Maud Delattre
email: maud.delattre@math.u-psud.fr

Valentine Genon-Catalot
email: valentine.genon-catalot@parisdescartes.fr

Adeline Samson
email: adeline.samson@parisdescartes.fr

Maximum likelihood estimation for stochastic differential equations with random effects Running headline: Mixed stochastic differential equations

Keywords: Asymptotic normality, consistency, maximum likelihood estimator, mixed-effects models, stochastic differential equations

published or not. The documents may come

Maximum likelihood estimation for stochastic differential equations with random effects

Introduction

Statistical analysis of data collected over time on a series of subjects requires to account for both the intra-individual variability, i.e. the variability occurring within the dynamics of each individual over time, and the variability existing between subjects. Modeling of such data goes through mixed-effects models which are very popular in the biomedical field [START_REF] Davidian | Nonlinear models to repeated measurement data[END_REF][START_REF] Pinheiro | Mixed-effect models in S and Splus[END_REF]. In mixed-effects stochastic differential equations (SDEs), the model for each individual set of data is given by a SDE, thus modeling the intraindividual variability in the data, and the parameters of each individual SDE are random variables, thus handling the variability between subjects. A major area of application for mixed effects SDEs is in pharmacokinetic/pharmacodynamic modeling, where they have been introduced as an alternative to the classical ODE-based models [START_REF] Ditlevsen | Stochastic vs. deterministic uptake of dodecanedioic acid by isolated rat livers[END_REF][START_REF] Overgaard | Non-linear mixed-effects models with stochastic differential equations: Implementation of an estimation algorithm[END_REF][START_REF] Donnet | Parametric inference for mixed models defined by stochastic differential equations[END_REF]. SDEs with random effects have also been proposed for neuronal data [START_REF] Picchini | Stochastic differential mixed-effects models[END_REF]. Maximum likelihood estimation (MLE) of the parameters of the random effects, also called population parameters, is generally not straightforward as the likelihood function can rarely be expressed in a closed-form. Approximations of the likelihood have been proposed, based on linearization [START_REF] Beal | Estimating population kinetics[END_REF] or Laplace's approximation [START_REF] Wolfinger | Laplace's approximation for nonlinear mixed models[END_REF]. Alternative methods have also been developed such as the SAEM algorithm [START_REF] Kuhn | Coupling an approximation version of em with an mcmc procedure[END_REF]. Maximum likelihood estimation in SDEs with random effects has been tackled in a few papers. [START_REF] Ditlevsen | Stochastic vs. deterministic uptake of dodecanedioic acid by isolated rat livers[END_REF] show that in the specific case of a mixed-effects Brownian motion with drift, the likelihood function can be explicitly derived, leading to explicit parameters estimators. For general mixed SDEs, approximations of the likelihood have been proposed [START_REF] Picchini | Stochastic differential mixed-effects models[END_REF][START_REF] Picchini | Practicle estimation of high dimensional stochastic differential mixed-effects models[END_REF]. For theoretical properties of the MLE in the context of mixed effects models, the main contribution to our knowledge is due to [START_REF] Nie | Strong consistency of the mle in nonlinear mixedeffects models with large cluster size[END_REF][START_REF] Nie | Strong consistency of the maximum likelihood estimator in generalized linear and nonlinear mixed-effects models[END_REF][START_REF] Nie | Convergence rate of the mle in generalized linear and nonlinear mixed-effects models: Theory and applications[END_REF] and covers the asymptotic properties of the MLE for the population parameters under several asymptotic frameworks, depending on whether the number of subjects and/or the number of observations per subject goes to infinity. Nie's results are nevertheless based on a series of technical assumptions, which may be uneasy to check. In the present work, we focus on mixed-effects SDEs with drift term depending on random effects and diffusion term without random effects. More precisely, we consider N real valued stochastic processes (X i (t), t ≥ 0), i = 1, . . . , N , with dynamics ruled by the following SDEs:

d X i (t) = b(X i (t), φ i )dt + σ(X i (t)) dW i (t), X i (0) = x i , i = 1, . . . , N, (1)
where (W 1 , . . . , W N ) are N independent Wiener processes, φ 1 , . . . , φ N are N i.i.d. R d -valued random variables, (φ 1 , . . . , φ N ) and (W 1 , . . . , W N ) are independent and x i , i = 1, . . . , N are known real values. The diffusion coefficient σ : R → R is a known real-valued function. The drift function b(x, ϕ) is a known function defined on R × R d and real-valued. Each process (X i (t)) represents an individual and the random vector φ i represents the random effect of individual i. We assume that the random variables φ 1 , . . . , φ N have a common distribution g(ϕ, θ)dν(ϕ) on R d , where θ is an unknown parameter belonging to a set Θ ⊂ R p and, for all θ, g(ϕ, θ) is a density w.r.t. a dominating measure ν on R d . Below, we denote by θ 0 the true value of the parameter. The process

(X i (t)
) is continuously observed on a time interval [0, T i ] with T i > 0 given. Our aim is to estimate the parameters θ of the density of the random effects from the observations {X i (t), 0 ≤ t ≤ T i , i = 1, . . . , N }. We introduce assumptions ensuring that the models (1) are well-defined together with the exact likelihood function. Then, we focus on the special case of one-dimensional linear Gaussian random effects, i.e. b(x, φ i ) = φ i b(x), where b is a known real function and φ i is Gaussian. It turns out in this case that the likelihood has a simple and explicit expression depending on θ and the sufficient statistics:

U i = Ti 0 b(X i (s)) σ 2 (X i (s)) dX i (s), V i = Ti 0 b 2 (X i (s)) σ 2 (X i (s)) ds, i = 1, . . . , N.
For the asymptotic study, the main difficulties are encountered to obtain specific moment properties of the random variables (U i , V i ), and to prove identifiability.

We prove the consistency and the asymptotic normality of the exact MLE as N tends to infinity and give the expression of the Fisher information matrix. The results are extended to Gaussian multidimensional linear random effects. The present likelihood theory is derived from continuous observations of the X ′ i s. In practice, one rather disposes of discrete observations on the time interval [0, T i ]. Thus we suggest to discretize the r.v.'s U i , V i in the expression of estimators and we show that under conditions on the discretization step, and thus on the number of observations per subject, the asymptotic properties of the estimates based on continuous observations are preserved. Our simulations are presented within the framework of discretely observed stochastic processes.

The paper is organized as follows. Section 2 introduces the notations and assumptions. In section 3, we make the likelihood function explicit. In sections 4 and 5, we show that the strong consistency and the asymptotic normality of the MLE when the model includes a Gaussian one-dimensional and a Gaussian multi-dimensional random effect respectively. The impact of discretization on the estimators is detailed in section 6. A simulation study is presented in section 7. Concluding remarks are given in section 8. Proofs are gathered in appendix.

Model, assumptions and notations

Consider N real valued stochastic processes (X i (t), t ≥ 0), i = 1, . . . , N , with dynamics ruled by (1). The processes (W 1 , . . . , W N ) and the r.v.'s φ 1 , . . . , φ N are defined on a common probability space (Ω, F, P). We introduce assumptions ensuring that the processes (1) are well defined and allowing to compute the exact likelihood of our observations. Consider the filtration (F t , t ≥ 0) defined by

F t = σ(φ i , W i (s), s ≤ t, i = 1, . . . , N ). As F t = σ(W i (s), s ≤ t) ∨ F i t , with F i t = σ(φ i , φ j , W j (s), s ≤ t, j = i) independent of W i , each process W i is a (F t , t ≥ 0)-Brownian motion. Moreover, the random variables φ i are F 0 - measurable. (H1) (i) The function (x, ϕ) → b(x, ϕ) is C 1 on R × R d ,
and such that:

∃K > 0, ∀(x, ϕ) ∈ R × R d , b 2 (x, ϕ) ≤ K(1 + x 2 + |ϕ| 2 ), (ii) The function σ(.) is C 1 on R and ∀x ∈ R, σ 2 (x) ≤ K(1 + x 2 ).
Under (H1), for all ϕ, the stochastic differential equation

d X ϕ i (t) = b(X ϕ i (t), ϕ)dt + σ(X ϕ i (t)) dW i (t), X ϕ i (0) = x i , (2)
admits a unique strong solution process (X ϕ i (t), t ≥ 0) adapted to the filtration (F t , t ≥ 0). Let C(R + , R) be the space of continuous functions on R + , endowed with the Borel σ-field associated with the topology of uniform convergence on compact sets. The distribution of X ϕ i (.) is uniquely defined on this space. Moreover, as x i is deterministic, for all integer k, all ϕ and all t ≥ 0,

sup s≤t E[X ϕ i (s)] 2k < +∞. (3)
For the observed processes, we have the following result.

Proposition 1. Under (H1), for i = 1, . . . , N , equation (1) admits a unique solution process (X i (t), t ≥ 0), adapted to the filtration (F t , t ≥ 0). Given that φ i = ϕ, the conditional distribution of (X i (t), t ≥ 0) is identical to the distribution of the process (X ϕ i (t), t ≥ 0). The processes

(X i (t), t ≥ 0), i = 1, . . . , N are independent. If for k ≥ 1, E|φ i | 2k < ∞, then for all T > 0, sup t∈[0,T ] E[X i (t)] 2k < ∞.

Likelihood

We introduce the canonical model associated with the observations. Let C Ti denote the space of real continuous functions (x(t), t ∈ [0, T i ]) defined on [0, T i ], endowed with the σ-field C Ti associated with the topology of uniform convergence on [0, T i ]. Under (H1), we introduce the distribution

Q x i ,Ti ϕ on (C Ti , C Ti ) of (X ϕ i (t), t ∈ [0, T i ]) given by (2). On R d × C Ti , let P i θ = g(ϕ, θ)dν(ϕ) ⊗ Q x i ,Ti ϕ
denote the joint distribution of (φ i , X i (.)) and let Q i θ denote the marginal distribution of (X i (t), t ∈ [0, T i ]) on (C Ti , C Ti ). From now on, we denote by (φ i , X i (.)) the canonical process of R d × C Ti . Let us consider the following assumptions.

