Siméon Valère

Bitseki Penda
email: bit-seki@cmap.polytechnique.fr

Mikael Escobar-Bach

Arnaud Guillin
email: guillin@math.univ-bpclermont.fr

Transportation cost-information and concentration inequalities for bifurcating Markov chains

Keywords: Transportation cost-information inequalities, Wasserstein distance, bifurcating Markov chains, deviation inequalities, geometric ergodicity

come   L'archive ouverte pluridisciplinaire

Introduction

Roughly speaking, a bifurcating Markov chain is a Markov chain indexed by a binary regular tree. This class of processes are well adapted for the study of populations where each individual in one generation gives birth

Bifurcating Markov chains

First we introduce some useful notations. Let T be a regular binary tree in which each vertex is seen as a positive integer different from 0. For r ∈ N, let

G r = 2 r , 2 r + 1, • • • , 2 r+1 -1 , T r = r q=0 G q ,
which denote respectively the r-th column and the first (r + 1) columns of the tree. The whole tree is thus defined by

T = ∞ r=0 G r .
A column of a given vertex n is G rn with r n = ⌊log 2 n⌋, where ⌊x⌋ denotes the integer part of the real number x.

In the sequel, we will see T as a given population in which each individual in one generation gives birth to two offsprings in the next one. This will make easier the introduction of different notions. The vertex n will denote the individual n and the ancestor of individuals 2n and 2n + 1. The individuals who belong to 2N (resp. 2N+1) will be called individual of type 0 (resp. type 1). The column G r and the first (r + 1) columns T r will denote respectively the r-th generation and the first (r + 1) generations. The initial individual will be denoted 1.

For each individual n, we look into a random variable X n , defined on a probability space (Ω, F, P) and which takes its values in a metric space (E, d) endowed with its Borel σ-algebra E. We assume that each pair of random variables (X 2n , X 2n+1 ) depends of the past values (X m , m ∈ T rn ) only through X n . In order to describe this dependance, let us introduce the following notion.

Definition 1.1 (T-transition probability, see ([29])). We call T-transition probability any mapping

P : E × E 2 → [0, 1] such that • P (•, A) is measurable for all A ∈ E 2 ,
• P (x, •) is a probability measure on (E 2 , E 2 ) for all x ∈ E.

In particular, for all x, y, z ∈ E, P (x, dy, dz) denotes the probability that the couple of the quantities associated with the children are in the neighbourhood of y and z given that the quantity associated with their mother is x.

For a T-transition probability P on E × E 2 , we denote by P 0 , P 1 the first and the second marginal of P , that is P 0 (x, A) = P (x, A × E), P 1 (x, A) = P (x, E × A) for all x ∈ E and A ∈ E. Then, P 0 (resp. P 1 ) can be seen as the transition probability associated to individual of type 0 (resp. type 1).

For p ≥ 1, we denote by B(E p ) (resp. B b (E p )), the set of all E pmeasurable (resp. E p -measurable and bounded) mappings f : E p → R. For f ∈ B(E 3 ), we denote by P f ∈ B(E) the function

x → P f (x) = S 2
f (x, y, z)P (x, dy, dz), when it is defined.

We are now in position to give a precise definition of bifurcating Markov chain.

Definition 1.2 (Bifurcating Markov Chains, see ( [START_REF] Guyon | Limit theorems for bifurcating Markov chains. application to the detection of cellular aging[END_REF])). Let (X n , n ∈ T) be a family of E-valued random variables defined on a filtered probability space (Ω, F, (F r , r ∈ N), P). Let ν be a probability on (E, E) and P be a Ttransition probability. We say that (X n , n ∈ T) is a (F r )-bifurcating Markov chain with initial distribution ν and T-transition probability P if

• X n is F rn -measurable for all n ∈ T, • L(X 1 ) = ν,
• for all r ∈ N and for all family

(f n , n ∈ G r ) ⊆ B b (E 3 ) E n∈Gr f n (X n , X 2n , X 2n+1 ) F r = n∈Gr P f n (X n ).
In the following, when unprecised, the filtration implicitly used will be F r = σ(X i , i ∈ T r ).

Remark 1.3. We may of course also consider in this work bifurcating Markov chains on a a-ary tree (with a ≥ 2) with no additional technicalities, but heavy additional notations. In the same spirit, Markov chains of higher order (such as BAR processes considered in [START_REF] Bitseki Penda | Deviation inequalities and moderate deviations for estimators of parameters in bifurcating autoregressive models[END_REF]) could be handled by the same techniques. A non trivial extension would be the case of bifurcating Markov chains on a Galton-Watson tree (see for example [START_REF]Bitseki Deviations inequalities for bifurcating markov chains on galton-watson tree[END_REF] under very strong assumptions), that we will consider elsewhere.

Transportation cost-information inequality

We recall that (E, d) is a metric space endowed with its Borel σ-algebra E. Given p ≥ 1, the L p -Wasserstein distance between two probability measures µ and ν on E is defined by

W d p (ν, µ) = inf d(x, y) p dπ(x, y) 1/p
, where the infimum is taken over all probability measures π on the product space E × E with marginal distributions µ and ν (say, coupling of (µ, ν)). This infimum is finite as soon as µ and ν have finite moments of order p.

When d(x, y) = 1 x =y (the trivial measure), 2W d 1 (µ, ν) = µ -ν T V , the total variation of µ -ν.
The Kullback information (or relative entropy) of ν with respect to µ is defined as

H(ν/µ) = log dν dµ dν, if ν ≪ µ +∞ else.
Definition 1.4 (L p -transportation cost-inequality). We say that the probability measure µ satisfies the L p -transportation cost-information inequality on (E, d) (and we write µ ∈ T p (C)) if there is some constant C > 0 such that for any probability measure ν,

W d p (µ, ν) ≤ 2CH(ν/µ).
This transportation inequality have been introduced by Marton [32, 33] as a tool for (Gaussian) concentration of measure property. The following result will be crucial in the sequel. It gives a characterization of L 1transportation cost-inequality in term of concentration inequality. It is of course one of the main tool to get deviation inequalities (via Markov inequality).

Theorem 1.5 ([11]). µ satisfies the L 1 -transportation cost-information inequality (say

T 1 ) on (E, d) with constant C > 0, that is, µ ∈ T 1 (C), if and only if for any Lipschitzian function F : (E, d) → R, F is µ-integrable and E exp (λ (F -F µ )) dµ ≤ exp λ 2 2 C F 2 Lip ∀λ ∈ R,
where F µ = E F dµ and

F Lip = sup x =y |F (x) -F (y)| d(x, y) < +∞.
In particular, we have the concentration inequality

µ (F -F µ ≤ -t) ∨ µ (F -F µ ≥ t) ≤ exp - t 2 2C F 2 Lip ∀t ∈ R.
In this work we will focus on transportation inequality T 1 mainly. There is now a considerable literature around these transportation inequalities. As a flavor, let us cite first the characterization of T 1 as a Gaussian integrability property [START_REF] Djellout | Transportation cost-information inequalities and applications to random dynamical systems and diffusions[END_REF] (see also [START_REF] Gozlan | Integral criteria for transportation-cost inequalities[END_REF]).

