Simple bounds for the convergence of empirical and occupation measures in 1-Wasserstein distance

We study the problem of non-asymptotic deviations between a reference measure and its empirical version, in the 1-Wasserstein metric, under the standing assumption that the measure satisfies a transport-entropy inequality. We extend some results of F. Bolley, A. Guillin and C. Villani with simple proofs. Our methods are based on concentration inequalities and extend to the general setting of measures on a Polish space. Deviation bounds for the occupation measure of a Markov chain are also given, under the assumption that the chain is contractive on the space of Lipschitz functions. Throughout the text, several examples are worked out, including the cases of Gaussian measures on separable Banach spaces, and laws of diffusion processes.

(1) δ Xi denote the empirical measure associated with the i.i.d. sample (X i ) 1≤i≤n , then with probability 1, L n ⇀ µ as n → +∞ (here the arrow denotes narrow convergence, or convergence against all bounded continuous functions over E). This theorem is known as the empirical law of large number or Glivenko-Cantelli theorem and is due in this form to Varadarajan [31]. Quantifying the speed of convergence for an appropriate notion of distance between probability measures is an old problem, with notable importance in statistics. For many examples, we refer to the book of Van der Vaart and Wellner [30] and the Saint-Flour course of P.Massart [24].
Our aim here is to study non-asymptotic deviations in 1-Wasserstein distance. This is a problem of interest in the fields of statistics and numerical probability. More specifically, we provide bounds for the quantity P(W 1 (L n , µ) ≥ t) for t > 0, i.e. we quantify the speed of convergence of the variable W 1 (L n , µ) to 0 in probability.
This paper seeks to complement the work of F.Bolley, A.Guillin and C.Villani in [7] where such estimates are obtained for measures supported in R d . We sum up (part of) their result here. Suppose that µ is a probability measure on R d for 1 ≤ p ≤ 2 that satisfies a T p (C) transportation-entropy inequality, that is W p (ν, µ) ≤ CH(ν|µ) for all ν ∈ P p (R d ) (see below for definitions). They obtain a non-asymptotic Gaussian deviation estimate for the p−Wasserstein distance between the empirical and true measures : P(W p (L n , µ) ≥ t) ≤ C(t) exp(−Knt 2 ). This is an effective result : the constants K and C(t) may be explicitely computed from the value of some square-exponential moment of µ and the constant C appearing in the transportation inequality.
The strategy used in [7] relies on a non-asymptotic version of (the upper bound in) Sanov's theorem. Roughly speaking, Sanov's theorem states that the proper rate function for the deviations of empirical measures is the entropy functional, or in other words that for 'good' subsets A ∈ P(E), P(L n ∈ A) ≍ e −nH (A|µ) where H(A|µ) = inf ν∈A H(ν|µ) (see [10] for a full statement of the theorem). In a companion work [5], we derive sharper bounds for this problem, using a construction originally due to R.M. Dudley [12]. The interested reader may refer to [5] for a summary of existing results. Here, our purpose is to show that in the case p = 1, the results of [7] can be recovered with simple arguments of measure concentration, and to give various extensions of interest.
• We would like to consider spaces more general than R d .
• We would like to encompass a wide class of measures in a synthetic treatment. In order to do so we will consider more general transportation inequalities, see below. • Another interesting feature is to extend the result to dependent sequences such as the occupation measure of a Markov chain. This is a particularly desirable feature in applications : one may wish to approximate a distribution that is unknown, or from which it is practically impossible to sample uniformly, but that is known to be the invariant measure of a simulable Markov chain.
Acknowledgements. The author thanks his advisor Patrick Cattiaux for suggesting the problem and for his advice. Arnaud Guillin is also thanked for enriching conversations.
In the remainder of this section, we introduce the tools necessary in our framework : transportation distances and transportation-entropy inequalities. In Section 2, we give our main results, as well as explicit estimates in several relevant cases. Section 3 is devoted to the proof of the main result. Section 4 is devoted to the proof of Theorem 2.5. In Section 5 we show how our strategy of proof can extend to the dependent case.

1.2.
A short introduction to transportation inequalities.

Transportation costs and Wasserstein distances.
We recall here basic definitions and propositions ; for proofs and a thorough account of this rich theory, the reader may refer to [32]. Define P p , 1 ≤ p < +∞, as the set of probability measures with a finite p-th moment. The p-Wasserstein metric W p (µ, ν) between µ, ν ∈ P p is defined by where the infimum is on probability measures π ∈ P(E × E) with marginals µ and ν. The topology induced by this metric is slightly stronger than the weak topology : namely, convergence of a sequence (µ n ) n∈N to a measure µ ∈ P p in the p-Wasserstein metric is equivalent to the weak convergence of the sequence plus a uniform bound on the p-th moments of the measures µ n , n ∈ N.
We also recall the well-known Kantorovich-Rubinstein dual characterization of W 1 : let F denote the set of 1-Lipschitz functions f : E → R that vanish at some fixed point x 0 . We have :