(H2) For i = 1, . . . , N , and for all ϕ, ϕ ′

Q x i ,Ti ϕ Ti 0 b 2 (X ϕ i (t), ϕ ′ ) σ 2 (X ϕ i (t))
dt < +∞ = 1.

(H3) For f = ∂b ∂ϕj , j = 1, . . . , d, there exist c > 0 and some γ ≥ 0 such that

sup ϕ∈R d |f (x, ϕ)| σ 2 (x) ≤ c(1 + |x| γ ).
Proposition 2. Assume (H1)-(H3) and let ϕ 0 ∈ R d .

• The distributions Q x i ,Ti ϕ are absolutely continuous w.r.t. Q i := Q x i ,Ti ϕ0 with density:

dQ x i ,Ti ϕ dQ i (X i ) = L Ti (X i , ϕ) = e ℓ T i (Xi,ϕ) with ℓ Ti (X i , ϕ) = Ti 0 b(X i (s), ϕ) -b(X i (s), ϕ 0 ) σ 2 (X i (s)) dX i (s)- Ti 0 b 2 (X i (s), ϕ) -b 2 (X i (s), ϕ 0 ) 2σ 2 (X i (s)) ds,
where (X i = X i (s), s ≤ T i ) denotes the canonical process of C Ti given by

(X i (s)(x) = x(s), s ≤ T i ).
• The function ϕ → L Ti (X i , ϕ) admits a continuous version Q i -a.s. and

(X i , ϕ) → L Ti (X i , ϕ) is measurable on (C Ti × R d , C Ti ⊗ B(R d )).
Remark 1. For a given drift function b(x, ϕ), it is often possible to check directly that ϕ → L Ti (X, ϕ) is continuous even if (H3) is not fulfilled. Then, the joint measurability follows.

To simplify notations, we assume that there is one value ϕ 0 such that b(x, ϕ 0 ) ≡ 0. Thus, we can choose the dominating measure

Q i = Q x i ,Ti ϕ0
which is the distribution of (2) with nul drift. Formula L Ti (X i , ϕ) simplifies into:

L Ti (X i , ϕ) = exp Ti 0 b(X i (s), ϕ) σ 2 (X i (s)) dX i (s) - 1 2 Ti 0 b 2 (X i (s), ϕ) σ 2 (X i (s)) ds . ( 4
)
By independence of the individuals,

P θ = ⊗ N i=1 P i θ is the distribution of (φ i , X i (.)), i = 1, . . . , N on the product space N i=1 R d × C Ti and Q θ = ⊗ N i=1 Q i θ is the dis- tribution of the whole sample (X i (t), t ∈ [0, T i ], i = 1, . . . , N ) on C = N i=1 C Ti . We now compute the density of Q θ w.r.t. Q = ⊗ N i=1 Q i .
We denote by E θ the expectation w.r.t. P θ .

Proposition 3. Assume (H1)-(H3).

• The probability measure Q i θ admits a density w.r.t. Q i equal to:

dQ i θ dQ i (X i ) = R d L Ti (X i , ϕ)g(ϕ, θ)dν(ϕ) := λ i (X i , θ). • The distribution Q θ on C = N i=1 C Ti admits a density given by dQ θ dQ (X 1 , . . . , X N ) = N i=1 λ i (X i , θ).
• The exact likelihood of the whole sample

(X i (t), t ∈ [0, T i ], i = 1, . . . , N ) is Λ N (θ) = N i=1 λ i (X i , θ). (5)
On this general expression, if we can check Nie (2006)'s assumptions, then weak consistency and asymptotic normality of the MLE will follow. However, these assumptions, even when the random effects are Gaussian, are uneasy.

Gaussian one-dimensional linear random effects

In this section, we consider model (1) with drift b(x, ϕ) = ϕb(x) where ϕ ∈ R, b(.), σ(.) are known functions. In this case, we simplify (H1)-(H2) and assume 6 that b, σ are C 1 and have linear growth, which implies Proposition 1. And, we assume that Ti

0 b 2 (X i (s))/σ 2 (X i (s))ds < ∞, Q x i ,Ti ϕ -a.s. for all ϕ. As ϕ → L Ti (X i , ϕ
) is obviously continuous, (H3) is not required. We also assume that, for i = 1, . . . , N , T i = T, x i = x so that the observed processes (X i (t), t ∈

[0, T ]), i = 1, . . . , N are i.i.d.. Let us introduce:

U i = T 0 b(X i (s)) σ 2 (X i (s)) dX i (s), V i = T 0 b 2 (X i (s)) σ 2 (X i (s)) ds, (6)
which are well defined under (H1)-(H2). Hence,

λ i (X i , θ) = R g(ϕ, θ) exp (ϕU i - ϕ 2 2 V i ) dν(ϕ). (7)

Exact likelihood

We propose here to model the random effects distribution by a Gaussian distribution N (µ, ω 2 ), and set θ = (µ, ω 2 ) ∈ R×(0, +∞) for the unknown parameters to be estimated. This choice leads to an explicit exact likelihood.

Proposition 4. Assume that g(ϕ, θ)dν(ϕ) = N (µ, ω 2 ). Then, λ i (X i , θ) = 1 (1 + ω 2 V i ) 1/2 exp - V i 2(1 + ω 2 V i ) µ - U i V i 2 exp U 2 i 2V i .
The conditional distribution, under P i θ , of φ i given X i is the distribution

N µ + ω 2 U i 1 + ω 2 V i , ω 2 1 + ω 2 V i .
Therefore, the logarithm of the likelihood function ( 5) is explicitely given by

L N (θ) = - 1 2 N i=1 log(1 + ω 2 V i ) - 1 2 N i=1 V i 1 + ω 2 V i µ - U i V i 2 + N i=1 U 2 i 2V i . (8)
The derivatives of the log-likelihood (8) are

∂ ∂µ L N (θ) = N i=1 U i 1 + ω 2 V i -µ V i 1 + ω 2 V i , ∂ ∂ω 2 L N (θ) = 1 2 N i=1 U i 1 + ω 2 V i -µ V i 1 + ω 2 V i 2 - V i 1 + ω 2 V i .
When ω 2 0 is known, we obtain the explicit estimator for µ 0 :

µ N = N i=1 Ui 1+ω 2 0 Vi N i=1 Vi 1+ω 2 0 Vi . ( 9
)
When both parameters are unknown, the maximum likelihood estimators of θ 0 = (µ 0 , ω 2 0 ) are given by the system:

µ N = N i=1 V i 1 + ω 2 N V i -1 N i=1 U i 1 + ω 2 N V i , N i=1 µ N - U i V i 2 V 2 i (1 + ω 2 N V i ) 2 = N i=1 V i 1 + ω 2 N V i .
Remark 2. Note that, when the effect φ i is non random and φ i ≡ µ 0 , the estimator of µ 0 is standardly given by:

µ N = N i=1 U i N i=1 V i , (10)
which corresponds to ω 2 0 = 0 in µ N .

Preliminary moments properties

For studying the maximum likelihood estimators of θ = (µ, ω 2 ), we need investigate properties of the following random variables:

γ i (θ) = U i -µV i 1 + ω 2 V i , I i (ω 2 ) = V i 1 + ω 2 V i . (11)
Indeed under Q θ , (γ i (θ), I i (ω 2 )) i=1,...,N are i.i.d. and the score function is

∂ ∂µ L N (θ) = N i=1 γ i (θ), ∂ ∂ω 2 L N (θ) = 1 2 N i=1 (γ 2 i (θ) -I i (ω 2 )). (12)
Evidently, 0 < I i (ω 2 ) ≤ 1/ω 2 is bounded. By the following lemma, which is crucial for the statistical study, we prove that γ i (θ) admits a finite Laplace transform, hence moments of any order.

Lemma 1. For all θ = (µ, ω 2 ) ∈ R × (0, +∞), and all u ∈ R,

E θ (exp (u U 1 1 + ω 2 V 1
)) < +∞.

We can now compute some useful moments of functions of γ 1 (θ), I 1 (ω 2 ).

Proposition 5. For all θ ∈ R × (0, +∞), the following relations hold:

E θ (γ 1 (θ)) = 0, E θ (γ 2 1 (θ)) = E θ (I 1 (ω 2 )), E θ (γ 3 1 (θ)) = 3E θ (γ 1 (θ) I 1 (ω 2 )), E θ γ 2 1 (θ) -I 1 (ω 2 ) 2 = 4E θ (γ 2 1 (θ)I 1 (ω 2 )) -2E θ (I 2 1 (ω 2 )).

Convergence in distribution of the normalized score function

Based on lemma 1 and Proposition 5, we can state:

Proposition 6. For all θ, under Q θ , as N tends to infinity, the random vector

1 √ N ∂ ∂µ L N (θ) ∂ ∂ω 2 L N (θ) = 1 √ N N i=1 γ i (θ) 1 2 N i=1 (γ 2 i (θ) -I i (ω 2 ))
converges in distribution to N 2 (0, I(θ)) and the matrix

- 1 N ∂ 2 ∂µ 2 L N (θ) ∂ 2 ∂µ∂ω 2 L N (θ) ∂ 2 ∂µ∂ω 2 L N (θ) ∂ 2 ∂ω 2 ∂ω 2 L N (θ)
converges in probability to I(θ) where

I(θ) = E θ (I 1 (ω 2 )) E θ (γ 1 (θ)I 1 (ω 2 )) E θ (γ 1 (θ)I 1 (ω 2 )) E θ (γ 2 1 (θ)I 1 (ω 2 )) -1 2 E θ (I 2 1 (ω 2 )) (13)
is the covariance matrix of the vector

γ 1 (θ) 1 2 (γ 2 1 (θ) -I 1 (ω 2 ))
.

The following corollary holds immediately.