Theorem 1.6 ([20]). µ satisfies the L 1 -transportation cost-information inequality (say T 1 ) on (E, d) if and only if there exists δ > 0 and x 0 ∈ E such that µ e δd 2 (x,x 0 ) < ∞,

and the constant of the Transportation inequality can be made explicit.

There is also a large deviations characterization [START_REF] Gozlan | A large deviation approach to some transportation cost inequalities[END_REF]. Recent striking results on transportation inequalities have been obtained for T 2 , namely that they are equivalent to dimension free Gaussian concentration [START_REF] Gozlan | A characterization of dimension free concentration in terms of transportation inequalities[END_REF], or to a restricted class of logarithmic Sobolev inequalities [START_REF] Gozlan | A new characterization of Talagrand's transport-entropy inequalities and applications[END_REF]. Se also [START_REF] Cattiaux | On quadratic transportation cost inequalities[END_REF] or [START_REF] Cattiaux | A note on Talagrand's transportation inequality and logarithmic Sobolev inequality[END_REF] for practical criterion based on Lyapunov type criterion and we refer for example to [START_REF] Gozlan | Transport inequalities. A survey. Markov Process[END_REF] or [START_REF] Villani | Optimal transport, volume 338 of Grundlehren der Mathematischen Wissenschaften[END_REF] for surveys on transportation inequality. One of the main aspect of transportation inequality is their tensorization property, i.e. µ ⊗n will satisfy some transportation measure if µ does (with dependence on the dimension n 2/p-1 ) . One important development was to consider such a property for dependent sequences such as Markov chains. In [START_REF] Djellout | Transportation cost-information inequalities and applications to random dynamical systems and diffusions[END_REF], Djellout et al., generalizing result of Marton [START_REF] Marton | A measure concentration inequality for contracting Markov chains[END_REF], have provided conditions under which the law of a homogeneous Markov chain (Y k ) 1≤k≤n on E n satisfies the L ptransportation cost-information inequality T p with respect to the metric

d lp (x, y) := n i=1 d(x i , y i ) p 1/p .
We will follow similar ideas here to establish the L p -transportation costinformation inequality for the law of a bifurcating Markov chain (X i ) 1≤i≤N on E N . This will allow us to obtain concentration inequalities for bifurcating Markov chains under hypotheses largely weaker than those of Bitseki et al. [START_REF] Bitseki Penda | Deviation inequalities, moderate deviations and some limit theorems for bifurcating Markov chains with application[END_REF]. It would also be tempting to generalize the approach of [START_REF] Guillin | Transportation-information inequalities for Markov processes[END_REF] to Markov chains and bifurcating Markov chains to get directly deviation inequalities for Markov chains, w.r.t. the invariant measure. However it would need to restrict to reversible Markov chains and thus not directly suited to bifurcating Markov chains and would thus recquire new ideas. Remark 1.7. There are natural generalizations of the T 1 inequality often denoted α -T 1 inequality, where α is a non negative convex lower semi continuous function vanishing at 0. We say that the probability measure µ satisfies α -T p (C) if for any probability measure ν α (W 1 (ν, µ)) ≤ 2C H(ν/µ).

The usual T 1 inequality is then the case where α(t) = t 2 . Gozlan [START_REF] Gozlan | Integral criteria for transportation-cost inequalities[END_REF] has generalized Bobkov-Götze's Laplace transform control [START_REF] Bobkov | Exponential integrability and transportation cost related to logarithmic sobolev inequalities[END_REF] and Djellout-Guillin-Wu [START_REF] Djellout | Transportation cost-information inequalities and applications to random dynamical systems and diffusions[END_REF] integrability criterion to this setting enabling to recover sub or super Gaussian concentration. The result of the following section can be generalized to this setting, however adding technical details and heavy notations. Details will thus be left to the reader.

Transportation cost-information inequalities for bifurcating Markov chains

Let (X i , i ∈ T) be a bifurcating Markov chain on E with T-probability transition P and initial measure ν. For p ≥ 1 and C > 0, we consider the following assumption that we shall call (H p (C)) in the sequel. (c) W d p (P (x, •, •), P (x, •, •)) ≤ q d(x, x), ∀x, x ∈ E and some q > 0.

It is important to remark that under (H p (C)), (c) we have that there exists r 0 and r 1 smaller than q such that for b = 0, 1

W d p (P b (x, •), P b (x, •)) ≤ r b d(x, x), ∀x, x ∈ E.
Note also that when P (x, dy, dz) = P 0 (x, dy)P 1 (x, dz), then these last two stability results in Wasserstein contraction implies (H p (C)), (c) with q ≤ (r p 0 +r p 1 ) 1/p (using trivial coupling). We may remark also that by (H p (C)), (b), P 0 and P 1 also satisfies (uniformly) a transportation inequality. Let us note that thanks to the Hölder inequality, (H p (C)) implies (H 1 (C)). We do not suppose here that q, r 0 and r 1 are strictly less than 1, and thus the two marginal chains, as well as the bifurcating one, are not a priori contractions. We are thus considering here both "stable" and "unstable" cases.

We then have the following result for the law of the whole trajectory on the binary tree.

Theorem 2.2. Let n ∈ N and let P be the law of (X i ) 1≤i≤Tn and denote N = |T n |. We assume Assumption 2.1 for 1 ≤ p ≤ 2. Then P ∈ T p (C N ) where

C N =          C N 2/p-1 (1-q) 2 if q < 1 C exp 2 -2 p N 2/p+1 if q = 1 C (N + 1) exp(q-1)r pN r p -1 2/p if q > 1.
Before the proof of this result, let us make the following notations. For a Polish space χ, we denote by M 1 (χ) the space of probability measures on χ.

For x ∈ E N , x i := (x 1 , • • • , x i ). For µ ∈ M 1 (E N ), let (x 1 , • • • , x N ) ∈ E N
be distributed randomly according to µ. We denote by µ i the law of x 2i+1 , and by µ i x 2i-1 the conditional law of (x 2i , x 2i+1 ) given x 2i-1 with the convention µ 1

x 0 = µ 1 , where x 0 = x 0 is some fixed point. In particular, if µ is the law of a bifurcating Markov chain with T-probability transition P , then

µ i x 2i-1 = P (x i , •, •).
For the convenience of the readers, we recall the formula of additivity of entropy (see for e.g. [START_REF] Villani | Optimal transport, volume 338 of Grundlehren der Mathematischen Wissenschaften[END_REF], Lemma 22.8).

Lemma 2.3. Let N ∈ N, let χ 1 , • • • , χ N be Polish spaces and P, Q ∈ M 1 (χ) where χ = N i=1 χ i . Then H(Q|P) = N i=1 χ H(Q i x i-1 |P i x i-1 )Q(dx)
where P i x i-1 and Q i x i-1 are defined in the same way as above. We can now prove the Theorem.

