Non-asymptotic variance bounds and deviation inequalities by optimal transport

The purpose of this note is to show how simple Optimal Transport arguments, on the real line, can be used in Superconcentration theory. This methodology is efficient to produce sharp non-asymptotic variance bounds for various functionals (maximum, median, $l^p$ norms) of standard Gaussian random vectors in $\R^n$. The flexibility of this approach can also provide exponential deviation inequalities reflecting preceding variance bounds. As a further illustration, usual laws from Extreme theory and Coulomb gases are studied.

It is well known that concentration of measure is an effective tool in various mathematical areas (cf. [8]). In a Gaussian setting, classical concentration results typically produce, for f : R n → R a Lipschitz function with Lipschitz constant f Lip , with γ n the standard Gaussian measure on R n . Another instance of concentration of measure is the Poincaré's inequality satisfied by γ n . Namely, for f ∈ L 2 (γ n ) smooth enough : (1.2) Var where | · | stands for the Euclidean norm on R n . As effective as (1.1) and (1.2) are, their generality can lead to sub-optimal bounds in some particular case. For instance, consider the 1-Lipschitz function on R n f (x) = max i=1,...,n x i . At the level of the variance, (1.2) gives with M n = max i=1,...,n X i where (X 1 , . . . , X n ) stands for a standard Gaussian random vector in R n , whereas it has been proven that Var(M n ) ≤ C/ log n with C > 0 a numerical constant. At an exponential level, (1.1) is not satisfying either. Indeed, it is well known in Extreme theory (cf. [15]) that M n can renormalized by some numerical constants, a n = √ 2 log n and b n = a n − log 4π+log log n 2an , n ≥ 1, such that Date: Note of April 11, 2018. 1 a n (M n − b n ) → Λ 0 in distribution, as n → ∞, where Λ 0 corresponds to the Gumbel distribution : Then, it is clear that the asymptotics of Λ 0 are not Gaussian but rather exponential on the right tail and double exponential on the left tail. It is now obvious that (1.1) and (1.2) lead to sub-optimal results for the function f (x) = max i=1,...,n x i . When such phenomenon happens it is referred as Superconcentration phenomenon (cf. [10]). This kind of phenomenon could be seen for different functionals of Gaussian random variables (and also, as we will see, for other laws of probability) and as been studied in [7,22,23,17,24]. . . .
The purpose of this note is to show how simple transport arguments on the real line can easily lead to weighted Poincaré's inequalities together with deviation inequalities which are relevant in Superconcentration theory. In particular, we will emphasize the fact that such results can be obtained by transporting the Exponential measure toward the measure of interest.
Let us describe the setting of our work before stating our main results. Let µ and ν be two probability measures on R. Assume that both of these measures are absolutely continuous with respect to the Lebesgue measure on R. More precisely, assume that there exists two smooth functions g : R → R and h : R → R such that dµ(x) = h(x)dx, dν(x) = g(x)dx Then, let X be a random variable with law µ and Y be a random variable with law ν. Denote by H (respectively by G) the cumulative distribution function of X (respectively Y ) and define the hazard function associated to the probability measure µ by Similarly, κ ν will be the hazard function associated to ν.
Besides, we will also assume that ν satisfies a Poincaré inequality on R with constant C ν > 0. That is to say, for f : R → R smooth enough, Remark. It is classical (cf. [16]) that ν n = ν ⊗ . . . ⊗ ν will also satisfy a Poincaré's inequality with the same constant C ν .
We will denote by T : R n → R n the transport map between µ n and ν n . It satisfies, for any Borelian function f : R n → R, and T (x 1 , . . . , x n ) = t(x 1 ), . . . , t(x n ) with t : R → R the monotone rearrangement map pushing ν toward µ (cf. section two).
In the sequel of this note (unless otherwise stated), Y = (Y 1 , . . . , Y n ) will stand for a random vector in R n with L(Y ) = ν n and X = (X 1 , . . . , X n ) for a random vector in R n with L(X) = µ n . Now, let us state our main results. Theorem 1.1. With the preceding notations, for any function f : R n → R smooth enough, n ≥ 1, we have As we will see, preceding Theorem can be used to obtain exponential deviation inequality for M n = max i=1,...,n X i .
and there exists ǫ n such that Then, for any t ≥ 0 and n ≥ 1, Remark. As it will be clear in the sequel, the arguments can also be performed for any other order statistics obtained from the random vector X.
To ease the understanding of our results, we give below an application of them when ν is the (symmetric) Exponential measure on R and µ is the standard Gaussian measure γ 1 on R. Proposition 1.1. For f : R n → R smooth enough and n ≥ 1, we have In particular, applied to (a smooth approximation of) f (x) = max i=1,...,n x i , we get, for every n ≥ 1, The following deviation inequality holds, for any n ≥ 1, Remark. Notice that preceding results improve upon classical concentration of measure (namely (1.1) and (1.2)) and can also be used for other functionals such as the Median.
Throughout all the article C will stand for a positive numerical constant which may change at each occurence.