Corollary 1. When ω 2 = ω 2 0 is known, the explicit estimator µ N (9) is consis- tent and √ N ( µ N -µ 0 ) converges in distribution under Q µ0 to N (0, 1/E µ0 (V 1 /(1+ ω 2 0 V 1 ))) where 1/E µ0 (V 1 /(1 + ω 2 0 V 1 )) ≥ ω 2 0 .
If φ 1 , . . . , φ N were observed, the MLE of µ 0 would be φ = 1

N (φ 1 + . . . + φ N ) which satisfies √ N ( φ -µ 0 ) ∼ N (0, ω 2 0 ).
As φ 1 , . . . , φ N are not observed, we obtain that the MLE µ N has a larger asymptotic variance.

Consistency and asymptotic normality

When both parameters µ 0 , ω 2 0 are unknown, we need to introduce additional assumptions to prove identifiability and consistency. Recall that Q is the distribution on C such that the canonical processes (X i (s), s ≤ T, i = 1, . . . , N ) are i.i.d. and X i satisfies the SDE with nul drift:

dX i (t) = σ(X i (t))dW i (t), X i (0) = x.
We assume that (H4) The function b(.)/σ(.) is not constant. Under Q, the random variable (H7) The matrix I(θ 0 ) is invertible (see ( 13)).

(U 1 , V 1 ) admits a density f (u, v) w.
Under smoothness assumptions on functions b, σ, assumption (H4) will be fulfilled by application of Malliavin calculus tools † . The case where b(.)/σ(.) is constant is rather simple and is treated separately in Section 7. Assumptions (H5)-(H7) are classical. We first state an identifiability result.

Proposition 7. Set K(Q 1 θ0 , Q 1 θ ) the Kullback information of Q 1 θ0 w.r.t. Q 1 θ . (i) Under (H1)-(H2) and (H4), Q 1 θ = Q 1 θ0 implies that θ = θ 0 . Hence, θ → K(Q 1 θ0 , Q 1 θ ) admits a unique minimum at θ = θ 0 . (ii) Under (H1)-(H2), the function θ → K(Q 1 θ0 , Q 1 θ ) is continuous on R × (0, +, ∞).
We are now able to prove the consistency and asymptotic normality of θN . Proposition 8.

1. Assume (H1)-( H2) and (H4)-(H5). Let θN be a maximum likelihood estimator defined as any solution of L N ( θN ) = sup θ∈Θ L N (θ).

Under Q θ0 , θN converges in probability to θ 0 . † If σ and f = b/σ are C ∞ , if their derivatives of any order greater than 1 are bounded, if σ is bounded below, and if the Lebesgue measure of the set of values x such that f (x)f ′ (x) = 0 is zero, then assumption (H4) holds.

Assume (H1)-(H2) and (H4)-(H7). The maximum likelihood estimator

satisfies, as N tends to infinity,

√ N ( θN -θ 0 ) → D N 2 (0, I -1 (θ 0 )).
Note that the consistency obtained here is a strong consistency in the sense that any solution of the likelihood equation is consistent.

5 Gaussian multidimensional linear random effects

In this section, we extend the previous results to multidimensional linear random effects. Let

φ i = (φ 1 i , . . . , φ d i ) ′ be a d-dimensional random vector and b(x) = (b 1 (x), . . . , b d (x)) ′ be a function R → R d . Consider the SDE d X i (t) = φ ′ i b(X i (t))dt + σ(X i (t)) dW i (t), X i (0) = x. (14)
We assume that b 1 (x), . .

θ = (µ, Ω) ∈ R d × S d (R).
To compute the likelihood, we introduce the random vectors

U i = T 0 b(X i (s)) σ 2 (X i (s)) dX i (s),
and the d × d random matrices

V i = T 0 b(X i (s))b ′ (X i (s)) σ 2 (X i (s)) ds.
The following assumption is now required.

(H8) For i = 1, . . . , N the matrix V i is positive definite Q i -a.s. and Q i θ -a.s. for all θ.

If the functions (b j /σ 2 ) are not linearly independent, (H8) is not true. Thus, (H8) can be interpreted as ensuring a well-defined dimension of the vector φ i .

We deduce the invertibility of matrices involved in the likelihood computation.

Lemma 2. Under (H8), the matrices

V i +Ω -1 , I d +V i Ω, I d +ΩV i are invertible Q i -a.s. and Q i θ -a.s. for all θ.
Then we can compute the likelihood.

Proposition 9. Under (H8), set R -1 i = (I d + V i Ω) -1 V i , we have λ i (X i , θ) = 1 det(I d + V i Ω) exp - 1 2 (µ -V -1 i U i ) ′ R -1 i (µ -V -1 i U i ) exp 1 2 U ′ i V -1 i U i .
The conditional distribution of φ i given X i is the Gaussian distribution

N d (I d + ΩV i ) -1 µ + (Ω -1 + V i ) -1 V i , (I d + ΩV i ) -1 Ω The likelihood is Λ N (θ) = N i=1 λ i (X i , θ).
The score function (respectively a d-vector and a d × d matrix) is given by:

∂ ∂µ L N (θ) = N i=1 γ i (θ), ∂ ∂Ω L N (θ) = 1 2 N i=1 (γ i (θ)γ ′ i (θ) -I i (Ω))
where θ = (µ, Ω) and :

γ i (θ) = (I d + ΩV i ) -1 (U i -V i µ), I i (Ω) = (I d + ΩV i ) -1 V i . (15)
When Ω 0 is known, the estimator for µ 0 is explicit.

Lemma 1 and Propositions 5 and 6 can be readily extended to the multidimensional case.

Proposition 10.

1. For all θ = (µ, Ω) ∈ R × (0, +∞), and all u ∈ R,

E θ (exp (u ′ (I d + ΩV i ) -1 U 1 )) < +∞.
2. For all θ ∈ R × (0, +∞), the following relations hold:

E θ (γ 1 (θ)γ ′ 1 (θ)) = E θ (I 1 (Ω)), E θ (γ 1 (θ)γ ′ 1 (θ)γ 1 (θ)) = 3E θ (I 1 (Ω) γ 1 (θ)), E θ (γ 1 (θ)) = 0, E θ (γ 1 (θ)γ ′ 1 (θ) -I 1 (Ω)) 2 = 4E θ (I 1 (Ω)γ 1 (θ)γ ′ 1 (θ))-2E θ (I 2 1 (Ω)).
3. For all θ, under Q θ , as N tends to infinity, the random vector

1 √ N ∂ ∂µ L N (θ) ∂ ∂Ω L N (θ) = 1 √ N N i=1 γ i (θ) 1 2 N i=1 (γ i (θ)γ ′ i (θ) -I i (Ω))
converges in distribution to N (0, I(θ)) where I(θ) is the covariance matrix of the vector

γ 1 (θ) 1 2 (γ 1 (θ)γ ′ 1 (θ) -I 1 (Ω))
which is also the limit of the observed Fisher information matrix.

The study of θ N = ( µ N , Ω N ) can be done as above.

Discrete data

In this section, we briefly discuss the case of discrete data. Let us assume that we observe synchronously the processes X i (t) at times

t n k = t k = k T n , k = 0, 1, . . . , n. To build estimators θ (n)
N based on these data, we simply replace the r.v.'s U i , V i , i = 1, . . . , N by their discretized versions:

U n i = n-1 k=0 b(X i (t k )) σ 2 (X i (t k )) (X i (t k+1 ) -X i (t k )), (16)
V n i = n-1 k=0 b 2 (X i (t k )) σ 2 (X i (t k )) (t k+1 -t k ). (17)
Looking at the expressions of (8) and its derivatives w.r.t. θ, it is enough to study the differences U i -U n i , V i -V n i . We can prove Lemma 3. Assume that b/σ is bounded and Lipschitz, σ(.) ≥ ǫ > 0, b and σ Lipschitz, then for all p ≥ 1 and all i = 1, . . . , N , there exists a constant C such that

E θ0 (|V i -V n i | p + |U i -U n i | p ) ≤ C n p/2 . We deduce Proposition 11. If n → +∞, then θ N -θ (n) N = o P θ 0 (1). If n = n(N ) → +∞ in such a way that n N → +∞, then √ N ( θ N -θ (n) N ) = o P θ 0 (1).

Simulation study

Several models are simulated. For each SDE model, 100 datasets are generated with N subjects on the same time interval [0, T ] and three experimental designs:

(N = 20, T = 5), (N = 50, T = 5) and (N = 50, T = 10). The empirical mean and variance of the MLE are computed from the 100 datasets. When possible,

N -1 I -1 N (θ 0
) is also computed and compared with the empirical variance.

When b(x) = c σ(x)

Let us consider the case where b(x) = c σ(x), with c = 0 known. Then we have

V i = c 2 T, U i = c T 0 dX i (s) σ(X i (s)) . ( 18
)
The estimators of µ 0 , ω 2 0 are simple and explicit:

µ N = 1 c 2 T N N i=1 U i = 1 c 2 T ŪN , ω 2 N = 1 (c 2 T ) 2 1 N N i=1 (U i -ŪN ) 2 -c 2 T .
Using that U i = c 2 T φ i + cW i (T ), an elementary study shows that µ N and ω 2 N are strongly consistent, that

√ N ( µ N -µ 0 ) has distribution N (0, ω 2 0 + 1/(c 2 T )) and that √ N ( ω 2 N -ω 2 0 ) converges in distribution to N (0, 2(ω 2 0 +1/(c 2 T )) 2
). The asymptotic variances of µ N and ω 2 N are increased in comparison with the case of non random effects. We stress the fact that, whatever the drift funtion b(.), when b(.)/σ(.) is constant, the estimators have the same distribution. Example 1. Consider a mixed-effects Brownian motion with drift

dX i (t) = φ i dt + σ dW i (t), X i (0) = 0, with φ i ∼ N (µ, ω 2
). This model is considered by [START_REF] Ditlevsen | Stochastic vs. deterministic uptake of dodecanedioic acid by isolated rat livers[END_REF] and the estimators are the same. We use two sets of population parameters: (µ = -1, ω 2 = 1) and (µ = 5, ω 2 = 1). All simulations are performed with a discretization step-size δ = 0.001 on [0, T ] and σ = 1 (σ known). Results, presented in Table 1, are satisfactory overall. Increasing N improves the accuracy of both estimates µ N and ω 2 N . For T = 5, both estimates are less biased when the number of subjects is 50 instead of 20. The variance of the estimates is also decreased with larger values of N . In general the empirical variances coincide with the values of the asymptotic variances. Apparently, increasing T does not have any significant impact on the properties of the estimates. Additional simulations with other values for µ 0 and ω 2 0 have basically shown that the properties of the estimates degrade when ω 2 0 takes bigger values.