Proof of the Theorem 2.2. Let Q ∈ M 1 (E N ). Assume that H(Q|P) < ∞ (trivial otherwise). Let ε > 0.
The idea is of course to do a conditionnement with respect to the previous generation, i.e. to G n-1 but we will do it sequentially by pairs. Conditionally to their ancestors, every pair of offspring of an individual is independent of the offspring of the other individuals for the same generation. Let i be a member of generation G j-1 , and define for a realization x on the tree T i (x) := (x 1 , ..., x |T j | ). By the definition of the Wasserstein distance, there is a coupling

π i y 2i-1 ,x 2i-1 of (Q i y 2i-1 , P i x 2i-1 ) such that A i := (d(y 2i , x 2i ) p + d(y 2i+1 , x 2i+1 ) p )dπ i y 2i-1 ,x 2i-1 ≤ (1 + ǫ)W d p Q i y 2i-1 , P i x 2i-1 p ≤ (1 + ǫ) W d p Q i y 2i-1 , P i y 2i-1 + W d p P i y 2i-1 , P i x 2i-1 p ≤ (1 + ǫ) W d p Q i y 2i-1 , P (y i , •, •) + W d p P (y i , •, •) , P (x i , •, •) p ,
where the second inequality is obtained thanks to the triangle inequality for the W d p distance and the equality is a consequence of the Markov property. By Assumption 2.1, and the convexity of the function x → x p , we obtain, for a, b > 1 such that 1/a + 1/b = 1,

A i ≤ (1 + ǫ) 2CH i (y 2i-1 ) + qd(y i , x i ) p ≤ (1 + ǫ) a p-1 2CH i (y 2i-1 ) p + b p-1 q p d p (y i , x i )
where

H i (y 2i-1 ) = H(Q i y 2i-1 |P i y 2i-1 )
. By recurrence, it leads to the finiteness of p-moments. Taking the average with respect to the whole law and summing on i, we obtain

|T n-1 | i=0 E(A i ) ≤ (1 + ε)   a p-1 (2C) p/2 |T n-1 | i=1 E H i (Y 2i-1 ) p/2   +   b p-1 q p |T n-2 | i=0 E(A i )   .
Letting ε goes to 0 + , we are led to

|T n-1 | i=0 E(A i ) ≤ N i=1 a p-1 (2C) p/2 E H i (Y i-1 ) p/2 +   b p-1 q p |T n-2 | i=0 E(A i )   .
Iterating the latter inequality, increasing some terms and thanks to Hölder inequality, we obtain

|T n-1 | i=0 E(A i ) ≤ N i=1   i j=1 h j   b p-1 q p N -i = N i=1 h i N -i j=0 b p-1 q p j ≤ N i=1 h 2/p i p/2    N i=1   N -i j=0 b p-1 q p j   2 2-p    2-p 2
where

h i = a p-1 (2C) p/2 E[H i (Y i-1 ) p/2 ]
. By the definition of the Wasserstein distance, the additivity of entropy and using the concavity of the function

x → x p/2 for p ∈ [1, 2], we obtain W d lp p (Q, P) p ≤ a p-1 (2CH (Q|P)) p/2    N i=1   N -i j=0 b p-1 q p j   2 2-p    2-p 2 ≤ a p-1 (2CH (Q|P)) p/2 N 1-p 2 N -1 j=0 b p-1 q p j .
When q < 1, we take b = q -1 , so that b p-1 q p = r < 1 and the desired result follows easily. When q ≥ 1, we take b = 1 + 1/N and the results follow from simple analysis and this ends the proof.

Remark 2.4. For q < 1, we then have that the constant C N of T 1 inequality for P increases linearly on the dimension N . However, for T 2 this constant is independent of the dimension as in the i.i.d. case.

Remark 2.5. As we will see in the next section, still when q < 1, Theorem 2.2 and Theorem 1.5 applied to

F (X 1 , • • • , X N ) = (1/N ) N i=1 f (X i ) (
where f is a Lipschitzian function defined on E) gives us deviation inequalities with a good order of N . But, when they are applied to

F (X 1 , • • • , X N ) = f (X N )
, deviation inequalities that we obtain does not furnish the good order of N when N is large. The same remark holds when

F (X 1 , • • • , X N ) = g(X n , X 2n , X 2n+1 ) with n ∈ {1, • • • , (N -N [2]
)} and g a Lipschitzian function defined on E 3 . As this last question is important for the L 1 -transportation cost-information inequality of the invariant measure of a bifurcating Markov chain, we give the following results. Proposition 2.6. Under (H 1 (C)), for any n ∈ T and x ∈ E

L(X n |X 1 = x) ∈ T 1 (c n ) where c n = C rn-1 k=0 r 2(k-a k ) 0 r 2a k 1 ; a 0 = 0
and for all k ∈ {1, • • • , r n -1}, a k is the number of ancestor of type 1 of X n which are between the r n -k + 1-th generation and the r n -th generation.

Before the proof, we introduce some more notations. Let n ∈ T. We denote by (z 1 , • • • , z rn ) ∈ {0, 1} rn the unique path from the root 1 to n. Then, for all i ∈ {1, • • • , r n }, z i is the type of the ancestor of n which is in the i-th generation and the quantities a k defined in the Proposition 2.6 are given by

a k = rn i=rn-k+1 z i .
For all k ∈ {1, • • • , r n }, we denote by P k and P -k the iterated of the transition probabilities P 0 and P 1 defined by

P k := P z 1 • • • • • P z k and P -k := P z rn-k • • • • • P zr n .
Proof of the Proposition 2.6. First note that since

W d 1 (ν, µ) = sup f : f Lip ≤1 S f dµ - S f dν , condition (c) of (H 1 (C)) implies that P b f Lip ≤ r b f Lip ∀b ∈ {0, 1}. Now let f be a Lipschitzian function defined on E. By (b)-(c) of (H 1 (C))
and Theorem 1.5, we have

P rn (e f ) ≤ P rn-1 exp P rn f + C f 2 Lip 2 .
Once again, applying Theorem 1.5, we obtain

P rn (e f ) ≤ P rn-2 exp P -1 f + C f 2 Lip 2 + C P zr n f 2 Lip 2 .
By iterating this method, we are led to

P rn (e f ) ≤ exp P -rn+1 f + (1 + r 2 zr n + r 2 zr n r 2 z rn-1 + • • • + rn i=2 r 2 z i ) C f 2 Lip 2 . Since 1+r 2 zr n +r 2 zr n r 2 z rn-1 +• • •+ rn i=2 r 2 z i = rn-1 k=0 r 2(k-a k ) 0 r 2a k 1 and P -rn+1 f = P rn f,
we conclude the proof thanks to Theorem 1.5.

The next result is a consequence of the previous Proposition.

