Bruno N Rémillard
email: bruno.remillard@hec.ca

Jean Vaillancourt
email: jean.vaillancourt@hec.ca

Detecting periodicity from the trajectory of a random walk in random environment

Keywords: Random walks, random environments, periodic functions, ergodic theory

For nearest neighbor univariate random walks in a periodic environment, where the probability of moving depends on a periodic function, we show how to estimate the period and the function. For random walks in non-periodic environments, we find that the asymptotic limit of the estimator is constant in the ballistic case, when the random walk is transient and the law of large numbers holds with a non zero limit. Numerical examples are given in the recurrent case, and the sub-ballistic case, where the random walk is transient but the law of large numbers yields a zero limit.

Introduction

Given no other information than the trajectory of a random walk in random environment (RWRE), one would like to estimate the law of the environment. While this is usually not feasible, there are simple instances where various modeling choices will generate the full distribution, at least asymptotically in some sense. Our purpose is solely to detect periodicity from observing a single trajectory of an RWRE.

All random variables are built on an ambient complete probability space (E, E, P). A random environment is a measurable two-sided sequence α(e) = α(•, e) ∈ (0, 1) Z indexed by e ∈ E and we write µ = P • α -1 for its distribution on the Borel subsets of [0, 1] Z . It will be convenient to write ρ i (e) = 1-α(i,e) α(i,e) and we will occasionally drop parameter e whenever there is no ambiguity in doing so. Let X = {X t } t≥0 be a (nearest neighbor) RWRE on Z, to wit

P e (∆X t = X t -X t-1 = 1| X t-1 = i) = P (X t = i + 1| X t-1 = i, E) (e) = α(i, e), (1.1)
with P e (∆X t = -1| X t-1 = i) = 1 -α(i, e), for every choice of i ∈ Z and positive integer t. This simply means that given a realization e of the environment and a starting point X 0 for the walk, the successive locations {X t : t ≥ 1} form a time homogeneous irreducible Markov chain on Z under P e and the conditional law of the whole walk is a probability P e on the power set over Z N known as the quenched law for the RWRE. The whole process is therefore encapsulated in the family of joint probability laws P µ defined by P µ (F × A) = A P e (F) µ(de), and its first marginal on Z N is known as the annealed law for the RWRE. Note that X is not in general a Markov process under P µ .

In order to use such models, it is necessary to be able to estimate the process A = {α(i)} i∈Z . Of course, this is not possible in general, but if there is some sort of parametric structure, estimation is possible. For instance Comets et al. [START_REF] Comets | Maximum likelihood estimator consistency for a ballistic random walk in a parametric random environment[END_REF] study the asymptotic distributions for an M-estimator of their choice that is very close to the MLE for the solution to (1.1) above. The authors consider that {α(i)} i∈Z are independent and identically distributed (iid) with a parameterized marginal distribution, in the so-called ballistic right-transient case E(log ρ 0 ) < 0 and E(ρ 0 ) < 1. The parameter of interest is estimated via a M-estimator of the sequence X 0 , . . . , X τ n , where τ n is the time of the first visit to n. The corresponding sub-ballistic case, i.e., E(log ρ 0 ) < 0 and E(ρ 0 ) ≥ 1, was studied in Falconnet et al. [START_REF] Falconnet | Maximum likelihood estimation in the context of a sub-ballistic random walk in a parametric random environment[END_REF],

while the recurrent case E(log ρ 0 ) = 0 is examined in Comets et al. [START_REF] Comets | Maximum likelihood estimator consistency for recurrent random walk in a parametric random environment with finite support[END_REF], when the distribution of the iid sequence {α(i)} i∈Z has finite support. Recently, Diel and Lerasle [START_REF] Diel | Non parametric estimation for random walks in random environment[END_REF] considered non-parametric estimators in the i.i.d. case.

Further generalization was achieved in the ballistic case by Andreoletti et al. [START_REF] Andreoletti | Hidden Markov model for parameter estimation of a random walk in a Markov environment[END_REF] where a tractable class of hidden Markov models is assumed for {α(i)} i∈Z . An alternative estimation procedure based on the moments of the random environment was proposed earlier on by Adelman and Enriquez [START_REF] Adelman | Random walks in random environment: What a single trajectory tells[END_REF], which also applies in more general contexts than those considered here. Even in these examples, where in each case {α(i)} i∈Z itself is a very simple stochastic process, the resulting RWRE X in (1.1) is still not a Markov process.

In the present paper, we shall first analyze the MLE for the unobserved environment A = {α(i)} i∈Z in (1.1) under the restricted context where this environment is known to be spatially periodic. We shall then investigate conditions on A under which the MLE obtained in the periodic case, allows for the detection of absence of periodicity.

Estimation of a periodic environment

Let P d be the set of all periodic functions on Z with values in (0, 1) and period d. For a given p ∈ P d , set

E p = {e j = p(• + j); 1 ≤ j ≤ d}.
The resulting environment {α(•, e j ) = p(• + j); j ≥ 0} is a deterministic sequence (save for the possibly random starting point e 0 ∈ E p ) with a unique invariant probability, namely the uniform distribution over the finite set E p , meaning that the probability of e j is 1/d, for every j ∈ S d = {1, . . . , d}. This particular case of a random environment leads to the definition of a random walk in a periodic environment (RWPE for short) introduced in Pyke [START_REF] Pyke | On random walks and diffusions related to Parrondo's games[END_REF]. We say that X is a RWPE if there exists p ∈ P d , for some d ∈ N, so that X = {X t } t≥0 is the (nearest neighbor) random walk defined by

P (∆X t = 1| X t-1 = i) = p(i), i ∈ Z, t ≥ 1. (2.1)
Of course, if p ∈ P d , then p ∈ P kd , for any k ∈ N. Clearly X is an irreducible Markov chain. The main aim of this section is to estimate the associated periodic function p and the smallest d 0 so that p ∈ P d 0 , using a single trajectory X 0 , . . . , X n of the RWPE X. The justification of using the class of RWPE instead of the more general RWRE models defined by (1.1) comes from a celebrated result due to Parthasarathy [START_REF] Parthasarathy | On the category of ergodic measures[END_REF] which says that the law µ of the stationary ergodic process A = {α(i)} i∈Z is the limit in distribution of a sequence of processes {α n (i) = p n (i

p (d) n ( j) = A (d) n, j / A (d) n, j + B (d) n, j , j ∈ {1, . . . , d}, (2.2)
where

A (d) n, j = n t=1 I{(X t-1 ) d = j, ∆X t = 1}, B (d) n, j = n t=1 I{(X t-1 ) d = j, ∆X t = -1}, and A (d) n, j + B (d) n, j = n t=1 I{(X t-1 ) d = j}. The associated log-likelihood is given by L n,d = d j=1 A (d) n, j log{p (d) n ( j)} + B (d) n, j log{1 -p (d) n ( j)} = - d j=1 A (d) n, j + B (d) n, j H p (d) n ( j) , (2.3)
where for any x ∈ (0, 1), n, j n → π j p( j), B (d) n, j n → π j {1p( j)} , so p (d) n ( j) → p( j), and L n,d n → L d = -d j=1 π j H{p( j)}.

H(x) = -x log x -(1 -x) log(1 -x) ≥ 0
The problem is to find d. The following example illustrates what one can expect.

Example 2.1. We generated a random walk of length n = 10000 satisfying (2.1), starting from X 0 = 100, with p = (0.099, 0.749, 0.749). The results of the estimation for d = 3 are p n = (0.0954, 0.7593, 0.7430) ∈ P 3 , with a log-likelihood of -4.6976 × 10 3 . Figure 1 shows the behavior of the log-likelihood for periods d ∈ {1, . . . , 10}. As expected, the first local maximum is reached at d = 3, with other local maxima at 6, 9, which are multiples of d = 3.

In addition, we plotted the theoretical value L 3 to show the convergence of L n,d /n to the former. n → ∞, for any i

Estimation of the least period

∈ S d , A (d) n,i n a.s. → m d j∈S d 0 , ( j) m =(i) m π j p( j), B (d) n,i n a.s. → m d j∈S d 0 , ( j) m =(i) m π j {1 -p( j)}, p (d) n (i) a.s. → p (d) (i) = j∈S d 0 , ( j) m =(i) m π j p( j) j∈S d 0 , ( j) m =(i) m π j , (2.4)
L n,d n a.s. → L d = - m d d i=1 j∈S d 0 , ( j) m =(i) m π j H p (d) (i) . (2.5)
In particular, if m = 1, then A

Behavior under non-periodic environments

What if the model is not a RWPE? From now on X = (X t ) t≥0 is a full fledged RWRE satisfying (1.1). We investigate next how our estimator behaves when the environment is no longer periodic. Henceforth the right shift operator on (0, 1) Z is denoted by T .

Fixed environments

For a fixed environment e, the asymptotic behavior for X is well-known [START_REF] Chung | Markov chains with stationary transition probabilities[END_REF]. For the sake of completeness, we formulate them in the following lemma. Recall that ρ k (e) = 1-α(k,e) α(k,e) and define (e) , C 0 (e) ≡ 1, and

S (e) = 1 + ∞ j=1 j k=1 ρ k (e), F(e) = 1 + ∞ j=1 j k=1 1 ρ -k
C j (e) =                  α(0,e) α( j,e) j k=1 1 ρ k (e)
for j > 0, 1-α(0,e) 1-α( j,e) j k=1 ρ -k (e) for j < 0.