[Table 1 about here.]

We have also considered simulations of the mixed-effects geometric Brownian motion where b(x) = x, σ(x) = σ x and the model where b

(x) = √ 1 + x 2 , σ(x) = σ √ 1 + x 2 .
On our simulated data, the true parameter values were correctly estimated.

General case

Example 2. Consider an Ornstein-Uhlenbeck process with one random effect

dX i (t) = φ i X i (t)dt + σ dW i (t), X i (0) = 0, with φ i ∼ N (µ, ω 2 ). We separate three situations: i) ω 2 is known (θ = µ), ii) µ is known (θ = ω 2 ), iii) both parameters are unknown (θ = (µ, ω 2 )). When ω 2
0 is known, we use the explicit expression of µ N . Otherwise, numerical optimization procedures are required for maximizing the log-likelihood with respect to µ and ω 2 . In situations i) and ii), the asymptotic variance of the estimate has an explicit expression and is computed. Each individual diffusion is simulated with a discretization step-size δ = 0.001 on [0, T ] and σ = 1. Several sets of parameter values are used: (µ = -5, ω 2 = 1) and (µ = 10, ω 2 = 1). Table 2 displays the results of the three inferences i), ii) and iii). Results highlight the accuracy of the estimates of both parameters µ and ω 2 whatever the design and the parameter values. In the three considered inference situations, increasing N leads to smaller bias of the parameter estimates and smaller variances. Moreover, we notice great similarities between N -1 I -1 N (θ 0 ) and the empirical variance of θ N , especially when N is large. Finally, we don't observe any significant impact of T neither on the bias nor on the variance of the parameter estimates.

[Table 2 about here.] Example 3. Consider an Ornstein-Uhlenbeck process with two random effects

dX i (t) = (-φ 1 i X i (t) + φ 2 i )dt + σ dW i (t), X i (0) = 0, with φ i = (φ 1 i , φ 2 i ) ′ ∼ N 2 (µ, Ω), µ = (µ 1 , µ 2 ) ′ and a diagonal matrix Ω with components (ω 2 1 , ω 2
2 ). For this model, assumption (H8) is satisfied. Indeed

V i = 1 σ 2 T 0 X 2 i (s)ds T 0 X i (s)ds T 0 X i (s)ds T .
Using the equality case in the Cauchy-Schwarz inequality, we see that det(

V i ) = 0 if and only if X i (t) ≡ cste on [0, T ] which is impossible. The estimation of θ = (µ 1 , µ 2 , ω 2 1 , ω 2
2 ) is obtained by optimizing numerically the log-likelihood. Each individual diffusion is simulated with a discretization step-size δ = 0.001 on [0, T ] and σ = 1. Several sets of parameter values are used: 3 displays the results of estimation. Parameters are well estimated although µ 2 has a larger bias when ω 2 2 is greater. Biases decrease when N increases. Again, the influence of T is small.

(µ 1 = 0.1, µ 2 = 1, ω 2 1 = 0.01, ω 2 2 = 1) and (µ 1 = 0.1, µ 2 = 1, ω 2 1 = 0.001, ω 2 2 = 1). Table
[Table 3

about here.]

Example 4. Consider the process with single random effect

dX i (t) = φ i X i (t)dt + σ 1 + X i (t) 2 dW i (t), X i (0) = 0,
with φ i ∼ N (µ, ω 2 ). We obtain the estimate for θ = (µ, ω 2 ) by numerical optimization of the log-likelihood with respect to µ and ω 2 . We simulate N individual diffusions with a discretization step-size δ = 0.001 on [0, T ], and several sets of parameter values are used: (µ = -1, ω 2 = 1) and (µ = 5, ω 2 = 1).

Table 4 displays the results of estimation. The results are satisfactory overall, although parameter ω 2 is estimated with larger bias than parameter µ. Bias and empirical variances of the estimates decrease when N becomes larger. T = 10 leads to better results (smaller bias) for the estimation of ω 2 than T = 5.

[Table 4 about here.]

Concluding remarks

In this paper, we have studied maximum likelihood estimation for i.i.d. observations of stochastic differential equations including a random effect in the drift term. When the drift term depends linearly on the random effect, we prove that the likelihood is given by a closed-form formula and that the exact MLE is strongly consistent and asymptotically Gaussian as the number of observed processes tends to infinity. For the clarity of exposure, we have considered only one-dimensional SDEs, but the theory can be done in the same way for multidimensional SDEs. For a drift term depending linearly on the random effects, the likelihood is still explicit although its formulae may be much more cumbersome. One could also include non random effects in the drift without much changes. the processes (X i (.)) are independent. As (x i , φ i ) is the initial condition, the moment result follows.

Proof of Proposition 2. Under (H1)-(H2), the first part is classical (see e.g. [START_REF] Lipster | Statistics of random processes I : general theory[END_REF].

To prove the continuity in ϕ, two kinds of terms are to be studied. The first is, for a given X i , the ordinary integral

ϕ → Ti 0 b 2 (X i (s), ϕ) σ 2 (X i (s)) ds. (19)
Using (H1)-(H3) and the continuity theorem for ordinary integrals, we obtain easily the continuity of ( 19).

Second, the stochastic integral

ϕ → Ti 0 b(X i (s), ϕ) σ 2 (X i (s)) dX i (s) := I 1 (ϕ) + I 2 (ϕ), (20)
with

I 1 (ϕ) = Ti 0 b(X i (s), ϕ) σ 2 (X i (s)) b(X i (s), ϕ 0 )ds, I 2 (ϕ) = Ti 0 b(X i (s), ϕ) σ 2 (X i (s)) (dX i (s) -b(X i (s), ϕ 0 )ds) .
The function ϕ → I 1 (ϕ) is studied using (H1)-(H3) and the continuity theorem for ordinary integrals again. For I 2 (ϕ), using the Burkholder-Davis-Gundy inequality, we get, using (H1)-( H3):

E Q i (I 2 (ϕ) -I 2 (ϕ ′ )) 2k ≤ C k E Q i Ti 0 (b(X i (s), ϕ) -b(X i (s), ϕ ′ )) 2 σ 2 (X i (s)) ds k ≤ C k |ϕ -ϕ ′ | 2k E Q i Ti 0 c 2 K(1 + |X i (s)| 2γ )(1 + |X i (s)|) 2 ds k ≤ C(k, γ)|ϕ -ϕ ′ | 2k Ti 0 (1 + E Q i (|X i (s)| 2(γ+1) )ds.
Using (3) and choosing 2k > d, the Kolmogorov criterion (see e.g. [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF] yields that ϕ → I 2 (ϕ) admits a continuous version Q i -a.s..

As X i → L Ti (X i , ϕ) is measurable for all ϕ and ϕ → L Ti (X i , ϕ) is continuous for all X i , the joint measurability can be proved as follows. For m ∈ N and

k = (k 1 , . . . , k d ) ∈ Z d , set B k,m = d i=1 [k i /2 m , (k i +
L m (X i , ϕ) = k∈Z d L Ti (X i , ϕ k,m )1 B k,m (ϕ).
Thus, L m (X i , ϕ) is jointly measurable. As L Ti (X i , ϕ) is continuous w.r.t. ϕ, L m (X i , ϕ) → m→+∞ L Ti (X i , ϕ). Hence, the result.

Proof of Proposition 3. For H a positive measurable function on C Ti , we have:

E Q i θ (H(X i )) = E P i θ (H(X i )) = E P i θ [E P i θ (H(X i )|φ i )]
. By Propositions 1 and 2, as L Ti (X i , ϕ) is the density of Q xi,Ti ϕ w.r.t. Q i , we get:

E P i θ (H(X i )|φ i = ϕ) = E Q x i ,T i ϕ (H(X i )) = E Q i (H(X i )L Ti (X i , ϕ)) .
Using the joint measurability of L Ti (X i , ϕ) w.r.t. (X i , ϕ), the Fubini theorem yields:

E Q i θ (H(X i )) = R d g(ϕ, θ)dν(ϕ)E Q i (H(X i )L Ti (X i , ϕ)) = E Q i H(X i ) R d g(ϕ, θ)L Ti (X i , ϕ)dν(ϕ).
Thus, the density of Q i θ w.r.t. Q i is computed as:

dQ i θ dQ i (X i ) = R d L Ti (X i , ϕ)g(ϕ, θ)dν(ϕ) := λ i (X i , θ).
The formula for the exact likelihood follows.

Proof of Proposition 4. We need compute first the joint density of (φ i , X i ) w.r.t. dϕ ⊗ dQ i :

exp ϕU i - ϕ 2 2 V i × 1 ω √ 2π exp - 1 2ω 2 (ϕ -µ) 2 .
Developping the exponent yields:

E i = - 1 2 ϕ 2 (V i + ω -2 ) -2ϕ(U i + ω -2 µ) - 1 2 ω -2 µ 2 . ( 21
)
Let us set:

m i = U i + ω -2 µ V i + ω -2 = µ + ω 2 U i 1 + ω 2 V i , σ 2 i = (V i + ω -2 ) -1 = ω 2 1 + ω 2 V i . (22)
Thus, the conditional distribution of φ i given X i is the Gaussian law N (m i , σ 2 i ). After some elementary algebra, we get:

E i = - 1 2σ 2 i (ϕ -m i ) 2 - 1 2 V i (1 + ω 2 V i ) -1 (µ -V -1 i U i ) 2 + 1 2 V -1 i U 2 i .
Thus, the result.