Corollary 2.7. Assume (H 1 (C)) and r := max{r 0 , r 1 } < 1. Then

L(X n |X 1 = x) ∈ T 1 (c ∞ ) and L((X n , X 2n , X 2n+1 )|X 1 = x) ∈ T 1 (c ′ ∞ )
where

c ∞ = C 1 -r 2 and c ′ ∞ = C 1 + (1 + q) 2 1 -r 2 . Proof. That L(X n |X 1 = x) ∈ T 1 (c ∞ ) is a direct consequence of Proposition 2.
6. It suffices to bound r 0 and r 1 by r.

In order to deal with the ancestor-offspring case (X n , X 2n , X 2n+1 ), we do the following remarks.

Let f : (E 3 , d l 1 ) → R be a Lipschitzian function. We have

P f Lip = sup x,x∈E
f (x, y, z)P (x, dy, dz) -f (x, y, z)P (x, dy, dz) d(x, x) .

Thanks to condition (c) of (H 1 (C)), we have the following inequalities f (x, y, z)P (x, dy, dz) -f (x, y, z)P (x, dy, dz)

≤ f Lip d(x, x) + W d l 1 1 (P (x, •), P (x, •)) ≤ (q + 1) f Lip d(x, x),
and then, P f Lip ≤ (q + 1) f Lip .

We recall that X 1 = x. We have

E [exp (f (X n , X 2n , X 2n+1
))] = P rn (P e f (x)). Now, from (H 1 (C)), the previous remarks and using the same strategy as in the proof of Proposition 2.6, we are led to

E [exp (f (X n , X 2n , X 2n+1 ))] ≤ exp P z 1 • • • P zr n P f (x) + C f 2 Lip 2 + C(1 + q) 2 f 2 Lip 2 rn-1 i=0 r 2i . Since P z 1 • • • P zr n P f (x) = E [f (X n , X 2n , X 2n+1 )] and rn-1 i=0 r 2i ≤ 1/(1-r 2 ), we obtain E [exp (f (X n , X 2n , X 2n+1 ))] ≤ exp E [f (X n , X 2n , X 2n+1 )] + c ′ ∞ with c ′
∞ given in the Corollary. We then conclude the proof thanks to Theorem 1.5.

Concentration inequalities for bifurcating Markov chains

3.1 Direct applications of the Theorem 2.2

We are now interested in the concentration inequalities for the additive functionals of bifurcating Markov chains. Specifically, let N ∈ N * and I be a subset of {1, • • • , N }. Let f be a real function on E or E 3 . We set or I = T n are relevant (see for e.g. [START_REF] Bitseki Penda | Deviation inequalities, moderate deviations and some limit theorems for bifurcating Markov chains with application[END_REF]). First, we will establish concentration inequalities when f is a real Lipschitzian function defined on E. For a subset I of {1, • • • , N }, let F I be the function defined on (E N , d lp ), p ≥ 1 by F I (x N ) = 1/(|I|) i∈I f (x i ) for all x N ∈ E N . Then F I is also a Lipschitzian function on (E N , d lp ) and we have F I Lip ≤ |I| -1/p f Lip . The following result is a direct consequence of Theorem 2.2. Proposition 3.1. Let N ∈ N * and let P be the law of (X i ) 1≤i≤N . Let f be a real Lipschitzian function on (E, d). Then, under (H p (C)) for 1 ≤ p ≤ 2,

M I (f ) = i∈I f (∆ i ) where ∆ i = X i if f is defined on E and ∆ i = (X i , X 2i , X 2i+1 ) if f is defined on E 3 .
P • F -1 I ∈ T p (C N |I| -2/p f 2 Lip )
where C N is given in the Theorem 2.2 and P • F -1

I

is the image law of P under F I . In particular, for all t > 0 we have

P F I (X N ) ≤ -t + E F I (X N ) ∨ P F I (X N ) ≥ t + E F I (X N ) ≤ exp - t 2 |I| 2/p 2C N f 2 Lip .
Proof. The first part is a direct consequence of Theorem 2.2 and Lemma 2.1 of [START_REF] Djellout | Transportation cost-information inequalities and applications to random dynamical systems and diffusions[END_REF]. The second part is an application of Theorem 1.5.

For the next concentration inequality, we assume that f is a real Lipschitzian function defined on (E 3 , d l 1 ), which means that

|f (x) -f (y)| ≤ f Lip 3 i=1 d(x i , y i ) ∀x, y ∈ E 3 .
We assume that N is a odd number. Let I be a subset of {1, • • • , (N -1)/2}. Now, we denote by F I the real function defined on (E N , d lp ) by F I (x N ) = (1/|I|) i∈I f (x i , x 2i , x 2i+1 ). For all x N , y N ∈ E N we have for some universal constant c

|F I (x N ) -F I (y N )| ≤ f Lip |I| i∈I (d(x i , y i ) + d(x 2i , y 2i ) + d(x 2i+1 , y 2i+1 )) ≤ c f Lip |I| 1/p d lp (x N , y N ).
F I is then a Lipschitzian function on (E N , d lp ) and

F I Lip ≤ c f Lip /|I| 1/p .
We then have the following result.

Proposition 3.2. Let N ∈ N * be a odd number and let P be the law of (X i ) 1≤i≤N . Let f be a real Lipschitzian function on (E 3 , d l 1 ). Then, under

(H p (C)) for 1 ≤ p ≤ 2, P • F -1 I ∈ T p (c C N |I| -2/p f 2 Lip )
where C N is given in the Theorem 2.2 and P • F -1

I

is the image law of P under F I . In particular, for all t > 0 we have

P F I (X N ) ≤ -t + E F I (X N ) ∨ P F I (X N ) ≥ t + E F I (X N ) ≤ exp - t 2 |I| 2/p 2cC N f 2 Lip .
Proof. The proof is a direct consequence of Theorem 2.2, Lemma 2.1 of [START_REF] Djellout | Transportation cost-information inequalities and applications to random dynamical systems and diffusions[END_REF] and Theorem 1.5.

Remark 3.3. The previous results applyed with p = 1 to the empirical means M Gn (f ) and M Tn (f ) (f being a real Lipschitzian function) give us relevant concentration inequalities, that is with the good order size of the index set, when q < 1. For example, for M Gn (f ) , it suffices to take N = |T n | and I = G n in the Propositions 3.1 and 3.2. But for q ≥ 1, the concentration inequalities obtained thanks to these results are not satisfactory. In the sequel, we will be interested in obtaining relevant concentration inequalities for the empirical means M Gn (f ) and M Tn (f ) when q ≥ 1.

Gaussian concentration inequalities for the empirical means

M Gn (f ) and M Tn (f )

Throughout this section, we will focus only in the case p = 1, and will assume (H 1 (C)). We set r = r 0 + r 1 .

