ( ϕ ( z )) = z for z such that ϕ ( z ) < ∞ (because ϕ is then continuous and increasing on [0 , z ]), thecondition (3.3) is implied by(3.5) ϕ (cid:0) | x − y | (cid:1) ≤ (cid:12)(cid:12) N ( | x | ) sgn x − N ( | y | ) sgn y (cid:12)(cid:12) for x, y ∈ A. In the next step we check that this is indeed satisfied.
Step 6 (checking the condition).
Let x = inf { x ≥ x − ∗ X ( x β ) } (if x = ∞ we simply do not have to consider Case 2 below). We consider three cases.We repeatedly use the fact that uN ( t ) ≥ N ( ut ) for u ≤ t ≥
0, which follows bythe convexity of N and the property N (0) = 0. Case 1. | x − y | ≤
1. Then ϕ (cid:0) | x − y | (cid:1) = ( x − y ) ≤
1, so (3.5) is trivially satisfied.
Case 2. | x − y | ≥ x . Then ϕ (cid:0) | x − y | (cid:1) = Λ ∗ X ( β | x − y | ) ≤ Λ ∗ X ( | x − y | / x , y are of the same sign, say x, y ≥
0, then N (cid:0) | x − y | / ≤ | N ( x ) − N ( y ) | and if x, y have different signs, we have N (cid:0)(cid:0) | x | + | y | (cid:1) / (cid:1) ≤ N ( | x | ) + N ( | y | ).By the convexity of N , for s, t ≥ N (cid:0) ( s + t ) / (cid:1) ≤ N ( s ) + 12 N ( t ) ≤ N ( s ) + N ( t )and N ( s/
2) + N ( t ) ≤ N ( s ) + N ( t ) ≤ ss + t N ( s + t ) + ts + t N ( s + t ) = N ( s + t ) . This finishes the proof of (3.5) in Case 2.
Case 3. ≤ | x − y | ≤ x . Then ϕ (cid:0) | x − y | (cid:1) = 2 | x − y | −
1. Consider twosub-cases:(i) x, y have different signs. Without loss of generality we may assume x ≥| y | ≥ ≥ y . Thus in order to obtain (3.5) it suffices to show that N ( x ) ≥ x + 2 | y | . Note that 1 ≤ x + | y | ≤ x , so x ≥ . Thus N ( x ) ≥ N (1 / x (3.1) ≥ x ≥ x + 2 | y | , which finishes the proof in case (i). ONVEX INFIMUM CONVOLUTION INEQUALITY 7 (ii) x, y have the same sign. Without loss of generality we may assume x ≥ y ≥ . Thus it suffices to show that 2( x − y ) ≤ N ( x ) − N ( y ). Note that due tothe assumption of Case 3 we have x ≥ x − y ≥ ≥ , so by the convexityof N we have N ( x ) − N ( y ) x − y ≥ N ( ) − N (0) − (3.1) ≥ ≥ (cid:3) Comparison of weak and strong moments
The goal of this section is to establish the comparison of weak and strong mo-ments with respect to any norm k · k for random vectors X with independentcoordinates having log-concave tails (Corollary 2.5). In view of Theorem 2.1 andRemark 2.3, it is enough to show Theorem 2.4.Our proof of Theorem 2.4 comprises three steps: first we exploit α -regularity ofmoments of X to control the size of its cumulant-generating function Λ X , secondwe bound the infimum convolution of the optimal cost function with the convex testfunction being the norm k · k properly rescaled, and finally by the property convexIC( β ) we obtain exponential tail bounds which integrated out give the desiredmoment inequality.We start with two lemmas corresponding to the first two steps described aboveand then we put everything together. Lemma 4.1.
Let p ≥ and suppose that the moments of a random vector X in R n grow α -regularly. If for a vector u ∈ R n we have kh u, X ik p ≤ , then Λ X ((2 eα ) − pu ) ≤ p. Proof.
Let k be the smallest integer larger than p . If αe kh u, X ik p ≤ /
2, then by α -regularity we haveΛ X ( pu ) ≤ ln (cid:16)X k ≥ E |h pu, X i| k k ! (cid:17) ≤ ln (cid:16) X ≤ k ≤ p p k kh u, X ik kp k ! + X k>p ( αk ) k kh u, X ik kp k ! (cid:17) ≤ ln (cid:16) X ≤ k ≤ p p k kh u, X ik kp k ! + X k>p (cid:0) αe kh u, X ik p (cid:1) k (cid:17) ≤ ln (cid:16) X ≤ k ≤ p p k kh u, X ik kp k ! + 2( αe kh u, X ik p ) k (cid:17) ≤ ln (cid:16) X ≤ k ≤ p p k kh u, X ik kp k ! + (2 αep kh u, X ik p ) k k ! (cid:17) ≤ ln (cid:16) X ≤ k ≤ k (2 αep kh u, X ik p ) k k ! (cid:17) ≤ αep kh u, X ik p ≤ p. Replace u with (2 eα ) − u to get the assertion. (cid:3) Lemma 4.2.