Proof of Lemma 1. For the proof, we set γ 1 (θ) = γ 1 and I 1 (ω 2 ) = I 1 (see ( 11)). Let l(X 1 , θ) = log λ 1 (X 1 , θ) and set θ(u) = (µ + u, ω 2 ). Developping

(U 1 -(µ + u)V 1 ) 2 , we get: l(X 1 , θ(u)) = l(X 1 , θ) + uγ 1 - u 2 2 I 1 .
Here, ∂ ∂µ l(X 1 , θ) = γ 1 and ∂ 2 ∂µ 2 l(X 1 , θ) = -I 1 . Taking exponential yields:

λ 1 (X 1 , θ) exp (uγ 1 ) = λ 1 (X 1 , θ(u)) exp ( u 2 2 I 1 ).
Integrating w.r.t. the dominating measure Q 1 , we obtain, as

I 1 ≤ 1/ω 2 , E θ exp (uγ 1 ) = E θ(u) exp ( u 2 2 I 1 ) ≤ exp ( u 2 2ω 2 ) < +∞. Now, as uµ ≤ (u + µ) 2 /4, E θ (exp (u U 1 1 + ω 2 V 1 )) ≤ E θ exp (uγ 1 ) exp ( (u + µ) 2 4ω 2 ) < +∞.
Proof of Proposition 5. We set γ 1 (θ) = γ 1 and I 1 (ω 2 ) = I 1 . Let θ = (µ, ω 2 ) and τ = (0, ω 2 ) and set

p 1 (θ) = λ 1 (X 1 , θ) λ 1 (X 1 , τ ) = dQ 1 θ dQ 1 τ = exp µ U 1 1 + ω 2 V 1 - µ 2 2 V 1 1 + ω 2 V 1 ,
so that C T p 1 (θ)dQ 1 τ = 1. Provided that we can interchange derivation w.r.t. µ and integration w.r.t. Q 1 τ , we have for j ≥ 1,

C T ∂ j p 1 ∂µ j (θ)dQ 1 τ = 0. ( 23
)
Before justifying the interchange of derivation and integration, let us compute the successive derivatives of p 1 (θ). We have:

∂p 1 ∂µ (θ) = γ 1 p 1 (θ), ∂ 2 p 1 ∂µ 2 (θ) = γ 2 1 -I 1 p 1 (θ), ∂ 3 p 1 ∂µ 3 (θ) = γ 3 1 -3γ 1 I 1 p 1 (θ), ∂ 4 p 1 ∂µ 4 (θ) = γ 4 1 -6γ 2 1 I 1 + 3I 2 1 p 1 (θ).
Therefore, (23) for j = 1, 2, 3, 4 imply the announced moments relations. It remains to justify the interchange of derivation and integration. Let us fix μ and ε > 0. For µ ∈ [με, μ + ε], we have the bound

| ∂p 1 ∂µ (θ)| ≤ | U 1 1 + ω 2 V 1 | + C ω 2 exp ((μ -ε) U 1 1 + ω 2 V 1 ) + exp ((μ + ε) U 1 1 + ω 2 V 1
) ,

where C = |μ + ε| + |μ -ε|. The upper bound is integrable w.r.t. Q 1 τ by Lemma 1 and independent of µ. Therefore, the interchange is justified. We proceed analogously for the other derivatives.

γ i (θ) = U i 1 + ω 2 0 V i 1 + (ω 2 0 -ω 2 )V i 1 + ω 2 V i -µ V i 1 + ω 2 V i . Thus, sup θ∈Θ |γ i (θ)| ≤ | U i 1 + ω 2 0 V i | 2 + ω 2 0 ω 2 + |µ| ω 2 . ( 29
)
Therefore, there is a constant C such that

E θ0 w N (η) ≤ C η E θ0 | U 1 1 + ω 2 0 V 1 | + ( U 1 1 + ω 2 0 V 1 ) 2 .
This leads to the consistency of θN .

For the second point, the proof follows the standard scheme. By the consistency and (H5), Q θ0 ( θN ∈

• Θ) → 1. For the proof, let θN,i , θ 0,i be the components of θN , θ 0 . Set U N (θ) = -(1/N )L N (θ) and denote by U ′ N,i , U ′′ N,ij the derivatives of U N w.r.t. θ i or θ i θ j . The Taylor formula writes:

0 = U ′ N,i ( θN ) = U ′ N,i (θ 0 ) + j=1,2 ( θN,j -θ 0,j )(U ′′ N,ij (θ 0 ) + R N ), with R N = 1 0 U ′′ N,ij (θ 0 + s( θN -θ 0 )) -U ′′ N,ij (θ 0 ) ds.
Using Propositions 8 and 6, it remains to check that R N tends in Q θ0 -probability to 0. For this, we compute the third derivatives of U N using ( 24)-( 25):

1 N ∂ 3 ∂µ 3 L N (θ) = 0, 1 N ∂ 3 ∂ω 2 ∂ω 2 ∂ω 2 L N (θ) = 1 N N i=1 3γ 2 i (θ)I 2 i (ω 2 ) -I 3 i (ω 2 ) , 1 N ∂ 3 ∂ 2 µ∂ω 2 L N (θ) = 1 N N i=1 I 2 i (ω 2 ), 1 N ∂ 3 ∂µ∂ω 2 ∂ω 2 L N (θ) = 2 N N i=1 γ i (θ)I 2 i (ω 2 ).
Using (29), we obtain, for C a constant depending on Θ

|R N | ≤ C | θN -θ 0 | 1 N N i=1 (1 + | U i 1 + ω 2 0 V i | + ( U i 1 + ω 2 0 V i ) 2 .
As we have proved the consistency, R N tends to 0. Hence the result.

Proof of Lemma 2. The matrix V i + Ω -1 is symmetric and satisfies, for all

x ∈ R d , x = 0, x ′ (V i + Ω -1 )x = x ′ V i x + x ′ Ω -1 x > 0 as V i and Ω -1 are positive definite. Thus V i + Ω -1 is positive definite hence invertible. Noting that I d + V i Ω = (Ω -1 + V i )Ω and I d + ΩV i = Ω(Ω -1 + V i ) yields that both matrices are invertible.
Proof of Proposition 9. We compute the joint density of (φ i , X i ) w.r.t. dϕ⊗dQ i :

exp ϕ ′ U i - 1 2 ϕ ′ V i ϕ × exp - 1 2 (ϕ -µ) ′ Ω -1 (ϕ -µ) .
Let

E i = - 1 2 ϕ ′ (V i + Ω -1 )ϕ -2ϕ ′ (U i + Ω -1 µ) - 1 2 µ ′ Ω -1 µ, (30)
and set:

m i = Σ i (U i + Ω -1 µ), Σ 2 i = (V i + Ω -1 ) -1 . (31)
Thus, the conditional distribution of φ i given X i is the Gaussian law N (m i , Σ 2 i ). Hence, we have

E i = - 1 2 (ϕ -m i ) ′ Σ -1 i (ϕ -m i ) - 1 2 (µ -V -1 i U i ) ′ R -1 i (µ -V -1 i U i ) + 1 2 U ′ i V -1 i U i .
Thus, the result.

Proof of Proposition 10. For the first point, we proceed as in Lemma 1. Let

γ 1 (θ) = γ 1 , I 1 (Ω) = I 1 , l(X 1 , θ) = log λ 1 (X 1 , θ) and θ(u) = (µ + u, Ω). Devel- opping ((µ + u) -V -1 1 U 1 ) ′ R -1 1 ((µ + u) -V -1 1 U 1 ) yields: l(X 1 , θ(u)) = l(X 1 , θ) + u ′ γ 1 - 1 2 u ′ I 1 u.
Thus λ 1 (X 1 , θ) exp (u ′ γ 1 ) = λ 1 (X 1 , θ(u)) exp ( 1 2 u ′ I 1 u).

Remark that u ′ I 1 u ≤ u ′ Ω -1 u as I 1 = Ω -1 -(ΩV 1 Ω + Ω) -1 . Integrating w.r.t.

Q 1 , we obtain,

E θ exp (u ′ γ 1 ) = E θ(u) exp ( 1 2 u ′ I 1 u) ≤ exp ( 1 2 u ′ Ω -1 u) < +∞.
Now, we can write, as u ′ I 1 µ = 1 4 ((u + µ) ′ I 1 (u + µ) -(uµ) ′ I 1 (uµ)) ≤ 1 4 (u + µ) ′ I 1 (u + µ):

E θ (exp (u ′ (I d + ΩV i ) -1 U 1 )) ≤ E θ exp (u ′ γ 1 ) exp 1 4 (u + µ) ′ Ω -1 (u + µ) < +∞.
This gives 1. The proof of the second point and third points is analogous to Proposition 5 and Proposition 6.

Proof of Lemma 3. We only treat the case p ≥ 2. Assumptions imply that b 2 /σ 2 is Lipschitz say with constant L. Using the Hölder inequality twice, we get Thus we obtain

E θ0 |V i -V (n) i | p ≤ L p T p-
E θ0 |V i -V (n) i | p ≤ C n p/2 .
For the difference U i -U n i , we have

U i -U n i = n-1 k=0 t k+1 t k b σ 2 (X i (s)) - b σ 2 (X i (t k )) dX i (s) = A 1 + A 2
where A 1 is a term analogous to the one already studied above and

A 2 = n-1 k=0 t k+1
t k b σ 2 (X i (s)) -b σ 2 (X i (t k )) σ(X i (s))dW i (s).

We introduce the process

H (n) s = n-1 k=0 1 ]t k ,t k+1 ] (s) b σ 2 (X i (s)) - b σ 2 (X i (t k )) σ(X i (s)) so that A 2 = T 0 H (n)
s dW i (s). We treat E θ0 |A 2 | p using the Burkholder-Davis-Gundy inequality and similar tools as above.

  r.t. the Lebesgue measure on R × (0, +∞) which is jointly continuous and positive on an open ball of R × (0, +∞). (H5) The parameter set Θ is a compact subset of R × (0, +∞). (H6) The true value θ 0 belongs to • Θ.