Transportation-entropy inequalities.
For a very complete overview of the subject, the reader is invited to consult the review [15]. More facts and criteria are gathered in Appendix A For µ, ν ∈ P(E), define the relative entropy H(ν|µ) as if ν is absolutely continuous relatively to µ, and H(ν|µ) = +∞ otherwise. Let α : [0, +∞) → R denote a convex, increasing, left-continous function such that α(0) = 0. Definition 1.1. We say that µ ∈ P p (E) satisfies a T p (C) inequality for some C > 0 if for all ν ∈ P p (E), We say that µ ∈ P(E) satisfies a α(T d ) inequality if for all ν ∈ P(E), Observe that T 1 (C) inequalities are particular cases of α(T d ) inequalities with α(t) = 1 C t 2/p . From here on, our focus will be on α(T d ) inequalities.

Results and applications
2.1. General bounds in the independent case. Let us first introduce some notation : if K ⊂ E is compact and x 0 ∈ K, we define the set F K of 1-Lipschitz functions over K vanishing at x 0 , which is is also compact w.r.t. the uniform distance (as a consequence of the Ascoli-Arzela theorem). We will also need the following definition : be a totally bounded metric space. For every δ > 0, define the covering number N (A, δ) of order δ for A as the minimal number of balls of radius δ needed to cover A.
We state our first result in a fairly general fashion.
Remark. With a mild change in the proof, one may replace in (6) the term t/2 by ct for any c < 1, with the trade-off of choosing a larger compact set, and thus a larger value of C t . For the sake of readability we do not make further mention of this.
The result in its general form is abtruse, but it yields interesting results as soon as one knows more about α. Let us give a few examples.
Corollary 2.4. Suppose that µ verifies the modified transport inequality (as observed in paragraph A.0.1, this is equivalent to the finiteness of an exponential moment for µ). Then, for t ≤ C/2, (observe that A(n, t) → C t when n → +∞).
Proof of Corollary 2.3. In this case, we have α(t) = 1 C t 2 , and so so that we get Proof of Corollary 2.4. Here, α(x) = 1 4 ( 1 + 4x C − 1) 2 , and one can get the bound By concavity of the square root function, for u ≤ 1, we have . This in turn gives the announced result.
Our technique of proof, though related to the one in [7], is based on different arguments : we make use of the tensorization properties of transportation inequalities as well as the estimates (20) in the spirit of Bobkov-Götze, instead of a Sanov-type bound. The notion that is key here is the phenomenon of concentration of measure (see e.g. [21]) : its relevance in statistics was put forth very explicitely in [24]. We may sum up our approach as follows : first, we rely on existing tensorization results to obtain concentration of W 1 (L n , µ) around its mean E[W 1 (L n , µ)], and in a second time we estimate the decay of the mean as n → +∞. Despite technical difficulties, the arguments are mostly elementary.
The next theorem is a variation on Corollary 2.3. Its proof is based on different arguments, and it is postponed to Section 4. We will use this theorem to obtain bounds for Gaussian measures in Theorem 2.5.

Remark.
As earlier, we could improve the factor 1/8C in the statement above to any constant c < 1/C, with the trade-off of a larger constant K t .

2.2.
Comments. We give some comments on the pertinence of the results above. First of all, we argue that the asymptotic order of magnitude of our estimates is the correct one. The term "asymptotic" here means that we consider the regime n → +∞, and the relevant tool in this setting is Sanov's large deviation principle for empirical measures. A technical point needs to be stressed : there are several variations of Sanov's theorem, and the most common ones (see e.g. [10]) deal with the weak topology on probability measures. What we require is a version of the principle that holds for the stronger topology induced by the 1-Wasserstein metric, which leads to slightly more stringent assumptions on the measure than in Theorem 2.2. With this in mind, we quote the following result from Wang [33] : Proposition 2.6. Suppose that µ ∈ P(E) satisfies e ad(x,x0) µ(dx) < +∞ for all a > 0 and some x 0 ∈ E, and a α(T d ) inequality. Then : • for all A ⊂ P(E) closed for the W 1 topology, Consider the closed set A = {ν ∈ P(E), W 1 (µ, ν) ≥ t}, then we have according to the above lim sup With Theorem 2.2 (and the remark following it), we obtain the bound lim sup for all c < 1, and since α is left-continuous, we indeed obtain the same asymptotic bound as from Sanov's theorem.
Let us come back to the non-asymptotic regime. When we assume for example a T 1 inequality, we get a bound in the form P(W 1 (L n , µ) ≥ t) ≤ C(t)e −Cnt 2 involving the large constant C(t). By the Kantorovich-Rubinstein dual formulation of W 1 , this amounts to simultaneous deviation inequalities for all 1-Lipschitz observables. We recall briefly the well-known fact that it is fairly easy to obtain a deviation inequality for one Lipschitz observable without a constant depending on the deviation scale t. Indeed, consider a 1-Lipschitz function f and a sequence X i of i.i.d. variables with law µ. By Chebyshev's bound, for θ > 0, According to Bobkov-Götze's dual characterization of T 1 , the term inside the log is bounded above by e Cθ 2 , for some positive C, whence P( 1 Thus, we may see the multiplicative constant that we obtain as a trade-off for the obtention of uniform deviation estimates on all Lipschitz observables.

