Bernard Bercu

Bruno Portier

Victor Vazquez

A DURBIN-WATSON SERIAL CORRELATION TEST FOR ARX PROCESSES VIA EXCITED ADAPTIVE TRACKING

Keywords: 2000 Mathematics Subject Classification. Primary: 62G05 Secondary: 93C40, 62F05, 60F15, 60F05 Durbin-Watson statistic, Estimation, Adaptive tracking control, Persistent excitation, Almost sure convergence, Asymptotic normality, Statistical test for serial autocorrelation. 1

We propose a new statistical test for the residual autocorrelation in ARX adaptive tracking. The introduction of a persistent excitation in the adaptive tracking control allows us to build a bilateral statistical test based on the well-known Durbin-Watson statistic. We establish the almost sure convergence and the asymptotic normality for the Durbin-Watson statistic leading to a powerful serial correlation test. Numerical experiments illustrate the good performances of our statistical test procedure.

Introduction

Model validation is an important and essential final step in the identification of stochastic dynamical systems. This validation step is often done through the analysis of residuals of the model considered. In particular, testing the non-correlation of the residuals is a crucial task since many theoretical results require independence of the driven noise of the systems. Moreover, non-compliance with this hypothesis can lead to misinterpretation of the theoretical results. For example, it is well known that for linear autoregressive models with autocorrelated residuals, the least squares estimator is asymptotically biased, see e.g. [START_REF] Bercu | A sharp analysis on the asymptotic behavior of the Durbin-Watson for the first-order autoregressive process[END_REF], [START_REF] Durbin | Testing for serial correlation in least-squares regression when some of the regressors are lagged dependent variables[END_REF], [START_REF] Nerlove | Use of the Durbin Watson statistic in inappropriate situations[END_REF], [START_REF] Proïa | Further results on the h-test of Durbin for stable autoregressive processes[END_REF], and therefore the estimated model is not the correct one. Consequently, to ensure a good interpretation of the results, it is necessary to have a powerful tool allowing to detect the possible autocorrelation of the residuals. The well-known statistical test of Durbin-Watson was introduced to deal with this question, and more specifically, for detecting the presence of a first-order autocorrelated noise in linear regression models [START_REF] Durbin | Testing for serial correlation in Least Squares regression I[END_REF], [START_REF] Durbin | Testing for serial correlation in Least Squares regression II[END_REF], [START_REF] Durbin | Testing for serial correlation in Least Squares regression III[END_REF], [START_REF] Durbin | Testing for serial correlation in least-squares regression when some of the regressors are lagged dependent variables[END_REF], firstly and for linear autoregressive models [START_REF] Bercu | A sharp analysis on the asymptotic behavior of the Durbin-Watson for the first-order autoregressive process[END_REF], [START_REF] Nerlove | Use of the Durbin Watson statistic in inappropriate situations[END_REF], [START_REF] Srivastava | Asymptotic distribution of Durbin Watson statistic[END_REF], [START_REF] Stocker | On the asymptotic bias of OLS in dynamic regression models with autocorrelated errors[END_REF], [START_REF] Proïa | Further results on the h-test of Durbin for stable autoregressive processes[END_REF], secondly.

To the best of our knowledge, no such serial correlation statistical test is available for controlled autoregressive processes. The aim of this paper is to carry out a serial correlation test, based on the Durbin-Watson statistic, for the ARX(p, 1) process given, for all n ≥ 0, by (1.1)

X n+1 = p k=1 θ k X n-k+1 + U n + ε n+1
where the driven noise (ε n ) is given by the first-order autoregressive process (1.2)

ε n+1 = ρ ε n + V n+1
and the control objective is the tracking of a given reference trajectory. More precisely, we shall propose a bilateral statistical test allowing to decide between the null hypothesis H 0 : ρ = 0 which ensures that the driven noise is not correlated, and the alternative one H 1 : ρ = 0 which means that the residual process is effectively first-order autocorrelated. The choice of the Durbin-Watson statistic, instead of any other statistical tests, is governed by its efficienty for autoregressive processes without control, see [START_REF] Bercu | A sharp analysis on the asymptotic behavior of the Durbin-Watson for the first-order autoregressive process[END_REF], [START_REF] Durbin | Testing for serial correlation in least-squares regression when some of the regressors are lagged dependent variables[END_REF], [START_REF] Nerlove | Use of the Durbin Watson statistic in inappropriate situations[END_REF], [START_REF] Proïa | Further results on the h-test of Durbin for stable autoregressive processes[END_REF].

In contrast with the recent work [START_REF] Bercu | On the asymptotic behavior of the Durbin-Watson statistic for ARX processes in adaptive tracking[END_REF], we propose to make use of a different strategy via a modification of the adaptive control law. This modification relies on the introduction of an additional persistent excitation. Since the pioneering works of Anderson [START_REF] Anderson | Exponential convergence and persistent excitation[END_REF] and Moore [START_REF] Moore | Persistency of excitation in extended least squares[END_REF], the concept of persistent excitation has been successfully developped in many fields of applied mathematics such as identification of complex systems, feedback adaptive control, etc. While it was not possible in [START_REF] Bercu | On the asymptotic behavior of the Durbin-Watson statistic for ARX processes in adaptive tracking[END_REF] to test the non correlation of the driven noise (ε n ), that is to test whether or not ρ = 0, the introduction of an additional persistent excitation term in the control law will be the key point to build our serial correlation test. Moreover, we wish to mention that all previous works devoted to non correlation test based on the Durbin-Watson statistic were only related to uncontrolled processes. Therefore, thanks to the persistent excitation, our statistical test is at our knowledge the first one in the context of linear processes with adaptive control.

The paper is organized as follows. Section 2 is devoted to the ARX process and to the persistently excited adaptive control law. In Section 3, we establish the asymptotic properties of the Durbin-Watson statistic as well as a bilateral statistical test for residual autocorrelation. Some numerical experiments are provided in Section 4. Finally, all technical proofs are postponed in the Appendices.

Model and excited adaptive tracking

We focus our attention on the ARX(p, 1) process given, for all n ≥ 0, by (2.1)

X n+1 = p k=1 θ k X n-k+1 + U n + ε n+1
where the driven noise (ε n ) is given by the first-order autoregressive process (2.2)

ε n+1 = ρ ε n + V n+1 .
We assume that the autocorrelation parameter satisfies |ρ| < 1 and the initial values X 0 , ε 0 and U 0 may be arbitrarily chosen. We also assume that (V n ) is a martingale difference sequence adapted to the filtration F = (F n ) where F n is the σ-algebra of the events occurring up to time n, such that, for all n ≥ 0, E V 2 n+1 |F n = σ 2 a.s. with σ 2 > 0. We denote by θ the unknown parameter of the ARX(p, 1) process,

θ t = (θ 1 , . . . , θ p ).
Our control strategy is to regulate the dynamic of the process (X n ) by forcing X n to track a bounded reference trajectory (x n ). We assume that (x n ) is predictable which means that for all n ≥ 1, x n is F n-1 -measurable. For the sake of simplicity, we also assume that

(2.3) n k=1 x 2 k = o(n) a.s.
In order to regulate the dynamic of the process (X n ) given by (2.1), we propose to make use of the adaptive control law introduced in [START_REF] Bercu | On the asymptotic behavior of the Durbin-Watson statistic for ARX processes in adaptive tracking[END_REF] together with additional persistent excitation. The strategy consists of using a control associated with a higher order model than the initial ARX(p, 1), and more precisely an ARX(p + 1, 2) model. The introduction of an additional excitation in the control law will be the key point to build our serial correlation test for the driven noise (ε n ), that is to test wether or not ρ = 0. Denote by (ξ n ) a centered exogenous noise with known variance ν 2 > 0, which will play the role of the additional excitation. We assume that (ξ n ) is independent of (V n ), of (x n ) and of the initial state of the system. One can observe that these assumptions are not at all restrictive as we have in our own hands the additional excitation (ξ n )

The excited adaptive control law is given, for all n ≥ 0, by (2.4)

U n = x n+1 -ϑ t n Φ n + ξ n+1
where ϑ n stands for the least squares estimator of the unknown parameter of the ARX(p+ 1, 2) model with uncorrelated driven noise (2.5)

X n+1 = ϑ t Φ n + U n + V n+1
where the new parameter ϑ ∈ R p+2 is related to θ and ρ by the identity

(2.6) ϑ =   θ 0 0   -ρ   -1 θ 1  
and the new regression vector is given by

Φ t n = (X n , . . . , X n-p , U n-1 ).
It is well-known that ϑ n satisfies the recursive relation

(2.7) ϑ n+1 = ϑ n + S -1 n Φ n X n+1 -U n -ϑ t n Φ n
where the initial value ϑ 0 may be arbitrarily chosen and

(2.8)

S n = n k=0 Φ k Φ t k + I p+2 .
As usual, the identity matrix I p+2 is added in order to avoid useless invertibility assumption. One can immediately see from (2.6) that the last component of the vector ϑ is -ρ. Consequently, we obtain an estimator of ρ by simply taking the opposite of the last coordinate of ϑ n which will be denoted by ρ n . In addition, one can also deduce from (2.6) that (2.9)

θ ρ = ∆ ϑ
where ∆ is the rectangular matrix of size (p + 1)×(p + 2) given by (2.10)

∆ =        1 0 • • • • • • • • • 0 1 ρ 1 0 • • • • • • 0 ρ ρ 2 ρ 1 0 • • • 0 ρ 2 • • • • • • • • • • • • • • • • • • • • • ρ p-1 ρ p-2 • • • ρ 1 0 ρ p-1 0 0 • • • • • • • • • 0 -1        .
Then, starting from (2.9) and replacing ρ by ρ n in (2.10), we can estimate θ by (2.11)

θ n = I p 0 ∆ n ϑ n
where ϑ n is given by (2.7) and

(2.12)

∆ n =        1 0 • • • • • • • • • 0 1 ρ n 1 0 • • • • • • 0 ρ n ρ 2 n ρ n 1 0 • • • 0 ρ 2 n • • • • • • • • • • • • • • • • • • • • • ρ p-1 n ρ p-2 n • • • ρ n 1 0 ρ p-1 n 0 0 • • • • • • • • • 0 -1        .
From the almost sure convergence of ϑ n to ϑ, we easily deduce the almost sure convergences of θ n and ρ n to θ and ρ, respectively.

A Durbin-Watson serial correlation test

We are in the position to introduce our serial correlation test based on the Durbin-Watson statistic which is certainly the most commonly used statistics for testing the presence of serial autocorrelation. Our goal is to test

H 0 : ρ = 0 vs H 1 : ρ = 0 .
For that purpose, we consider the Durbin-Watson statistic [START_REF] Bercu | A sharp analysis on the asymptotic behavior of the Durbin-Watson for the first-order autoregressive process[END_REF], [START_REF] Durbin | Testing for serial correlation in Least Squares regression I[END_REF], [START_REF] Durbin | Testing for serial correlation in Least Squares regression II[END_REF], [START_REF] Durbin | Testing for serial correlation in Least Squares regression III[END_REF], [START_REF] Durbin | Testing for serial correlation in least-squares regression when some of the regressors are lagged dependent variables[END_REF] given, for all n ≥ 1, by

(3.1) D n = n k=1 ( ε k -ε k-1 ) 2 n k=0 ε 2 k
where the residuals ε k are defined, for all 0 ≤ k ≤ n, by (2.11) and ϕ t n = (X n , . . . , X n-p+1 ). The initial value ε 0 may be arbitrarily chosen and we take ε 0 = X 0 .