  . , b d (x) are such that b(x, ϕ) = d j=1 ϕ j b j (x) satisfies (H1)-(H2) and that (φ i , i = 1, . . . , N ) are i.i.d. Gaussian vectors, with expectation vector µ and covariance matrix Ω ∈ S d (R) where S d (R) is the set of positive definite symetric matrices. The parameter to be estimated is

  1)/2 m [. These sets are disjoint and for all m, R d = ∪ k∈Z d B k,m . Let ϕ k,m = (k i /2 m , i = 1, . . . , d) and 19 set:

  True parameter N = 20, T = 5 N = 50, T = 5 N = 50,

  TrueN = 20, T = 5 N = 50, T = 5 N = 50,

  E θ0 |X i (s) -X i (t k )| p ds. Now, for 0 ≤ t ≤ t + h ≤ T , we have X i (t + h) -X i (t) = Using φ i b(X i (s)) ≤ φ 2 i /2 + b 2 (X i (s))/2 and the Hölder inequality, we get|X i (t+h)-X i (t)| p ≤ C φ 2p i h p + (s))dW i (s)| p .We use that b and σ have linear growth, the Burkholder-Davis-Gundy inequality andM k = sup s∈[0,T ] E θ0 |X i (s)| k < ∞ for all k, then E θ0 |X i (t+h)-X i (t)| p ≤ C h p E θ0 φ 2p i + h p (1 + M 2p ) + h p/2 (1 + M p ) ≤ Ch p/2 .

	t+h		t+h
	φ i b(X i (s))ds +	σ(X i (s))dW i (s).
	t		t
		t+h			t+h
	t	b 2p (X i (s))ds h p-1 + |	t	σ(X
	1	n-1	t k+1
		k=0	t k

i

Table 3 :

 3 Example 3: Mixed Ornstein-Uhlenbeck model with two random effects. Empirical mean, variance (in brackets) are computed from 100 repeated simulated datasets for different values of (N , T ) and different parameter values.

		095 (0.082)	0.102 (0.059)	0.102 (0.028)
	µ 2 = 1 ω 2 1 = 0.01 ω 2 2 = 1	1.007 (0.275) 0.010 (0.014) 0.919 (0.526)	1.020 (0.193) 0.010 (0.009) 0.956 (0.342)	0.984 (0.175) 0.010 (0.005) 1.029 (0.278)
	µ 1 = 0.1	0.123 (0.073)	0.104 (0.047)	0.106 (0.019)
	µ 2 = 1 ω 2 1 = 0.001 ω 2 2 = 1	1.085 (0.287) 0.002 (0.004) 1.095 (0.488)	1.010 (0.166) 0.002 (0.004) 1.005 (0.353)	1.022 (0.171) 0.001 (0.002) 1.024 (0.253)

Table 4 :

 4 Example 4. Empirical mean and variance (in brackets) of the estimates µ N and ω 2 N computed from 100 datasets for different values of (N , T ) and different parameter values.

Acknowledgements. We thank Arnaud Gloter for his helpful contribution on assumption (H4).

The corresponding author for negotiations concerning the manuscript is Adeline Samson (adeline.samson@parisdescartes.fr) Laboratoire MAP5 Universite Paris Descartes 45 rue des St Peres 75006 Paris France

Proof of Proposition 6. We set γ 1 (θ) = γ 1 and I 1 (ω 2 ) = I 1 . Let us compute the second derivatives of the loglikelihood. Using that ∂γ i /∂µ = -I i , ∂γ i /∂ω 2 = -γ i I i , ∂I i /∂ω 2 = -I 2 i (ω 2 ), we obtain

We use the simple law of large numbers, the standard central limit theorem and Proposition 5 to conclude.

Proof of Proposition 7.

, by the continuity of the functions f θ (u, v), f θ0 (u, v), the equality holds everywhere. As f (u, v) is positive on an open ball B of R × (0, +∞), we deduce that, on the ball B, the following equality holds:

The left-hand side is a function of v only while the right-hand side is a function of (u, v). This is only possible if ω = ω 0 and µ = µ 0 . As 4)-( 8)). We have

Rearranging terms, we get:

Let us prove that this r.v. has finite expectation under E θ0 . We have the upper bound:

Introducing the function h(x) = x -1log x, which is defined on (0, +∞) and non-negative, we get the lower bound:

Thus,

For the second term, we write:

which has finite E θ0 -expectation by Lemma 1. For the last terms, we only need

For this, we remark that:

Therefore,

By Lemma 1, the right-hand side has finite E θ0 -expectation. The function

R × (0, +∞), using inequalities ( 26)-( 27)-( 28 This gives (ii).

Remark 3. It is worth noting that, although we have an explicit expression of K(Q 1 θ0 , Q 1 θ ), we cannot prove directly, using this expression, that

Following the usual standard proof of consistency of minimum contrats estimators (see e.g. van der [START_REF] Van Der Vaart | of Cambridge Series in Statistical and Probabilistic Mathematics[END_REF], it remains to study the continuity modulus of -(1/N )L N (θ) defined by:

We simply use w N (η) ≤ η sup θ∈Θ ∇L N (θ)/N and bound the score function (12). By (H5), we have Θ ⊂ [µ, µ] × [ω 2 , ω 2 ] with µ < µ, 0 < ω 2 < ω 2 . We have:

Tables N computed from 100 datasets for three designs and two sets of parameters (µ, ω 2 ). The exact asymptotic variance diag N -1 I -1 N (θ 0 ) is also computed. Estimation of µ when ω 2 0 is known, of ω 2 when µ 0 is known and simultaneous estimation of µ and ω 2 , for different values of (N , T ) and different parameter values. Empirical mean, variance (in brackets) and estimated value of the asymptotic variance diag N -1 I -1 N (θ 0 ) are computed from 100 repeated simulated datasets.


Martine Vanhove
email: vanhove@vjf.cnrs.fr

Llacan

TOWARDS A SEMANTIC MAP OF THE OPTATIVE IN BEJA (NORTH-CUSHITIC)

INTRODUCTION

The verbal system of Beja, the sole North-Cushitic language, possesses several indicative and modal verb paradigms whose precise semantic values and syntactic uses are still ill-known. The fact that almost each linguist developed its own terminology (with fortunately some commonalities), does not help clarify the situation and the analyses are often different and controversial for some TAM. This paper focuses on one of the Beja paradigms, the so-called Optative, as labeled by [START_REF] Roper | An Elementary Handbook for the Use of Sudan Government Officials[END_REF], in its negative form on the basis of spontaneous narrative data that I recorded in Sinkat (Sudan) between 2004 and 2007. 1 Their analysis, which confirmed in their great lines previous studies, nevertheless brought to light particular syntactic uses and modal values of capacity and necessity that have never been mentioned before.

Section 2 presents a brief overview of the main Indicative (2.1) and Mood (2.2) paradigms in order to give the reader the basic knowledge required to better understand the morphology and the semantics of the Optative within the verbal system. Section 3 is dedicated to the analysis of the Optative Negative as an optative in independent clauses (3.1), as a dependent verb form in relative (3.2), completive (3.3) and conditional (3.4) clauses, and as a modality marker of capacity and necessity in exclamatory utterances (3.5). Section 4 proposes a tentative semantic map of the Optative Negative.

THE MAIN VERB PARADIGMS

The following sketch is based on my own analysis of the Beja verbal system (for further details, see VANHOVE forth.), as I understood it from the above-mentioned data and other data that I collected between 2001 and 2011. My terminology differs to some extent from the previous ones 2 partly because I consider the whole system as more aspect oriented than previous researchers, and most probably also partly because of dialectal differences. 1 The data consist in a one-hour pilot corpus sound indexed with the transcription, fully glossed and translated. The Beja corpus, to be freely accessible online at http://corpafroas.tge-adonis.fr/ Home.html in the fall of 2012, is part of the CorpAfroAs project (grant ANR-06-CORP-018-02, coordinator Amina Mettouchi). I am grateful to Ahmed Abdallah, my main informant and language assistant in Sinkat, and to his family for their willingness to share their knowledge as well as their home with me during each of my stays. My gratefulness also goes to their relatives in Khartoum, Yacine Ahmed Hamid and his family who host me with so much kindness, and Mohamed-Tahir Hamid Ahmed, also my colleague at Ahlia University, for his patient help and explanations about the Beja language and culture. And last but not least, to the Llacan which has been supporting my research on the Beja language since the beginning. 2 ALMKVIST (1981-85), REINISCH (1893-94), [START_REF] Roper | An Elementary Handbook for the Use of Sudan Government Officials[END_REF], [START_REF] Hudson | A Structural Sketch of Beja[END_REF][START_REF] Hudson | The Non-Semitic Languages of Ethiopia[END_REF], [START_REF] Dahl | Sketch of the TMA system of Beja[END_REF], [START_REF] Morin | Des paroles douces comme la soie[END_REF], [START_REF] Appleyard | Beja morphology[END_REF], GRAGG (forth.).

Together with four other Cushitic languages, Afar, Saho, Somali and South-Agaw, Beja has two morphological verb classes. The prefix-inflecting or "strong verb" class 1 (V1), historically the oldest one, is characterized by prefixed indexes (also infixed for disyllabic verbs) for all finite Indicative paradigms and by a stem with ablaut patterns. Like in Semitic Arabic, the root of this verb class is consonantal (one, two or three consonants), and the vowels of the stem vary with TAM. Beja is the Cushitic language where this verb type is by far the largest, and V1 contains the majority of the verbs (56%, according to COHEN 1988: 275). Suffix-inflecting or "weak verb" class 2 (V2), the innovative verb class, has its indicative paradigms conjugated with suffixes and the stem is invariable for all TAM, like in most Cushitic languages. Verb class assignment is synchronically arbitrary; however most borrowings from Arabic, the contact language, are integrated as V2 verbs.