Tools and proofs of the main results
2.1. Basics facts. First, let us expose the elementary tools from Optimal Transport, on the real line, that will be needed in the sequel. We want to highlight the fact that we will mostly choose (in practice) ν as the Exponential measure on R + (or as the symmetric Exponential mesure on R) from which we will improve some concentration properties satisfied by the measure of interest µ. However, when stated, we will not specify the measure µ and ν in our results.
Recall that the monotone transport from ν to µ (cf. [25] for more details) is obtained by an application t : R → R such that, for every x ∈ R, Which leads, after differentiation, to the following equality Then, the application T : R n → R n defined by T (x) = t(x 1 ), . . . , t(x n ) , for every x = (x 1 , . . . , x n ) ∈ R n transports ν n on µ n . In particular, for any f : R n → R smooth enough, The following Lemma (cf. [16]) will also be useful in the sequel.
The preceding Lemma will be combined with Harris's negative association inequality (cf. [8]) in order to prove the deviation inequality from Theorem 1.2.
Now, let us state Harris's result. Recall that a fonction f : R n → R is considered to be non-increasing (respectively non-decreasing) if it is non-increasing, (respectively non-decreasing) in each coordinates while the others are fixed.
[Harris] Let f : R n → R be a non-decreasing function and g : R n → R be a non-increasing function . Let X 1 , . . . , X n be independant random variables and set X = (X 1 , . . . , X n ). Then Remark. As we will explain in details later, Harris' negative association was a crucial argument in [7] when they studied order statistics.

Variance bounds.
We give below the proof of Theorem 1.1.
Proof. Since T transports ν n on µ n , we have Then, one can apply the Poincaré's inequality, satisfied by the measure ν n , to the function f • T : Besides, relation (2.2) yields that Remark. As we will see on the examples, the important step will be to estimate the behaviour of the transport map t in order to get some relevant bound on the variance of various functionals.
Notice that this approach is reminiscent of some previous work of Barthe and Roberto [3] or Gozlan [14] on the so-called weighted Poincaré's inequalities on the real line. Although our approach is similar in nature, the method of Barthe and Roberto relies on Hardy's inequality whereas ours is based on monotone rearrangement argument on the real line. Our methodology is very similar to Gozlan's work [14] (in his article the transport map T is denoted by ω −1 ).

2.3.
Deviation inequality. Now, let us prove Theorem 1.2 with the combination of Theorem 1.1 together with Lemma 2.1 and Proposition 2.1.
Recall that, given an i.i.d. sample X 1 , . . . , X n with common law µ we define M n , n ≥ 1, as For any θ > 0, apply Theorem 1.1 to, a suitable approximation of, the function e θf with f (x) = max i=1,...,n x i and notice that the partial derivatives with M n = max i=1,...,n X i . Then, under the hypothesis of Theorem 1.2, use Harris's inequality (2.1). Thus, The conclusion follows easily with Lemma 2.1.