(3.2) ε k = X k -U k-1 -θ t n ϕ k-1 with θ n given by
On the one hand, we would like to emphasize that it is not possible to perform this statistical test if the control law is not persistently excited [START_REF] Bercu | On the asymptotic behavior of the Durbin-Watson statistic for ARX processes in adaptive tracking[END_REF]. On the other hand, one can notice that it is also possible to estimate the serial correlation parameter ρ by the least squares estimator

(3.3) ρ n = n k=1 ε k ε k-1 n k=1 ε 2 k-1
which is certainly the more natural estimator of ρ. The Durbin-Watson statistic D n is related to ρ n by the linear relation (3.4) where the remainder ζ n plays a negligeable role. The almost sure properties of D n and ρ n are as follows.

Theorem 3.1. Assume (V n ) has a finite conditional moment of order > 2. Then, ρ n converges almost surely to ρ

(3.5) (ρ n -ρ) 2 = O log n n a.s.
In addition, D n converges almost surely to D = 2(1ρ). Moreover, if (V n ) has a finite conditional moment of order > 4, we also have

(3.6) D n -D 2 = O log n n a.s.
Proof. The proofs are given in Appendix A.

Let us now give the asymptotic normality of the Durbin-Watson statistic which will be useful to build our serial correlation test.

Theorem 3.2. Assume that (V n ) has finite conditional moments of order > 2. Then, we have

(3.7) √ n(ρ n -ρ) L -→ N 0, τ 2
where the asymptotic variance τ 2 is given by

τ 2 = (1 -ρ 2 ) (σ 2 + ν 2 )(ν 2 + σ 2 ρ 2(p+1) ) (σ 2 -ν 2 ) -(p + 1)σ 2 ρ 2p + (p -1)σ 2 ρ 2(p+1) 2 + σ 2 (ν 2 + σ 2 ρ 2(p+1) ) 4 -(4p + 3)ρ 2p + 4pρ 2(p+1) -ρ 2(2p+1) . (3.8)
Moreover, if (V n ) has finite conditional moments of order > 4, we also have

(3.9) √ n( D n -D) L -→ N 0, 4τ 2 .
Proof. The proofs are given in Appendix B.

Remark 3.1. We now point out the crucial role played by the additional excitation in the control law given by (2.4). It follows from (3.8) that if ρ = 0, then τ 2 reduces to

τ 2 = σ 2 + ν 2 ν 2 .
Consequently, if ν 2 = 0 i.e. there is no persistent excitation, then this variance explodes. Therefore, the persistent excitation allows to investigate the important case ρ = 0 and more generally to stabilize the asymptotic variance of the Durbin-Watson statistic.

We are now in the position to test whether or not the serial correlation parameter ρ = 0. According to Theorem 3.1, we have under the null hypothesis H 0 ,

lim n→∞ D n = 2
a.s.

In addition, we clearly have from (3.9) that under H 0 ,

(3.10) n 4τ 2 D n -2 2 L -→ χ 2
where χ 2 stands for a Chi-square distribution with one degree of freedom. It remains to accurately estimate the asymptotic variance τ 2 . It is not hard to see that (3.11)

lim n→∞ 1 n n k=0 X 2 k = σ 2 + ν 2 a.s.
Consequently, as ν 2 is known, it immediately follows from (3.11) that

σ 2 n = 1 n n k=1 X 2 k -ν 2
converges almost surely to σ 2 . Hence, we can propose to make use of

τ 2 n = (1 -ρ 2 n ) ( σ 2 n + ν 2 )(ν 2 + σ 2 n ρ 2(p+1) n ) ( σ 2 n -ν 2 ) -(p + 1) σ 2 n ρ 2p n + (p -1) σ 2 n ρ 2(p+1) n 2 + σ 2 n (ν 2 + σ 2 n ρ 2(p+1) n ) 4 -(4p + 3)ρ 2p n + 4pρ 2(p+1) n -ρ 2(2p+1) n . (3.12)
Therefore, our bilateral statistical test relies on the following results.

Corollary 3.1. Assume that (x n ) and (V n ) have finite conditional moments of order > 4. Then, under the null hypothesis H 0 : "ρ = 0",

(3.13) T n = n 4 τ 2 n D n -2 2 L -→ χ 2
In addition, under the alternative hypothesis H 1 : "ρ = 0",

(3.14) lim n→∞ T n = +∞ a.s.
From a practical point of view, for a significance level α where 0 < α < 1, the acceptance and rejection regions are given by A = [0, a α ] and R =]a α , +∞[ where a α stands for the (1α)-quantile of the Chi-square distribution with one degree of freedom. The null hypothesis H 0 will be accepted if T n ≤ a α , and will be rejected otherwise.

Let us now make a few comments. First of all, under H 0 , we already saw that τ 2 reduces to (σ 2 + ν 2 )/ν 2 . It can be estimated by ( σ 2 n + ν 2 )/ν 2 . Therefore, it is also possible to consider the test statistic associated with

(3.15) T n = n 2 ν 2 4( σ 2 n + ν 2 ) D n -2 2 .
Intuitively, one may think that the statistical test based on T n is more efficient under H 0 since we do not estimate the parameter ρ, but less powerful under H 1 . This point will be examined in Section 4. Next, the acceptance of H 0 after our statistical test procedure should lead to a change of control law. As a matter of fact, if we accept ρ = 0, the driven noise (ε n ) is not correlated. It means that we can implement the usual control law [START_REF] Aström | Adaptive Control[END_REF] associated with model (2.1) given, for all n ≥ 0, by

U n = x n+1 -θ t n ϕ n
where θ n stands for the standard least-squares estimator associated with (2.1). Finally, the test provided by Corollary 3.1 may be of course extended if we replace zero by any ρ 0 ∈ R with |ρ 0 | < 1 in the null hypothesis. To be more precise, we are able as in [START_REF] Bercu | On the asymptotic behavior of the Durbin-Watson statistic for ARX processes in adaptive tracking[END_REF] to test H 0 : ρ = ρ 0 versus H 1 : ρ = ρ 0 . We wish to mention that the asymptotic variance τ 2 is smaller than the one obtained in [START_REF] Bercu | On the asymptotic behavior of the Durbin-Watson statistic for ARX processes in adaptive tracking[END_REF].

Numerical Experiments

This section is devoted to the application of our Durbin-Watson serial correlation test. Although this test has several potential of being applied in concrete situations, a large search in the literature did not offer any one. We then consider artificial models for illustrative purposes and for studying the empirical level and power of our test for sample sizes from small to moderate, that is n = 50, 100, 200, 500, 1000 and 2000.

In order to keep this section brief, we restrict ourself to the three explosive models in open-loop

X n+1 = 3 2 X n + U n + ε n+1 (4.1) X n+1 = -X n + 2X n-1 + U n + ε n+1 (4.2) X n+1 = X n + 1 2 X n-1 + 1 4 X n-2 + U n + ε n+1 (4.3)
where the driven noise (ε n ) is given by (2.2) and (V n ) is a sequence of independent and identically distributed random variables with N (0, 1) distribution. The control law U n is given by (2.4) where, for the sake of simplicity, the reference trajectory x n = 0 and the persistent excitation (ξ n ) s a sequence of independent and identically distributed random variables with N (0, ν 2 ) distribution.

For each model, we based our numerical simulations on N = 1000 realizations of sample size n. We use a short learning period of 100 time steps. This learning period allows us to forget the transitory phase. The level of significance is set to α = 5%. For the statistical tests based on T n and T n , we are interested in the empirical level under H 0 to be compared to the theoretical level 5%, and the empirical power under H 1 , to be compared with 1.

First of all, let us study the effect of the variance ν 2 of the exogenous noise (ξ n ) on the behavior of the statistical test under H 0 . n 50 100 200 500 1000 2000 ν = 0.5 T n 0.9% 1.6% 2% 3% 2.9% 4.9% T n 0% 0.1% 0.5% 2.1% 2.4% 4.4% ν = 1 T n 2.5% 2.5% 3.3% 4.4% 4.4% 4.9%

T n 1.3% 1.3% 2.5% 4.1% 4% 4.8% ν = 2 T n 5.2% 4.1% 4.9% 5.3% 4.7% 4.6%

T n 3.7% 3.7% 4.1% 5.1% 4.7% 4.6% ν = 3 T n 5.8% 5.1% 5.6% 4.3% 4.3% 4.8%

T n 4.5% 4.6% 5.1% 4.2% 4.3% 4.7% Table 1. Model (4.1). Percentage of rejections of our test under H 0 (to be compared to the 5% theoretical level).

It is clear from Table 1, where one can find the results obtained for different values of ν, that the variance of the persistent excitation (ξ n ) in the control law plays a crucial role. Indeed, one can observe that if it is too small, then the empirical level of the test is bad for sample sizes from small to moderate n ≤ 1000. Of course, a high value of ν 2 improves the performance of the test under H 0 , but degrades the performance of the tracking. The value ν = 2 realizes a good compromise and allows a good calibration of the test under H 0 . n 50 100 200 500 1000 2000 Model (4.1) T n 5.2% 4.1% 4.9% 5.3% 4.7% 4.6%

T n 3.7% 3.7% 4.1% 5.1% 4.7% 4.6% Model (4.2) T n 5.9% 3% 3.9% 4.6% 4.8% 5.2%

T n 4.7% 2.5% 3.7% 4.5% 4.8% 5.1% Model (4.3) T n 4.8% 4.7% 4.1% 5.2% 4.9% 6%

T n 3.8% 3.5% 3.9% 5% 4.9% 5.9%

Table 2. Percentage of rejections of our test under H 0 (to be compared to the 5% theoretical level). ν = 2.

One can find in Table 2 the percentage of rejections of our test under H 0 for the three different models (4.1) to (4.3). The empirical levels of the test are close to the 5% theoretical level even for small sample sizes. Both statistical tests based on T n and T n are comparable even if the test statistic T n systematically tends to less reject H 0 than the test statistic T n .

Let us now study the empirical power of our statistical test. One can find in Tables 3 to 5 the results obtained for each of the three models (4.1) to (4.3). As expected, it is difficult to reject H 0 when ρ = 0.05 for small sample sizes or ρ = 0.1 to a lesser extent. However, the test performs pretty well as the percentage of correct decisions increases with the sample size. We further observe, as expected, that for a fixed value of the sample size n, the higher the value of ρ is, the more the percentage of correct decisions increases. We also notice that for a fixed value of ρ, the empirical power increases with the sample size. In conclusion, the test performs very well under H 1 . Moreover, higher values of the order p does not degrade the performances of our statistical test. Finally, one can realize that for small sample sizes, the statistical test based on T n is less powerful than the one associated with T n . We also wish to mention that, by symmetry, the performance of our statistical tests are the same for negative values of ρ.

n
n

Conclusion

Thanks to the introduction of a persistent excitation in the control law used to regulate an ARX(p,1) process, we were able to propose a non correlation test for the driven noise based on the well-known Durbin-Watson statistic. In addition, we have shown through a simulation study on artificial models the efficiency of our statistical test procedure.