Let k · k be a norm on R n and let X be a random vector with valuesin R n and moments growing α -regularly. For β > , p ≥ , and x ∈ R n we have (cid:0) Λ ∗ X ( · /β ) (cid:3) a k · k (cid:1) ( x ) ≥ a k x k − p, where a = p (2 eαβσ k·k ,X ( p )) − . M. STRZELECKA, M. STRZELECKI, AND T. TKOCZ
Proof.
For f ( x ) = a k x k with positive a being arbitrary for now we bound theinfimum convolution as follows (cid:0) Λ ∗ X ( · /β ) (cid:3) f (cid:1) ( x ) = inf y sup z (cid:8) β − h y, z i − Λ X ( z ) + a k x − y k (cid:9) = inf y sup u (cid:8) (2 eαβ ) − p h y, u i − Λ X ((2 eα ) − pu ) + a k x − y k (cid:9) ≥ inf y sup u : kh u,X ik p ≤ (cid:8) (2 eαβ ) − p h y, u i − p + a k x − y k (cid:9) , where in the last inequality we have used Lemma 4.1. Choose u = σ k·k ,X ( p ) − v with k v k ∗ ≤ h y, v i = k y k . Then clearly kh u, X ik p ≤ ∗ X ( · /β ) (cid:3) f ( x ) ≥ inf y (cid:8) (2 eαβσ k·k ,X ( p )) − p k y k − p + a k x − y k (cid:9) . If we now set a = p (2 eαβσ k·k ,X ( p )) − , then by the triangle inequality we obtainthe desired lower bound (cid:0) Λ ∗ X ( · /β ) (cid:3) a k · k (cid:1) ( x ) ≥ a k x k − p. (cid:3) Proof of Theorem 2.4.
Let f ( x ) = a k x k with a = p (2 eαβσ k·k ,X ( p )) − as in Lemma4.2. Testing the property convex IC( β ) with f and applying Lemma 4.2 yields E e a k X k E e − a k X k ≤ e p . By Jensen’s inequality we obtain that both E e a ( k X k− E k X k ) and E e a ( −k X k + E k X k ) are bounded above by e p . Thus Markov’s inequality implies the tail bound P (cid:0) a (cid:12)(cid:12) k X k − E k X k (cid:12)(cid:12) > t (cid:1) ≤ e − t e p ≤ e − t/ , t ≥ p. Consequently, a p E (cid:12)(cid:12) k X k − E k X k (cid:12)(cid:12) p = Z ∞ pt p − P (cid:0) a (cid:12)(cid:12) k X k − E k X k (cid:12)(cid:12) > t (cid:1) dt ≤ (2 p ) p + 2 Z ∞ pt p − e − t/ dt = (2 p ) p + 2 · p p Γ( p ) ≤ p ) p . Plugging in the value of a gives the result (we can take C = 4 √ e < (cid:3) An example
Let X be a symmetric random variable defined by P ( | X | > t ) = T ( t ), where(5.1) T ( t ) := 1 [0 , ( t ) + ∞ X k =1 e − k [2 k , k +1 ) ( t ) , t ≥ , or, in other words, let | X | have the distribution(1 − e − ) δ + ∞ X k =2 (cid:0) e − k − − e − k (cid:1) δ k . Let us first show that the moments of X grow 3-regularly, but X does not satisfyIC( β ) for any β < ∞ (we also prove a slightly stronger statement later).Let Y be a symmetric exponential random variable. Then Y has log-concavetails, so the moments of Y grow 1-regularly (see Remark 2.3). Moreover, if X and Y are constructed in the standard way by the inverses of their CDFs on theprobability space (0 , | Y | ≤ | X | ≤ | Y | + 2 . ONVEX INFIMUM CONVOLUTION INEQUALITY 9