The inflexion system indexes person (1 st , 2 nd and 3 rd ), gender (F and M, in 2 nd and 3 rd persons singular only), number (SG and PL), and TAM. There are three main Indicative finite forms with aspect-temporal values: Imperfective, Perfective, Narrative,3 to which a fourth paradigm, the Perfect, morphologically different from the above mentioned TAM has to be added (see Section 2.1 below), and two Mood paradigms, Imperative and Optative (Section 2.2). Several non-finite forms, i.e. converbs, and complex predicates with various auxiliary verbs complement the TAM system, too numerous to be presented within the limits of a short overview and of no particular importance for the purpose of this article.

INDICATIVE FINITE PARADIGMS

Tables 1 and2 below present the main inflexion morphemes of three of the four Indicative finite paradigms in the base form for V1 and V2. 4 The IPFV markers of V1 in Table 1 are those of monosyllabic verbs. Table 2 provides the patterns for disyllabic V1.

Indicative IPFV PFV NAR V1 SG PL SG PL SG PL 1 an-ˈCiːC neː-ˈCiC a-ˈCiC n(ii)-ˈC C iː-ˈCiC niː-ˈCiC 2M ˈtin-CiːC-a ˈteː-CiC-na ˈti-CiC-a ˈti-CiC-na ˈtiː-CiC-a ˈtiː-CiC-na F ˈtin-CiːC-i ˈti-CiC-i ˈtiː-CiC-i 3M in-ˈCiːC ˈeː-CiC-na i-ˈCiC ˈi-CiC-na iː-ˈCiC ˈiː-CiC-na 3F tin-ˈCiːC ti-ˈCiC tiː-ˈCiC V2 1 -ˈani -nej/-naj -ˈan -na -i -ni 2M -ˈtinija -teːn(a) -ta -taːna -tija -tiːna 2F -ˈtiniː -t a j -tiː 3M -ˈiːni -eːn(a) -ija -ˈijaːn -i -iːna) 3F -ˈtini -ta -ti
Table 1: Indicative inflexion morphemes Disyllabic verbs in the base form insert the SG.IPFV marker -an-after the first root consonant of the stem: -C1anC2iːC3-, as shown in Table 2.

Indicative IPFV PFV NAR V1 SG PL SG PL SG PL 1 a-ˈCanCiːC neː-CaˈCiC a-ˈCCiC neː-CaˈCiC iː-ˈCCiC niː-ˈCCiC 2M ˈCanCiːC-a teː-ˈCaCiC-na ˈti-CCiC-a teː-ˈCaCiC-na ˈtiː-CCiC-a ˈtiː-CCiC-na 2F ˈCanCiːC-i ˈti-CCiC-i ˈtiː-CCiC-i 3M CanˈCiːC eː-ˈCaCiC-na i-ˈCCiC eː-ˈCaCiC-na iː-ˈCCiC ˈiː-CCiC-na 3F CanˈCiːC ti-ˈCCiC tiː-ˈCCiC Table 2: Indicative inflexion morphemes of disyllabic V1
The prefixed index of 1SG.IPFV is usually elided before initial laryngeals (Ɂ and h) for both syllabic types.

The fourth finite Indicative paradigm, the Perfect, is built with the verb-noun form with the nominalizer suffix -a(ː)5 (its stem is that of the Imperative for V1), followed by the indefinite article (a portmanteau morpheme which also marks gender and accusative case), and the suffixed nominal copula: ʃʔag-aː-b-u 'he has carried', ʃʔag-aː-t-u 'she has carried'.

MOODS

Moods are morphologically different from the Indicative. In the Imperative, the flexional morphemes are suffixed for both verb classes. The Optative Negative8 is characterized by flexional prefixes for both verb classes, the same verb stems as the Prohibitive (Cii(C) or CaCiiC for V1), and makes use of an obligatory variable particle ba= / bi=, which precedes the inflexion morphemes. It is followed by a set of suffixes for monosyllabic V1 and for all V2; disyllabic V1 have no suffixes. The vocalic variation of the negative marker is due to the regressive assimilation of the 1SG index a-; the underlying form of the negative particle is thus bi=. The paradigms are detailed in The final vowel of 2SG.M and 3PL of V2 is elided before clitic pronouns.

5.

V1 monosyllabic: bi=i-dʔiː-na 'let them not do!' 6.

V1 disyllabic: bi=t-katiːm 'let her not arrive!' 7.

V2: ba=a-kʷinh-aːj 'Let me not shout!'; bi=i-hass-ej 'let him not cross!'

Note that some of the Optative Negative verbs in the data used in the following Section are either irregular, often mono-consonantal, V1 and V2 verbs or derived V1 verbs whose stems are different from the base form.

THE OPTATIVE NEGATIVE

As already mentioned by ROPER, HUDSON and APPLEYARD, in addition to its optative negative function, this verb paradigm is used to negate a verb in embedded clauses, more precisely relative, completive, conditional clauses, as well as the Future auxiliary verb di 'say' in all clause types. The analysis of the pilot corpus also showed that the Optative Negative can express modal internal-participant10 values of necessity and capacity in independent clauses. All these different values and uses are detailed in the following sub-sections.

THE OPTATIVE NEGATIVE VALUE

As described in e.g. ROPER (1928: 51-52 & 62, 67), the basic prototypical value of the Optative Negative in independent clauses is precisely the negation of the Optative Affirmative form as in ( 8) and ( 9).

IN RELATIVE CLAUSES

The syntax of relative clauses depends on definiteness criteria and on the syntactic role of the head. Several strategies, which may combine, are possible:

1. No relative marker. This is the rule with subject heads and only optional with oblique indefinite heads (and for focused cleft constructions).

2. With oblique definite heads, (i) a proclitic conjunction, which derives from the definite article, i= / wi= / ji= (M), t= / ti= (F), is added to the verb (or the whole relative clause; together with the indefinite article -t for the feminine); or (ii) an enclitic conjunction follows the verb: =eː (neuter), =eːb (M), =eːt (F); (iii) in addition to (ii), the dummy noun toː-na (lit. 'the thing') may precede or follow the verb of the relative clause (or the whole relative clause); (iv) the enclitic and proclitic conjunctions may combine.

Relative clauses in Beja are head external, and in the canonical order relative clauses are embedded within the matrix clause. 12In affirmative relative clauses, the four finite main Indicative paradigms can be used. Whatever the corresponding affirmative paradigm, in negative relative clauses, only the Optative Negative paradigm can be used; the Indicative Negative particle ka= / ki= cannot occur in this context. If this syntactic constraint has been noticed since the early works about Beja, it does not tell the whole story, at least in the data studied here. In all the examples of the spontaneous data used for the purpose of this paper this verb tense expresses in addition a modality of capacity which has never been recorded before. The pilot corpus only provided examples with an oblique head.

Example (10) below is a relative clause with an indefinite head (mhiːn 'place'). Example (11) illustrates the case with the proclitic relative conjunction and a definite head (oː-kʷan 'the flood).

oːkʷan ibiːhassej isinneːk oː-kʷan i=bi=i-hass-ej

i-sin-n=eːk DEF.SG.M.ACCflood REL.M=NEG.OPT=3SG.M-cross(V2)-OPT 3M-wait\PFV-PL=if 'When it found the flooding river that cannot be crossed' (BEJ_MV_NARR_12_witch_144) Example ( 12) is with the neuter enclitic relative marker. Note that the dummy noun =na clitic to the verb form has the function of an indefinite negative pronoun, head of the relative clause.

12. oːkna hoːj biːbariːneːna oːkna hoːj bi=i-bar-iːn=eː=na PRO.REFL.SG.M.ACC13 3ABL NEG.OPT=3-have\OPT-PL=REL=thing (V1.IRG) kithaːj eːn ki=t-haːj NEG.IPFV=3SG.F-be_there\PFV 'There was really nothing that they could not have from it.' (BEJ_MV_NARR_02_farmer_323) Example ( 13) has two successive relative clauses. The first one is in fact a cleft focus ('once it is not a donkey that I rode'; lit. 'once I am not this donkey') with a definite head (oː-meːk 'the donkey') and has no overt relative marker. In the second one ('a camel that even three men could not ride'), the enclitic masculine conjunction is used with an indefinite head (kaːm 'camel'). Note that the modal value of capacity occurs only in the relative clause. 'Once, it is not this donkey that I rode but a camel, he said, that even three men could not ride at all.' (BEJ_MV_NARR_03_camel_035-041)

IN COMPLETIVE CLAUSES

The syntax of completive clauses with cognition and perception verbs is similar to that of the relative clauses with the enclitic conjunctions. In negative utterances, the use of the Optative Negative is also obligatory. Similarly to relative clauses, the modal reading of capacity is also present.

Examples ( 14) and ( 15) have the masculine enclitic conjunction. The two completive clauses in (15) present an alternative, the first one affirmative with the Imperfective, the second one negative with the Optative Negative. Recall that for the Future Negative tense as in ( 14), the auxiliary verb di 'say' is obligatorily in the Optative form even in matrix and independent clauses. Examples ( 16) and ( 17) have in addition to the relative marker the dummy noun toː-na 'the thing'. When it follows the verb ( 16), the enclitic relative conjunction is feminine, when it precedes it (17), the enclitic is neutral.

ɖaːbi biːdiːjeːb hiːsan ɖaːb-i bi=i-diː=jeːb

PARTICIPANT-INTERNAL MODALITIES IN INDEPENDENT CLAUSES

Beja has no modal verbs such as 'can', 'may', 'must' or 'need', if one excepts the recent loan from Arabic agdir / adgir 'can'. Participant-internal modalities are usually retrievable only from the context. We already saw that in relative and completive clauses (Sections 3.2 & 3.3) the Negative Optative brings a modal value of capacity. The data of the pilot corpus show usages of the Optative Negative, rare but unrecognized so far, in exclamatory independent utterances, for the expression of capacity as well as of necessity, which both pertain to the domain of participant-internal modalities, and are not directive ones as the Imperative, the Prohibitive and the Optative. 15For the internal-participant modality of capacity,16 the Optative Negative is used alone (as for the optative value in dependent clauses).