Of course, many questions remain open. In particular, the extension of our results to ARX(p, q) processes where q > 1, would be a very attractive challenge for the control community. Even though our test is a potentially useful tool, we have seen in the literature that ARX(p, 1) models are often too simple for being applied to real physical models. However, the study of such models is much more difficult to handle. On the one hand, it will be necessary to make an additional assumption of strong controllability, see [START_REF] Bercu | A new concept of strong controllability via the Schur complement for ARX models in adaptive tracking[END_REF], [START_REF] Bercu | On the usefulness of persistent excitation in ARX adaptive tracking[END_REF]. On the other hand, the asymptotic variance given by (3.8) will be much more complicated as well as its estimate given by (3.12).

Finally, this work may be seen as a first step towards a serial correlation test for ARX(p, q) processes. The implementation of our statistical test within a real physical model will allow to fully validate its efficiency.

where J p = (I p 0 p ), T is the vector of R p given by T t = (1, ρ, . . . , ρ p-1 ) and ∇ is the rectangular matrix of size (p + 1)×(p + 2) given by

∇ =        1 0 • • • • • • • • • 0 1 ρ 1 0 • • • • • • 0 ρ -ξ 1 ρ 2 ρ 1 0 • • • 0 ρ 2 -ξ 2 • • • • • • • • • • • • • • • • • • • • • ρ p-1 ρ p-2 • • • ρ 1 0 ρ p-1 -ξ p-1 0 0 • • • • • • • • • 0 -1       
where, for all 1 ≤ k ≤ p -1, ξ k is the weighted sum

ξ k = k i=1 ρ k-i θ i .
We already saw that (Z n ) is a martingale with predictable quadratic variation given, for all n ≥ 1, by

Z n = σ 2 n-1 k=0 Φ k Φ t k Φ k ε k Φ t k ε k ε 2 k .
Hence, we deduce once again from Lemma A.1 that

lim n→∞ 1 n Z n = Z a.s.
where Z is the positive-definite symmetric matrix given by

Z = σ 4 σ -2 Λ ζ ζ t η
where ζ is the vector of R p+2 such that ζ t = (1, ρ, . . . , ρ p , ̺ p ) with

̺ p = -ηρ 2 - p i=1 ρ i θ i and η = 1 1 -ρ 2 .
As (Z n ) satisfies the Lindeberg condition, we deduce from the central limit theorem for multidimensional martingales given e.g. by Corollary 2.1.10 in [START_REF] Duflo | Random Iterative Models[END_REF] where the asymptotic variance τ 2 is given by τ 2 = (1ρ 2 ) 2 (σ -2 C t ΛC + 2C t ζ + η). It follows from tedious but straighforward calculations that τ 2 coincides with the expansion given by (3.8). Finally, as

(B.5) D n -D = -2(ρ n -ρ) + R n

  0, Z which, via the martingale decomposition (B.2) and Slutsky's lemma, leads to(B.3) √ n ϑ nϑ ρ nρ L -→ N 0, AZA ′ .Therefore, we immediately obtain from (B.3) that(B.4) √ n(ρ nρ) L -→ N 0, τ 2

Table 3 .

 3 Model (4.1). Percentage of correct decisions of our test under H 1 .

	50	100	200	500	1000 2000
	ρ = 0.05 T n 6.4% 7.1% 9.6% 18.8% 30.6% 50.3%
	9% 18.4% 30.5% 50.2% ρ = 0.1 T n 11.1% 17.4% 25.9% 56.8% 81.9% 98.2% T n 4.8% 6.2%
	T n 8.9% 15.2% 24.7% 56.2% 81.6% 98.1% ρ = 0.2 T n 32% 47.5% 77.6% 98.9% 100% 100%
	T n 10.7% 24.1% 49.5% 91.2% 99.5% 100% ρ = 0.3 T n 57% 83% 97.7% 100% 100% 100%
	T n 51.7% 81.7% 97.5% 100% 100% 100% ρ = 0.4 T n 78.8% 97.2% 99.7% 100% 100% 100%
	T n 73.4% 96.5% 99.7% 100% 100% 100%

Table 4 .

 4 Model (4.2). Percentage of correct decisions of our test under H 1 .

	50	100	200	500	1000 2000
	ρ = 0.05 T n 5.2% 6.2% 11% 17.6% 28.5% 54.6%
	T n 4.5% 5.6% 10.7% 17.4% 28.3% 54.6% ρ = 0.1 T n 11.5% 15% 25.3% 53% 78.1% 97.3%
	T n 9.7% 14.1% 24.6% 52.5% 78.1% 97.3% ρ = 0.2 T n 28.5% 45.8% 70.9% 98.3% 99.9% 100%
	T n 24.4% 44.1% 69.6% 98.3% 99.9% 100% ρ = 0.3 T n 53.1% 80.6% 97% 100% 100% 100%
	T n 48.4% 79.1% 96.9% 100% 100% 100% ρ = 0.4 T n 77.4% 96.4% 100% 100% 100% 100%
	T n 74% 95.9% 100% 100% 100% 100%

Table 5 .

 5 Model (4.3). Percentage of correct decisions of our test under H 1 .

	n	50	100	200	500	1000 2000
	ρ = 0.05 T n 6.9%	7%	9.6% 16% 28.5% 50.6%
	T n 6.1% 6.6% 9.4% 15.9% 28.5% 50.5% ρ = 0.1 T n 11% 14.2% 23.8% 53.6% 83.5% 98%
	T n 9.6% 12.6% 23.6% 53% 83.1% 98% ρ = 0.2 T n 29.7% 46.2% 77.1% 97.7% 100% 100%
	T n 25.9% 44.6% 76.3% 97.7% 100% 100% ρ = 0.3 T n 52.5% 81.1% 97.4% 100% 100% 100%
	T n 48.8% 79.2% 97.3% 100% 100% 100% ρ = 0.4 T n 75.9% 96.9% 100% 100% 100% 100%
	T n 73.3% 96.6% 100% 100% 100% 100%

Acknowledgements. The authors would like to thanks the anonymous reviewers for their constructive comments which helped to improve the paper substantially.

PROOFS OF THE ALMOST SURE CONVERGENCE RESULTS

The almost sure convergence results rely on the following keystone lemma.

Lemma A.1. Assume that (V n ) has a finite conditional moment of order > 2. Then, we have

where Λ is the symmetric square matrix of order p + 2 given by

with L = (σ 2 + ν 2 )I p+1 where I p+1 stands for the identity matrix of order p + 1, K is the line vector of R p+1

ρ k and H is the positive real number given by

The proof of Lemma A.1 is left to the reader as it follows exactly the same lines as the one of Theorem 4.1 in [START_REF] Bercu | On the usefulness of persistent excitation in ARX adaptive tracking[END_REF]. Denote by S the Schur complement of L in Λ,

.

We deduce from (A.2) that

Consequently, whatever the value of the parameter ρ with |ρ| < 1, det(Λ) = 0 which means that the matrix Λ is always invertible. The almost sure convergence of the least squares estimator ϑ n of the parameter ϑ associated with the ARX(p + 1, 2) process given by (2.5) is as follows.

Theorem A.1. Assume that (V n ) has a finite conditional moment of order > 2. Then, ϑ n converges almost surely to ϑ,

Proof. We deduce from (2.7) and (2.8) that

The sequence (M n ) is a locally square-integrable (p + 2)-dimensional martingale with increasing process

Then, it follows from the strong law of large numbers for martingales given e.g. in Theorem 4.3.16 of [START_REF] Duflo | Random Iterative Models[END_REF] that

Therefore, we clearly obtain (A.5) from (A.1) and (A.7).

We immediately deduce from Theorem A.1 the almost sure convergence of the least squares estimators θ n and ρ n to θ and ρ.

Corollary A.1. Assume that (V n ) has a finite conditional moment of order > 2. Then, θ n and ρ n both converge almost surely to θ and ρ,

Proof of Theorem 3.1. The proof of Theorem 3.1 relies on Corollary A.1. It is left to the reader inasmuch as it follows essentially the same lines as those in Appendix C of [START_REF] Bercu | On the asymptotic behavior of the Durbin-Watson statistic for ARX processes in adaptive tracking[END_REF].

Appendix B

PROOFS OF THE ASYMPTOTIC NORMALITY RESULTS

We shall now prove Theorem 3.2. First of all, we obtain from (3.3) that (B.1)

where

As in [START_REF] Bercu | On the asymptotic behavior of the Durbin-Watson statistic for ARX processes in adaptive tracking[END_REF], we deduce from (A.6) and (B.1) the martingale decomposition

where (Z n ) is the locally square-integrable (p + 3)-dimensional martingale given by

In addition, it follows from Lemma A.1 that the sequences (A n ) and (B n ) converge almost surely to A and B given by

where 0 p+2 stands for the null vector of R p+2 and Λ is the matrix given by (A.2). Moreover, the vector C belongs to R p+2 with

where the remainder R n is negligeable which means that

we obtain (3.9) from (B.4) and (B.5), which achieves the proof of Theorem 3.2.


Mouez Fodha
email: mouez.fodha@gmail.com.

Thomas Seegmuller
email: thomas.seegmuller@univ-amu.fr.

Hiroaki Yamagami
email: yamagami@econ.seikei.ac.jp.

Environmental policies under debt constraint

Keywords: JEL Classi…cation: Q5, H23, H63 Environmental tax reform, Debt, Public emission abatement, Double dividend

This article analyzes the consequences of environmental tax policies when the government imposes a constraint on stabilizing public debt. A public sector of pollution abatement is …nanced by taxation and by issuing public debt. Considering a simple overlapping-generations model, the tax reform stimulates steady-state investment. Then, the environmental quality and the aggregate consumption increase if and only if (i) pollution abatement is large enough and (ii) there is underaccumulation of the per capita capital stock. This arises if environmental taxation allows a decrease of either income taxation or debtoutput ratio.

Introduction

The growing environmental concerns motivate the developed countries to adapt their tax structure by introducing new taxes on pollutants. France, following the Scandinavian countries, has planned to adopt a carbon tax in the next few years. The French Environmental Protection Agency (ADEME) is entirely …nanced by revenues of taxes on pollutants, called General Tax on Polluting Activities. One of the advantages of the environmental tax is that it provides a public revenue which can be recycled. This is the reason why it is often preferred to subsidies or emission quotas. Several authors like [START_REF] Parry | Pollution Taxes and Revenue Recycling[END_REF] or [START_REF] Poterba | Global Warming Policy: A Public Finance Perspective[END_REF] argued that this revenue recycling could reduce or even annihilate the gross cost of the implementation of an environmental tax. The revenues of these taxes are used to limit the economic distortions of the reform by reducing other taxes, or alternatively, these revenues are allocated to pollution abatement programs. However, whatever the government's decision about the use of the tax revenues, public engagements in the environmental protection are often constrained by …scal objectives which impose to control public de…cits and public debt. Therefore, a pre-existing high level of public debt can be an obstacle for the launching of new environmental protection programs. This is basically the case in Europe during the global debt crisis of 2007-2010.

Accordingly, we study the impacts of environmental policies under a debt stabilization constraint, when public actions to protect the environment are at least partially …nanced by public funds. Can public debt be an obstacle for the …nancing of environmental policies? Reversely, could the environmental tax reduce e¢ ciently the public debt burden, and protect the environment simultaneously?