Therefore, for p ≥ q ≥ k X k p ≤ k Y k p + 2 ≤ pq k Y k q + 2 ≤ pq k X k q (we used the fact that | X | ≥ X grow 3-regularly.On the other hand, for every h > t > P ( | X | ≥ t + h ) = P ( | X | ≥ t ) . Therefore by [2, Theorem 1] there does not exist a constant C such that the pair( X, ϕ ( · /C )), where ϕ ( x ) = x {| x |≤ } + ( | x | − / {| x | > } , satisfies the convexinfimum convolution inequality. But, by symmetry and the 3-regularity of momentsof X ,Λ X ( s ) ≤ ln (cid:16) X k ≥ s k E X k (2 k )! (cid:17) ≤ ln (cid:16) X k ≥ s k (3 k ) k (cid:0) E X (cid:1) k (2 k )! (cid:17) ≤ ln (cid:16) X k ≥ s k (3 e/ k (cid:0) E X (cid:1) k (cid:17) = ln (cid:16) X k ≥ (cid:0) e s E X / (cid:1) k (cid:17) . Thus for some
A, ε > X ( s ) ≤ As for | s | ≤ ε and 2 Aε ≥
1. HenceΛ ∗ X ( t ) ≥ sup | s |≤ ε { st − As } = A t {| t |≤ Aε } + ( ε | t | − Aε )1 {| t | > Aε } = 2 Aε ϕ (cid:0) t/ (2 Aε ) (cid:1) ≥ ϕ (cid:0) t/ (2 Aε ) (cid:1) . We conclude that X cannot satisfy IC( β ) for any β . Remark . Let us also sketch an alternative approach. Take a, c > b ∈ R , anddenote ϕ ( x ) = min { x , | x |} , f ( x ) = f a,b ( x ) = a ( x − b ) + for x ∈ R . One can checkthat (cid:0) f (cid:3) ϕ ( c · ) (cid:1) ( x ) = x ≤ b,c ( x − b ) if b < x ≤ b + 1 /c,c ( x − b ) if x > b + 1 /c, if a > c . It is rather elementary but cumbersome to show that for any c > a > b ∈ R such that (1.1) is violated by the test function f . We omitthe details.In fact, the above example shows that even a slightly stronger statement is true:for vectors with independent coordinates with α -regular growth of moments thecomparison of weak and strong moments of norms does not hold with the constant1 at the first strong moment. More precisely, let X , X , . . . be independent randomvariables with distribution given by (5.1). We claim that there does not exist any K < ∞ such that(5.2) (cid:0) E max i ≤ n | X i | p (cid:1) /p ≤ E max i ≤ n | X i | + K sup k t k ≤ (cid:16) E (cid:12)(cid:12) n X i =1 t i X i (cid:12)(cid:12) p (cid:17) /p holds for every p ≥ n ∈ N (note that we chose the ℓ ∞ -norm as our norm).We shall estimate the three expressions appearing in (5.2).We have(5.3) sup k t k ≤ (cid:16) E (cid:12)(cid:12) n X i =1 t i X i (cid:12)(cid:12) p (cid:17) /p ≤ sup k t k ≤ n X i =1 | t i |k X i k p = k X k p (this inequality is in fact an equality). Since the moments of X grow 3-regularly,the last term in (5.2) is bounded by e Kp for some e K < ∞ .To estimate the remaining two terms we need the following standard fact. Lemma 5.2.
For independent events A , . . . , A n , (1 − e − ) (cid:16) ∧ n X i =1 P ( A i ) (cid:17) ≤ P (cid:16) n [ i =1 A i (cid:17) ≤ ∧ n X i =1 P ( A i ) . In particular, for i.i.d. non-negative random variables Y , . . . , Y n , (1 − e − ) Z ∞ h ∧ n P ( Y > t ) i dt ≤ E max i ≤ n Y i ≤ Z ∞ h ∧ n P ( Y > t ) i dt. Proof.