(3.1) Lemma 3.1. Let X = (X t ) t≥0 satisfy (1.1) for some fixed environment α(•, e) ∈ (0, 1) Z . For any starting point X 0 , one and only one of the following four statements holds.

• If S (e) < ∞ and F(e) < ∞ then P e lim inf n→∞

|X n | = +∞ = 1. • If S (e) < ∞ and F(e) = ∞ then P e lim n→∞ X n = +∞ = 1. • If S (e) = ∞ and F(e) < ∞ then P e lim n→∞ X n = -∞ = 1. • If S (e) = ∞ and F(e) = ∞ then P e lim inf n→∞ X n = -∞, lim sup n→∞ X n = +∞ = 1.
Further, if j∈Z C j (e) < ∞, then S (e) = ∞, F(e) = ∞ and X has a unique invariant measure π j (e) = C j (e)π 0 (e), with π 0 (e) = 1/ j∈Z C j (e).

In the case where X is positive recurrent, prescribed by the finiteness of j∈Z C j (e) < ∞, the ergodic theorem for Markov chain (X t , X t+1 ) t≥0 , which is also irreducible on {( j, k) ∈ Z 2 ; kj = ±1}, and positive recurrent as well, directly implies the following result, proven in Appendix A.4. Proposition 3.1. Let π be the unique invariant probability measure of the Markov chain X associated with an arbitrary but fixed environment α satisfying j∈Z C j (e) < ∞ in (3.1). As n → ∞, there holds, for every integer pair n ; d ≥ 1}, which exhibits a telltale seesaw motion under the hypothesis of periodicity, as shown in Figure 1, will instead display a tendency to increase in d from approximatelylog 2 ∼ -.693 to an upper bound L ∞ when the environment is no longer periodic but such that the simple random walk X on the whole line is positive recurrent. The convexity of -H yields the inequalities L n,1 ≤ L n,d ≤ L n,kd for any n ≥ 1, d ≥ 1, k > 1 and all environments, recurrent or not, hence any departure from periodicity in the environment is likely to be detected through this tendency to increase, simply by graphing the empirical log-likelihoods (L n,kd ) k≥1 , for a fixed d.

1 ≤ i ≤ d < ∞ and any starting point X 0 , p (d) n (i, e) a.s. → α (d) (i, e) = j∈Z, ( j) d =i π j (e)α( j, e) j∈Z, ( j) d =i π j (e), (3.2)

Random environments

From now on, assume that the sequence α(i, •) is stationary and ergodic with respect to the measure µ and T is a Let us return to (1.1) and write X n = n t=1 {∆X t -2α(X t-1 ) + 1}n + 2 n t=1 α(X t-1 ). Now, ξ t = ∆X t -2α(X t-1 ) + 1 is a bounded martingale difference sequence, so 1 n n t=1 ξ t converges to 0 almost surely, as n → ∞. Hence the limiting behavior of X n /n is the same as the limiting behavior of -1

+ 2 n n t=1 α(X t ). It is shown in Alili [2, Theorem 4.1] that
X n /n converges µ almost surely, an extension of the original iid case due to Solomon [START_REF] Solomon | Random walks in a random environment[END_REF].

Invariant measure for T X t e

In order to find the limit of n t=1 α(X t ) n, consider the Markov chain T X t e on E. Its Markov operator T is given by

T h(e) = α(0, e)h(T e) + {1 -α(0, e)}h(T -1 e), (3.5)
for any bounded measurable function h. Setting X x t (e) for the chain starting at x from environment e, then X x t (e) = x + X 0 t (T x e), so α(X x t , e) = α(X 0 t (T x e), T x e). The associated Markov operator, denoted T x , satisfies

T x h(e) = E h(T X x 1 e) = T h(T x e).
If λ is an invariant measure for T , the invariant measure λ x for T x is λ x (A) = λ(T x A).

Hence 1 n n t=1 α(X x t-1 , e) should converge to E λ x {α(x)} = E λ {α(0)}.
Hence the limit is independent of x even if α(x) is not necessarily stationary with respect to λ. If λ is a stationary distribution for this Markov chain that is absolutely continuous with respect to µ, its density φ satisfies φ = T φ, where T is the adjoint operator of T on L 1 (µ) given by T φ(e) = α(0, T -1 e)φ(T -1 e) + {1 -α(0, T e)}φ(T e).

(

Suppose that E(S ) < ∞. Then dλ = v{1 + ρ 0 (e)}S (e)dµ is an invariant ergodic measure for T X t e, so in this case X n /n converges almost surely to -1 + 2E λ (α 0 ) = -1 + 2vE(S ) = v, where α 0 (e) = α(0, e). Next, if E(F) < ∞,

d λ = ṽ{1 + ρ -1
0 (e)}F(e)dµ is an invariant ergodic measure for T X t e, so in this case X n /n converges almost surely to

-1 + 2Eλ(α 0 ) = 1 -2ṽE(F) = -ṽ. If E(S ) = E(F) = +∞, then X n /n converges almost surely to 0, which is equivalent to 1 n n t=1
α(X t ) converging almost surely to 1/2. That T is an ergodic operator goes back to Kozlov [START_REF] Kozlov | The method of averaging and walks in inhomogeneous environments[END_REF], where a proof is sketched. We prove it in Appendix A.5.

The following section will be used for computing some important limiting values such as E λ (α 0 ).

Distribution of α 0 under the invariant measure λ in the ballistic case

Suppose we are in the right ballistic case, i.e., E(S ) < ∞. Then, the (unique) invariant measure λ has density vS /α 0 with respect to µ,

and v = 1 2E(S )-1 . It follows that E λ (α 0 ) = vE(S ) = 1 1+ρ .
What is the distribution of α 0 under λ? In the iid case, S is independent of α 0 and E(S ) = 1 1-ρ , so v = 1-ρ 1+ρ , and E(1

+ ρ 0 ) = E(1/α 0 ) = 1 + ρ.
Note also that E(log ρ 0 ) < 0 and E(ρ 0 ) = ρ < 1. For any bounded measurable function H on (0, 1),

E λ {H(α 0 )} = vE {H(α 0 )(1 + ρ 0 )S } = vE(S )E H(α 0 ) α 0 = 1 1 + ρ E H(α 0 ) α 0 .
Hence the density of the distribution of α 0 under λ, with respect to the distribution of α 0 under µ is 1 (1+ρ)x . In particular, if α 0 has density g(x) under µ, then it has density g(x)

(1+ρ)x under λ. Suppose now that we are in the left ballistic case, i.e., E(F) < ∞. Then, the (unique) invariant measure λ has density vF/(1 -α 0 ) with respect to µ, and ṽ = 1 2E(F)-1 . In the iid case, i.e., E(log ρ 0 ) > 0 and E(1/ρ 0 ) = ρ < 1. Then under the (unique) invariant measure λ, Eλ(α 0 ) = ρ 1+ ρ . Also, F is independent of α 0 and E(F) = 1 1-ρ , so ṽ = 1-ρ 1+ ρ , and E(1 + 1/ρ 0 ) = E 1 1-α 0 = 1 + ρ. For any bounded measurable function H on (0, 1),

E λ {H(α 0 )} = ṽE {H(α 0 )(1 + 1/ρ 0 )F} = ṽE(F)E H(α 0 ) 1-α 0 = 1 1+ ρ E H(α 0 ) 1-α 0 .
Hence the density of the distribution of α 0 under λ, with respect to the distribution of α 0 under µ is 1 (1+ ρ)(1-x) . In particular, if α 0 has density g(x) under µ, then it has density g(x) (1+ ρ)(1-x) under λ. In this case, X n /n converges also surely to 2Eλ(α 0 ) -1 = ρ-1 ρ+1 = -ṽ. One can ask now if there is an invariant measure when E(S ) = E(F) = +∞ which is not treated by Alili [START_REF] Alili | Asymptotic behaviour for random walks in random environments[END_REF]. We can give a complete answer to this question in the periodic case.

Invariant measure in the periodic case

Suppose that E p = {T k p; k ∈ S d }, for some p ∈ P d . We noted earlier that (X t ) d forms an irreducible Markov chain with a unique ergodic distribution π = π(p, X 0 ), one for each fixed e ∈ E p . The unique solution of (3.6) is φ(T k e) = dπ k (e), where π(e) is the unique stationary distribution of (X t ) d . As a result, E λ (α 0 ) = d k=1 π k (p)α(k, p). In fact the density of λ with respect to µ should be cφ(e), where φ(e) = {1 + ρ 0 (e)} {1 + ρ 1 (e)

+ • • • + ρ 1 (e) • • • ρ d-1 (e)},
and one gets that c = d/ d k=1 φ(T k e) . This means that we have a closed-form expression for the invariant measure π, namely, for any k ∈ {1, . . . , d}, π k (e) = φ(T k e)/ d j=1 φ(T j e) . Note that the invariant measure exists even in the recurrent case, i.e., if E(S ) = E(F) = +∞. In this case, E(log ρ 0 ) = 0 means that 1

d d k=1 log(ρ k ) = 0, so ρ 1 • • • ρ d = 1.