To analyze the interactions between environmental policies and public debt, this article considers an overlapping generations model à la [START_REF] Diamond | National Debt in a Neoclassical Model[END_REF] with an environmental intergenerational externality. Pollution emission occurs through production processes which deteriorate the environmental quality, harming the welfare of future generations. Public expenditure for pollution abatement are …nanced by taxation and debt. Moreover, a debt stabilizing constraint imposes a constant level of debt per output.

We take into account both the e¢ ciency and the intergenerational distributional aspects of environmental taxes: like Bovenberg and Heijdra (1998), we examine whether a revenue-neutral increase in the pollution tax compensated by a change of the labor tax can yield a double dividend and whether a higher pollution tax can be Pareto welfare improving by bene…ting all generations.

We show that the steady-state levels of the capital stock and of the public pollution abatement are the key factors that explain the consequences of the tax reform. Namely, if the capital stock is low and the public pollution abatement is large enough, an increase of the environmental tax, compensated by a decrease of the income tax, will increase both the environmental quality and the aggregate households'consumption. During the convergence to the steady-state, these bene…ts are no longer available for the …rst generation that has to implement the policy, but the policy may be welfare improving in the long run. Finally, an increase of the environmental taxation budgetbalanced by a variation of the debt-output ratio may also increase the environmental quality and the aggregate consumption. Hence, the …scal policy may improve the aggregate consumption and the environmental quality while reducing the debt-output ratio. Our …ndings con…rm the empirical results of Raush (2013). Using an OLG dynamic general-equilibrium model of the U.S. economy, Raush (2013) shows that when a carbon tax is employed to consolidate public debt, the environmental policies allow the possibility of sustained welfare gains for future generations.

Whether an environmental tax reform can be designed without negatively a¤ecting the economic welfare has given rise to a huge literature on the double dividend issue. [START_REF] Terkla | The E¢ ciency Value of E-uent Tax Revenues[END_REF], [START_REF] Parry | Pollution Taxes and Revenue Recycling[END_REF], or [START_REF] Poterba | Global Warming Policy: A Public Finance Perspective[END_REF] …rst had the intuition that the recycling of the revenue of an environmental tax could reduce or even eliminate the gross cost of its implementation. As governments use the revenues from pollution taxes to decrease other distortionary taxes, environmental taxes may lead to a double dividend, according to Goulder's de…nition, by improving the environmental quality and achieving a less distortionary tax system [START_REF] Goulder | Environmental Taxation and the "Double Dividend": A Reader's Guide[END_REF]). Baumol and Oates (1988), [START_REF] Pearce | The Role of Carbon Taxes in Adjusting To Global Warming[END_REF] and [START_REF] Oates | Pollution Charges As a Source of Public Revenues, Resources of the Future Discussion Paper[END_REF] suggested that these e¢ ciency gains could be a powerful argument in favor of environmental taxation.

Beside these potential e¢ ciency properties, environmental decisions have an impact on the welfare of both current and future generations, since environmental quality is a public good shared by di¤erent generations. These intergenerational issues on environmental externalities and taxation have been widely studied. [START_REF] John | An Overlapping Generations Model of Growth and Environment[END_REF] and John et alii (1995) examine the e¤ect of an environmental tax which revenues …nance a public pollution abatement sector. Bovenberg and Heijdra (1998) examine the e¤ects of a green tax on polluting capital when the tax revenue is redistributed by lumpsum intergenerational transfers. More generally, this literature concludes that a double dividend can be obtained at the expense of equity [START_REF] Proost | The Double Dividend and the Role of Inequality Aversion and Macroeconomic Regimes[END_REF], Bovenberg and van der Ploeg (1996), Bosello et alii (2001), Chiroleu-Assouline andFodha (2005), ( 2006) and ( 2014)).

Nevertheless, in these previous studies, government cannot fund pollution abatement programs by issuing public debt. In consequence, they only consider tax …nancing schemes. Debt …nancing has been introduced in dynamic models with environmental concerns (Bovenberg and Heijdra (1998); Heijdra et alii (2006), Fernandez et alii (2010)), but these contributions focus on a di¤erent issue than ours. Instead of using debt to …nance a share of pollution abatement, debt policy makes possible to redistribute welfare gains from future to existing generations. In our model, the role of the public debt is twofold: as usual, it redistributes welfare among young and old generations, but …rst of all, it …nances the public pollution abatement sector. Hence, the redistribution properties of the public debt are limited by the environmental actions of the government. Fodha and Seegmuller ( 2012) and ( 2014) analyze the consequences of some environmental tax reforms under a public debt stabilization constraint. In [START_REF] Fodha | A Note on Environmental Policy and Public Debt Stabilization[END_REF], the households can invest in private pollution abatement activities, in addition to public abatement. [START_REF] Fodha | Environmental quality, public debt and economic development[END_REF] consider the impacts of pollution on life expectancy. These articles point out the crucial role of the public debt on the dynamics of capital stock and environmental quality, and to reach the optimal allocation. In this paper, we rather analyze the impacts of the tax reforms on the capital stock and the environmental quality, while maintaining the debt to output ratio constant.

The rest of this paper is organized as follows. Section 2 presents an OLG model in which environmental externalities are provided as a by-product of production, and the government issues debts and imposes taxes on personal income and production for …nancing public emission abatement. Section 3 de…nes the intertemporal equilibrium and examines the multiplicity and the stability of steady states. Section 4 studies a possibility of the double dividend when the increase of the environmental tax is balanced through a decrease of the income tax. Section 5 considers the intergenerational distributive issues. Section 6 presents the consequences of an environmental tax reform balanced by the debt-output ratio. The …nal section provides the conclusions.

The model

We consider an overlapping generation model with discrete time (t = 0; 1; :::; +1), capital accumulation, and environmental quality which degrades with production, but may be improved by public abatements. These government expenditures are …nanced by environmental taxation on production, labor income taxes, or public debt.

Household

At each period, a new generation is born. There is no population growth and population size of a generation is normalized to N > 0. Individuals live for two periods. They have preferences over their consumption bundle when young (c t ) and old (d t+1 ), and environmental quality when young (E t ) and old (E t+1 ). E t is an externality for the household. The life-cycle utility is given by:

ln c t + v (E t ) + (ln d t+1 + v (E t+1 ))
where 2 (0; 1) is the discount factor, > 0 the relative weight of the environmental quality and v (:) measures the welfare gains from the environmental quality. The young born in period t inelastically supplies one unit of labor and receives real wage (w t ). A personal income tax ( w t ) is imposed on the real wage and the after-tax income is shared between present consumption and savings (s t ). When old, the household is retired and entirely consumes the remunerated savings (r t+1 s t ) where r t+1 is the real interest rate.1 Budget constraints of an individual born in period t are given by:

c t + s t = (1 w t )w t ; d t+1 = r t+1 s t
Then, the savings function is derived as:

s t = 1 + (1 w t )w t
Because labor is inelastically supplied, the income tax does not distort labor market.

Firms

A representative …rm produces the unique …nal good using a Cobb-Douglas technology:

Y t = K t L 1 t ;
where Y t , L t , and K t are output, labor, and capital stock, respectively. The intensive production function is given by y t = k t , where k t and y t are per worker capital stock and output. Production process emits pollutions as byproducts and, therefore, the government imposes an environmental tax ( e ) on its product sales. Pro…ts write (1 e )Y t w t L t r t K t . The …rst order conditions for pro…t maximization are:

w t = (1 e )(1 )k t ; r t = (1 e ) k 1 t :

Government

The government imposes taxes on income and sales. Moreover, debt (B t ) is issued in order to …nance a share of government spending for emission abatement (G t ). The government budget constraint is:

B t = r t B t 1 ( w t w t N + e Y t ) + G t :
with B 1 > 0 given.

To avoid explosive debt path, we assume that the government spendingoutput G t =Y t and debt-output B t =Y t ratios are constant over time, i.e. equal to g 0 and 0, respectively (see also [START_REF] De La Croix | A theory of economic growth: dynamics and policy in overlapping generations[END_REF]). Therefore, the government budget constraint is:

Y t = r t Y t 1 ( w t w t N + e Y t ) + gY t :

Environmental quality

Pollution emission occurs through polluting production processes while the government spends on public emission abatement. Because environmental quality evolves in opposite direction than pollution, its law of motion is given as:

E t+1 = (1 )E t + G t Y t , with E 0 given,
where > 0, > 0, and 2 (0; 1) are e¢ ciency parameters measuring the pollution emission from production, the public emission abatement, and the capacity to converge to the natural environmental quality in the absence of any pollution ‡ow.

Equilibrium, steady states and dynamics

The labor market equilibrium is given as N = L t , for all t. Therefore, environmental quality per young household e t satis…es:

e t+1 = (1 )e t + ( g )k t
and the government budget constraint rewrites:

y t = r t y t 1 ( w t w t + e y t ) + gy t :
Since g, and e are kept constant over time, the government must adjust the income tax rate to balance the government budget:

w t w t = r t y t 1 ( e + g)y t : (1)
The market-clearing condition for capital market is described as:

k t+1 = s t y t (2)
De…ning z t+1 = k t+1 =k t as an investment factor, equation (2) can be rewritten as:

z t+1 = (z t ) = ( + e ) 1 + 1 + 1 e z t (3)
where 1 g. By direct inspection of this equation, we immediately see that the following assumption is required to have positive values of z t :

Assumption 1 (i) > 0 and (ii) ( + e ) > :

We are now able to de…ne an intertemporal equilibrium:

De…nition 1 Under Assumption 1, an intertemporal equilibrium is characterized as a sequence of investment factors (z t ) 1 t=1 , such that (3) is satis…ed, given z 0 > 0.

Dynamics are driven by a one-dimensional dynamic system, where z t is a predetermined variable. Note that z 0 given implies two initial conditions k 0 and k 1 . In fact, the second initial condition comes from the initial condition on debt B 1 > 0 and the constant debt-output ratio B t = Y t . Given the sequence (z t ), we are able to determine (k t ) de…ned by k t+1 = z t+1 k t . Finally, given (k t ), one deduces the dynamics of e t .

The steady-state investment factors are solutions to:

P (z) = z 2 ( + e ) 1 + z + (1 e ) 1 + = 0 (4)
Note that the corresponding stationary level of capital (k) and environmental quality (e) per capita are given by:

k = z 1 1 (5) e = g z 1 (6)
Steady states and dynamics are determined as follows:

Proposition 1 Let Assumption 1 be satis…ed and such that ( + e ) = + 2 p (1 + )(1 e ). When 2 [0; ), there are two steady states, an unstable one z and a stable one z, given as:

z = ( + e ) p f ( + e ) g 2 4(1 + )(1 e ) 2(1 + ) (7) z = ( + e ) + p f ( + e ) g 2 4(1 + )(1 e ) 2(1 + ) ( e ; g; ) (8)
When = , a saddle-node bifurcation occurs and no steady state exists when > .

Proof. The existence of two steady states requires that the discriminant of P (z) must be positive, that is, f ( + e ) g 2 > 4

(1 + )(1 e ). Under Assumption 1, this condition can be reduced to f ( + e ) g > 2 p

(1 + )(1 e ). This de…nes an upper bound lower than ( + e ) (Assumption 1). When < there exist two steady states, given by (7) and (8). When = , the two steady states merge, and disappear for > . We deduce the stability properties from the features of (3). Since

lim zt!0 (z t ) = 1; lim zt!+1 (z t ) = ( + e ) 1 + > 0; (z t ) 0 > 0; (z t ) 00 < 0
the lower steady state is unstable, whereas the larger one is stable (see Figure 1).