Examples of application.
For practical purposes, it is important to give the order of magnitude of the multiplicative constant C t depending on t. We address this question on several important examples in this paragraph.
Example 2.7. Denote θ(x) = 32x log [2 (32x log 32x − 32x + 1)]. In the case E = R d , the numerical constant C t appearing in Theorem 2.2 satisfies : where C d only depends on d. In particular, for all t ≤ 1 2a , there exist numerical constants C 1 and C 2 such that Remark. The constants C d , C 1 , C 2 may be explicitely determined from the proof. We do not do so and only state that C d grows exponentially with d.
Proof. For a measure µ ∈ P(R d ), a convenient natural choice for a compact set of large measure is a Euclidean ball. Denote Next, the covering numbers for B R are bounded by : Using the bound (22) of Proposition B.2, we have This concludes the proof for the first part of the proposition. The second claim derives from the fact that for x > 2, there exists a numerical constant k such that θ(x) ≤ kx log x.
Example 2.7 improves slightly upon the result for the W 1 metric in [7]. One may wonder whether this order of magnitude is close to optimality. It is in fact not sharp, and we point out where better results may be found.
In the case d = 1, W 1 (L n , µ) is bounded above by the Kolmogorov-Smirnov divergence sup x∈R |F n (x) − F (x)| where F n and F denote respectively the cumulative distribution functions (c.d.f.) of L n and µ. As a consequence of the celebrated Dvorestky-Kiefer-Wolfowitz theorem (see [25], [30]), we have the following : if µ ∈ P(R) has a continuous c.d.f., then The behaviour of the Wasserstein distance between empirical and true distribution in one dimension has been very thoroughly studied by del Barrio, Giné, Matran, see [9].
In dimensions greater than 1, the result is also not sharp. Integrating (7), one recovers a bound of the type E(W 1 (L n , µ)) ≤ Cn −1/(d+2) (log n) c . Looking into the proof of our main result, one sees that any improvement of this bound will automatically give a sharper result than (7). For the uniform measure over the unit cube, results have been known for a while. The pioneering work in this framework is the celebrated article of Ajtai, Komlos and Tusnády [1]. M.Talagrand [29] showed that when µ is the uniform distribution on the unit cube (in which case it clearly satisfies a T 1 inequality) and d ≥ 3, there exists c d ≤ C d such that Sharp results for general measures are much more recent : as a consequence of the results of F. Barthe and C. Bordenave [3], one may get an estimate of the type EW 1 (L n , µ) ≤ cn −1/d under some polynomial moment condition on µ.