Behavior of the estimator

Next, consider the Markov chain X t = ((X t ) d , T X t e) on S d × E, and suppose that there are m closed classes E 1 , . . . , E m . Its Markov operator Ť is given, for any bounded measurable h on S d × E, by Ť h( j, e) = α(0, e)h( j + 1, T e) + {1 -α(0, e)}h( j -1, T -1 e). ϕ k ( j, e)µ(de) = 1. Here it is assumed that ϕ k ( j, e) = 0 whenever ( j, e) E k . If φ solves (3.6), then ϕ k ( j, e) = φ(e)I E k ( j, e) satisfies (4.2). In particular, if m = 1, i.e., the Markov chain X is irreducible, then ϕ( j, e) = φ(e) solves (4.1) if φ solves (3.6). We can also prove uniqueness in this case.

Theorem 4.1. Assume that all powers of T are ergodic. Then ϕ( j, e) = φ(e) solves (4.1) if φ solves (3.6) and it is the unique invariant ergodic measure for the Markov chain X.

Remark 4.1. A sufficient condition for all powers of T to be ergodic is that T is weak mixing [5, p. 16], i.e., for any measurable sets A, B, lim n→∞

1 n n-1 k=1 µ A ∩ T -k B -µ(A)µ(B) = 0.
If T is weak mixing then T k is also weak mixing for any k, yielding that T k is ergodic.

Using the ergodic theorem, we have the following interesting result.

Corollary 4.1. Under the conditions of Theorem 4.1, for any ( j, e), A (d) n, j n a.s.

→ 1 d E{ϕ( j, •)α 0 } = 1 d E λ (α 0 ), B (d) n, j n a.s. → 1 d E {ϕ( j, •)(1 -α 0 )} = 1 d E λ (1 -α 0 ), so p (d) n ( j) a.s.
→ E λ (α 0 ). Also, L n,d /n a.s.

→ -H {E λ (α 0 )}.

Remark 4.2. Suppose that d 0 is the least k so that T k = I. Then we are in the periodic case, and it follows from the proof of the theorem that any harmonic function is constant on the closed classes

E i = ( j, T i+ j-1+ d p); j ∈ S d , = 0, . . . , (d 0 /m -1) = ( j, T i+ j-1+ m p); j ∈ S d , = 0, . . . , (d 0 /m -1) , i ∈ {1, . . . , m}, with m = (d, d 0 )
. In this case the Markov chain is ergodic on each class E i , with invariant density proportional to φ. Let φ be defined as in Section 3.3. Set d = d 0 ×d/m. Then p ∈ P d. As a result, since µ is the uniform measure on E p = {T p, . . . , T d 0 p}, one gets for ( j, e) = j, T i+ j-1+ m p ∈ C i , ϕ i ( j, e) = md 0 φ(e)

d 0 r=1 φ(T r p) = md 0 π i+ j-1+ m ,
where π k = π k (p) is the unique invariant measure of the Markov chain (X t ) d 0 defined by (2.1). One then recovers the results of Proposition 2.2 using the ergodic theorem for the Markov chain X on E i , i ∈ S d .

Numerical experiments

We provide three illustrations when the α(i) are iid with uniform distribution on [a, b]. First, we consider a rightballistic case, where the assumptions of Theorem 4.1 are met. In the other two cases, where X n /n → 0, we consider a sub-ballistic case and a recurrent case. More precise rates of convergence are known to exist [START_REF] Diel | Non parametric estimation for random walks in random environment[END_REF]. n with d ∈ {1, . . . , 5}, and using a simulated trajectory of length 10 7 , one obtains that p (d) n (i) = .5954 ± .0005 for all i ∈ {1, . . . , d}. These results are coherent with Corollary 4.1. The estimation of the distribution of α 0 and the graph of the log-likelihoods are displayed in Figure 2. The graph of the likelihoods seems to vary but if we look closely at the scale, there is no significant variability.

Sub-ballistic iid case

Suppose again that the α(i) are iid, with a uniform distribution over the interval (a, b). It then follows that The trajectory and the graph of the log-likelihoods are displayed in Figure 3. According to Alili's result, the RWRE is transient with X n → +∞, but X n /n → 0 as n → ∞, explaining the slow convergence of X n . In fact, according to Kesten et al. [START_REF] Kesten | A limit law for random walk in a random environment[END_REF] and Diel and Lerasle [START_REF] Diel | Non parametric estimation for random walks in random environment[END_REF], X n /n κ converges in law to a non-trivial distribution, where κ = .6802. Here we see that the log-likelihoods behave erratically, compared to the periodic case or the ballistic case.

E(log ρ 0 ) = 1 b-a a log a + (1 -a) log(1 -a) -b log b -(1 -b) log(1 -b) . Also, E(S ) < ∞ iff ρ = E(ρ 0 ) = -1 + 0.

Recurrent iid case

Here we generated the so-called Sinai's random walk of length 10 7 satisfying (1.1), starting from X 0 = 0, with iid uniform probabilities over (0, 1). We are the recurrent case so X n /n → 0. However, as shown in Sinai [START_REF] Sinai | The limit behavior of a one-dimensional random walk in a random environment[END_REF],

X n /(log n) 2 converges in law to a non-trivial distribution. The trajectory and the graph of the log-likelihoods are displayed in Figure 4. Clearly the RWRE X in (1.1) is not a stationary process in this case. For more on those recurrent cases affording tractability, see Andreoletti [START_REF] Andreoletti | On the estimation of the potential of Sinai's RWRE[END_REF] and the references therein. Again, the log-likelihoods behave erratically.

Conclusion

To distinguish between cases, one should first plot the graph of X t /t, for t ∈ {1, . . . , n}. It it goes to 0, we are in the sub-ballistic or the recurrent cases. If the limit is not 0, we can use the likelihoods behavior to distinguish between the periodic and ballistic cases. For a RWPE, we can detect the period and estimate the associated probabilities, while in the ballistic non-periodic case, the limiting distribution of the probability estimators is constant, as well as the log-likelihoods. i = j mod (m), then there is a unique T (i, j) ∈ {1, . . . , d} so that x = T (i, j) mod ( d). If ( ji) m 0, then there is no solution T (i, j). Set a = d/m and b = d 0 /m. Then (a, b) = 1. For a given i ∈ {1, . . . , d} such that (i) m = β ∈ {1, . . . , m}, then for any j = (l -1)m + β, l ∈ {1, . . . , b}, there is a solution T (i, j). As a result, for any i ∈ {1, . . . , d}, A → π j p( j), since the invariant measure for the Markov chain {(X t ) d : t ≥ 0} satisfies πl = π j = π j /a whenever l j mod (d 0 ). As result, A (d) n,i a.s.

→ 1 a j∈S d 0 , ( j) m =(i) m π j p( j). Similarly, B (d) n,i a.s.

→ 1 a j∈S d 0 , ( j) m =(i) m π j {1p( j)}. The almost sure convergence of p (d) n (i) and L n,d /n then follows. Finally, to complete the proof, note that for any i, { j ∈ S d 0 , ( j) 1 = (i) 1 } = S d 0 . ( j) = p (d 0 ) n ( j). Hence the "randomness" of d 0,n is not important in the limit.

First, assume thatProposition 2 . 2 .Figure 1 :

 221 Figure 1: Plot of L n,d /n (solid line) and L 3 (dashed line), for d ∈ {1, . . . , 10}.

Theorem 2 . 1 .

 21 π j p( j) and p(d) n (i) converges almost surely to d 0 j=1 π j p( j), which are both independent of i ∈ S d .For the estimation of d 0 , we propose the following method: for d = 1, . . ., assume that the model defined by (2.1) holds with d 0 = d and estimate p (d) and L d according to (2.2)-(2.3). d0 is then the first local maximum of L d . Then we set p n = p d0 . We prove in Appendix A.3 that this method works. Suppose p ∈ P d 0 and d 0 is minimal. Then for any d ∈ N, L d < L d 0 whenever (d, d 0 ) < d 0 . If (d, d 0 ) = d 0 , then L d = L d 0 .

  L n,d (e) n a.s. → L d (e) = -d i=1 j∈Z, ( j) d =i π j (e)H α (d) (i, e) . (3.3) Furthermore, for any d ≥ 1 and k > 1, we have log 2 = L 1 (e) ≤ L d (e) ≤ L kd (e) ≤ L ∞ (e) = -j∈Z π j (e)H {α( j, e)} ≤ 0. (3.4) Remark 3.1. Comparing these results with Proposition 2.2 and Theorem 2.1 shows that the sequence of log-likelihoods for large sample estimates {p (d)

µTheorem 3 . 1 .

 31 measure preserving transformation. The asymptotic behavior of X is completely determined by the expectation of log ρ 0 , as proven in[START_REF] Alili | Asymptotic behaviour for random walks in random environments[END_REF] Theorem 2.1]. Suppose that u = E log ρ 0 is well defined, with u ∈ [-∞, +∞]. If u > 0, then µ a.s., F(e) < ∞, S (e) = ∞, and for any i ∈ Z, P e i (lim n→∞ X n = -∞) = 1. If u < 0, then µ a.s., S (e) < ∞, F(e) = ∞, and for any i ∈ Z, P e i (lim n→∞ X n = +∞) = 1. If u = 0, then µ a.s., F(e) = ∞, S (e) = ∞, and for any i ∈ Z, P e i lim inf n→∞ X n = -∞, lim sup n→∞ X n = +∞ = 1.