The con…guration where there are two steady states ( < ) is represented in Figure 1. 2 The lower steady state z is unstable, while the higher one z is stable. Therefore, for z t lower than z, the economy is relegated to a poverty trap, where z t decreases to 0. Otherwise, the economy converges to the steady state z. Note that since k t+1 = z t+1 k t , the convergence of the investment factor to a stationary value corresponds to the convergence of the capital stock k t to its steady state level.

By direct inspection of (4), we see that without debt ( = 0), the trap disappears. The dynamics become z t+1 = ( + e ) 1+ and may be explicitly given by a sequence of per capita capital (k t ) satisfying k t+1 = ( + e ) 1+ k t .

Therefore, the higher steady state k = ) is globally stable, whereas the lower steady state k = 0 is unstable. There is no more any poverty trap. Indeed, the trap comes from a crowding-out e¤ect due to the existence of public debt.

( + e ) 1+ 1= (1
Using (3), we also deduce that a larger public spending-output ratio reinforces the level of the trap. Indeed, we can show that @z=@g < 0, @z=@g > 0 and @ (z t )=@g < 0 for all z t > 0. A larger income taxation is needed to balance the budget, implying a lower saving. For g > = , the associated level of environmental quality always raises at the low steady state, whereas it raises at the high steady state if the e¢ ciency of public spending is large enough.

In the next sections, we will focus on the e¤ect of …scal policy on a steady state. Because z t is a predetermined variable, we focus on the stable steady state z.3

Environmental tax reform balanced by labor income taxation

We are interested in the e¤ect of an increase of environmental taxation, given that public spending-output and debt-output ratios are constant. Therefore, income taxation will vary to balance the budget, modifying the level of the investment factor. We will focus on a possible improvement of both environmental quality and macroeconomic variables, i.e. capital accumulation, aggregate consumption. As a preliminary step, we examine the e¤ects of such an increase of environmental taxation on the investment factor and the labor income tax rate.

Proposition 2 Let Assumption 1 be satis…ed and assume 2 [0; ). Following a raise of the environmental tax rate e , the investment factor z increases, while the labor income tax rate w evaluated at this steady state decreases.

Proof. The e¤ect on the steady state investment factor is derived by di¤er-entiating (8):

dz d e d =dg=0 = @ ( ) @ e = (z + ) p f ( + e ) g 2 4 (1 e )(1 + ) > 0 (9)
The steady state income tax rate is derived as a function of other policy instruments from the government budget constraint (1):

w = 1 z + g e (1 e )(1 ) (10)
Using z = ( e ; g; ), it can be de…ned as w ( e ; g; ). By di¤erentiating (10) with respect to e , we obtain:

d w d e dg=d =0 = 1 (1 )(1 e ) 1 g + 1 e + (1 e ) z 2 @ ( ) @ e < 0 (11)
The environmental tax, in principle, imposes additional costs on polluting behavior, which reduces the steady state investment factor. However, recycling revenues provided from the increase of the environmental tax rate leads to lower income tax rates. Because the latter e¤ect is greater than the former, this environmental tax reform will increase the steady state investment factor. Alternatively, income taxation is more harmful to investment or capital accumulation than environmental taxation. Considering the government budget, note that the decrease of w comes from two direct e¤ects and a general equilibrium e¤ect. The …rst one is explained by the increase of government revenue coming from a larger environmental tax rate. The second direct e¤ect goes through the fact that a higher environmental tax rate directly decreases the interest rate. Finally, the general equilibrium e¤ect goes in the same direction: a higher level of capital induces a decrease in the interest rate. This leads to a smaller amount of debt reimbursement in the future and, thereby, lowering the income tax rate.

We focus now more speci…cally on the possible improvement of both environmental quality and macroeconomic variables of the model. Beside an increase in the amount of environmental quality per capita (i.e. de=d e > 0), we are interested in an increase in total amount of consumption per capita C c + d (i.e. dC=d e > 0), called in the "macroeconomic e¤ect". Using the resource constraint y = c + d + k + gy, C is given by:

C = (1 g)k k (12)
Given the government policy, k = k g maximizes C, where:

k g = [ (1 g)] 1 1 (13)
From ( 5), the stationary investment factor corresponding to k g is z g = (1 g). This allows us to show: Proposition 3 Let Assumption 1 be satis…ed and assume 2 [0; ). The environmental tax reform produces positive environmental and macroeconomic e¤ects if and only if (i) the public emission abatement is large enough (g > = ), and (ii) there exists under-accumulation at the stable steady state (k < k g ).

Proof. Di¤erentiating (6) with d = dg = 0, we get:

de d e d =dg=0 = ( g ) z dk d e d =dg=0
From ( 5) and ( 9), the positive environmental e¤ect is obtained if and only if the public emission abatement is large enough, that is, g > = .

Di¤erentiating now (12) with d = dg = 0, and using ( 5) and ( 13), one obtains:

dC d e d =dg=0 = (z) 1 1 z g z 1 dz d e d =dg=0
From (9), the macroeconomic e¤ect is obtained if and only if z < z g .

The environmental tax reform cuts the personal income tax, allowing a larger level of capital per capita. This raises aggregate consumption when there is under-accumulation, explaining the macroeconomic e¤ect. Note that the requirement of under-accumulation of capital seems to be quite realistic, since this is equivalently ensured by a not too low real interest rate, which is experienced by most developed countries in the last decades. Recall that under-accumulation also means dynamic e¢ ciency, which is a feature supported by the …ndings of Abel et alii (1989). Public emission abatements play an important role for the environmental e¤ect. The public spendingoutput ratio g or e¢ ciency of public emission abatements has to be large enough to ensure negative pollution ‡ows. In this case, environmental quality is positive at the steady state and positively varies with the level of capital.

At this stage, public debt has no yet an explicit role on our results. However, since the macroeconomic e¤ect occurs if and only if there is underaccumulation of capital, we now discuss its implication on the level of debtouput ratio .

Because @ ( e ; g; )=@ < 0 for all < , there exists at most a unique debt-output ratio = g that corresponds to the maximized level of total consumption per capita:

( e ; g; g ) = z g = (1 g): (14)
We deduce that there is under-accumulation if and only if > g . Then, considering the three cases, (i) g 0, (ii) g , and (iii) g 2 (0; ), we clarify in which con…guration both positive macroeconomic and environmental e¤ects are ful…lled:

Proposition 4 Let Assumption 1 be satis…ed and assume 2 [0; ). There is under-accumulation, and both positive macroeconomic and environmental e¤ects may apply, if one of the following conditions is satis…ed:

(i) e 6 1 1 g [ (1 + )] ^ e ;
(ii) 1+ < 4 (1 ) and 2 ( g ; ).

Proof. When g 0, there is under-accumulation for all 2 [0; ). However, because @ ( e ; g; )=@ < 0, g 0 is equivalent to ( e ; g; g ) > ( e ; g; 0). Using ( 8) and ( 14), we deduce case (i) of the proposition.

To prove case (ii), we begin by determining . From Proposition 1, we recall that is the lowest root of:

[ ( + e ) ] 2 4 (1 + )(1 e ) = 0 , 2 2 [ + e + 2 (1 + )(1 e )] + [ ( + e )] 2 = 0
We deduce that:

= [ + e + 2 (1 + )(1 e )] 2 p (1 + )(1 e )[ + e + (1 + )(1 e )] (15)
Note that g < is equivalent to ( e ; g; g ) > ( e ; g; ). Using ( 8) and ( 14), this inequality rewrites > ( + e ) 2(1 + ) (1 g). Substituting ( 15), one obtains P (1 e ) > 0, with:

P (1 e ) 2 (1 e ) 2 + (1 g)[2 (1 + ) ](1 e ) + (1 + )(1 g) 2
The discriminant of this polynomial of degree 2 is given by 1), it is negative, which shows that P (1 e ) > 0 for all e . This ensures that g < . Therefore, for 2 ( g ; ), the steady state is characterized by under-accumulation. Finally, note that of course, the con…guration where g is not relevant to get under-accumulation.

3 (1 g) 2 [ 4 (1 )(1 + )]. When 1+ < 4 (
In con…guration (i) of Proposition 4, under-accumulation requires a suf-…ciently low environmental tax rate, i.e. e 6 ^ e . Following Proposition 2, this implies a su¢ ciently large labor income tax rate. As we have seen, this last one has a dominant e¤ect on savings, and therefore capital accumulation. This explains that a low e may ensure under-accumulation.

The second con…guration of Proposition 4 is also of special interest. Our result requires a su¢ ciently large level of debt-output ratio. Therefore, debt plays a role. It is useful to notice that the saving rate =(1+ ) is smaller than 1=2, while under a standard parametrization, the capital share in income belongs to (1=4; 1=2). In this case, the inequality 1+ < 4 ( 1) is ful…lled and con…guration (ii) of Proposition 4 may apply. Note that if e > ^ e and su¢ ciently close to ^ e , the debt-output ratio does not need to be so large since g tends to zero.

Because the environmental tax reform positively a¤ects the steady-state investment factor z, the conditions for the double dividend di¤er from those in the literature. [START_REF] Ono | Environmental-tax …nanced social security tax cuts and the double dividends[END_REF] considers an environmental tax reform that cuts the social security tax in the absence of public emission abatement. Therefore, the environmental dividend is produced only when the capital per capita decreases. Moreover, the non-environmental dividend is obtained because there is over-accumulation at the steady state and capital per worker decreases.

Distributive issues

We now focus on the distributive issues of the tax reform balanced by labor income taxation. We investigate both the intergenerational (i.e. between generations) and the intragenerational (i.e. between life-cycle consumptions) impacts of the policies. First, we analyze if following the implementation of the …scal reform at a given period, the total amount of consumption increases during all the dynamic path converging to the stable steady state. Second, we study if the tax reform implies an increase of consumption of both young and old at the steady state.

We start by extending our analysis considering an intertemporal equilibrium which is no more stationary, but converges to the stable steady state. The question we address is the following. Assuming that the steady state z satis…es the positive e¤ects of the …scal reform and we apply the …scal reform at some given date, should the …scal reform be detrimental for some generations and for the consumption at some periods? Can we determine which generations will bene…t from the …scal reform? These issues are solved in the following proposition:

Proposition 5 Let Assumption 1 be satis…ed and assume that 2 [0; ). Furthermore, consider that Propositions 3 and 4 are satis…ed, and z 0 > z. Following an increase of e at date t 0 , the positive e¤ects of the …scal reform fails at t = t 0 , but there exists t 1 > t 0 such that for all t > t 1 , they occur.

Proof. Assume that 0 increases permanently at t = t 0 . By direct inspection of equation (3) and Figure 1, we deduce that z t raises for all t > t 0 + 1. Since k t+1 = z t+1 k t , the same happens for k t for all t > t 0 + 1. At t = t 0 , we have c t + d t = (1 g)k t k t+1 = k t (1 g z t+1 ). Since k t is predetermined and z t+1 increases, aggregate consumption falls, which means that the positive e¤ects of the …scal reform fail. Considering now that t

> t 0 + 1, d(c t + d t ) = (1 g) k 1 t dk t dk t+1 .
For t su¢ ciently large, namely t > e t, the capital stock is characterized by under-accumulation, 4 i.e. we have (1 g) k 1 t > 1, which implies that d(c t + d t ) > dk t (1 dk t+1 dkt ). Since at a stable equilibrium with z t = z we have k t+1 = zk t , there exists t > b t such that dk t+1 =dk t < 1. Therefore, when t > t 1 = maxf e t; b tg, d(c t + d t ) > 0 because dk t > 0 for all t > t 0 + 1. Since e t+1 = ( 1)e t + ( g )k t also raises, the positive e¤ects of the …scal reform apply for all t > t 1 .