A first bound for Standard Brownian motion.
We wish now to illustrate our results on an infinite-dimensional case. A first natural candidate is the law of the standard Brownian motion, with the sup-norm as reference metric. The natural idea that we put in place in this paragraph is to choose as large compact sets the α-Hölder balls, which are compact for the sup-norm. However the remainder of this paragraph serves mainly an illustrative purpose : we will obtain sharper results, valid for general Gaussian measures on (separable) Banach spaces, in paragraph 2.3.4.
To this end we use the fact that the Wiener measure is also a Gaussian measure on C α (see [2]). Therefore Lemma D.1 applies : denote On the other hand, according to Corollary C.2, m α and s α are bounded by C α . And Lemma D.3 shows that choosing a = √ 2 log 2/3 ensures Ee a sup t |Bt| ≤ 2. Elementary computations show that for t ≤ 144/ √ 2 log 2, we can pick to comply with the requirement in (8).
Bounds for the covering numbers in α-Hölder balls are computed in [6] : We recover the (unpretty !) bound We quote their result from [17].
is a standard m-dimensional Brownian motion. We assume that b and σ are locally Lipschitz and that for all For each starting point x it has a unique non-explosive solution denoted (X t (x) t≥0 and we denote its law on C([0, 1], R d ) by P x . Theorem 2.9 ( [17]). Assume the conditions above. There exists C depending on A and B only such that for every x ∈ R d , P x satisfies a T 1 (C) inequality on the space C([0, 1], R d ) endowed with the sup-norm.
We will now state our result. A word of caution : in order to balance readability, the following computations are neither optimized nor made fully explicit. However it should be a simple, though dull, task for the reader to track the dependence of the numerical constants on the parameters.
From now on we make the simplifying assumption that the drift coefficient is globally bounded by B (this assumption is certainly not minimal).
. For all 0 < α < 1/2 there exist C α and c depending only on A, B, α and d, and such that for t ≤ c, Proof. The proof goes along the same lines as the Brownian motion case, so we only outline the important steps. First, there exists a depending explicitely on A, B, d such that E Px e a X. ∞ ≤ 2 : this can be seen by checking that the proof of Djellout-Guillin-Wu actually gives the value of a Gaussian moment for µ as a function of A, B, d, and using standard bounds. Corollary C.3 applies for α < 1/2 and p such that The conclusion is reached again by using estimate (9) The small ball function of a Gaussian Banach space (E, µ) is the function We can associate to the couple (E, µ) their Cameron-Martin Hilbert space H ⊂ E, see e.g. [20] for a reference. It is known that the small ball function has deep links with the covering numbers of the unit ball of H, see e.g. Kuelbs-Li [19] and Li-Linde [22], as well as with the approximation of µ by measures with finite support in Wasserstein distance (the quantization or optimal quantization problem), see Fehringer's Ph.D. thesis [13], Dereich-Fehringer-Matoussi-Scheutzow [11], Graf-Luschgy-Pagès [16]. It should thus come as no surprise that we can give a bound on the constant K t depending solely on ψ and σ. This is the content of the next example.
Example 2.11. Let (E, µ) be a Gaussian Banach space. Denote by ψ its small ball function and by σ its weak variance. Assume that t is such that ψ(t/16) ≥ log 2 and t/σ ≤ 8 √ 2 log 2. Then for some universal constant c.
A bound for c may be tracked in the proof. Proof.
Step 1. Building an approximating measure of finite support. Denote by K the unit ball of the Cameron-Martin space associated to E and µ, and by B the unit ball of E. According to the Gaussian isoperimetric inequality (see [20]), for all λ > 0 and ε > 0, 1 λK+εB µ the restriction of µ to the enlarged ball. As proved in [5], Appendix 1, the Gaussian measure µ satisfies a T 2 (2σ 2 ) inequality, hence a T 1 inequality with the same constant. We have On the other hand, denote k = N (λK, ε) the covering number of λK (w.r.t. the norm of E). Let x 1 , . . . , x k ∈ K be such that union of the balls B(x i , ε) contains λK. From the triangle inequality we get the inclusion Choose a measurable map T : λK + εB → {x 1 , . . . , x k } such that for all x, |x − T (x)| ≤ 2ε. The push-forward measure µ k = T # µ ′ has support in the finite set {x 1 , . . . , x k }, and clearly Choose ε = t/16, and 2π is the tail of the Gaussian distribution (we have used the fact that Φ −1 +Υ −1 = 0, which comes from symmetry of the Gaussian distribution).
With some elementary majorizations, this ends the proof.
We can now sharpen the results of Proposition 2.8. Let γ denote the Wiener measure on C([0, 1], R d ) endowed with the sup-norm, and denote by σ 2 its weak variance. Let λ 1 be the first nonzero eigenvalue of the Laplacian operator on the ball of R d with homogeneous Dirichlet boundary conditions : it is well-known that the small ball function for the Brownian motion on R d is equivalent to λ 1 /t 2 when t → +∞. for t small enough.

2.4.
Bounds in the dependent case : occupation measures of contractive Markov chains. The results above can be extended to the convergence of the occupation measure for a Markov process. As an example, we establish the following result.
Let π denote its invariant measure. Let (X i ) i≥0 denote the Markov chain associated with P under X 0 = 0.
Remark. The result is close to the one obtained in the independent case, and, as stressed in the introduction, it holds interest from the perspective of numerical simulation, in cases where one cannot sample uniformly from a given probability distribution π but may build a Markov chain that admits π as its invariant measure.
Remark. We comment on the assumptions on the transition kernel. The first one ensures that the T 1 inequality is propagated to the laws of X n , n ≥ 1. As for the second one, it has appeared several times in the Markov chain literature (see e.g. [17], [26], [18]) as a particular variant of the Dobrushin uniqueness condition for Gibbs measures. It has a nice geometric interpretation as a positive lower bound on the Ricci curvature of the Markov chain, put forward for example in [26]. Heuristically, this condition implies that the Markov chains started from two different points and suitably coupled tend to get closer.