(4. 1 )

 1 Let u d be the uniform distribution on {1, . . . , d}. It is then easy to check that if λ(k) is a stationary distribution for this Markov chain on E k which is absolutely continuous with respect to u d ⊗ µ, then its density ϕ k satisfies, for any ( j, e) ∈ E k , ϕ k = Ť ϕ k , where Ť is the adjoint operator of Ť defined by Ť ϕ k ( j, e) = α(0, T -1 e)ϕ k ( j -1, T -1 e) + {1 -α(0, T e)}ϕ k ( j + 1, T e), µ a.s.

4. 1 . 1 . 1 log ( 13 / 11 )

 1111311 Right-ballistic iid caseSuppose the α(i) are iid, with a uniform distribution over the interval (a, b). Then E(ρ k ) = ρ = -1 + log(b/a) b-a < 1 is required for the right-ballistic case. Using the results of Section 3.2.2, the density of α 0 under λ is then 1 x log(b/a) I(a < x < b). The density of the invariant measure λ for T is φ(e) = 1-ρ 1+ρ S (e) α(0,e) , where S (e) = 1 + ∞ n=1 n k=1 ρ k (e). In particular E λ [α(0)] = 1 1+ρ . As an example, take a = .55, b = .65. Hence, ρ = -1 + 10 log (65/55) ≈ .6705, ≈ 0.5986085. Choosing estimator p (d)

Figure 2 :

 2 Figure 2: Estimation of the distribution of α 0 (left panel) and graph of the log-likelihoods for period d ∈ {1, . . . , 20} (right panel).

Figure 3 :

 3 Figure 3: Trajectory of the RWRE (left panel) and graph of the log-likelihoods for period d ∈ {1, . . . , 20} (right panel).

Figure 4 :

 4 Figure 4: Trajectory of the RWRE (left panel) and graph of the log-likelihoods for period d ∈ {1, . . . , 20} (right panel).

  0 , ( j) m =(i) m I (X t-1 ) d = i, (X t-1 ) d 0 = j, ∆X t = 1 = j∈S d 0 , ( j) m =(i) m A ( d)n,T (i, j) . Now, as n → ∞, it follows from Proposition 2.1 that A ( d) n,T (i, j) /n a.s.

Appendix A. 3 . 1 Fromπππππ

 31 Proof of Theorem 2.Proposition 2.1, L n,d 0 /n a.s.→ L d 0 = -d 0 j=1 π j H{p( j)}. Introduce the following notation for the relative entropy. For any x, y ∈ (0, 1), set h(x|y) = x log(x/y) + (1x) log{(1x)/(1y)}. Then h(x|y) ≥ 0 with equality iff x = y. We will show thatL d 0 > L d if m = (d, d 0 ) < d 0 . Set a = d/m and b = d 0 /m. It follows from (2.4) that for any i = β + lm, with β ∈ {1, . . . , m} and l ∈ {0, . . . , a -1}, p (d) (β + lm) = p (d) (β) = b-1 k=0 π km+β p(km + β) b-1 k=0 π km+β .It then follows from the previous equation and (2.5) that km+β p(d) (β) log p (d) (β) + 1p (d) (β) log 1p (d) (β) km+β p (d) (β) log p (d) (β) + 1p (d) (β) log 1p (d) (β) .Using the previous computations, one getsL d 0 = m β=1 b-1 k=0 π km+β p(km + β) log {p(km + β)} + {1p(km + β)} log {1p(km + β)} . Hence, L d 0 -L d = lm+β p(lm + β) log{p(lm + β)}log p (d) (β) lm+β {1p(lm + β)} × log{1p(lm + β)}log 1p (d) (β) lm+β h p(lm + β)|p (d) (β) ≥ 0,with equality iff m = d 0 . This shows that L d < L d 0 for all d ∈ {1, . . . , 2d 0 -1} \ {d 0 }. Hence, if n is large enough, it follows from Proposition 2.2 that L n,d < L n,d 0 for all d ∈ {1, . . . , 2d 0 -1} \ {d 0 }. This also explains the local maxima pictured in Figure 1. Note that d 0,n → d 0 implies d 0,n = d 0 when n is large enough so p (d 0,n ) n

  + •)} i∈Z , where . Henceforth we use the notation (x) d = i to mean that x = i mod (d), the residual class modulo d. Note that the stochastic process {(X t ) d } t≥0 valued in S d is a Markov chain only when d is a multiple of d 0 . It is this fact which requires some extra care in the treatment of periodic environments.

	p n ∈
	2.1. Estimation of p with d known
	If p ∈ P d then, assuming (X t ) n t=0 satisfies (2.1), the maximum likelihood estimator (MLE) p (d) n of p is given by

d≥1

P d

  is the well-known Boltzmann entropy function for the Bernoulli distribution with parameter x. Note that for any multiple of d, one should get a consistent estimator as well, since p ∈ P d entails that p ∈ P kd , for any k ∈ N.

	The following result, proven in Appendix A.1, shows that the
	MLE is consistent.
	Proposition 2.1. Suppose p ∈ P d and let π be the unique invariant probability measure of the irreducible Markov
	chain (X t ) d on S d , associated with p. Then, as n → ∞, for any j ∈ S d , µ almost surely, A (d)

Acknowledgments

Partial funding in support of this work was provided by the Natural Sciences and Engineering Research Council of Canada (Grants 04430-2014, 04083-2017), and the Fonds québécois de la recherche sur la nature et les technologies (2015-PR-183236). Appendix A. Proofs of the main results

Appendix A.1. Proof of Proposition 2.1 Suppose j ∈ S d 0 is given. Then

I{(X t-1 ) d 0 = j}α(X t-1 ). Now, ξ t I{(X t-1 ) d 0 = j}{I(∆X t = 1) -α(X t-1 )} is a bounded martingale difference sequence, so 1 n n t=1 ξ t converges to 0 almost surely, as n → ∞. Hence the limiting behavior of A (d 0 ) n, j /n is the same as the limiting behavior of

). Similarly, the limiting behavior of B (d 0 ) n, j /n is the same as the limiting behavior of

→ p( j)π( j). Similarly,

→ {1p( j)}π( j).

Thus p (d 0 ) n ( j) converges almost surely to p( j). The almost sure convergence of L n,d 0 /n follows. Irreducible Markov chain X = (X t ) t≥0 satisfies (1.1) for some arbitrary deterministic environment α(•, e) ∈ (0, 1) Z , periodic or not. By definition any invariant positive measure π will satisfy the usual balance equation, which is written, for all j ∈ Z, as π j+1 = π j+1 α( j + 1, e) + π j α( j, e), from which there ensues j∈Z π j α( j, e) = 1 2 j∈Z π j as soon as either sum converges. This also implies both L 1 =log 2 and the validity of (3.1) since j∈Z C j (e) < ∞ holds. The convergence of j∈Z C j (e) < ∞ ensures that the Markov chain (X t , X t+1 ) t≥0 on {( j, k) ∈ Z 2 ; kj = ±1} is also irreducible and positive recurrent with unique invariant probability measure Π = (Π j,k ) given by

The ergodic theorem for irreducible Markov chain (X t , X t+1 ) t≥0 immediately yields the asymptotic behavior provided in (3.2) for the sequence of statistics p (d) n and in (3.3) for the sequence of likelihoods L n,d defined by (2.3). The convexity of -H successively yields the inequalities L d ≤ L kd and L 1 ≤ L d , for all d ≥ 1 and k > 1, through an application of Jensen's inequality: Hence h(T e) = h(e) µ a.-s. so h is constant by the ergodic property of T . This is a necessary and sufficient condition for a Markov chain to be ergodic [5, p. 14].

Appendix A.6. Proof of Theorem 4.1

Suppose h is a bounded harmonic function for Ť . Then we have the analog of (A.2). As a result, h( j + 1, T e) = h( j, e) µ-a.-s. Moreover, d j=1 h( j, e) is a bounded harmonic function for T so it is constant. Next, set z(e) = (h(1, e), . . . , h(d, e)) . Then Az(T e) = z(e), where A is a permutation matrix on {1, . . . , d}, with A i j = 1 iff j = i + 1 mod d, while A -1 i j = 1 iff j = i -1 mod d, i, j ∈ {1, . . , d}. In particular, A d = I. It then follows that z(T d e) = z(e). Since T d is ergodic, it follows that z is constant so h is constant as well. It then follows from Brown [5, p.14] that the Markov chain X is ergodic with unique invariant measure having density φ with respect to m d × µ.