One may note that aggregate consumption decreases at some periods, because consumption when old does. In fact, at t = t 0 , consumption when young raises because, given the level of capital, the increase of the environmental tax rate reduces the labor income tax rate, which enhances the after-tax income to consume. In contrast, consumption when old goes down, because a higher environmental tax rate reduces the interest rate, i.e. the remunerated saving. At the following periods, the increase of c t is reinforced by the raise of capital, which pushes up the wage and pushes down labor income taxation. The interest rate becomes even lower because of the larger capital accumulation. After some periods, this mechanism allows to get a positive e¤ect on aggregate consumption.

This intergenerational issue on environmental externalities and taxation has already been widely studied [START_REF] John | An Overlapping Generations Model of Growth and Environment[END_REF], John et alii (1995), [START_REF] Howarth | Climate Change And Overlapping Generations[END_REF], Fisher and van [START_REF] Fisher | Pollution and economic growth[END_REF]). The main result of all these studies is that environmental taxation implies such a welfare loss for the older generations experiencing the …scal reform that its implementation can not be wished because the generation which would decide it would also bear the heaviest burden. This result of the literature originates in the fact that balanced environmental …scal reforms have generally not been considered. We show here that this negative result for the political feasibility can be generalized to the balanced-budget reform case.

We now focus on the distributive e¤ect of the policy between consumptions when young and old at the stable steady state. Should the …scal reform not only improve aggregate consumption but also both c and d? This is an important issue because if this occurs, utility for consumption of young and old consumers increase. Moreover, since under Proposition 4, the environmental tax reform increases utility for environmental quality, this leads to a double dividend.

Proposition 6 Let Assumption 1 be satis…ed and assume that 2 [0; ). Furthermore, consider that Propositions 3 and 4 are satis…ed. Following an increase of e , the consumption of young c is increasing at the stable steady state. For < 2 1 and e < 1 , there exists e > 0, such that for all < e , the consumption of old d is increasing in e too.

Proof. At the stable steady state, consumption when young c is given by:

c = 1 1 + k + e + (1 e ) z
Because @z=@ e > 0 (see Proposition 2) and k = z 1=(1 ) , we deduce that @c=@ e > 0. Using the fact that saving is equal to k + y, consumption when old d is equal to:

d = (1 e ) z 1 + z 2 1 1
For = 0, we have @z=@ e = z=( + e ). We deduce that:

@d @ e = z 1 (1 )( + e ) 2 (1 )
e Therefore, for = 0, @d=@ e > 0 if and only if

< 2 1 , or equivalently 1 > g > 1 2 1
, and e < 1 . In this case, by continuity, there exists e > 0, such that for all < e ,5 we have @d=@ e > 0.

If this proposition applies, welfare is increasing. Indeed, following an increase of the environmental tax rate, environmental quality, consumptions when young and old become larger. We obtain a double-dividend. Note that this requires a su¢ ciently low debt-output ratio. Otherwise, the increase of the environmental tax rate that decreases the return of assets, directly and through the raise of capital, implies a too large decrease of the remunerated debt. In this case, remunerated saving, i.e. consumption when old, decreases.

Environmental tax reform balanced by debtoutput ratio

In our model, the government issue debt to …nance current de…cits. Instead of assuming that a larger environmental tax rate may be used to reduce income taxation, the environmental policy can also be used to modify the debt-output ratio. We will address this issue now and investigate if such a policy may induce both the positive environmental and macroeconomic e¤ects at the stable steady state.

To be more speci…c, we consider that following an increase of e , the government budget is balanced by a modi…cation of , taking the labor income tax rate as constant. Di¤erentiating (10) with d w = dg = 0, the policy change is described as:

d d e d w =dg=0 = @ ( )=@ e @ ( )=@ (16)
Using ( 6) and ( 12), we note that debt-output ratio a¤ects aggregate consumption and environmental quality only through the investment rate z. Therefore, using Proposition 3, the positive environmental and macroeconomic e¤ects of the …scal reform may be obtained if z is increasing following the new policy.

Proposition 7 Let Assumption 1 be satis…ed and assume 2 [0; ). Consider an increase of environmental taxation budget-balanced by a variation of the debt-output ratio. If e > 1 (1 g) = , there is b 2 (0; ), such that z is increasing in e if < b . This goes through a decrease of . In this case, the environmental tax reform produces positive environmental and macroeconomic e¤ects if and only if (i) the public emission abatement is large enough (g > = ), and (ii) there exists under-accumulation at the stable steady state (k < k g ).

Proof. To investigate the e¤ect of the increase of e on z, we …rst note that: where @ ( )=@ e > 0 is given by (9) and @ ( )

@ = z + (1 e ) p f ( + e ) g 2 4 (1 e )(1 + ) < 0 (17)
Therefore, a su¢ cient condition to have dz=d e > 0 is d =d e < 0. Using ( 11) and ( 16), we deduce that d =d e and @ ( )=@ have the same sign. To …nd it, a derivative of (10) with respect to the debt-output ratio gives:

@ ( ) @ = 1 1 z 1 z @ ( ) @ 1 (1 e )(1 ) (18)
Using ( 17) and ( 18), @ ( )=@ < 0 is equivalent to ( ) < 1, with:

( ) (1 e ) z " 1 + 1 + (1 e ) =z p [ ( + e ) ] 2 4
(1 e )(1 + )

#

We can easily show that 0 ( ) > 0. Moreover, using (8), we have:

( ) = +1 (0) = (1 + )(1 e ) ( + e )
Therefore, if (0) < 1, there exists b 2 (0; ) such that ( ) < 1 for all < b . Using Proposition 3, we deduce the proposition. 6Under Proposition 3, the positive environmental and macroeconomic effects occur if the …scal reform raises z (or k). Here, a larger environmental tax rate implies a variation of the debt-output ratio, the income tax rate staying constant. We show that when a larger e implies a lower debt-output ratio , capital accumulation raises. Since the income tax rate is constant, the debtoutput ratio modi…es k or z through the level of the crowding-out e¤ect only. This con…guration is especially interesting because the …scal policy allows to improve aggregate consumption and environmental quality by reducing the debt-output ratio. Regarding the debt sustainability constraints faced by many countries today, this …scal reform gives rise to a third dividend.

Conclusion

This paper examines the e¤ects of environmental tax reform in an overlapping generations model by taking into account a debt stabilization constraint and public pollution abatement. We show that, when the budget-neutral reform allows a decrease in the income tax, the steady state investment factor increases. This result implies an increase in the pollution emission, the (…rst) environmental dividend cannot be obtained in the absence of public abatement. On the other hand, the second (i.e. economic) dividend is obtained when the economy is characterized by under-accumulation of the per worker capital stock. Finally, an increase of the taxation budget-balanced by a variation of the debt-output ratio may also increase the environmental quality and the aggregate consumption. Hence, the …scal policy allows to improve aggregate consumption and environmental quality by reducing the debt-output ratio.

Figure 1 :

 1 Figure 1: Dynamics with two steady states.

We assume complete depreciation of capital. Since the period length is quite long in overlapping generations model with two-period lived households, this assumption is not restrictive.

Recall that, as it is clear from the proof of the proposition, lower than satis…es Assumption 1.

To clarify notations, x will denote in the following the value of the variable x t evaluated at the steady state z.

This is always the case if the sequence of (k t ) is increasing though time.

Note that this condition is compatible with case (i) of Proposition 4 and case (ii) for e su¢ ciently close to ^ e :

Note that in order to ensure under-accumulation, the conditions obtained in this proposition should be in accordance with Proposition 4. Case (i) of Proposition 4 is satis…ed for 1 (1 g) = < e 6 ^ e : This requires > (1 )=(1 + ), which is satis…ed for su¢ ciently close to 1. Case (ii) of Proposition 4 is ful…lled under the same condition on , but e larger and close to ^ e :

8 References A. Abel, G. Mankiw, L. Summers and R. Zeckhauser, Assessing Dynamic E¢ ciency: Theory and Evidence, Review of Economic Studies, 56(1), 1-19 (1989). W.J. Baumol and W.E. Oates, "The Theory of Environmental Policy", Cambridge University Press, 2nd edition, (1988). F. Bosello, F. Carraro, Carlo and M. Galeotti, The double dividend issue: modeling strategies and empirical …ndings, Environment and Development Economics, vol. 6(01), 9-45 (2001). A.L. Bovenberg and B.J. Heijdra, Environmental Tax Policy and Intergenerational Distribution, Journal of Public Economics, 67, 1-24 (1998). A.L. Bovenberg and F. van der Ploeg, Optimal Taxation, Public Goods and Environmental Policy with Involuntary Unemployment, Journal of Public Economics, 62 (1), 59-83 (1996). M. Chiroleu-Assouline and M. Fodha, Double Dividend with Involuntary Unemployment: E¢ ciency and Intergenerational Equity, Environmental and Resource Economics, 31(4), 389-403 (2005).


Nicole El Zoghby
email: nicole.el-zoghby@hds.utc.fr

Véronique Cherfaoui
email: veronique.cherfaoui@hds.utc.fr

Thierry Denoeux
email: thierry.denoeux@hds.utc.fr

Evidential Distributed Dynamic Map for Cooperative Perception in VANets

In this paper, we present a distributed approach to build a dynamic map in the context of VANets (Vehicular Ad hoc Networks). It is based on the principle of cooperative perception where vehicles work as a team in order to extend their field of view. Each vehicle is equipped with sensors allowing it to detect its environment and to build its map, denoted by local map. It receives messages from other vehicles containing mobile objects detected in their surroundings. The algorithm of distributed dynamic map builds a map of the dynamic environment including objects in the sensor's field of view as well as those sent by other vehicles. This algorithm is developed under the belief functions framework. The implementation of such an application is complex and needs many treatments: temporal and spatial alignment, object association, fusion of messages and data dissemination. This approach has been validated by simulation on scenario involving several vehicles in traffic situation.

I. INTRODUCTION

Nowadays, the research field of intelligent vehicles systems benefit from the emerging inter-vehicle communication to perform cooperation. Vehicles are equipped with sensors for surroundings' detection, absolute localization system for localization and wifi antenna for broadcasting information. Different applications can be considered such as cooperative localization and perception.

Cooperative localization has been treated in several domains such as robots cooperation and intelligent vehicles applications. Cooperative localization approaches differ with the data exchanged in the network. We can distinguish two classes of approaches: in the first one, vehicle sends only its own detected data ( [START_REF] Roumeliotis | Distributed multirobot localization[END_REF], [START_REF] Martinelli | Multi-robot localization using relative observations[END_REF], [START_REF] Karam | Localization of a group of communicating vehicles by state exchange[END_REF], [START_REF] Tischler | Enhanced environmental perception by inter-vehicle data exchange[END_REF]), while in the second approach, it sends what it receives from others or it combines its own local data with received data ( [START_REF] Li | Cooperative multi-vehicle localization using split covariance intersection filter[END_REF], [START_REF] Curn | Data incest in cooperative localisation with the common past-invariant ensemble kalman filter[END_REF]). The latter case is a distributed method where the same information can be combined several times. This phenomenon is called data incest and can be avoided with an appropriate treatment. A review on cooperative localization can be found in [START_REF] Curn | Data incest in cooperative localisation with the common past-invariant ensemble kalman filter[END_REF].