Applications
In this section, we provide some applications, in different mathematical areas, of Theorem 1.1 and Theorem 1.2.
3.1. Extreme Theory. We refer to [15,13] for more details about Extreme Theory. Recall that, given a probability measure µ (absolutely continuous with respect to the Lebesgue measure) and an i.i.d. sample X 1 , . . . X n with L(X 1 ) = µ, it is a classical fact that one can find renormalizing constants a n and b n such that a n (M n − b n ) (where M n = max i=1,...,n X i ) converges in distribution as n → ∞ and the limiting distributions are now fully caracterized. We will show that our main results are revelant to produce non-asymptotic variance bounds and deviation inequality in accordance to Extreme Theory. Let us begin at the level of the variance.
3.1.1. Non-asymptotics variance bounds. Let us start with a pedagogical example from the Weibull's domain of attraction. To do so, we choose ν as the standard Exponential measure on R + (that is to say H(x) = 1 − e −x if x ≥ 0, H(x) = 0 otherwise). Then, Theorem 1.1 yields the following Corollary Corollary 3.1. If Y follows a standard Exponential distribution on R + then, for any function f : R n → R smooth enough and every n ≥ 1, where X 1 , . . . , X n are independant random variables with distribution µ.
In particular, for (any smooth approximation of ) where M n = max i=1,...,n X i and C > 0 is a numerical constant.
In particular, if µ stands for the uniform measure on [0, 1] we have Proof. The first part is a straightforward application of Theorem (1.1). Now, If µ stands for the uniform measure on [0, 1] we have κ µ (x) = 1 x∈[0,1] It is now an easy task to show that the preceding inequality is sharp. Indeed, for any t ∈ [0, 1], P(M n ≤ t) = t n this implies that the maximum M n admits t → nt n−1 1 [0,1] as density with respect to the Lebesgue measure. Thus, . The same estimates also imply that (1) Recall that, n(M n − 1) converge in law toward the Weibull distribution. So, the preceding bound is the correct order of the variance of M n .
(2) More generally, if µ stands for the Beta law with parameter a, b > 0, it is not difficult to show that, for every x ∈ [0, 1], Notice that if a = b = 1 we recover the estimates for the uniform measure. When a > 0 and b > 0 it seems hard to achieve the expected bound (of order n −b ) on the variance from the preceding estimate of κ µ .
(3) It is also possible to send the standard exponential measure toward the Paréto distribution (which belongs to the Fréchet domain of attraction), however this leads to a trivial bound which is not really relevant.
Now, let us focus on the domain of attraction of the Gumbel distribution. To this task, we will transport the symmetric Exponential mesure (on R) ν towards strictly log-concaves measure µ (on R) (the standard Gaussian measure for instance).
Recall that ν admits the following density g(x) = 1 2 e −|x| with respect to the Lebesgue and admits G( x > 0 as a cumulative distribution function. Elementary calculus yields that Thus, Theorem 1.1 implies the following Corollary Corollary 3.2. If Y follows the symmetric Exponential distribution on R then, for any functions f : R n → R smooth enough, where X = (X 1 , . . . , X n ) has for distribution µ n .
Remark. Here, the constant 4 stands for the Poincaré constant of the symmetric Exponential measure (cf. [2]).
To illustrate the preceding Corollary, we will need a technical Lemma. This one is a precise estimation of the behaviour of the transport function which will permit to obtain relevant bounds for the variance of the maximum of symmetric (strictly) log-concave measure dµ( Then, the following holds with C α > 0 a numerical constant only depending on α.
Proof. We would like to bound, for any x ∈ R, the following ratio Let A > 0 be sufficiently large. For x > A, the equation (3.5) is easily bounded by standard estimates (cf. [1]), we get , For x belonging to the compact [0, A], there exists C > 0 such that |t ′ •t −1 (x)| ≤ C.
To sum up, So, it is enough to bound from above t −1 (x) when x < −A in order to conclude. Using the symmetry of the law µ, we obtain Thus, for x < −A, .
Similarly, when −A ≤ x ≤ 0, we also obtain that |t ′ • t −1 (x)| ≤ C Finally, all of this can be rewritten as follows If V is the quadratic potential associated to the standard Gaussian measure, we obtain, thanks to Lemma 3.1 and Corollary 3.2, the following result (as announced in the introduction).  Var In particular, applied to (a smooth approximation of ) f (x) = max i=1,...,n x i , we get, for every n ≥ 1, Remark. Notice that inequality (3.7) has been already obtained, in dimension one, in [6,5].
Proof. Indeed, for the function maximum, ∂ i f = 1 Ai , i = 1, . . . , n with A i = {X i = max j=1,...,n X j } and, again, observe that (A i ) i=1,...,n is a partition of R n . Therefore, n with X 1 a Gaussian standard random variable. Then, we can use the following estimate (cf. [17] (Lemma 2.5) or the appendix in [10]) to bound the preceding quantity : for any t ≥ 0, Thus, Var(M n ) ≤ C log n . Remark. Let us make few remarks on what preceed.
(1) As mentionned in the introduction, √ 2 log n(M n − b n ) converge, when n → ∞, in law toward the Gumbel distribution (the precise value of b n is irrelevant here but can be found in [13,15]). So, the preceding Corollary gives a non-asymptotic variance bound of the maximum in accordance with Extreme theory. Besides, such a bound is classically obtained by hypercontractive and interpolation arguments (cf. [10]). Here, we provide an alternative proof based on Optimal Transport arguments.
(2) Let us further notice that the scheme of proof can also be performed for the function f (x) = Med(x 1 , . . . , x n ), n ≥ 1, which correspond to the correct order of magnitude of the variance of the median (cf. [7]). Notice that, as far as we know, such bounds can not be obtained by hypercontractive arguments.
Since, if X 1 stands for a random variables with law µ, we can proceed as the Gaussian case. Indeed, P(X 1 ≥ t) ∼ 1 t α −1 e −t α /α as t → ∞. In particular, for t large enough, this yields that P(X 1 ≥ t) ≥ 1 2t α−1 e −t α /α . Remark. Following the proof (when α = 2 ) in [15], it can be easily proved that a n (M n − b n ) → Λ 0 , in law, when n → ∞, with a n = α(log n) 2(α−1)/α et b n = (log n) 1/α − log(αZ)+ α−1 α log log n (log n) (α−1)/α . Therefore, Corollary gives a non-asymptotic bound of the variance of the maximum reflecting this convergence result. We want to highlight the fact that such bound is another example of the Superconcentration phenomenon. Nevertheless, as far as we know, such estimates can not be obtained by hypercontractive methods (when α > 2) as the Gaussian case.