Proof of Theorem 2.2
The starting point is the following result, obtained by Gozlan and Leonard ([14], see Chapter 6) by studying the tensorization properties of transportation inequalities.
Then µ ⊗n ∈ P(E n ) verifies a α ′ (T d ⊕n ) inequality, where α ′ (t) = 1 n α(nt). Hence, for all Lipschitz functionals Z : E n → R (w.r.t. the distance d ⊕n ), we have the concentration inequality Let X i be an i.i.d. sample of µ. Recalling that the distance d ⊕n on E n (as a supremum of 1 n -Lipschitz functions), the following ensues : Therefore, we are led to seek a control on E[W 1 (L n , µ)]. This is what we do in the next lemma. Let δ > 0 and K ∈ E be a compact subset containing x 0 . Let N δ denote the covering number of order δ for the set F K of 1-Lipschitz functions on K vanishing at x 0 (endowed with the uniform distance).
The following holds : Proof. We denote by F the set of 1-Lipschitz functions f over E such that f (x 0 ) = 0. Let us denote we have for f, g ∈ F : When f : E → R is a measurable function, denote by f | K its restriction to K. Notice that for every g ∈ F K , there exists f ∈ F such that f | K = g. Indeed, one may set and check that f is 1-Lipschitz over E. By definition of N δ , there exist functions g 1 , . . . , g N δ ∈ F K such that the balls of center g i and radius δ (for the uniform distance) cover F K . We can extend these functions to functions f i ∈ F as noted above.
Consider f ∈ F and choose f i such that |f − f i | ≤ δ on K : The right-hand side in the last line does not depend on f , so it is also greater than W 1 (L n , µ) = sup F Ψ(f ).
We pass to expectations, and bound the terms on the right. We use Orlicz-Hölder's inequality with the pair of conjugate Young functions (for definitions and a proof of Orlicz-Hölder's inequality, the reader may refer to [27], Chapter 10). We get It is easily seen that 1 K c τ = 1/σ(1/µ(K c )). And we assumed that a is such that E a,1 = exp ad(x, x 0 )dµ ≤ 2, so d(x, x 0 ) τ * ≤ 1/a. Altogether, this yields .
Also, if X 1 , . . . , X n are i.i.d. variables of law µ, as seen above. Putting this together yields the inequality The remaining term can be bounded by a form of maximal inequality. First fix some i and λ > 0 : we have In the last line, we have used estimate (20). Using Jensen's inequality, we may then write So minimizing in λ we have Bringing it all together finishes the proof of the lemma.
We can now finish the proof of Theorem 2.2.

Proof of Theorem 2.5
In this section, we provide a different approach to our result in the independent case. As earlier we first aim to get a bound on the speed of convergence on the average W 1 distance between empirical and true measure. The lemma below provides another way to obtain such an estimate. Lemma 4.1. Let µ k ∈ P(E) be a finitely supported measure such that |Supp µ k | ≤ k. Let D(µ k ) = Diam Supp µ k be the diameter of Supp µ k . The following holds : Proof. Let π opt be an optimal coupling of µ and µ k (it exists : see e.g. Theorem 4.1 in [32]), and let (X i , Y i ), 1 ≤ i ≤ n, be i.i.d. variables on E × E with common law π opt .
Let L n = 1/n n i=1 δ Xi and L k n = 1/n n i=1 δ Yi . By the triangle inequality, we have . With our choice of coupling for L n and L k n it is easily seen that EW 1 (L n , L k n ) ≤ W 1 (µ, µ k ) Let us take care of the last term. We use Lemma 4.2 below to obtain that Observe that the variables nL k n (x i ) follow binomial laws with parameter µ k (x i ) and n. We get : (the last inequality being a consequence of the Cauchy-Schwarz inequality).
Proof. We build a coupling of µ and ν that leaves as much mass in place as possible, in the following fashion : set f (x i ) = µ(x i )∧ν(x i ) and λ = k i1 f i . Set q(x i ) = f i /λ, and define the measures Finally, build independent random variables X 1 ∼ µ 1 , Y 1 ∼ ν 1 , Z ∼ q and B with Bernoulli law of parameter λ. Define It is an easy check that X ∼ µ, Y ∼ ν. Thus we have the bound and this concludes the proof.
Proof of Theorem 2.5. As stated earlier, we have the concentration bound P(W p (L n , µ) ≥ t + EW p (L n , µ)) ≤ e −nt 2 /C . The proof is concluded by arguments similar to the ones used before, calling upon Lemma 4.1 to bound the mean.