We present a distributed approach that consists of sending the result of data fusion of own vehicle detection with received messages. It is a cooperative perception approach where vehicles cooperate in order to extend their field of view (constrained by sensor's limitations and occultation) and to reduce the false alarms. The purpose of this method is not to improve the accuracy of objects localization but to extend vehicle's perception range by reinforcing confidence in objects existence. As in [START_REF] Aeberhard | Object existence probabilty fusion using dempster-shafer theory in a highlevel sensor data fusion architecture[END_REF] and [START_REF] Munz | Using dempster-shafer-based modeling of object existence evidence in sensor fusion systems for advanced driver assistance systems[END_REF], we use the belief functions framework to manage uncertainty of object existence.

Recent research works have addressed cooperative perception in the robotics and intelligent vehicles fields. Merino et al. [START_REF] Merino | A cooperative perception system for multiple uav s: Application to automatic detection forest fires[END_REF] have developed a cooperative perception system for heterogeneous multi-UAV. Their system considers heterogeneous sensors. They presented the system architecture and experimental results on automatic forest fire detection and localization. Li and Nashashibi [START_REF] Li | Multi-vehicle cooperative perception and augmented reality for driver assistance: A possibility to see through front vehicle[END_REF] have presented a method of cooperative perception for augmented reality application. Their method was applied on an example with two vehicles. The idea is to transform the occulted part of the first vehicle to a perception of the rear vehicle based on the 3D perspective. The authors estimate the relative pose between two reference vehicles in order to have a common reference for vehicle perception, allowing them to transform perceptions of vehicles in 3D perspective. Their method has been tested on experimental vehicles. Raugh et al [START_REF] Rauch | Car2xbased perception in a high-level fusion architecture for cooperative perception systems[END_REF] have established a cooperative perception system. Car2X-based perception is modeled as a virtual sensor in order to integrate it into a high-level sensor data fusion architecture. Temporal and spatial alignments are performed to improve vehicles state accuracy. Their method is validated using experimental data. In [START_REF] Kim | Cooperative perception for autonomous vehicle control on the road: Motivation and experimental results[END_REF], authors used cooperative perception for vehicle control on the road. The proposed cooperative systems had been implemented on self-driving vehicle and manned vehicles.

We consider the concept of Local Dynamic Map (LDM) developed in the Safespot project [START_REF]The safespot project[END_REF]. This map includes static and dynamic information in the surroundings of a vehicle. This information is updated periodically. The LDM is divided into four levels. We are interested in the fourth level including the mobile elements in the scene (egovehicle, other vehicles, pedestrians, trucks, . . . ). The aim is to exchange these maps (fourth level only) between vehicles in order to increase the field of view of each one of them. Each vehicle sends its current pose and the list of objects. We present in this paper the algorithm that allows combining the maps exchanged between vehicles. This algorithm is an extension of the distributed data fusion algorithm presented in [START_REF] Zoghby | Distributed data fusion for detecting sybil attack in vanets[END_REF]. It is developed under the belief functions framework. This framework seems appropriate for its ability to model uncertainties and to combine data using adapted operator that considers data dependency.

For sake of simplicity, in what follows, we replace the notion of local dynamic map by the dynamic map DM since the vehicle will have two types of information: local and distributed.

This paper is organized as follows. The problem is stated in Section II. In Section III, we present our distributed dynamic map algorithm based on the belief functions framework. Implementation and results are presented in Section IV. Finally, Section VI concludes the paper.

II. PROBLEM STATEMENT

A. Dynamic Map and vehicle's state

We consider that each vehicle V j detects with its sensors a map denoted by DM j . The vehicle map contains a list of objects {O i } j (where i is the identity of the object). Each vehicle has a confidence in the existence of the detected object, this confidence is represented by m i .

The vehicle V j is ego-localized and knows its state represented by X Vj = (x j , y j , ϕ j , v j , P j ) where x j , y j represent the vehicle's absolute position, ϕ j its orientation, v j the vehicle's velocity and P j represents the covariance matrix.

Objects {O i } j are represented by different attributes in the ego vehicle reference frame such as:

• p i , v i and c i position, velocity and class of object i, • σ pi and σ vi uncertainties of position and velocity or the covariance matrix represented by P i , • m ci the mass function that represents the of belonging to class c i . This information depends on the type of the sensor used to detect the scene. The class mass function is only used in the association. • m i the mass function of the object existence. The aim is to evaluate the confidence in the objects detected in the scene. We are interested in objects' existence. The frame of discernment is: Ω = {O, N O} where "O" represents the objects (vehicle, truck, bus, . . . ) and "NO" non-objects (false alarms due to the detection of the road, sidewalk, . . . ). The confidence on the object existence is represented by mass functions as in the following:

m i (O) = a, m i (N O) = b, m i (Ω) = 1 -a -b.
(1)

The mass functions assign a degree of belief to the different parts of Ω where m ( Ω) represents the uncertainty. The computation of these mass functions is detailed in section IV-A.

B. Local and distributed map

To implement a distributed algorithm, each vehicle V j must have two types of information: local and distributed. The local map, denoted by DLM , is what the vehicle detects with its own sensor and contains the mass it gives to the detected objects. The vehicle keeps this information and does not send it as it is. It combines it with the received messages in order to establish the distributed map. As for the distributed map, we distinguish between DDM (Dynamic Distributed Map) and DP M (Dynamic Public Map). The DDM is a DM updated with maps received from others. The DP M is the result of the combination of the distributed map and the local map. The DP M is sent over the network.

To exploit messages, each vehicle should transform received data in a global reference frame, execute a temporal alignment and associate the detected objects. Objects are associated using the algorithm developed in [START_REF] Zoghby | Optimal object association from pairwise evidential mass functions[END_REF]. After these different steps, the vehicle can update its distributed map. This is possible in VANets thanks to the GPS pose and GPS common time.

III. DISTRIBUTED DYNAMIC MAP ALGORITHM

Algorithm 1 presents the Distributed Dynamic Map algorithm based on the distributed data fusion algorithm developed in [START_REF] Zoghby | Distributed data fusion for detecting sybil attack in vanets[END_REF]. This algorithm is applied by the receiver when it receives a message containing the sender's map. In this algorithm, the sender information is indexed by s and the receiver information by r.

The principle of this approach is that when a vehicle receives a message, it updates its distributed map with the received map using the cautious rule [START_REF] Denoeux | Conjunctive and disjunctive combination of belief functions induced by nondistinct bodies of evidence[END_REF]. Each vehicle can combine the same information many times as it is coming from independent sources. This is called data incest problem provoked by cycles of data dissemination. To avoid this problem, the idempotent cautious rule is used. It is also an associative, commutative rule and defines an order relationship on the weights. After that, it combines its distributed map with its local map by the Dempster's rule [START_REF] Dempster | Upper and lower probabilities induced by a multivalued mapping[END_REF]. This rule is based on conjunctive operator and must be used when sources are independent (this is the case here). The result is used to update the public map. The vehicle sends its public map over the network. The convergence of this algorithm had been demonstrated in [START_REF] Ducourthial | Self-stabilizing distributed data fusion. Stabilization, Safety, and Security of Distributed Systems[END_REF]. It was shown, in this latter paper, that this algorithm is robust to errors. This property is ensured by the fact that the received information has to be discounted before the combination. The algorithm is detailed in the following subsections. A. Update the DDM of receiver with message Discounting: In order to ensure the algorithm convergence [START_REF] Ducourthial | Self-stabilizing distributed data fusion. Stabilization, Safety, and Security of Distributed Systems[END_REF], each receiver has to discount the received data. This operation tends to give less confidence in received information. For this aim, the receiver applies a discounting on the sender DP M s . This discounted map is denoted by α DP M s . It also adds the sender S to its map. The discounting is applied on the existence mass functions as follows:

α m(A) = α.m(A) where A = Ω, α m(Ω) = (1-α)+α.m(Ω). (2)
Prediction: A temporal alignment is needed before updating the receiver distributed map with the sender's one. A transmission delay is assumed in this approach. The temporal alignment is to predict these two distributed maps at time t ′ of the treatment with a prediction model with constant velocity. The prediction function ( DP M s ← prediction(DP M s ), DDM r ← prediction(DDM r )) allows the maps prediction along these equations:

O            x(t ′ ) = x(t) + v x (t) * ∆t, y(t ′ ) = y(t) + v y (t) * ∆t, P (t ′ ) = F.P (t).F T + Q, m (t ′ ) c ← m (t) c , v(t ′ ) = v(t). (3a) m m (t ′ ) ← α ′ m (t) . ( 3b
)
where ∆t = t ′ -t, t ′ is the prediction time and t is the time when the map is F is the model matrix that relates the state at time t ′ with the state at time t and Q is the covariance matrix. The existence mass functions of the objects are discounted in terms of time. The discounting factor is represented by α ′ where α ′ = exp (-∆t) . Fusion: The predicted maps are associated using the association algorithm detailed in [START_REF] Zoghby | Optimal object association from pairwise evidential mass functions[END_REF] that takes into account the objects position, velocity and class. Algorithm 2 presents the F usionCautious function that allows considering three cases when fusing objects:

• if objects (O s , O r ) are associated, the receiver existence mass is updated with the cautious rule [START_REF] Denoeux | Conjunctive and disjunctive combination of belief functions induced by nondistinct bodies of evidence[END_REF],

m (t) r = m (t) r ∧ m (t) s ; (4)
• if the sender has not detected an object already detected by the receiver (O r is not associated), the object existence mass function is discounted as in equation 2;

• if the sender has detected an object not detected by the receiver, this object is added to DDM r . The dynamic public map DP M r is the result of combination of the distributed map (DDM r ) and local map (DLM r ) of the receiver. The DLM r is built by the receiver embedded sensors. This map is then combined to the distributed map. For this aim, both maps are associated before the update step. The result is saved in DP M r that will be sent to other vehicles. Algorithm 3 shows the three cases that should be taken into account for the objects' fusion:

• if objects (O r , O r l ) are associated, O r is updated with O r l ,
where O r l is the object detected by the receiver in its local map. The mass functions are combined in this case by the Dempster's rule [START_REF] Dempster | Upper and lower probabilities induced by a multivalued mapping[END_REF] as follows:

m (t) r = m (t) r l ⊕ m (t) r ; (5)
• if O r l is not associated, we keep it in the DP M r ;

• if the object O r of the distributed map DDM r is not associated and should be in the receiver's field of view, we delete it because it is considered as a false alarm. If it is not in the field of view, we add it to DP M r .

Finally, DP M r is sent over the network. We should note that the local map in this algorithm is injected once, because the objects are supposed to be tracked in time by intelligent sensors.

IV. IMPLEMENTATION

The Distributed Dynamic Map algorithm have been validated on simulated data created by A. Houenou [START_REF] Houenou | A track-totrack association method for automotive perception systems[END_REF]. We have simulated several scenarios with different vehicles, some of them are equipped with an intelligent camera (field of view: maxrange = 60m and angle = 45 • ), a localization system and a wifi antenna. Vehicles, equipped with a wifi antenna, share their distributed dynamic map. They move along a multi-lane road and detect their surroundings.