Deviation inequalities.
It is possible to use the preceding variance bounds to immediately obtain deviation inequalities thanks to Theorem 1.2.
Proposition 3.2. The following deviation inequality holds, for any n ≥ 1, Remark.
(1) Concerning Extreme theory, notice that this Theorem is only relevant if µ belongs to the domain of attraction of the Gumbel distribution. Indeed, the right tail of the Gumbel distribution behaves like t → e −t (whereas the left tail goes faster to 0 with the following asymptotic : t → e −e t ).
(3) Similar results can be also be obtained if one replace the maximum by another order statistics.

3.2.
Variance of l p , p ≥ 2 norm of standard Gaussian vector. As a further illustration of our approach, we propose to recover some variance's bounds of l p -norms, p ≥ 1, of a standard Gaussian vector, obtained in [17]. The proof will be based on Proposition 3.1. We will adopt the following notations : given a vector x = (x 1 , . . . , x n ) ∈ R n we denote by x p p = n i=1 |x i | p its norm.
In the article of Paouris et al. [17], the authors have noticed that the variance of X p is not precisely estimated by classical concentration theory. More precisely, classical tools from the theory of concentration of measure such as Poincaré's inequality or the isoperimetric Gaussian inequality yields the following bound Var( X p ) ≤ max(n 2/p−1 , 1), p ≥ 1.
According to [17], this bound is only optimal when 1 ≤ p ≤ 2. The authors of [17] improved this bound by using precise estimates of moments of Gaussian functionnals together with logarithmic Sobolev inequality (through the so-called Talagrand's inequality). More precisely, Theorem 3.1 (Paouris,Valettas, Zinn ). Let X be a standard Gaussian vector on R n then Var( X p ) ≤ C 2 p p n 2/p−1 , 2 < p ≤ c log n, C/ log n, p > c log n, with C, c > 0 some numerical constants which are independant of n and p.
Here, we propose to recover Proposition 3.1 with Proposition 3.1. We will only deal with the second assertion of the Proposition (the first part can be proved with similar arguments). Proof. Let δ > 0 be a parameter to be choosen later and denote by B ∞ (0, δ) = {x ∈ R n , x ∞ < δ}. Thus, We recall the following relations between l p and l q norms, for p < q, which will be freely used in the sequel, On one hand, since p < 2(p − 1), On the other hand, since p < 2(p − 2), δ)) Furthermore, notice that the following upper bound is satisfied So far we have obtained, Then, choose δ = √ 2 log n (with n large enough) to conclude. Indeed, we have In other terms which is the result.
3.3. Coulomb gazes. This section exposes another application of our main results in another mathematical area. We want to highlight that, in this section, the factors µ i , i = 1, . . . , n (from the product measure µ 1 ⊗ . . . µ n ) will not assumed to be identical. This difference justifies the separation of this section from the others.
Now, let us introduce few notions about Coulomb gazes and the results obtained by Chafaï and Péché in [9]. Let us consider a gas of charged particules {z 1 , . . . , z n } on the complex plane C, confined individually by the external field Q and experiencing a Coulomb pair repulsive interaction. This corresponds to the probability distribution C n with density proportional to (3.11) (z 1 , . . . , z n ) ∈ C n → n j=1 e −nQ(zj ) 1≤j<l≤n |z j − z k | β with β > 0 is a fixed parameter and where Q is a fixed smooth function. We will focus on the particular case where β = 2 and Q(z) = V (|z|) with V (t) = t α , t ≥ 0, α ≥ 1. We are interested in the study of (3.12) |z| (1) ≥ . . . ≥ |z| (n) the order statistics of the moduli of the Coulomb gas. Notice that |z| (1) = max 1≤k≤n |z k |.
In their article, the authors proved the following representation formula with R (1) ≥ . . . ≥ R (n) the order statistics associated to independent random variables R 1 , . . . , R n where R k , for k = 1, . . . , n, has a density proportional to Remark. More precisely, the case β = 2 and V (r) = r 2 has been proved by Rider in [18]. Chafaï and Péche extended Rider's results when β = 2 and V statisfies some convexity assumption together with some decay conditions at infinity.