Nina De Montmollin

Daniel Dusser

Christine Lorut

Jérémie Dion

Nathalie Costedoat- Chalumeau

Luc Mouthon

Guillaume Chassagnon

Marie-Pierre Revel

MD Xavier Puéchal
email: xavier.puechal@aphp.fr

Tracheobronchial involvement of relapsing polychondritis

Keywords: relapsing polychondritis, respiratory infections, tracheal stenosis, tracheobronchomalacy, endoscopy

Recent studies show that relapsing polychondritis patients with tracheobronchial involvement are distinct from others in terms of clinical characteristics, therapeutic management, and disease evolution. Tracheobronchial involvement affects 20 to 50% of patients and may reveal the disease. It should be sought at the time of diagnosis and at each follow-up visit. Respiratory impairment is confirmed by computed tomography (CT) of the chest, including the cervical portion of the trachea, with end-inspiratory and dynamic expiratory scans, and pulmonary function tests. These investigations should be performed, even in asymptomatic patients, at the time of diagnosis, and repeated as necessary during follow-up. Bronchoscopy and a fortiori endoscopic intervention should be considered with caution and performed only by expert endoscopists after careful evaluation of the risks and benefits of such procedures, which can lead to damage or perforation of the airways and bronchospasm. Early detection and management of tracheobronchial involvement in relapsing polychondritis has significantly improved the prognosis of patients, especially with the development of interventional fiberoptic bronchoscopy. However, relapsing polychondritis-related morbidity and mortality are still elevated, particularly in tracheobronchial disease.

Highlights

• Recent evidence shows that relapsing polychondritis patients with associated tracheobronchial involvement are distinct from others in terms of clinical characteristics, therapeutic management, and disease evolution.

• Pulmonary function tests should be routinely performed at diagnosis and during follow up on a yearly basis to detect and evaluate the progression of tracheobronchial involvement.

• A chest computed tomography scan including the cervical portion of the trachea and dynamic expiratory computed tomography should be performed, even in asymptomatic patients, at the time of diagnosis and repeated as necessary during the evolution of the disease.

• Overall survival has improved considerably, partially due to the availability of interventional treatments, but related morbidity and mortality are still elevated.

• Therapeutic management is not based on trial data, but on expert recommendations, and involves a close collaboration between pulmonologists and rheumatologists or internists.

Relapsing polychondritis (RP) is a rare systemic disease which can be disabling and lifethreatening. It is characterized by recurrent bouts of inflammation followed, in some cases, by degeneration and deformation involving the cartilage of the ears, nose, larynx, and tracheobronchial tree [START_REF] Trentham | Relapsing polychondritis[END_REF][START_REF] Michet | Relapsing polychondritis. Survival and predictive role of early disease manifestations[END_REF][START_REF] Mcadam | Relapsing polychondritis: prospective study of 23 patients and a review of the literature[END_REF]. Other structures that consist of proteoglycans can also be involved, such as the eyes, heart, and inner ear. There may also be manifestations of vasculitis.

A recent cluster analysis identified three separate phenotypes: a hematological form (in 10% of cases), a respiratory form (in 25% of cases), and a mild form with a good prognosis (approximately 65% of cases) (4). Tracheobronchial involvement is the hallmark of the respiratory phenotype of RP, which was classically considered to carry a poor prognosis. In this large series, patients with such a phenotype received more intensive treatment and suffered from more infections but no patient died of airway collapse or obstruction due to RP, which has been a significant cause of mortality in older series. Another study confirmed that patients with lower respiratory tract involvement have different characteristics [START_REF] Shimizu | Relapsing polychondritis patients were divided into three subgroups: patients with respiratory involvement (R subgroup), patients with auricular involvement (A subgroup), and overlapping patients with both involvements (O subgroup), and each group had distinctive clinical characteristics[END_REF].

These results show that patients with associated tracheobronchial involvement are distinct from others in terms of clinical characteristics, therapeutic management, and disease evolution [START_REF] Dion | Relapsing polychondritis can be characterized by three different clinical phenotypes: analysis of a recent series of 142 Patients[END_REF].

Here, we review the current knowledge of respiratory tract involvement and its management in RP, with special emphasis on the most recent data.

EPIDEMIOLOGY

RP chiefly affects middle-aged adults, with a slight female predominance, but cases have been reported in the very young and very elderly [START_REF] Belot | Pediatric-onset relapsing polychondritis: case series and systematic review[END_REF]. All ethnic groups are affected. In the US, the estimated incidence of RP is 3.5 per million inhabitants per year [START_REF] Luthra | Relapsing polychondritis[END_REF] and the prevalence among Department of Defense beneficiaries is 4.5 per million inhabitants [START_REF] Mathew | Relapsing polychondritis in the Department of Defense population and review of the literature[END_REF]. A more recent population-based cohort study found an incidence of 0.71 per million inhabitants per year [START_REF] Hazra | Incidence and mortality of relapsing polychondritis in the UK: a population-based cohort study[END_REF]. The prevalence of respiratory impairment is 20-50% [START_REF] Trentham | Relapsing polychondritis[END_REF][START_REF] Dion | Relapsing polychondritis can be characterized by three different clinical phenotypes: analysis of a recent series of 142 Patients[END_REF][START_REF] Rafeq | Pulmonary manifestations of relapsing polychondritis[END_REF][START_REF] Ernst | Relapsing polychondritis and airway involvement[END_REF] during RP. Respiratory symptoms often lead to diagnosis (50% of cases). In the study of Dion et al., involving 142 patients, the tracheobronchial cluster represented one quarter of patients, was characterized by a younger age, and could be isolated [START_REF] Dion | Relapsing polychondritis can be characterized by three different clinical phenotypes: analysis of a recent series of 142 Patients[END_REF]. A negative correlation was found in two studies between tracheobronchial and auricular involvement [START_REF] Shimizu | Organ involvement pattern suggests subgroups within relapsing polychondritis: comment on the article by Dion et al[END_REF][START_REF] Dion | Reply[END_REF].

CLINICAL MANIFESTATIONS OF TRACHEOBRONCHIAL INVOLVEMENT

Respiratory involvement is inaugural in 10% of cases [START_REF] Puéchal | Relapsing polychondritis[END_REF][START_REF] Wallaert | Respiratory involvement in relapsing polychondritis. Clinical, functional, endoscopic, and radiographic evaluations[END_REF]. Airway involvement occurs early during the course of the disease, at a mean of 2.5 years after diagnosis (4).

Laryngeal involvement is manifested by pain and soreness of the larynx, above the thyroid gland. Dysphonia, aphonia, or hoarseness of the voice are possible. Laryngeal involvement may result in laryngomalacia or laryngeal stenosis, responsible for inspiratory dyspnea [START_REF] Puéchal | Relapsing polychondritis[END_REF][START_REF] Suyama | Clinical Images: Arytenoid chondritis[END_REF].

Tracheobronchial involvement leads to inflammation or even destruction of the cartilage, which can result in tracheo-and bronchomalacia. These are responsible for tracheobronchial obstruction, manifested by progressively increasing inspiratory dyspnea, cough, stridor, tracheal pain, and even respiratory distress. There may also be edema and soreness of the tracheal cartilage.

In addition, costosternal chondritis can also affect breathing, occasionally even resulting in complete cartilaginous destruction. Costosternal chondritis is present in 35% of patients [START_REF] Puéchal | Relapsing polychondritis[END_REF].

RP can lead to sudden death (due to laryngeal or tracheal spasm) or acute edema during an inflammatory attack. However, this is now very rare since the improvement of treatment and interventional endoscopic procedures [START_REF] Rafeq | Pulmonary manifestations of relapsing polychondritis[END_REF]. There is also an elevated frequency of lower respiratory tract infections, which are often associated with bronchiectasis.

RP is associated with autoimmune diseases in 22% to 30% of cases, which may also have their own respiratory component [START_REF] Michet | Relapsing polychondritis. Survival and predictive role of early disease manifestations[END_REF][START_REF] Mcadam | Relapsing polychondritis: prospective study of 23 patients and a review of the literature[END_REF][START_REF] Dion | Relapsing polychondritis can be characterized by three different clinical phenotypes: analysis of a recent series of 142 Patients[END_REF].

DIAGNOSTIC AND CLASSIFICATION CRITERIA

There are no validated classification or diagnostic criteria for RP. The diagnosis of RP is based on clinical evidence and physician experience [START_REF] Trentham | Relapsing polychondritis[END_REF][START_REF] Puéchal | Relapsing polychondritis[END_REF][START_REF] Sharma | Relapsing polychondritis: a review[END_REF]. The most commonly used criteria are the Michet criteria (2), but they are for classification. Patients are classified as having RP if they fulfill two major criteria or one major and two minor criteria. Laryngotracheal involvement is a major criterion of this classification. It is now commonly accepted that the exclusion of differential diagnoses, particularly granulomatosis with polyangiitis (Wegener's), which can mimic RP, is crucial. Thus, positive anti-neutrophil cytoplasm antibodies (ANCA) with anti-PR3 specificity, involvement of the lung parenchyma, posterior non-cartilaginous tracheal wall, kidney, or destructive ear nose throat (ENT) lesions classify the patient as having granulomatosis with polyangiitis and not RP, with therapeutic consequences. A delay in the diagnosis is common, with a median time from first symptoms to diagnosis ranging from 1.9 to 3.2 years [START_REF] Trentham | Relapsing polychondritis[END_REF][START_REF] Mathew | Relapsing polychondritis in the Department of Defense population and review of the literature[END_REF][START_REF] Hazra | Incidence and mortality of relapsing polychondritis in the UK: a population-based cohort study[END_REF]. The next steps, currently ongoing, consist of developing RP classification or even diagnostic criteria and improving its categorization into subsets, a prerequisite for future therapeutic trials [START_REF] Piette | News on relapsing polychondritis: the patient's experience[END_REF][START_REF] Ferrada | An international consensus exercise to develop candidate items for classification criteria in relapsing polychondritis[END_REF].