Proofs in the dependent case
Before proving Theorem 2.12, we establish a more general result in the spirit of Lemma 3.2.
As earlier, the first ingredient we need to apply our strategy of proof is a tensorization property for the transport-entropy inequalities in the case of nonindependent sequences. To this end, we restate results from [17], where only T 1 inequalities were investigated, in our framework. For x = (x 1 , . . . , x n ) ∈ E n , and 1 ≤ i ≤ n, denote x i = (x 1 , . . . , x i ). Endow E n with the distance d 1 (x, y) = n i1 d(x i , y i ). Let ν ∈ P(E n ), the notation ν i (dx 1 , . . . , dx i ) stands for the marginal measure on E i , and ν i (.|x i−1 ) stands for the regular conditional law of x i knowing x i−1 , or in other words the conditional disintegration of ν i with respect to ν i−1 at x I−1 (its existence is assumed throughout).
The next theorem is a slight extension of Theorem 2.11 in [17]. Its proof can be adapted without difficulty, and we omit it here.
Theorem 5.1. Let ν ∈ P(E n ) be a probability measure such that (1) For all i ≥ 1 and all inequality, and (2) There exists S > 0 such that for every 1-Lipschitz function Then ν verifies the transportation inequalityα(T d ) ≤ H with In the case of a homogeneous Markov chain (X n ) n∈N with transition kernel P (x, dy), the next proposition gives sufficient conditions on the transition probabilities for the laws of the variables X n and the path-level law of (X 1 , . . . , X n ) to satisfy some transportation inequalities. Once again the statement and its proof are adaptations of the corresponding Proposition 2.10 of [17].
Then there exists a unique invariant probability measure π for the Markov chain associated to P , and the measures P n (x, .) and π satisfy α Moreover, under these hypotheses, the conditions of Theorem 5.1 are verified with S = r 1−r so that the law P n of the n-uple (X 1 , . . . , Proof. The first claim is obtained exactly as in the proof of Proposition 2.10 in [17], observing that the contraction condition 2 is equivalent to and also to This entails that whenever f is 1-Lipschitz, P n f is r n -Lipschitz. Now, by condition 1, we have P n (e sf ) ≤ P n−1 exp sP f + α ⊛ (s) ≤ exp sP n f + α ⊛ (s) + . . . + α ⊛ (r n s) .
As α ⊛ is convex and vanishes at 0, we have α ⊛ (rt) ≤ rα ⊛ (t) for all t ≥ 0. Thus, It remains only to check that 1 1−r α ⊛ is the monotone conjugate of α ′ and to invoke Proposition A.3.
Moving on to the final claim, since the process is homogeneous, to ensure that (17) is satisfied, we need only show that for all k ≥ 1, for all f : is r 1−r -Lipschitz. We show the following : if g : E 2 → R is such that for all x 1 , x 2 ∈ E the functions g(., x 2 ), resp. g(x 1 , .), are 1-Lipschitz, resp. λ-Lipschitz, then the function It follows easily by induction that the function has Lipschitz norm bounded by 1 + r + . . . r k ≤ 1 1−r . Hence the function x → f k (x 1 )P (x, dx 1 ) has Lipschitz norm bounded by r 1−r . But this function is precisely x → E [f (X 1 , . . . , X k )|X 0 = x] and the proof is complete.
We are in position to prove the analogue of Lemma 3.2 in the Markov case. Lemma 5.3. Consider the Markov chain associated to a transition kernel P as in Proposition 5.2. Let P n denote the law of the Markov path (X 1 , . . . , X n ) associated to P under X 0 = x 0 . Introduce the averaged occupation measure π n = E Pn (L n ) and the invariant measure π. Let m 1 = d(x, x 0 )π(dx).
Suppose that there exists a > 0 such that for all i ≥ 1 E a,i = e ad(x,x0) P i (dx) ≤ 2.
Let δ > 0 and K ∈ E be a compact subset containing x 0 . Let N δ denote the metric entropy of order δ for the set F K of 1-Lipschitz functions on K vanishing at x 0 (endowed with the uniform distance). Also define σ : [0, +∞) → [1, +∞) as the inverse function of x → x ln x − x + 1 on [1, +∞).
In order to take care of the second term, we will use the same strategy (and notations) as in the independent case. Introduce once again a compact subset K ⊂ E and a covering of F K by functions f 1 , . . . , f N δ suitably extendend to E. With the same arguments as before, we get As before we can use Orlicz-Hölder's inequality to recover the bound .
And likewise, and we have the same bound as above.
As for the last term remaining : it will be possible to use the maximal inequality techniques just as in the proof of Theorem 2.2, provided that we can find bounds on the terms E [exp λΨ(f j )], where this time This is a 1 n -Lipschitz function on E n . Since P n satisfies aα(T d ) ≤ H inequality, we have exp λF j dP n ≤ exp λ F j dP n + nα ⊛ ( λ n (1 − r) ) .

But this is exactly the bound
. We may then proceed as in the independent case and obtain ) .
For any Lipschitz function f : E n → R (w.r.t. d 1 ), we have the concentration inequality Remembering that E n ∋ x → W 1 (L x n , π n ) is 1 n -Lipschitz, we get the bound Thanks to the triangular inequality W 1 (L n , π n ) ≥ W 1 (L n , π) − W 1 (π n , π), it holds that This in turn leads to an estimate on the deviations, under the condition that we may exhibit a compact set with large measure for all the measures P i . We now move on to the proof of Theorem 2.12.
With a as in the theorem, the above ensures that e a|x| P i (dx) ≤ 2.
Let B R denote the ball of center 0 and radius R : We have chosen K = B R and δ = t/8. Working as in the proof of Proposition 2.7, when t ≤ 2/a, we can bound log N δ by where C d is a numerical constant depending on the dimension d. Plugging the above estimates in (19) and using the inequality (u − v) 2 ≥ u 2 /2 − v 2 gives the desired result.