The intelligent camera is a tracking system that provides different types of information such as object's identity, its relative position x and y and its covariance matrix P x , its relative velocity v x and v y and the covariance matrix P v , object's age and class. This provided information allows computing mass functions and building the dynamic maps.

This algorithm needs experimental setup. In the following, the discounting factor is fixed to 0.8 empirically and it is the same for all the senders. It may depend on the number of hop, the confidence in sender. The association parameters are chosen as detailed in [START_REF] Zoghby | Optimal object association from pairwise evidential mass functions[END_REF].

A. Mass functions computations

The object's age provided by the sensor of vehicle V j presents the number of times the object O i has been detected. The object's existence mass function is computed in this manner:

m(O) = a = β.(1 -e -k.age(Oi) ), m(N O) = b = β.(e -k.age(Oi) ), m(Ω) = 1 -a -b = (1 -β), (6)
where β = 0.9 represents the sensor's reliability and k is a positive coefficient k = 0.1. The more the sensor detects the object, the more the existence mass function is enhanced. The simulator does not provide the class mass function m c . The class frame of discernment is Ω c = {V, N V }, where V represents the fact that the object is a vehicle and N V not a vehicle. We a mass 0.9 to {V } or {N V } depending on the sensor's decision (the object is a vehicle or not) and a mass 0.1 to {V, N V }.

B. Spatial and temporal alignment

Spatial alignment: Each vehicle detects objects in its local perception's reference frame and builds its dynamic local map. Sensor and vehicle reference frames are the same in the simulator, no need for spatial alignment between these two frames. The frame transformation is done on two levels:

• when a vehicle receives a message, it transforms objects in the global frame. • when the local map is fused with the distributed map, the local map is transformed in the global frame before combining it with the distributed map. Temporal alignment: We consider in this application that sensors have common time. Sensor detects objects each ∆t = 0.1s. For simplification, we consider that the vehicle will send messages at the same frequency. The transmission time of message is not controlled but it is bounded. It is assumed that data are dated to the time of calculation. Objects are predicted using the function prediction() and fused at the present time. Whatever the time of arrival of the messages, all data is synchronized and processed at the same time. As shown in Algorithm 1, when a vehicle receives a message, it predicts at the reception time its DDM r (t) and the received message. It fuses these two information at time t.

These temporal and spatial alignments are possible with real data by using a GPS pose and time.

C. Scenario

We have created a scenario with 4 vehicles

{V 0 , V 1 , V 2 , V 3 }. Vehicles {V 0 , V 1 , V 2 }
are equipped with a camera and wifi antenna while V 3 is not equipped. Equipped vehicles (V 0 , V 1 , V 2 ) can exchange messages. V 0 follows V 1 for 15s, V 3 overtakes V 0 at 1.4s and V 1 at 3.2s. V 2 at 3s comes from the other side. Figure 1 shows the scenario at time 2.7s. This figure represents the ground truth (GT ) of the scenario.

Sensors detect objects each 0.1s, each vehicle builds its DLM , updates its DDM with available data (received Fig. 1: Scenario at time= 2.7s messages) and sends its DP M at the same frequency. We tested the scenario for 15s. Figure 2 shows the result at the instant 2.7s. The first column shows the DLM and the second column presents the DP M (sent distributed map). The field of view of each vehicle detecting the scene is drawn in red. DLM: V 0 detects V 1 and V 3 , this detection is represented by the black triangles in the DLM V 0 . Vehicles V 1 and V 2 detect nothing. Vehicles V 0 , V 1 , V 2 communicate with each other.

DPM: At each message reception, the receiving vehicle adds the sender vehicle to its list. The DP M of each vehicle is represented by black squares. Vehicles V 1 and V 2 do not detect other vehicles, but add them when receiving their messages. For example, V 1 adds V 0 and V 2 to its DDM when receiving a message from them, and adds V 3 to its list since it was detected by V 0 . The set of ground truth vehicles are reported as colored dots in the DP M , to simplify the verification.

V. RESULTS

Different tests have been implemented to evaluate the application of the distributed dynamic map. The first test concerns the evolution of the existence mass on a non equipped object. We also show the performances by comparing the rate of true positive (T P ), false positive (F P ) (non detection) and false negative (F N ) (false alarms) of the detections of the three vehicles V 0 , V 1 and V 2 in the distributed and local maps. Results are shown in the following.

a) Evolution of the existence mass: To make the decision concerning the object existence after updating the distributed map, we calculate the pignistic probability BetP (N O) = m(N O)+m(Ω)/2. This object does not exist if its BetP (N O) is superior to a predefined threshold ǫ. The object is then deleted from the distributed map.

Figure 3 shows the evolution of the existence mass of the vehicle V 3 in the local and distributed maps of the vehicles V 0 , V 1 and V 2 . V 3 is not detected at all times by all other vehicles. We should note that in this test V 0 , V 1 and V 2 can communicate with each other all the time. The blue curve represents the mass on the existence of the distributed map of each vehicle. V 0 detects V 3 first at instant 2s. It does not keep it in its distributed map until the pignistic probability reach the threshold ǫ. For this reason, the mass function of V 0 does not evolve at time 2.5s. Afterwards, V 0 sends its distributed map to V 1 and V 2 . This explains the appearance of V 3 in the distributed map of V 1 between 2.5 and 3.8s and in the map of V 2 between 2.5 and 3s. At these times, the mass function of the local map is empty (i.e. m(Ω) = 1, m(V 3 ) = 0 and m(N V 3 ) = 0), V 1 and V 2 didn't detect V 3 . All vehicles keep V 3 in their distributed maps while others detect it. This figure shows the difference between the distributed and local maps. The local map is limited at the vehicle detection while the distributed map increases the field of view of the vehicles.

Case of a defective sensor: In this simulation, V 0 does not detect V 3 between 1.9s and 5.3s. In this case, we search to evaluate the influence of the non detection of a sensor on the vehicle network. We notice the increase of the non detection of V 0 in the local map. The effect is more important on the rate of F P of V 0 . Table I shows a light variation at the level of recall of the DLM and the precision of the DP M of V 0 . Change of the range of communication The change of the range of communication has an influence on the distributed maps of the vehicles. The ranges of communications have been changed according to the following:

• range 1: vehicles have a short range, so that vehicles V 0 and V 1 , which are very close, cannot communicate. • range 2: the range is increased so that vehicles V 0 and V 1 can always communicate and V 3 can exchange its map when it approaches the other vehicles. • range 3: the range 3 is bigger than the previous two ranges, which corresponds to the case where all vehicles receive all sent messages. The table III shows the influence of the change of the range on the performance of the method in this scenario. We notice the increase of the recall of each vehicle when the range increases, as well as a weak variation of the precision. Results of vehicle V 0 decrease with range 3 due to false alarms. Send its DLM or its DPM?: As mentioned in the section I, vehicles can send either their own detections, either the data sent by other vehicles, as received or combined with their own data. In this section, we compare the change of type of the data sent in the messages. Two types of messages are sent: first, vehicles send messages containing their local maps, and then messages containing their distributed maps. The exchange of the distributed maps allows increasing the good detection V P for the vehicles and the field of view of the vehicles. This phenomenon is highlighted in the table IV which shows the increase of the recall in the case of sending a message containing the DP M . Results are reported on one scenario. These different tests were done on several scenarios, results were similar to those presented here.

VI. CONCLUSION

In this paper, we presented a distributed fusion algorithm for distributed dynamic map application. This application is based on the cooperative perception concept, which supposes that vehicles cooperate to improve their field of view. Each vehicle is equipped with sensors allowing it to detect objects in its surroundings, and a wifi antenna in order to communicate with other vehicles. The distributed fusion algorithm allows constructing a dynamic map of the environment, involving objects in the sensor's field of view, as well as the ones sent by other vehicles. The distributed fusion combines the confidence in the existence of objects due to appropriated operators, according to the source of the data. This generates a distributed dynamic map offering an increased perception of the environment. The implementation of such application needs temporal and spatial alignment of the exchanged data, as well as matching of objects by the association algorithm developed in [START_REF] Zoghby | Optimal object association from pairwise evidential mass functions[END_REF]. The distributed dynamic map was validated by simulation implying many vehicles. Future works are concentrated on implementing this approach on the airplug software distribution [1] and validating it on real experimentations data. Results will be reported in future publications.

Algorithm 2 : 4 For each associated objects Os and Or 5 ms 6 For each Or not associated 7 Or ← α Or 8 For each Os not associated 9 DDMr

 2456789 FusionCautious 1 Input: DP Ms, DDMr; 2 Output:DDMr; 3 Association( DP Ms, DDMr) ; ← DDMr + Os B. Compute the receiver DP M with its DLM

Algorithm 3 : 5 For 6 Or ← Or l 7 m 8 For Or l not associated 9 DP 10 For each Or not associated 11 if inF ieldof V iew then 12 delete

 356789101112 FusionDempster 1 Input: DLMr,DDMr; 2 Output:DP Mr ; 3 DP Mr ← DDMr 4 Association(DDMr, DLMr) ; each associated objects Or and Or l Mr ← DP Mr + Or l

Fig. 2 :

 2 Fig. 2: Scenario at 2.7s: first column is DLM for each vehicle, the second is DP M .

Fig. 3 :

 3 Fig. 3: The existence mass of V 3 , m(V 3 ), detected by the others vehicles.

TABLE I :

 I Precision and recall in the case of defective sensor Case of a defective wifi antenna: This example illustrates the case where V 2 has a problem with its wifi antenna. It does not receive messages sent by other vehicles for 7.5s. This problem decreases the good detection of V 2 (V P ) and increases the rate of non detection (F P ). TableIIshows the influence of the problem on the recall of V 2 .

		V 0		V 1		V 2
		Prec	Rec	Prec	Rec	Prec	Rec
	DLM	1	0.53	1	0.34	1	0.34
	DPM	1	0.79	1	0.81	0.99	0.81

TABLE II :

 II Precision and recall in the case of a defective wifi antenna

		V 0		V 1		V 2
		Prec	Rec	Prec	Rec	Prec	Rec
	DLM	1	0.59	1	0.34	1	0.34
	DPM	0.93	0.73	0.97	0.77	0.97	0.57

TABLE III :

 III Precision and recall in the case of changing the communication range

			V 0		V 1		V 2
			Prec	Rec	Prec	Rec	Prec	Rec
		range	1	0.59	1	0.34	1	0.34
	DLM	range 1	1	0.59	1	0.34	1	0.4
	DPM	range 2	0.98	0.83	1	0.64	0.98	0.47
		range 3	0.94	0.79	0.98	0.81	0.99	0.83

TABLE IV :

 IV Precision and recall for sending messages containing local map and distributed map

			V 0		V 1		V 2
			Prec	Rec	Prec	Rec	Prec	Rec
	DLM		1	0.59	1	0.34	1	0.34
	DPM	Msg DLM Msg DPM	0.98 0.98	0.63 0.83	0.99 1	0.58 0.64	0.98 0.98	0.45 0.47

ACKNOWLEDGMENT

This work was carried out in the framework of the Labex MS2T, which was funded by the French Government, through the program "Investments for the future" managed by the National Agency for Research (Reference ANR-11-IDEX-0004-02).


Delphine Battistelli

Thierry Charnois

Jean-Luc Minel

Charles

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