The main goal of this subsection is to broaden the range of application of deviation inequalities of M Gn (f ) and M Tn (f ) to cases where r > 1, namely when it is possible that one of the two marginal Markov chains is not a strict contraction. The transportation inequality of Theorem 2.2 is a very powerful tool to get deviation inequalities for all lipschitzian functions of the whole trajectory (up to generation n), and may thus concern for example Lipschitzian function of only offspring generated by P 0 or P 1 . Consequently, to get "consistent" deviation inequalities, both marginal Markov chains have to be contractions in Wasserstein distance. However when dealing with M Gn (f ) or M Tn (f ), we may hope for an averaging effect, i.e. if one is not a contraction and the other one a strong contraction it may in a sense compensate. Such averaging effect have been observed at the level of the LLN and CLT in [START_REF] Guyon | Limit theorems for bifurcating Markov chains. application to the detection of cellular aging[END_REF][START_REF] Saporta | Random coefficients bifurcating autoregressive processes[END_REF] but only asymptotically. Our purpose here will be then to show that such averaging effect will also affect deviation inequalities.

We will use, directly inspired by Bobkov-Götze's Laplace transform control, what we call Gaussian Concentration property: for κ > 0, we will say that a random variable X satisfies GC(κ) if

E [exp (t (X -E [X]))] ≤ exp κt 2 /2 ∀t ∈ R.
Using Markov's inequality and optimization, this Gaussian concentration property immediately implies that

P(X -E(X) ≥ r) ≤ e -r 2 2κ .
We may thus focus here only on the Gaussian concentration property (GC).

Proposition 3.4. Let f be a real Lipschitzian function on E and n ∈ N.

Assume that (H 1 (C)) holds. Then M Gn (f ) satisfies GC(γ n ) where

γ n =      2C f 2 Lip |Gn| 1-(r 2 /2) n+1 1-r 2 /2 if r = √ 2 2C f 2 Lip (n+1) |Gn| if r = √ 2.
We recall that here r = r 0 + r 1 .

Remark 3.5. One can observe that for r < √ 2, the previous inequalities are on the same order of magnitude that the inequalities obtained thanks to Proposition 3.1 with q < 1. For r < 2 the above inequalities remain relevant since we just have a negligible loss with respect to |G n |. But for r ≥ √ 2, these inequalities are not significant (see the same type of limitations at the CLT level in [START_REF] Saporta | Random coefficients bifurcating autoregressive processes[END_REF]).

Proof. Let f be a real Lipschitzian function on E, n ∈ N and t ∈ R. We have

E exp t2 -n i∈Gn f (X i ) = E   exp   t2 -n i∈G n-1 (P 0 + P 1 )f (X i )   × E   exp   t2 -n i∈G n-1 (f (X 2i ) + f (X 2i+1 ) -(P 0 + P 1 )f (X i ))   F n-1     .
Thanks to the Markov property, we have

E   exp   t2 -n i∈G n-1 (f (X 2i ) + f (X 2i+1 ) -(P 0 + P 1 )f (X i ))   F n-1   = i∈G n-1 P exp t2 -n (f ⊕ f -(P 0 + P 1 )f ) (X i )
where

f ⊕ f is the function on E 2 defined by f ⊕ f (x, y) = f (x) + f (y).
We recall that from (H 1 (C)) we have P (x, •, •) ∈ T 1 (C) for all x ∈ E. Now, thanks to Theorem 1.5, we have

i∈G n-1 P exp t2 -n (f ⊕ f -(P 0 + P 1 )f ) (X i ) ≤ i∈G n-1 exp t 2 C f ⊕ f 2 Lip 2 × 2 2n . Since f ⊕ f Lip ≤ 2 f Lip , we are led to E exp t2 -n i∈Gn f (X i ) ≤ exp 2 2 t 2 2 n-1 C f 2 Lip 2 × 2 2n × E   exp   t2 -n i∈G n-1 (P 0 + P 1 )f (X i )     .
Doing the same for E[exp(t2 -n i∈G n-1 (P 0 + P 1 )f (X i ))] with (P 0 + P 1 )f replacing f and using the inequality

(P 0 + P 1 )f ⊕ (P 0 + P 1 )f Lip ≤ 2r f Lip , we are led to E exp t2 -n i∈Gn f (X i ) ≤ E   exp   t2 -n i∈G n-2 (P 0 + P 1 ) 2 f (X i )     × exp 2 2 t 2 C f 2 Lip 2 n-1 2 × 2 2n exp 2 2 t 2 C f 2 Lip r 2 2 n-2 2 × 2 2n
.

Iterating this method and using the inequalities

(P 0 + P 1 ) k f ⊕ (P 0 + P 1 ) k f Lip ≤ 2r k f Lip ∀k ∈ {1, • • • , n -1},
we obtain

E exp t2 -n i∈Gn f (X i ) ≤ exp 2 2 t 2 C f 2 Lip 2 × 2 2n n-1 k=0 r 2k 2 n-k-1 × E exp t2 -n (P 0 + P 1 ) n f (X 1 ) . Since E [t2 -n (P 0 + P 1 ) n f (X 1 )] = E t2 -n i∈Gn f (X i ) = t2 -n ν(P 0 +P 1 ) n f , we obtain E exp t2 -n i∈Gn f (X i ) -ν(P 0 + P 1 ) n f ≤ exp 2 2 t 2 C f 2 Lip 2 × 2 2n n-1 k=0 r 2k 2 n-k-1 × E exp t2 -n ((P 0 + P 1 ) n f (X 1 )) -ν(P 0 + P 1 ) n f .
Thanks to (H 1 (C)), we conclude that

E exp t2 -n i∈Gn f (X i ) -ν(P 0 + P 1 ) n f ≤ exp 2 2 t 2 C f 2 Lip 2 × 2 2n n k=0 r 2k 2 n-k-1
and the results of the Proposition then follow from this last inequality.

For the ancestor-offspring triangle (X i , X 2i , X 2i+1 ), we have the following result which can be seen as a consequence of the Proposition 3.4.

Corollary 3.6. Let f be a real Lipschitzian function on

E 3 and n ∈ N. Assume that (H 1 (C)) holds. Then M Gn (f ) satisfies GC(γ ′ n ) where γ ′ n =      2C(1+q) 2 f 2 Lip r 2 |Gn| 1-(r 2 /2) n+2 1-r 2 /2 if r = √ 2 2C(1+q) 2 f 2 Lip (n+2) |Gn| if r = √ 2.
Proof. Let f be a real Lipschitzian function on E 3 , n ∈ N and t ∈ R. We have

E exp t2 -n i∈Gn f (X i , X 2i , X 2i+1 ) = E exp t2 -n i∈Gn P f (X i ) × E exp t2 -n i∈Gn (f (X i , X 2i , X 2i+1 ) -P f (X i )) F n .
By the Markov property and thanks to the Proposition 2.2 and the Theorem 1.5, we have

E exp t2 -n i∈Gn (f (X i , X 2i , X 2i+1 ) -P f (X i )) F n ≤ exp t 2 C f 2 Lip 2 n 2 × 2 2n
. Now, using P f instead of f in the proof of the Proposition 3.4 and using the fact that P f Lip ≤ (1 + q) f Lip and

E 2 -n i∈Gn f (X i , X 2i , X 2i+1 ) = E 2 -n i∈Gn P f (X i ) = 2 -n ν(P 0 +P 1 ) n P f,
we are led to

E exp t2 -n i∈Gn f (X i , X 2i , X 2i+1 ) -ν (P 0 + P 1 ) n P f ≤ exp 4t 2 C(1 + q) 2 f 2 Lip 2 2 × 2 n n k=-1 r 2 2 k .
The results then follow by easy calculations.