IMAGING

The main imaging method used to diagnose RP is computed tomography (CT), with endinspiratory and dynamic expiratory scans. A chest CT scan, including an expiratory acquisition, should be routinely performed at diagnosis, even in the absence of respiratory symptoms. It should be repeated as necessary during the evolution of the disease. The cervical portion of the trachea should be included. Thus, the acquisition should start below the vocal cords. Expiratory CT scan images can be acquired at a very low dose of X-rays to minimize the radiation dose. This is made possible by newer image reconstruction modes (iterative reconstruction), allowing the suppression of background noise in the images.

Several typical CT scan patterns may be encountered [START_REF] Ernst | Relapsing polychondritis and airway involvement[END_REF][START_REF] Lee | Relapsing polychondritis: prevalence of expiratory CT airway abnormalities[END_REF]:

1) A thickening of the tracheal or tracheobronchial wall, defined by a wall thickness >2 mm, with or without calcifications, is observed in 40% of cases. Sparing of the posterior membranous part of the tracheal wall, which does not contain cartilage, is highly typical 21). The administration of iodinated contrast medium is not required for the demonstration of tracheal or bronchial wall thickening.

Classification criteria of relapsing polychondritis according to Michet et al (2).

Major

2) Fixed lumen narrowing and/or tracheobronchial obstruction, including subglottic stenosis. Stenosis is defined as at least a 25% reduction in the diameter of the lumen (Figures 4,5).

3) Tracheobronchomalacia detected on dynamic expiratory CT, which should be systematically performed. It is characterized by excessive collapse of the tracheal or bronchial lumen during expiration (Figure 6B). Such excessive collapse may only affect the trachea, the main bronchi, or both. 5) Air trapping during dynamic expiration, which can be due to malacia [START_REF] Brillet | CT imaging features of pulmonary involvement in connective tissue disorders[END_REF] or fixed tracheobronchial stenoses (Figure 5). Air trapping is defined by the presence of lung areas that fail to increase in attenuation after full expiration relative to full inspiration. It can exhibit a lobular, segmental, lobar, or even pulmonary distribution. On inspiratory CT images, the corresponding areas may show mosaic perfusion. Mosaic perfusion consists of heterogeneity of the lung parenchyma attenuation, with hypodense (abnormal) areas of vascular rarefaction, and hyperdense (normal) lung areas, where more vessels are visible.

Mosaic perfusion can be detected without administration of a contrast agent and is due to hypoxic vasoconstriction in poorly ventilated areas (Figure 5). 6/ Bronchiectasis, frequently observed in our experience, may be due to either direct cartilaginous injury, the consequence of recurrent infections, or both (Figure 7).

Apart from bronchial abnormalities, there is no lung parenchymal involvement during RP [START_REF] Rafeq | Pulmonary manifestations of relapsing polychondritis[END_REF]. In summary, some abnormalities on dynamic expiratory CT images (malacia and air trapping) may not be visible on inspiratory CT images [START_REF] Lee | Relapsing polychondritis: prevalence of expiratory CT airway abnormalities[END_REF]. Malacia and air trapping are the result of cartilage inflammation and are the unique CT abnormalities detectable at early stages of RP [START_REF] Rafeq | Pulmonary manifestations of relapsing polychondritis[END_REF][START_REF] Lee | Relapsing polychondritis: prevalence of expiratory CT airway abnormalities[END_REF][START_REF] Brillet | CT imaging features of pulmonary involvement in connective tissue disorders[END_REF]. Early detection is important, because treatment may change the prognosis.

Post-processing of CT images with 3D rendering (virtual bronchoscopy) may facilitate the visualization of stenosis and size reductions for interventional endoscopists, but their added value for the detection of bronchial abnormalities has not been demonstrated [START_REF] Rafeq | Pulmonary manifestations of relapsing polychondritis[END_REF].

Magnetic resonance imaging (MRI) is considered to be more sensitive than CT for the distinction between fibrosis and inflammation. Fibrosis is characterized by low signal intensity on T1-and T2-weighted images, whereas inflammation is characterized by high signal intensity on T2-weighted images and enhancement following gadolinium injection [START_REF] Thaiss | Imaging diagnosis in relapsing polychondritis and correlation with clinical and serological data[END_REF]. However, MRI is limited for the evaluation of the airway wall by its low spatial resolution compared to CT. Fluorodeoxyglucose (FDG) positron emission tomography (PET)/CT scan may reveal active cartilaginous lesions at any stage of the disease [START_REF] Yamashita | Utility of fluorodeoxyglucose positron emission tomography/computed tomography for early diagnosis and evaluation of disease activity of relapsing polychondritis: a case series and literature review[END_REF][START_REF] Zhang | Airway involvement of relapsing polychondritis revealed by 18F-fluoride PET/CT[END_REF][START_REF] Wang | 18F-FDG PET/CT is an ideal imaging modality for the early diagnosis of relapsing polychondritis: a case report[END_REF], with the limitation of very low spatial resolution. PET/CT may also reveal unsuspected cartilaginous involvement (Figure 6).

Although its performance has not been fully evaluated, follow-up of standard uptake value (SUV) can be useful to distinguish between active and fibrotic lesions, especially for patients with isolated tracheobronchial involvement.

PULMONARY FUNCTION TESTING

Pulmonary function testing must be performed at diagnosis and repeated annually to detect airway impairment and estimate its progression. It should not be performed during a superimposed infection. Spirometry, volume-flow curves, and the measurement of upper airway resistance may show early involvement in asymptomatic patients [START_REF] Rafeq | Pulmonary manifestations of relapsing polychondritis[END_REF][START_REF] Gibson | Respiratory complications of relapsing polychondritis[END_REF]. Obstructive disease (FEV1/FVC <70%) and reduced exercise capacity (walk test desaturation and decreased distance walked) should be sought.

Obstructive disorders during the course of RP may be inspiratory and/or expiratory and reversible or incomplete. The residual volume (RV) can also increase without distension (normal total lung capacity (TLC) and RV/TLC) [START_REF] Krell | Pulmonary function in relapsing polychondritis[END_REF]. Reduced inspiratory flow (the plateau of the inspiratory curve on the volume-flow curve), however, indicates extra-thoracic involvement but the correlation is not always good. Reduced expiratory flow indicates intrathoracic obstruction (15,27) (Figure 8). There is no ventilation-perfusion disorder associated with RP nor diffusion or gas exchange abnormality, and the alveolar-capillary gradient and dead space are normal. Elasticity is also not decreased (no decrease in TLC).

The initial reduction in the size of the respiratory tract is due to inflammatory edema, but later results from collapse related to malacia and, finally, fibrous stenosis. There are therefore three presentations of obstruction: fixed obstruction with a double plateau volume flow curve (expiratory and inspiratory involvement), variable or extra-thoracic dynamic obstruction, corresponding to the upper airways (extra-thoracic trachea or larynx, inspiratory involvement), and variable or dynamic intra-thoracic obstruction (expiratory involvement). The most common presentation is fixed obstruction due to stenosis.

The decrease in maximum ventilation is mainly due to a reduction in forced expiratory flow. When severe, it can lead to hypercapnia; hypoxemia is much rarer and associated with alveolar hypoventilation [START_REF] Rafeq | Pulmonary manifestations of relapsing polychondritis[END_REF].

FIBROSCOPY

Fibroscopy should be performed in patients with respiratory symptoms, especially when interventional endoscopy is necessary. It can evaluate mucositis, define the severity (narrowing or stenosis), and allow the dynamic assessment of possible collapse (expiratory maneuver or coughing). However, it should be performed with care because of increased morbidity and mortality associated with the procedure: risk of acute flare-up of the disease, leading to edema of the respiratory or supraglottic airways, bronchospasm, bleeding, respiratory distress, or even death. It should only be performed by an experienced interventional endoscopist with trained personnel and an emergency airway and tracheostomy kit nearby [START_REF] Rafeq | Pulmonary manifestations of relapsing polychondritis[END_REF]. The correlation with dynamic CT for stenosis or malacia is good. For example, in one study [START_REF] Rafeq | Pulmonary manifestations of relapsing polychondritis[END_REF], 22 patients had both examinations, of which 17 had lesions by fibroscopy comparable to those detected by CT. Fibroscopy may not detect thickening but is more sensitive for the detection of edema or stenosis in a narrow area.

PATHOPHYSIOLOGY OF RESPIRATORY IMPAIRMENT

An immune origin, involving humoral and cellular immunity, is suspected [START_REF] Puéchal | Relapsing polychondritis[END_REF][START_REF] Sharma | Relapsing polychondritis: a review[END_REF][START_REF] Arnaud | Pathogenesis of relapsing polychondritis: a 2013 update[END_REF], as:

1) there is an associated autoimmune disease in up to 30% of cases, 2) chondritis lesions contain a CD4 + T-cell lymphocyte infiltrate and plasmocytes, as well as immune deposits, 3) autoantibodies directed against type II collagen are detected in approximately 30% of cases, or against other types of minor collagen (IX and XI) or cartilage proteins, such as oligomeric proteins of the cartilaginous matrix (COMP) and matrilin-1, 4) a specific T-cell response against collagen II peptides, which represents 95% of cartilage collagen, or specific to matrilin-1 is sometimes observed, 5) there is an association with Class-II MHC, 6) high-dose glucocorticoids (GCs) are generally effective, and especially 7) RP-like symptoms are reproduced in animal models after immunization against type II collagen or matrilin-1.