Appendix A. Some facts on transportation inequalities
A crucial feature of transportation inequalities is that they imply the concentration of measure phenomenon, a fact first discovered by Marton ([23]). The following proposition is obtained by a straightforward adaptation of her famous argument : Moreover, let X be a r.v. with law µ. For all 1-Lipschitz functions f : E → R and all r ≥ r 0 , we have where m f denotes a median of f . Bobkov and Götze ([4]) were the first to obtain an equivalent dual formulation of transportation inequalities. We present it here in a more general form obtained by Gozlan and Leonard (see [15]), in the case when the transportation cost function is the distance. Then µ satisfies the α(T d ) inequality for all ν ∈ P(E) with finite first moment if and only if for all f : E → R 1-Lipschitz and all λ > 0, (20) e λ(f (x)− f dµ) µ(dx) ≤ e α ⊛ (λ) .
A.0.1. Integral criteria. An interesting feature of transportation inequalities is that some of them are characterized by simple moment conditions, making it tractable to obtain their existence. In [17], Djellout, Guillin and Wu showed that µ satisfies a T 1 inequality if and only if exp[a 0 d 2 (x 0 , y)]µ(dy) < +∞ for some a 0 and some x 0 . They also connect the value of a 0 and of the Gaussian moment with the value of the constant C appearing in the transportation inequality. More generally, Gozlan and Leonard provide in [14] a nice criterion to ensure that a α(T d ) inequality holds. We only quote here one side of what is actually an equivalence : Theorem A.4. Let µ ∈ P(E). Suppose there exists a > 0 with e ad(x0,x) µ(dx) ≤ 2 for some x 0 ∈ E, and a convex, increasing function γ on [0, +∞) vanishing at 0 and Then µ satisfies the α(T d ) inequality for all ν ∈ P(E) with finite first moment, with One particular instance of the result above was first obtained by Bolley and Villani, with sharper constants, in the case when µ only has a finite exponential moment ( [8]), Corollary 2.6). Their technique involves the study of weighted Pinsker inequalities, and encompasses more generally costs of the form d p , p ≥ 1 (we give only the case p = 1 here).
Appendix B. Covering numbers of the set of 1-Lipschitz functions In this section, we provide bounds for the covering numbers of the set of 1-Lipschitz functions over a precompact space.
Note that these results are likely not new. However, we have been unable to find an original work, so we provide proofs for completeness.
Let (K, d) be a precompact metric space, and let N (K, δ) denote the minimal number of balls of radius δ necessary to cover K. Let x 0 ∈ K be a fixed point, and let F denote the set of 1-Lipschitz functions over K vanishing at x 0 . This is also a precompact space when endowed with the metric of uniform convergence. We denote by N (F , δ) the minimal number of balls of radius δ necessary to cover F . Finally, we set R = max x∈K d(x, x 0 ).
Our first estimate is a fairly crude one. .
Proof. For simplicity, write n = N (K, ε). Let x 1 , . . . , x n be the centers of a set of balls covering K. For any f ∈ F and 1 ≤ i ≤ n, we have For any n-uple of integers k = (k 1 , . . . , Consider f ∈ F . Let l i = ⌊ f (xi) ε ⌋ and l = (l 1 , . . . , l n ). Then the function f l defined above exists and |f (x i ) − f l (x i )| ≤ ε for 1 ≤ i ≤ n. But then for any x ∈ K there exists i, 1 ≤ i ≤ n, such that x ∈ B(x i , ε), and thus This implies that F is covered by the balls of center f k and radius 3ε. As there are at most (2 + 2⌊ R ε ⌋) n choices for k, this ends the proof.
However, this bound is quite weak : as one can see by considering the case of a segment, for most choices of a n-uple, there will not exist a function in F satisfying the requirements in the proof. With the extra assumption that K is connected, we can get a more refined result. .
Remark. The simple idea in this proposition is first to bring the problem to a discrete metric space (graph), and then to bound the number of Lipschitz functions on this graph by the number of Lipschitz functions on a spanning tree of the graph.
Proof. In the following, we will denote n = N (K, ε) for simplicity. Let x i , 1 ≤ i ≤ n be the centers of a set of n balls B 1 , . . . B n covering K. Consider the graph G built on the n vertices a 1 , . . . , a n , where vertices a i and a j are connected if and only if i = j and the balls B i and B j have a non-empty intersection.
Lemma B.3. The graph G is connected. Moreover, there exists a subgraph G ′ with the same set of vertices and whose edges are edges of G, which is a tree.
Proof. Suppose that G were not connected . Upon exchanging the labels of the balls, there would exist k, 1 ≤ k < n, such that for i ≤ k < j the balls B i and B j have empty intersection. But then K would be equal to the disjoint reunion of the sets k i=1 B i and n j=k+1 B j , which are both closed and non-empty, contradicting the connectedness of K.
The second part of the claim is obtained by an easy induction on the size of the graph.
Introduce the set A of functions g : {a 1 , . . . , a n } → R such that g(a 1 ) = 0 and |g(a i ) − g(a j )| = 4ε whenever a i and a j are connected in G ′ . Using the fact that G ′ is a tree, it is easy to see that A contains at most 2 n elements.
Define a partition of K by setting C 1 = B 1 , C 2 = B 2 \C 1 , . . ., C n = B n \C n−1 (remark that none of the C i is empty since the B i are supposed to constitute a minimal covering). Also fix for each i, 1 ≤ i ≤ n, a point y i ∈ C i (choosing y 1 = x 1 ). Notice that C i is included in the ball of center y i and radius 2ε, and that d(y i , y j ) ≤ 4ε whenever a i and a j are connected in G (and therefore in G ′ ).
To every 1-Lipschitz function f : K → R we associate T (f ) : {a 1 , . . . , a n } → R defined by T (f )(a i ) = f (y i ). For any x ∈ K, and f 1 , f 2 ∈ F , we have the following : where i is such that x ∈ C i . We now make the following claim : Assume for the moment that this holds. As there are at most 2 n functions in A, it is possible to choose at most 2 n 1-Lipschitz functions f 1 , . . . , f 2 n vanishing at x 1 such that for any 1-Lipschitz function f vanishing at x 1 there exists f i such that |T (f ) − T (f i )| ≤ 8ε. Using the inequality above, this implies that the balls of center f i and radius 12ε for the uniform distance cover the set of 1-Lipschitz functions vanishing at x 1 .
Finally, consider f ∈ F . We may write ) and observe that on the one hand, f − f (x 1 ) is a 1-Lipschitz function vanishing at x 1 , and that on the other hand, |f (x 1 )| ≤ R. Thus the set F is covered by the balls of center f i + 4kε and radius 16ε, There are at most (2 + 2⌊ R 4ε ⌋)2 n such balls, which proves the desired result.
We now prove Lemma B.4.
Proof. Let us use induction again. If K = B 1 then T (f ) = 0 and the property is straightforward. Now if K = C 1 ∪. . .∪C n , we may assume without loss of generality that a n is a leaf in G ′ , that is a vertex with exactly one neighbor, and that it is connected to a n−1 . By hypothesis there existsg : {a 1 , . . . , a n−1 } → R such that |g(a i )−g(a j )| = 4ε whenever a i and a j are connected in G ′ , and |g(a i )−f (a i )| ≤ 4ε for 1 ≤ i < n. Set g =g on {a 1 , . . . , a n−1 }, and • g(a n ) = g(a n−1 ) + 4ε if f (y n ) − g(a n−1 ) < 0, • g(a n ) = g(a n−1 ) − 4ε otherwise. Since |f (y n ) − g(a n−1 )| ≤ |f (y n ) − f (y n−1 )| + |f (y n−1 ) − g(a n−1 )| ≤ 8ε it is easily checked that |f (y n ) − g(a n )| ≤ 4ε. The function g belongs to A and our claim is proved.