For the subtree T n , we have the following result.

Proposition 3.7. Let f be a real Lipschitzian function on E and n ∈ N.

Assume that (H 1 (C)) holds. Then M Tn (f ) satisfies GC(τ n ) where

τ n =              2C f 2 Lip (r-1) 2 |Tn| 1 + 1-(r 2 /2) n+1 1-r 2 /2 if r = √ 2, r = 1 2C f 2 Lip (r-1) 2 |Tn| (r 2 (n + 1) + 1) if r = √ 2 2C f 2 Lip |Tn| 2 |T n | -n+1 2 if r = 1.
Proof. Let f be a real Lipschitzian function on E and n ∈ N. Note that

E i∈Tn f (X i ) = ν n m=0 (P 0 + P 1 ) m f .
We have

E exp t |T n | i∈Tn f (X i ) = E   exp   t |T n | i∈T n-2 f (X i )   × exp   t |T n | i∈G n-1 (f + (P 0 + P 1 ) f ) (X i )   × E   exp   t |T n | i∈G n-1 (f (X 2i ) + f (X 2i+1 ) -(P 0 + P 1 )f (X i ))   F n-1     .
As in the proof of Proposition 3.4, we have

E   exp   t |T n | i∈G n-1 (f (X 2i ) + f (X 2i+1 ) -(P 0 + P 1 )f (X i ))   F n-1   ≤ exp 2 2 Ct 2 f 2 Lip 2 n-1 2|T n | 2 .
This leads us to

E exp t |T n | i∈Tn f (X i ) ≤ exp 2 2 Ct 2 f 2 Lip 2 n-1 2|T n | 2 ×E   exp   t |T n | i∈T n-2 f (X i )   exp   t |T n | i∈G n-1 (f + (P 0 + P 1 ) f ) (X i )     .
Iterating this method, we are led to

E exp t |T n | i∈Tn f (X i ) ≤ exp   2 2 t 2 C f 2 Lip 2|T n | 2 n-1 k=0 k l=0 r l 2 2 n-k-1   × E exp t |T n | n m=0 (P 0 + P 1 ) m f (X 1 )
and we then obtain thanks to (a) of (H 1 (C)) and Theorem 1.5

E exp t |T n | i∈Tn f (X i ) -ν n m=0 (P 0 + P 1 ) m f ≤ exp   2 2 t 2 C f 2 Lip 2|T n | 2 n k=0 k l=0 r l 2 2 n-k-1   .
In the last inequality we have used n m=0

(P 0 + P 1 ) m f Lip ≤ n k=0 r k f Lip .
The results then easily follows.

For the ancestor-offspring triangle we have the following results which can be seen as a consequence of the Proposition 3.7.

Corollary 3.8. Let f be a real Lipschitzian function on E 3 and n ∈ N. Assume that (H 1 (C)) holds. Then M Tn (f ) satisfies GC(τ ′ n ) where τ ′ n =                2 3 C(1+q) 2 f 2 Lip |Tn| 1 + 1 (r-1) 2 1 + r 2 1-(r 2 /2) n+1 1-r 2 /2 if r = √ 2, r = 1 2 3 C(1+q) 2 f 2 Lip |Tn| 1 + 1+r 2 (n+1) (r-1) 2 if r = √ 2 2 3 C(1+q) 2 f 2 Lip |Tn| 2 2|T n | -n+1 2 if r = 1.
Proof. Let f be a real Lipschitzian function on E 3 and n ∈ N. By Hölder inequality and using the fact that

E i∈Tn f (∆ i ) = E i∈Tn P f (X i ) ,
we have

E exp t |T n | i∈Tn f (∆ i ) -E i∈Tn f (∆ i ) ≤ E exp 2t |T n | i∈Tn (f (∆ i ) -P f (X i )) 1/2 × E exp 2t |T n | i∈Tn P f (X i ) -E i∈Tn P f (X i ) 1/2 .
We bound the first term of the right hand side of the previous inequality by using the same calculations as in the first iteration of the proof of Corollary 3.6. We then have

E exp 2t |T n | i∈Tn (f (∆ i ) -P f (X i )) 1/2 ≤ exp 2t 2 C f 2 Lip |T n | 2|T n | 2 .
For the second term, we use the proof of the Proposition 3.7 with P f instead of f . We then have

E exp 2t |T n | i∈Tn P f (X i ) -E i∈Tn P f (X i ) 1/2 ≤ exp   2 3 t 2 R(1 + q) 2 f 2 2|T n | 2 n k=0 k l=0 r l 2 2 n-k-1   .
The results then follow by easy analysis and this ends the proof.

Deviation inequalities towards the invariant measure of the randomly drawn chain

All the previous results do not assume any "stability" of the Markov chain on the binary tree, whereas for usual asymptotic theorem the convergence is towards mean of the function with respect to the invariant probability measure of the random lineage chain. To reinforce this asymptotic result by non asymptotic deviation inequality, it is thus fundamental to be able to replace for example E(M Tn (f )) by some asymptotic quantity. This random lineage chain is a Markov chain with transition kernel Q = (P 0 + P 1 )/2. We shall now suppose the existence of a probability measure π such that πQ = π. We will consider a slight modification of our main assumption and as we are mainly interested in concentration inequalities, let us focus in the p = 1 case:

Assumption 3.9 (H ′ 1 (C)). (a) ν ∈ T 1 (C); (b) P b (x, •) ∈ T 1 (C), ∀x ∈ E, b = 0, 1 ; (c) W d 1 (P (x, •, •), P (x, •, •)) ≤ q d(x, x), ∀x,
x ∈ E and some q > 0. And for r 0 , r 1 > 0 such that r 0 +r 1 < 2, for b = 0, 1,

W d 1 (P b (x, •), P b (x, •)) ≤ r b d(x, x), ∀x, x ∈ E.
Under this assumption, using the convexity of W 1 (see [START_REF] Villani | Optimal transport, volume 338 of Grundlehren der Mathematischen Wissenschaften[END_REF]), we easily see that

W 1 (Q(x, •), Q(x, •)) ≤ r 0 + r 1 2 d (x, x) , ∀x, x
ensuring the strict contraction of Q, and then the exponential convergence towards π in Wasserstein distance, namely (assuming that π has a first moment)

W 1 (Q n (x, •), π) ≤ r 0 + r 1 2 n d(x, y)π(dy).
Let us show that we may now control easily the distance between E(M Tn (f )) and π(f ). Indeed, we may first remark that

E   k∈Gn f (X k )   = ν(P 0 + P 1 ) n f
so that assuming that f is 1-lipschitzian, and by the dual version of the Wasserstein distance

E(M Tn (f )) -π(f ) = 1 |T n | n j=1 E   k∈G j (f (X k ) -π(f )   = 1 |T n | n j=1 2 j ν P 0 + P 1 2 j (f -π(f )) ≤ 1 |T n | n j=1 2 j W 1 (νQ j , π) ≤ 1 |T n | n j=1 (r 0 + r 1 ) j ≤ c n := c r 0 +r 1 2 n+1 if r 0 + r 1 = 1 c n 2 n+1 if r 0 + r 1 = 1
for some universal c, which goes to 0 exponentially fast as soon as r 0 +r 1 < 2 which was assumed in (H ′ 1 (C)). We may then see that for r > c n P M Tn (f ) -π(f ) > r ≤ P M Tn (f ) -E(M Tn (f )) > r -c n and one then applies the result of the previous subsection.