Hansson et al. demonstrated that antibodies against matrilin-1 were associated with tracheobronchial involvement in 69% of cases and were able to bind cartilage structures in vivo [START_REF] Hansson | The occurrence of autoantibodies to matrilin 1 reflects a tissue-specific response to cartilage of the respiratory tract in patients with relapsing polychondritis[END_REF]. They developed a mouse model in which tracheobronchial and nasal chondritis, (without ear or articular involvement), dependent on the complement system, B cells, and class-II MHC, were induced after immunization against matrilin-1 (30,31).

Thus, according to current data, RP is a probable autoimmune disease, with specific immunization against cartilage structures and matrillin-1 shown to be responsible for the tracheobronchial phenotype.

The mechanism of respiratory involvement, including tracheal or bronchial obstruction, depends on the stage of the disease: inflammatory edema occurs during the active phase, followed by malacia, resulting from cartilage destruction, and then stenosis due to fibrous replacement of the impaired cartilage [START_REF] Rafeq | Pulmonary manifestations of relapsing polychondritis[END_REF]. Bronchial cartilage inflammation may also impair mucociliary function [START_REF] Trentham | Relapsing polychondritis[END_REF]. The mechanism of bronchial dilation has been the subject of debate, but probably involves several causes. It may be due to inflammatory destruction of the cartilage of the tracheobronchial axis of the proximal bronchi but also recurrent bronchial infections resulting from the inefficiency of bronchial drainage, when coughing, because of bronchial collapse.

HISTOLOGY

Histological specimens of the anterior tracheal cartilage are sometimes obtained during fibroscopy but there is no histological specificity. If performed, there is perichondral lymphoplasmocytic cellular infiltration of the cartilage, with the loss of basophilic staining of the cartilaginous matrix, corresponding to the loss of proteoglycans. The cartilage is gradually replaced by fibrous tissue. These abnormalities evolve from the periphery to the center and predominate at the junction between the cartilage and connective tissue.

TREATMENT OF RELAPSING POLYCHONDRITIS RESPIRATORY DISEASE

Therapeutic management is empirical. It is not based on therapeutic trial data but mainly the physician's experience and the data from a few case series and involves a collaboration between the pulmonologist and the rheumatologist or internist. The goal of treatment is to control symptoms and maintain airway stability [START_REF] Rafeq | Pulmonary manifestations of relapsing polychondritis[END_REF].

General treatment.

General treatment is the same as for other forms of the disease.

GCs are an important component of RP treatment and show good efficacy but often highdose steroid dependence. Doses of up to 1 mg/kg/day are sometimes prescribed, depending on the severity of the disease. High doses of intravenous methylprednisolone may be necessary if the oral route is not sufficient or if there is acute respiratory distress. GCs are used to control acute airway flare-ups and reduce their severity, duration, and frequency [START_REF] Lahmer | Relapsing polychondritis: an autoimmune disease with many faces[END_REF]. They are sometimes associated with antibiotics because coexistence with an infection is sometimes difficult to rule out. Most patients require long-term treatment with GCs for months or even years [START_REF] Trentham | Relapsing polychondritis[END_REF][START_REF] Rafeq | Pulmonary manifestations of relapsing polychondritis[END_REF].

Oral GCs are very often associated with immunosuppressive agents for their steroidsparing effect. Immunosuppressants (methotrexate, cyclophosphamide, azathioprine, or mycophenolate mofetil) should be considered if there is involvement of a vital organ, insufficient response to GCs, high-dose steroid dependence, or as a first line agent to reduce GC doses. The choice of the agent is empirical. Methotrexate is widely used and appears to be effective [START_REF] Trentham | Relapsing polychondritis[END_REF].

Biologics have been tried in this often-steroid dependent disease [START_REF] Lekpa | Biologics in relapsing polychondritis: a literature review[END_REF][START_REF] Moulis | Efficacy and safety of biologics in relapsing polychondritis: a French national multicentre study[END_REF][START_REF] Kingdon | Relapsing polychondritis: a clinical review for rheumatologists[END_REF]. Some authors have advocated that current data from case reports support the use of infliximab for RP with pulmonary involvement, with early therapy being associated with improved outcomes [START_REF] Kingdon | Relapsing polychondritis: a clinical review for rheumatologists[END_REF].

However, their use is not based on any therapeutic prospective trial and comparison of the effectiveness of the various agents is difficult between the different series. Adalimumab and infliximab have been the most commonly used. During the first six months, the rates of response and complete response were less than two-thirds and approximately one-third, respectively, with only a modest reduction in median daily GC dose [START_REF] Moulis | Efficacy and safety of biologics in relapsing polychondritis: a French national multicentre study[END_REF]. Although adalimumab appeared to have the highest rate of persistence, almost all patients had to discontinue the TNF inhibitor due to insufficient efficacy (24-35%), loss of efficacy (28-30%), or adverse drug reactions, the most frequent being infections (20-15%).

Local treatment. Techniques for local treatment should only be performed in highly specialized centers with a thoracic surgery service nearby. They consist of interventional endoscopic treatment most often using rigid bronchoscopy and include thermocoagulation, laser, balloon dilation or bronchoscopes of increasing size, and tracheobronchial prosthesis.

For focal bronchial stenosis, the proposed treatment is balloon dilatation or bronchoscopes of increasing size, high frequency thermocoagulation, or laser. In case of recurrence, a prosthesis may be proposed.

For more diffuse stenosis with a malacia component, a tracheobronchial prosthesis may be proposed, consisting of a Montgomery tube for high glottic stenosis, a bifurcated prosthesis for tracheobronchial involvement, or a straight prosthesis for bronchial involvement. These stents also make it possible to evaluate the improvement of symptoms upon resolution of the obstruction and thus select patients who could benefit from extensive surgery, such as tracheobronchoplasty [START_REF] Rafeq | Pulmonary manifestations of relapsing polychondritis[END_REF].

Silicone prostheses are most often used because they are more durable and can be removed if necessary but their insertion is sometimes limited by technical difficulties. In case of failure of silicone prosthesis, a covered metal prosthesis can be discussed [START_REF] Oryoji | Sudden Respiratory failure due to tracheobronchomalacia by relapsing polychondritis, successfully rescued by multiple metallic stenting and tracheostomy[END_REF].

Tracheostomy may be necessary for severe subglottic involvement.

The complications of interventional endoscopy include the risk of acute flare-up, perforation, prosthesis migration, increased mucous secretion, with risk of obstruction of the prosthesis, bronchospasm, and increased cough [START_REF] Rafeq | Pulmonary manifestations of relapsing polychondritis[END_REF][START_REF] Chapron | Bronchial rupture related to endobronchial stenting in relapsing polychondritis[END_REF].

The use of these local therapeutic interventions must be considered when medical treatments fail or for patients awaiting the effect of the medical treatment who have major bronchial involvement associated with a high risk of morbidity and mortality. Local management reduces respiratory symptoms in most patients [START_REF] Ernst | Relapsing polychondritis and airway involvement[END_REF].

The decision of local or general treatment or a combination is made on a case by case basis after a multidisciplinary evaluation involving the pulmonologist, the operator, the radiologist, and the rheumatologist or internist. In a study by Dion et al., 15% of patients with tracheobronchial involvement required local endoscopic treatment (4).

Other treatments. Respiratory physiotherapy plays an important role in bronchial drainage, which improves ventilation and prevents secondary infections.

There are few studies on inhaled treatments, which are not recommended. One clinical case, however, reported the efficacy of high doses of inhaled fluticasone in a patient with tracheal inflammatory exacerbations, which reduced the oral GC dose [START_REF] Tsuburai | Use of inhaled fluticasone propionate to control respiratory manifestations of relapsing polychondritis[END_REF].

For patients with moderate to severe tracheobronchial involvement, particularly those with malacia, screening for obstructive sleep apnea or hypercapnia can be performed, as they may benefit from noninvasive positive-pressure ventilation, which can be an alternative to interventional endoscopy in some cases [START_REF] Yamaguchi | Successful treatment with noninvasive positive-pressure ventilation based on the prediction of disease onset using CT and respiratory function tests in an elderly patient with relapsing polychondritis[END_REF][START_REF] Riha | Obstructive sleep apnoea/hypopnoea as the initial presentation of relapsing polychondritis[END_REF].

The prevention of infectious complications, associated or not with bronchiectasis, also needs to be emphasized. Such measures can include pneumococcal and influenza vaccination, antibiotics, surveillance of the nature of bacterial colonization and treatment of bronchial superinfections, and low-dose azithromycin prophylaxis for symptomatic and superinfected bronchiectasis.

One of the most challenging issues is the differential diagnosis between flare-ups and respiratory tract infections in those patients with a chronic obstructive respiratory disease.

Furthermore, infections and flare-ups can coexist or succeed one another by a few days [START_REF] Piette | News on relapsing polychondritis: the patient's experience[END_REF].

EVOLUTION AND PROGNOSIS

The evolution of the disease is progressive, consisting of flare-ups, leading to possible permanent destruction of the cartilage [START_REF] Trentham | Relapsing polychondritis[END_REF]. Some symptoms may persist between flare-ups.