Appendix C. Hölder moments of stochastic processes
We quote the following result from Revuz and Yor's book [28] (actually the value of the constant is not given in their statement but is easily tracked in the proof).
Corollary C.3. Let X t be the solution on [0, T ] of dX t = σ(X t )dB t + b(X t )dt with σ, b : R → R locally Lipschitz and satisfying the following hypotheses : • sup x | trσ(x) t σ(x)| ≤ A, • sup x |b(x)| ≤ B.
Proof. We first apply Itô's formula to the function |X t − X s | 2 : this yields Using the elementary inequality x ≤ 1/2(1 + x 2 ), we get Gronwall's lemma entails E|X t − X s | 2 ≤ (A + B)e BT |t − s| Likewise, applying Itô's formula to |X t − X s | 4 , we get and by Gronwall's lemma E|X t − X s | 4 ≤ 1 2 (6A + 2B)(A + B)e 3BT |t − s| 2 . By an easy recurrence, following the above, one may show that E|X t − X s | 2p ≤ C(A, B, T, p)|t − s| p .
To conclude it suffices to call on Theorem C.1.  Note that the weak variance σ 2 is bounded by the variance s 2 defined as s 2 = E(sup t |B t |) 2 (here and hereafter sup t |B t | refers to the supremum on [0, 1]). In turn we can give a (quite crude) bound on s : Remember the well-known fact that sup t B t , − inf t B t and |B 1 | have the same law, so that where τ u is the stopping time inf{t, |B t | = u}. It is a simple exercise to compute Ee −λ 2 τu/2 = 1/ cosh(λu) ≤ 2e −λu . This yields e a x ∞ γ(dx) ≤ 2ae λ 2 /2 +∞ 0 e (a−λ)u du = 2ae λ 2 /2 λ − a .
We can choose λ = 3a to get e a x ∞ γ(dx) ≤ e 9a 2 /2 . In turn it implies the desired result.