Application to nonlinear bifurcating autoregressive models

The setting will be here the case of the nonlinear bifurcating autoregressive models. It has been considered as a particular realistic model to study cell aging [START_REF] Stewart | Aging and death in an organism that reproduces by morphologically symmetric division[END_REF], and the asymptotic behavior of parametric estimators as well as non parametric estimators has been considered in an important series of work, see e.g. [START_REF] Basawa | Extensions of the bifurcating autoregressive model for cell lineage studies[END_REF][START_REF] Basawa | Inference for the extended bifurcating autoregressive model for cell lineage studies[END_REF][START_REF] Basawa | Non-Gaussian bifurcating models and quasilikelihood estimation[END_REF][START_REF] Basawa | Least-squares estimation for bifurcating autoregressive processes[END_REF][START_REF] Basawa | Maximum likelihood estimation for a firstorder bifurcating autoregressive process with exponential errors[END_REF][START_REF] Guyon | Limit theorems for bifurcating Markov chains. application to the detection of cellular aging[END_REF][START_REF] Bercu | Asymptotic analysis for bifurcating autoregressive processes via a martingale approach[END_REF][START_REF] Delmas | Detection of cellular aging in Galton-Watson process[END_REF][START_REF] Saporta | Asymmetry tests for bifurcating auto-regressive processes with missing data[END_REF][START_REF] Saporta | Parameters estimation for asymmetric bifurcating autoregressive processes with missing data[END_REF][START_REF] Bitseki Penda | Deviation inequalities and moderate deviations for estimators of parameters in bifurcating autoregressive models[END_REF] (and for example in the random coefficient setting in [START_REF] Saporta | Random coefficients bifurcating autoregressive processes[END_REF]).

We will then consider the following model where to simplify the state space E = R, where L(X 1 ) = µ 0 satisfies T 1 and we recursively define on the binary tree as before

X 2k = f 0 (X k ) + ε 2k X 2k+1 = f 1 (X k ) + ε 2k+1 (4.1)
with the following assumptions: It is then easy to deduce that under these two assumptions, we perfectly match with the previous framework. Denoting P 0 and P 1 as previously, we see that (H ′ 1 ) is verified, with the additional fact that P = P 0 ⊗ P 1 . We will do the proof for P 0 , being the same for P 1 . The conclusion follows for P by conditional independence of X 2k and X 2k+1 . Let us first prove that P 0 (x, •) satisfies T 1 . Indeed P 0 (x, •) is the law of f 0 (x) + ε 2k , and we have thus to verify the Gaussian integrability property of Theorem 1.6. To this end, consider x 0 = f (x), and choose δ ε of condition (No) to verify the Gaussian integrability property. We have thus that P 0 satisfies T 1 (C P ). We prove now the Wasserstein contraction property. P 0 (x, •) is of course the law of f 0 (x) + ε k . Here ε k denotes a generic random variable and thus the law of P 0 (y, •) is the law of f 0 (y)+ε k and an upper bound of the Wasserstein distance between P 0 (x, •) and P 0 (y, •) can then be obtained by the coupling where we really choose the same noise ε k for the realization of the two marginal laws so that Let f be any Lipschitz function such that f

Lip ≤ 1 S f (z)P 0 (x, dz) - S f (z)P 0 (y, dz) = E [f (f 0 (x) + ε 1 ) -f (f 0 (y) + ε 1 )] ≤ f Lip |f 0 (x) -f 0 (y)|.
By the Monge-Kantorovitch duality expression of the Wasserstein distance, one has then

W 1 (P 0 (x, •), P 0 (y, •)) ≤ |f 0 (x) -f 0 (y)| ≤ f 0 Lip |x -y|.
Thus under (NL) and (No), our model fits in the framework of the previous section with q = f 0 Lip + f 1 Lip , r 0 = f 0 Lip and r 1 = f 1 Lip . We will be interested here in the non parametric estimation of the autoregression functions f 0 and f 1 , and we will use Nadaraya-Watson kernel type estimator, as considered in [START_REF] Bitseki Penda | Asymptotic behavior of nonparametric estimators in nonlinear bifurcating autoregressive models[END_REF]. Let K be a kernel satisfying the following assumption. Let us also introduce as usual a bandwidth h n which will be taken to simplify as h n := |T n | -α for some 0 < α < 1. The Nadaraya-Watson estimators are then defined as for x ∈ R

f 0,n (x) := 1 |T n |h n k∈Tn K X k -x h n X 2k 1 |T n |h n k∈Tn K X k -x h n f 1,n (x) := 1 |T n |h n k∈Tn K X k -x h n X 2k+1 1 |T n |h n k∈Tn K X k -x h n .
Let us focus on f 0 , as it will be exactly the same for f 1 and fix x ∈ R. We will be interested here in deviation inequalities of f 0,n (x) with respect to f (x). One has to face two problems. First it is an autonormalized estimator. It will be dealt with considering deviation inequalities for the numerator and denominator separately and reunite them. Secondly (x, y) → K(x)y is in fact not Lipschitzian in general state space, so that the result of the previous section for deviation inequalities of Lipschitzian function of ancestor-offspring may not be applied directly. Let us tackle this problem. By definition

f 0,n (x) -f (x) = 1 |T n |h n k∈Tn K X k -x h n [f 0 (X k ) -f 0 (x) + ε 2k ] 1 |T n |h n k∈Tn K X k -x h n := N n + M n D n .
where

N n := k∈Tn K X k -x h n [f 0 (X k ) -f 0 (x)], M n := k∈Tn K X k -x h n ε 2k , D n = k∈Tn K X k -x h n . Denote also Ñn = N n /(|T n |h n ), Mn = M n /(|T n |h n ), Dn = D n /(|T n |h n ).
Let us remark that D n and M n completely enter the framework of Proposition 3.7. We may thus prove Proposition 4.4. Let us assume that (N L), (N o) and (Ker) holds, and

q = f 0 Lip + f 1 Lip < √ 2.
Let us also suppose that α < 1/4. Then for all r > 0 such that r > E( Ñn )/E( Dn ), there exists constants C, C ′ , C ′′ > 0 such that