Overall RP survival has improved considerably. It was 55% at 10 years in 1986 (2), 94% at 8 years in 1998 [START_REF] Trentham | Relapsing polychondritis[END_REF], and 83% at 10 years in the most recent study [START_REF] Dion | Relapsing polychondritis can be characterized by three different clinical phenotypes: analysis of a recent series of 142 Patients[END_REF]. The improved prognosis is partially due to the availability of interventional treatments, but RP-related morbidity and mortality are still high, particularly in tracheobronchial disease. In the most recent study, these patients received more intensive treatment, were prone to infections, and were frequently admitted to the intensive care unit (4). However, their survival (although impaired) was not significantly less than that of patients with the mildest phenotype and tracheobronchial involvement no longer appears to be a factor of excess mortality in multivariate analysis ( 4).

An activity score, the relapsing polychondritis disease activity index (RPDAI), has been developed based on clinical and biological data [START_REF] Arnaud | The Relapsing Polychondritis Disease Activity Index: development of a disease activity score for relapsing polychondritis[END_REF]. It consists of 27 weighted items, including respiratory chondritis, with or without respiratory failure.

Damage is defined as irreversible manifestations of the disease that are unlikely to benefit from treatment escalation, ascertained by clinical assessment, and present for at least three months. A damage score has also recently been finalized and includes three respiratory items: permanent chest wall deformities, dysphonia, and obstructive syndrome (laryngeal, tracheal, or bronchial) [START_REF] Mertz | The relapsing polychondritis damage index (RPDAM): Development of a disease-specific damage score for relapsing polychondritis[END_REF].

CONCLUSIONS

Over the past few years, it has become clear that there are distinct forms of RP in terms of clinical characteristics, therapeutic management, and disease evolution. Overall, RP survival has improved considerably. However, more complete characterization of the different subsets and their pathophysiological features are needed to establish more appropriate therapeutic strategies based on the patients' disease characteristics and prognosis.

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  Ocular inflammation (conjunctivitis, keratitis, episcleritis, uveitis) -Hearing loss, vestibular dysfunction -Seronegative inflammatory arthritis (Figures 1,2,3) (

Figure 1 .

 1 Figure 1. Unenhanced CT scan, showing thickening of the anterior and lateral wall of the

Figure 2 .

 2 Figure 2. Another example of anterior and lateral wall thickening of the trachea at a lower

Figure 3 .

 3 Figure 3. Extensively calcified anterior and lateral wall thickening of the trachea at the level

Figure 4 .

 4 Figure 4. A) Unenhanced CT scan showing thickening of the main bronchi wall in a 36-year-

Figure 5 .

 5 Figure 5. A) Unenhanced CT scan, showing wall thickening of the entire trachea and both

Figure 6 .

 6 Figure 6. CT scans of a patient with relapsing polychondritis and respiratory involvement.

Figure 7 .

 7 Figure 7. Unenhanced CT scan, demonstrating bronchiectasis of both lower lobes. The

Figure 8 .

 8 Figure 8. Limitation of expiratory flow (upper part of the curve), corresponding to an intra-

Funding The authors have not declared a specific grant for this research from any funding agency in the public, commercial or not-for-profit sectors.

Contributors N.d.M. and X.P. wrote the initial draft. D.D., C.L., J.D., N.C.C., L.M., G.C., and M.P.R. reviewed the manuscript and made substantial contributions to the discussion of the content. All authors reviewed and/or edited the manuscript before submission and agreed to its publication.

Competing interests None declared.

Patient consent Not required.

Provenance and peer review


Adrien Lee-Bion

MD-MSc Benjamin Menahem
email: menahem-b@chu-caen.fr

MD-MSc Yannick Le Roux

Mdnicolas Contival

SADI-S Single Anastomosis Duodeno Ileal bypass -Sleeve: Technique chirurgicale

SADI-S Single Anastomosis Duodeno Ileal bypass -Sleeve: Technique chirurgicale.

Le SADI-S est une intervention de chirurgie bariatrique malabsorptive qui fait partie des dérivations bilio-pancréatiques. Elle peut être réalisée en première intention mais est aussi indiquée en seconde temps après échec de Sleeve Gastrectomy chez les patients présentant une super-obésité (IMC supérieur à 50). Le patient est installé en position semi-assise, jambes écartées, l'opérateur se place entre les jambes. Le trocart optique est positionné à 3 travers de main de la pointe xiphoïde, légèrement latéralisé à droite. Puis, 3 trocarts opérateurs de 12 mm sont positionnés, un en flanc droit, les deux autres à gauche au-dessus et en dessous de l'ombilic. Un trocart de 5mm est positionné en sous costal gauche.

Figure 2 : Exposition et test d'ascension

Le premier temps consiste à vérifier que l'ascension de l'anse intestinale au niveau du premier duodénum est aisée. Pour ce temps, l'opérateur se place à gauche du patient. L'anse commune est mesurée à 2,5 m à partir de la valvule de Bauhin puis on met en place un clip sur le mésentère en regard pour repérer la position de la future anastomose. Le premier duodénum est disséqué à la thermofusion à sa face inférieure pour le déjanter du pancréas. Puis, libération de son bord supérieur en prenant garde à l'intégrité du pédicule hépatique. La dissection est la moins étendue possible de manière à ne pas dévasculariser le duodénum. Mais la mobilisation du cadre duodénale doit être suffisante pour réaliser une anastomose sans tension. Une anastomose duodéno-iléale termino-latérale manuelle par deux hémi-surjets de fil résorbable 2/0 est réalisée. Cette anastomose est calibrée sur une sonde nasogastrique Ch 18. L'intervention se termine par un test au bleu puis la SNG est retirée en fin de procédure.

Conclusion :

Le SADI-S est une nouvelle technique innovante en chirurgie bariatrique. Trois pièges sont à retenir :

-La plaie des voies biliaires -Les hémorragies lors de la mobilisation du cadre duodénal -Le risque ischémique de D1 lors de la mobilisation et de la section-agrafage.

Références:

1. Sánchez-Pernaute A, Herrera MA, Pérez-Aguirre ME, Talavera P, Cabrerizo L, Matía P, et al.

Single anastomosis duodeno-ileal bypass with sleeve gastrectomy (SADI-S). One to three-year follow-up. Obes Surg. 2010; 20: 1720-6. doi: 10.1007/s11695-010-0247-3.

Single Anastomosis Duodeno-Ileal Bypass -Sleeve Gastrectomy: Surgical

Technique Single Anastomosis Duodeno-Ileal Bypass -Sleeve Gastrectomy (SADI-SG) is a malabsorptive bariatric surgical procedure, one of several variations of different bilio-pancreatic diversions. It can be performed as a first-line bariatric procedure but is also indicated as a second-stage procedure after failure of Sleeve Gastrectomy (SG) in patients with super-obesity (Body Mass Index (BMI) greater than 50 Kg/M 2 . The first step is to assess the mobility of the intestinal loop to see whether it can be easily brought up to the level of the duodenal bulb (D1). During this phase, the operator moves to the patient's left. A 2.5-meter length of distal ileum is measured from the ileo-cecal valve and then a clip is placed on the mesentery at this level to identify the position of the duodeno-ileal anastomosis. The first portion of the duodenum is dissected using thermofusion along its lower edge to free it from the underlying pancreas. Its upper edge is then freed taking care to preserve the integrity of the hepatic pedicle. The encircling dissection is kept as small as possible to avoid devascularizing the duodenum. But D1 must be sufficiently mobilized to allow performance of the duodeno-ileal anastomosis without tension.

Figure 1 :

 1 Figure 1 : Installation du patient et position des trocarts

Figure 3 :

 3 Figure3: Gastrolyse et Sleeve Gastrectomy Réalisation de la gastrolyse de la grande courbure à la thermofusion puis Sleeve Gastrectomy avec préservation de l'antre, calibrée par un tube de 36 French.

Figure 4 :

 4 Figure 4 : Dissection du premier duodénum

Figure 5 :

 5 Figure 5 : Ligature du pédicule gastro-épiploïque droit Le pédicule gastro-épiploïque droit est sectionné par thermofusion après dissection minutieuse.

Figure 6 :

 6 Figure 6 : Section duodénale post-pylorique Après mise sur lac, le premier duodénum est sectionné par un agrafage vertical à environ 3 cm en aval du pylore. On vérifie alors la bonne vascularisation du D1.

Figure 7 :

 7 Figure 7 : Adossement duodéno-iléal On s'assure de l'absence de torsion mésentérique et l'anse est ascensionnée vers le duodénum. Après adossement de l'anse iléale au duodénum, une duodénotomie et une entérotomie sont réalisées sur environ 10-15 mm.

Figure 8 :

 8 Figure 8 : Anastomose duodéno-iléale

Figure 1 :

 1 Figure 1: Patient installation and trocar placement

Figure 2 :

 2 Figure 2: Exposure and assessment of small intestinal mobility

Figure 3 :

 3 Figure 3: Gastrolysis and Sleeve Gastrectomy

Figure 4 :

 4 Figure 4: Dissection of D1

Figure 5 :

 5 Figure 5: Ligature of the right gastroepiploic pedicle

Figure 6 :

 6 Figure 6: Post-pyloric duodenal transection

Figure 7 :

 7 Figure 7: Duodeno-ileal approximation

Figure 8 :

 8 Figure 8: Duodeno-ileal anastomosis