We will see that it is not difficult to get a non-asymptotic upper bound on the variance of |z (1) |, together with a deviation inequality for our main results. A crucial step is the representation formula (3.12) of |z (1) | : where R 1 , . . . , R n are independent random variables and R k , for any k = 1, . . . , n, has a density proportionnal to Then, it is possible to transport the standard Exponential measure on R n + toward the measure µ 1 ⊗ · · · ⊗ µ n with µ k = L(R k ) for any k = 1, . . . , n. Notice then, for every k = 1, . . . , n, that µ k is log-concave on R + with potential So it is not difficult to prove (thanks to the estimates from [1]) that with C α > 0 a numerical constant. Thus, Proposition 3.1 yields C α n log n Also, Theorem 1.2 immediatly gives the following deviation inequality where C α > 0 is a numerical constant that does not depend on n. In other words, we have obtained a non asymptotic deviation inequality together with a variance bounds which are in accordance with Theorem 3.3. That is to say, we have proven Proposition 3.4. Let {z 1 , . . . , z n } be a Coulomb gazes with density proportional to with Q = V (|z|) and V (t) = t α , α ≥ 1. Then, for any n > 1, the following holds with C α > 0 a numerical constant, independent of n, and P n log n |z (1) with C α > 0 a numerical constant independent of n.

Remarks and comparison with existing literature
In this section, we will briefly explain how stronger functional inequalities can be used to reach the right asymptotic of the left tail in the Gumbel's domain of attraction. Then, we will compare our main results with the existing literature.

4.1.
Few words on isoperimetric inequalities. As we have already seen, the transport of the Exponential measure (toward a measure µ n ) permit to improve some concentration's properties of the measure µ n . This phenomenon as already been observed by Talagrand in [21]. He used the isoperimetric inequality (involving a mixture of l 1 and l 2 balls) satisfied by the (symmetric) Exponential measure µ n to improve the isoperimetric inequality satisfied by the standard Gaussian measure. More precisely, such improvement can be seen on the following concentration inequality Remark.
(1) This type of inequality recently appeared in [22] for more general Gaussian measure.
(2) This gives the correct asymptotic behaviour (with respect to Extreme Theory) of the right tail of the maximum. However, the asymptotic behaviour of the left tail, in (4.1), is still sub-obtimal.
The symmetry of the (two sided) Exponential measure on R, through Talagrand's isoperimetric inequality, seems to not make any distinctions between the left tail from the right and only gives a exponential decay. In [4], Bobkov studied a different isoperimetric problem (with the standard Exponential measure and uniform enlargements B ∞ instead). The lack of symmetry of the (standard) Exponential measure can be used to achieve the correct decay of the left tail on the maximum (in the Gumbel's domain of attraction).
More precisely, Bobkov proved the following Theorem.
Theorem 4.1 (Bobkov). Let ν n stands for the (standard) Exponential measure on R + . Then, for every non empty ideal A ⊂ R n + such that ν n (A) = ν n (B ∞ ) = and every r ≥ 0, the following inequality holds : in other words,
(2) If n → ∞ and ν n (A) = p is constant (with respect to n), the right hand side of the preceding inequality decreases and converges toward a double exponential. That is to say ν n (A + rB ∞ ) ≥ exp(−e −r log(1/p)).
As presented in [4], it possible to achieve the following deviations inequalities for a measure µ n by transporting the Exponential measure ν n . Theorem 4.2. (Bobkov) Let X 1 , . . . , X n be i.i.d. random variables with L(X 1 ) = µ ∈ F 0 and set M n = max i=1,...,n X i . Then, for every p, 0 < p < 1, every t ≥ 0, with m p stands for the quantile of order p of M n and C, c > 0 are numerical constants.
Remark. In [4], there is some workable conditions which describe the set of measure F 0 . For instance Gamma measure or absolute value of standard Gaussian measure belong to F 0 .
In particular, if we choose p such that p 1/n = F −1 (1 − 1/n), m p corresponds to the renormalizing term used in Extreme theory. For instance, for the the Gamma measure, Bobkov's Theorem yields Proposition 4.1. Let X 1 , . . . , X n be i.i.d Gamma random variables. Set M n = max i=1,...,n X i , then for every t ≥ 0 and every n ≥ 1 P(M n − log n ≥ t) ≤ Ce −ct and P(M n − log n ≤ −t) ≤ Ce −e ct with C, c > 0 are numerical constants.
These non-asymptotic deviations inequalities express the correct tail behaviour of the maximum of Gamma random variables (which belongs to the Gumbel's domain of attraction). Furthermore, such inequalities imply that P(|M n − log n| ≥ t) ≤ Ce −ct , which can be integrated to recover the fact (that can be easily obtained from Poincaré's inequality) that Var(M n ) ≤ C.
All of this should be obtained for the maximum of absolutes values of independent and identically distributed standard Gaussian random variables. The details are left to the reader. Recall that such kind of inequality as already been obtained by Schetchtman in [19].