P | f 0,n (x) -f (x)| > r ≤ 2 exp -C(rE( Dn ) -E( Ñn )) 2 |T n |h 2 n +2 exp -C ′ (rE( Dn ) -E( Ñn )) 2 |T n |h 2 n 1 + C ′′ r 2 h 2 n .
Proof. Remark first that, by (Ker), K is Lipschitz continuous so that y → K( y-x hn ) is also lipschitzian with constant K Lip /h n . The mapping y → K( y-x hn )(f 0 (y) -f 0 (x)), as K has a compact support and f 0 is Lipschitzian, is also Lipschitzian with constant R K Lip f 0 Lip + f 0 Lip K ∞ . We can then use Proposition 3.7 to get deviation inequalities for D n . For all positive r there exists a constant L (explicitly given through Proposition 3.7), such that

P(|D n -E(D n )| > r|T n |h n ) ≤ 2 exp -Lr 2 |T n |h 4 n / K 2 Lip .
For N n + M n we cannot directly apply Proposition 3.7 due to the successive dependence of X k at generation n and ε 2k of generation n -1. But as we are interested in deviation inequalities, we may split the deviation coming from each term. For N n , it is once again a simple application of Proposition 3.7,

P(|N n -E(N n )| > r|T n |h n ) ≤ 2 exp -Lr 2 |T n |h 2 n (R K Lip f 0 Lip + f 0 Lip K ∞ ) 2 .
Note that ε 2k is independent of X k , and centered so that E(M n ) = 0, and satisfies a transportation inequality. Note also that K is bounded. By simple conditioning argument, we may control the Laplace transform of M n quite simply. We then have for all positive r

P(|M n | > r|T n |h n ) ≤ 2 exp -r 2 |T n |h 2 n K 2 ∞ 2Cε .
However, we cannot use directly these estimations as the estimator is autonormalized. Instead Remark now to conclude that K((y -x)/h n )(f (y)-f (x))+ K((y -x)/h n ) is (R K Lip f 0 Lip + f 0 Lip K ∞ + r K Lip /h n )-Lipschitzian, and we may then proceed as before. We refer to [START_REF] Bitseki Penda | Asymptotic behavior of nonparametric estimators in nonlinear bifurcating autoregressive models[END_REF] for quantitative versions of these limits.

Remark 4.6. Of course this non parametric estimation is in some sense incomplete, as we would have liked to consider a deviation inequality for sup x | f0,n (x) -f 0 (x)|. The problem is somewhat much more complicated here, as the estimator is self normalized. However, it is a crucial problem that we will consider in the near future. For some ideas which could be useful here, let us cite the results of [START_REF] Bolley | Quantitative concentration inequalities for empirical measures on non-compact spaces[END_REF] for (uniform) deviation inequalities for estimators of density in the i.i.d. case, and to [START_REF] Fournier | On the rate of convergence in Wasserstein distance of the empirical measure[END_REF] for control of the Wasserstein distance of the empirical measure of i.i.d.r.v. or of Markov chains.

Remark 4.7 (Estimation of the T-transition probability). We assume that the process has as initial law, the invariant probability ν. We denote by f the density of (X 1 , X 2 , X 3 ). For the estimation of f , we propose the estimator f n defined by

f n (x, y, z) = 1 |T n |h n k∈Tn K x -X k h n K y -X 2k h n K z -X 2k+1 h n .
An estimator of the T-probability transition is then given by P n (x, y, z) = f n (x, y, z) Dn .

For x, y, z ∈ R, one can observe that the function G defined on R 3 by

G(u, v, w) = K x -u h n K y -v h n K z -w h n
is Lipschitzian with G Lip ≤ ( K 2 ∞ K Lip )/h n . We have P n (x, y, z) -P (x, y, z) = f n (x, y, z) -f (x, y, z) Dn + f (x, y, z)(ν(x) -Dn ) ν(x) Dn .

Now using the decomposition

f n (x, y, z) -f (x, y, z) = f n (x, y, z) -E f n (x, y, z)

+ E f n (x, y, z) -f (x, y, z)
and the convergence of E f n (x, y, z) to f (x, y, z), we obtain a deviation inequality for | P n (x, y, z) -P (x, y, z)| similar to that obtained at the Proposition 4.4.

When the density g ε of (ε 2 , ε 3 ) is known, another strategy for the estimation of the T-transition probability is to observe that P (x, y, z) = g ε (yf 0 (x), z -f 1 (x)). An estimator of P (x, y, z) is then given by P n (x, y, z) = g ε (y -f 0,n (x), z -f 1,n (x)) where f 0,n and f 1,n are estimators defined above. If g ε is Lipschitzian, we have

| P n (x, y, z) -P (x, y, z)| ≤ g ε Lip | f 0,n (x) -f 0 (x)| + | f 1,n (x) -f 1 (x)|
and the deviation inequalities for | P n (x, y, z) -P (x, y, z)| are thus of the same order that those given by the Proposition 4.4.

Assumption 2 . 1 (

 21 H p (C)).

  (a) ν ∈ T p (C); (b) P (x, •, •) ∈ T p (C), ∀x ∈ E ;

  We also consider the empirical mean M I (f ) over I defined by M I (f ) = (1/|I|)M I (f ) where |I| denotes the cardinality of I. In the statistical applications, the cases N = |T n | and I = G m (for m ∈ {0, • • • , n})

Assumption 4 . 1 (

 41 NL). f 0 and f 1 are Lipschitz continuous function. Assumption 4.2 (No). (ε k ) k≥1 are centered i.i.d.r.v. and for all k ≥ 0, ε k have law µ ε and satisfy for some positive δ ε , µ ε e δεx 2 < ∞. Equivalently, µ ε satisfies T 1 (C ε ).

Assumption 4 . 3 (

 43 Ker). The function K is non negative, has compact support [-R, R], is Lipschitz continuous with constant K Lip and such that K(z)dz = 1.

P

  f 0,n (x) -f (x) > r ≤ P( Ñn + Mn > r Dn )≤ P Ñn -E( Ñn ) -r( Dn -E( Dn )) + Mn > rE( Dn ) -E( Ñn ) ≤ P Ñn -E( Ñn ) -r( Dn -E( Dn ) > (rE( Dn ) -E( Ñn ))/2 +P Mn > (rE( Dn ) -E( Ñn ))/2

Remark 4 . 5 .

 45 In order to get fully practical deviation inequalities, let us remark thatE Dn = 1 |T n |h n n m=0 2 m µ 0 Q m H -→ n→+∞ ν(x)where H(y) = K((y -x)/h n ), ν(•) is the invariant density of the Markov chain associated to a random lineage andE Ñn = 1 |T n |h n n m=0 2 m (µ 0 Q m (Hf 0 ) -f 0 (x)µ 0 Q m H) -→ n→+∞ 0.

Acknowledgements S.V. Bitseki Penda sincerely thanks Labex Mathématiques Hadamard. A. Guillin was supported by the French ANR STAB project.