The upper bound is just the union bound. The lower bound follows from deMorgan’s laws combined with independence and the inequalities 1 − x ≤ e − x and1 − e − y ≥ (1 − e − ) y for x ∈ R , y ∈ [0 , (cid:3) Fix m ≥ e m − ≤ n < e m . Then1 ∧ nT ( t ) = ( < t < m ,nT ( t ) if t ≥ m . By the above lemma, E max i ≤ n | X i | ≤ Z m dt + n Z ∞ m T ( t ) dt = 2 m + n ∞ X j = m e − j (2 j +1 − j )= 2 m + n ∞ X j = m e − j j ≤ m + ne − m m ∞ X j =0 (2 e − m ) j = 2 m + ne − m m − e − m . Set θ = θ ( m, n ) = ne − m ∈ [ e − m − , E max i ≤ n | X i | ≤ m (cid:16) θ − e − m (cid:17) . Similarly, E max i ≤ n | X i | p ≥ (1 − e − ) Z ∞ ∧ T ( t /p ) dt = (1 − e − ) h Z mp dt + n Z ∞ mp T ( t /p ) dt i = (1 − e − ) h mp + n ∞ X j = m e − j (cid:0) ( j +1) p − jp (cid:1)i . Hence(5.5) E max i ≤ n | X i | p > (1 − e − ) ne − m (cid:0) ( m +1) p − mp (cid:1) = (1 − e − ) θ mp (2 p − . Putting (5.3), (5.4), and (5.5) together, we see that (5.2) would imply(1 − e − ) /p θ /p m (2 p − /p ≤ m (cid:16) θ − e − m (cid:17) + e Kp for every p ≥ m ≥
2, and θ ∈ [ e − m − ,
1) of the form ne − m , n ∈ N . Take p = 1 /θ and θ ∼ /m to get(1 − e − ) θ θ θ (2 /θ − θ ≤ θ − e − m + e K m θ . Since θ → m θ → ∞ as m → ∞ this inequality yields 2 ≤
1, which isa contradiction. Hence inequality (5.2) cannot hold for all p ≥ n ∈ N . Acknowledgments
We thank Rados law Adamczak and Rafa l Lata la for posing questions which ledto the results presented in this note.
ONVEX INFIMUM CONVOLUTION INEQUALITY 11
References
1. R. Adamczak, R. Lata la, A. E. Litvak, K. Oleszkiewicz, A. Pajor, and N. Tomczak-Jaegermann,
A short proof of Paouris’ inequality , Canad. Math. Bull. (2014), no. 1, 3–8.MR 31507102. N. Feldheim, A. Marsiglietti, P. Nayar, and J. Wang, A note on the convex infimum convolu-tion inequality , to appear in Bernoulli, preprint (2015), arXiv:1505.00240 .3. N. Gozlan, C. Roberto, P.M. Samson, Y. Shu, and P. Tetali,
Characterization of a class ofweak transport-entropy inequalities on the line , to appear in Ann. Inst. Henri Poincar´e Probab.Stat., preprint (2015), arXiv:1509.04202v2 .4. N. Gozlan, C. Roberto, P.M. Samson, and P. Tetali,
Kantorovich duality for general transportcosts and applications , to appear in J. Funct. Anal., preprint (2014), arXiv:1412.7480v4 .5. O. Gu´edon, P. Nayar, and T. Tkocz,
Concentration inequalities and geometry of convex bodies ,Analytical and probabilistic methods in the geometry of convex bodies, IMPAN Lect. Notes,vol. 2, Polish Acad. Sci. Inst. Math., Warsaw, 2014, pp. 9–86. MR 33290566. R. Lata la,
On some problems concerning log-concave random vectors , to appear in IMAVolume “Discrete Structures: Analysis and Applications”, Springer.7. R. Lata la and M. Strzelecka,
Comparison of weak and strong moments for vectors with inde-pendent coordinates , preprint (2016), arXiv:1612.02407v1 .8. ,
Weak and strong moments of ℓ r -norms of log-concave vectors , Proc. Amer. Math.Soc. (2016), no. 8, 3597–3608. MR 35037299. R. Lata la and T. Tkocz, A note on suprema of canonical processes based on random variableswith regular moments , Electron. J. Probab. (2015), no. 36, 17. MR 333582710. R. Lata la and J. O. Wojtaszczyk, On the infimum convolution inequality , Studia Math. (2008), no. 2, 147–187. MR 244913511. B. Maurey,
Some deviation inequalities , Geom. Funct. Anal. (1991), no. 2, 188–197.MR 109725812. Krzysztof Oleszkiewicz, Precise moment and tail bounds for Rademacher sums in terms ofweak parameters , Israel J. Math. (2014), no. 1, 429–443. MR 327344713. G. Paouris,
Concentration of mass on convex bodies , Geom. Funct. Anal. (2006), no. 5,1021–1049. MR 227653314. Y. Shu and M. Strzelecki, A characterization of a class of convex log-Sobolev inequalities onthe real line , preprint (2017), arXiv:1702.04698 . Institute of Mathematics, University of Warsaw, Banacha 2, 02–097 Warsaw, Poland.
E-mail address : [email protected] Institute of Mathematics, University of Warsaw, Banacha 2, 02–097 Warsaw, Poland.
E-mail address : [email protected] Mathematics Department, Princeton University, Fine Hall, Princeton, NJ 08544-1000USA.
E-mail address ::