4.2.
Comparison with existing literature. In this section we compare our main results with recent articles which produce Superconcentration for i.i.d. random variables by other means.

4.2.1.
Renyi's representation and order statistics. The authors of [7] combined three different arguments to bound the variance (or to obtain deviation inequalities) of order statistics from a sample of i.i.d. random variables. More precisely, let X 1 , . . . X n be real i.i.d. random variables. Denote the associated order statistics by In their article [7], the authors obtained the following result Their scheme of proof is based on Renyi's representation formula (cf. [13]), which allow one to express order statistics in terms of renormalized sums of i.i.d Exponential random variables. They combined this representation with Efron-Stein's inequality (cf. [8]) and Harris's negative association (to do so they must assume that the function κ µ is non-increasing) in order to bound from above the variance of X (k) , k = 1, . . . , n.
The major drawback of this approach is that it can only be performed on order statistics. Our method seems to be more fexible and allows one to recover (from the measure ν) Poincaré's inequality (for the measure of interest µ) when the transport map is Lipschitz. It is also clear that the hypothesis (non-increasing) on the function κ µ is not necessary to obtain upper bound on the variance. We have shown that this argument can only be used to reach exponential deviation inequalities. On this matter, Berstein's type of deviation inequality from [7] is more precise than ours, but it does not give back a relevant bound on the variance after integration. It is also surprising that the authors [7] did not deal with the more classical standard Gaussian case (without absolute value).

4.2.2.
Hypercontractive approach and semigroup interpolations. The comparison with the hypercontractive approach is straightforward. On one hand the hypercontractive approach can be used to deal with correlated Gaussians vectors (cf. [10,22,23]). On the other hand, the hypercontractive method can not reach any decay faster than 1/ log n and can only provide an exponential decay at the level of concentration inequalities. For instance, it does not seem possible to show, with hypercontractive arguments, that neither the variance of the Median of a standard Gaussian sample is of order 1/n nor to obtain the right order of the fluctuations of log-concave measure with potential V (x) = |x| α when α > 2 (notice also that hypercontractivity is not satisfied when 0 < α < 1).

4.2.3.
Comparison with Talagrand's inequality. This section's purpose is to compare Proposition 3.1 with the following result. Remark. This inequality was originally proved in [20] and have been a major tool in Superconcentration theory (cf. [10,22,23]).
To this task, it is enough to deal with the dimension one case. Such inequalities are not comparable as it can be seen on the following functions f M and f ǫ . and, for every 0 < ǫ < 1, consider the function f , defined by f ǫ (x) = |x| ǫ + 1, |x| ≤ ǫ 0, |x| > ǫ, Then, it is enough to choose ǫ = 1/2n, n ≥ 1.