21. kak meːk hasamaːbi bani ʔaːdam han kak meːk hasam-a-b-i bani ʔaːdam han how donkey pass_by-NMLZ-INDF.M.ACC-COP.3SG son Adam also biːhassejt bi=i-hass-ej-t NEG.OPT=3SG.M-pass(V2)-OPT-COORD 'How a donkey could have threaded its way through it? A human being cannot even pass! And…' (BEJ_MV_NARR_05_eritrea_174)

For the internal-participant modality of necessity, and auxiliary verb is added, namely the verb ak 'be', here in the Perfect form.

  .F.ACC NEG.OPT=3SG.F-arrive\OPT(V1) place uːmeːk ingad uː-meːk i-ngad DEF.SG.M.NOM-donkey 3SG.M-stop\PFV 'The donkey stopped in a place where nothing can arrive' (BEJ_MV_NARR_05_eritrea_180-181)11 The references in between brackets are those of the pilot corpus.

  13. doːr oːmeːk oːn baːkaj kaːm doːr oː-meːk oːn ba=a-kaj kaːm time DEF.SG.M.ACC-donkey PROX.SG.M.ACC NEG.OPT=1SG-be\OPT camel V1.IRG han amaːbi ini mheːj nafara han ʔam-aː-b-i ini mheːj nafar-a also ride\REFL-NMLZ-INDF.M.ACC-COP.1SG say\PFV.3SG.M three person-PL nuːn naːt bitʔameːb nuːn naː-t bi=t-ʔam=eːb only thing-INDF.F.ACC 14 NEG.OPT=3SG.F-ride\REFL(V1)=REL.M

Figure 1 :

 1 Figure 1: Tentative semantic map of the Optative Negative

  The Optative 6 Affirmative paradigm, with either suffixes or prefixes depending on the verb class, is based on the Narrative paradigm of each verb class, with the obligatory addition of an invariable proclitic particle, baː=, 7 homophonous with the SG.M & PL prohibitive particle.

		'arrive' (V1)	'eat' (V2)
		SG	PL	SG	PL
	1	baː=iː-ktim	baː=niː-ktim	baː=tam-i	baː=tam-ni
	2M baː=tiː-ktim-a baː=tiː-ktim-na 2F baː=tiː-ktim-i	baː=tam-tija baː=tam-tiːn(a) baː=tam-tiː
	3M baː=iː-ktim 3F baː=tiː-ktim	baː=iː-ktim-na	baː=tam-i baː=tam-ti	baː=tam-iːn(a)
		Table 3: Optative Affirmative paradigm

They are identical in the singular and differentiate gender: -a (M), -i (F). V1 and V2 suffixes are slightly different in the plural (with no gender distinction): V1 -na, V2-aːna. 1. V1: dif-a 'go (SG.M)!'; dif-i 'go (SG.F)!'; dif-na 'go (PL)!' 2. V2: jhak-a 'get up (SG.M)!'; jhak-i 'get up (SG.F)!'; jhak-aːna 'get up (PL)!'; For the Prohibitive, a proclitic negative Mood particle baː= (SG.M & PL) / biː= (SG.F) precedes the stem for the two verb classes, also both conjugated with suffixes. The V1 stems undergo ablaut and their vocalic patterns are different from those of the Imperative: CiiC and CaCiiC (which are otherwise nominal patterns, see VANHOVE, forth.). The suffixes are the same as the Imperative for each verb class, except the feminine morpheme of V2 which becomes -ej. 3. V1: baː=diːf-a 'don't go (SG.M)!', biː=diːf-i 'don't go (SG.F)!', baː-diːf-na 'don't go (PL)!' 4. V2: baː=raːt-a 'don't ask (SG.M)!', biː=raːt-ej 'don't ask (SG.F)!', baː=raːt-aːna 'don't ask (PL)!'

Table 4

 4

	below: 9
		SG	V1mono	V2	PL	V1mono	V2
	1	ba=a-stem		-aːj / ej	bi=n-stem		-aːj / ej
	2M bi=t-stem -a 2F bi=t-stem -i	-aːja / ej bi=t-stem -na -aːj / ej	-eːna
	3M bi=i-stem 3F bi=t-stem		-aːj / ej -aːj / ej	bi=i-stem -na	-eːna

Table 4 :

 4 Optative Negative paradigms

  The data used for this article only provided examples with V2 and non in the 2 nd persons:

	8.	tidirʔa han baːʃagaːmaːj diːt ti-dirʔa han ba=a-ʃaga-am-aːj	diːt
		DEF.F-field also NEG.OPT=1SG-work(V2)-REFL.PASS-OPT say\CVB.ANT
		'After he had told himself: Let me not work in the field anymore!'
			(BEJ_MV_NARR_02_farmer_142) 11
	9.	harʔiːsiː bitijʔaheːbaj anit harʔi=isi bi=t-jʔ-a=heːb=aj	ani
		after=1SG.ABL NEG.OPT=3SG.F-come-OPT=1SG.ACC=INTSF say\PFV.1SG
		V2.IRG
		'Don't let it come from behind me! I told myself'	(BEJ_MV_NARR_05_eritrea_322-326)

Called "conditional" (ROPER, MORIN), "aorist" (APPLEYARD, GRAGG; also in some of my previous works), "Pluquamperfect" (REINISCH).

The stems and indexes of derived verbs and a small verb class in the base form do not conform to these patterns. It is out of the aim of this paper to provide them all. For a still valid overview see[START_REF] Roper | An Elementary Handbook for the Use of Sudan Government Officials[END_REF]. Clear synthetic and more modern overviews are provided in[START_REF] Appleyard | Beja morphology[END_REF] and GRAGG (forth.). See also VANHOVE (forth.).

This form is called "particip" (REINSICH), "gerund" (ROPER, MORIN) or "past participle" (HUDSON, APPLEYARD). As it is highly polyfunctional, I preferred to gloss the -a suffix with a more neuter term: "nominalizer".

REINISCH gives the label "optativ" to another form, taken up from MUNTZIGER (1864), but unrecorded in other sources.

The short a given by APPLEYARD, following HUDSON, is a dialectal variant (Northern variety).

This form is called "bound negative" by HUDSON.

APPLEYARD (2007: 470) analyzes the marker as b[i]-+ the incorporation of the personal marking. Consequently his parsing is different.

The terminology is that of[START_REF] Auwera | Modality's semantic map[END_REF] and concerns part of the non-epistemic modalities.

Beja is a "canonical" SOV language, rather consistently head final, and the matrix clause usually follows the dependent clause.

The reflexive pronouns are often used as enunciative particles with an assertive value.

The dummy noun is also a negative particle.

Which AUWERA et al. (1998[START_REF] Hengeveld | Illocution, mood, and modality[END_REF], consider as belonging to the domain of illocutionary type rather than of modality.

Which AUWERA et al. (1998: 82) prefer to call participant-internal possibility.

OPT, Optative PASS, passive PFV, Perfective PL, plural PRO, pronoun PROX, proximal REFL, reflexive REL, relative S, subject SG, singular TAM, tense, aspect, mood V, verb V1, verb class 1 V2, verb class 2 VOC, vocative

GEN, genitive IMP, Imperative INDF, indefinite INT, intensive INTSF, intensifier IPFV, imperfective IRG, irregular LOC, locative M, masculine mono, monosyllabic NAR, Narrative NEG, negative NMLZ, nominalizer NOM, nominative O, object

The different uses of the Optative Negative form show a cline of modal values, be it synchronic or diachronic, whose directionality is not easy to grasp yet in order to draw its semantic map, and is open for further study. Still, one can already notice that all these values fall within the domain of participant-internal modality. 17 The Future tense with the auxiliary di 'say' 18 has intentional and volitional modal values [START_REF] Morin | Des paroles douces comme la soie[END_REF]VANHOVE forth.) which are linked to the semantic field of the verb 'say' in Beja, which includes the intention and volition domains as well as the quotative domain. In my opinion, these two modal values also belong to the domain of internal-participant modality, just as the other modal values of optative, capacity and necessity, because they correspond to the characterization provided by AUWERA et al. (1998: 80): "[internal-participant modality] refers to a kind of possibility or necessity internal to a participant engaged in a state of affairs." Tentatively, the semantic map of the Optative Negative (with no arrows yet, thus the position of the different values has to considered as arbitrary) may look as in Figure 1 below. 17 I depart from [START_REF] Auwera | Modality's semantic map[END_REF] who explicitly exclude volition and optative from the domains of the participant-internal modality (as well as participant-external and epistemic modalities). The Beja data makes me tend to refuse such restrictions, hence their inclusion in the tentative semantic map. 18 Called "Permissive" by APPLEYARD, following HUDSON. What they call "Future" is in fact more precisely a "Potential" (see [START_REF] Ahmed | Contrastive negation in Beja: the auxiliary verb rib[END_REF]) without speaker's or participant's involvement.


Martine Vanhove

Thomas Stolz

Introduction

 [START_REF] Ferguson | The Ethiopian language area[END_REF][START_REF] Ferguson | The Ethiopian language area[END_REF]studied a series of linguistic features (phonetic, morphological, syntactic and lexical) in Semitic, Cushitic and Omotic languages to establish the so-called Ethiopian language area (a misleading label since it concerns Ethiopia, Eritrea, Somalia, Northern Kenya and Eastern Sudan; [START_REF] Zaborski | Ethiopian language macroarea[END_REF] proposed instead the term 'North-East African Language Macro-Area'). Among others, [START_REF] Tosco | Is there an 'Ethiopian language area'?[END_REF] reviewed all these features and questioned their choice and the validity of the isogloss bundles. He concluded on the non-existence of an Ethiopian language area and that unilateral contact is the rule in this zone, somehow following the lines developed in [START_REF] Leslau | The influence of Cushitic on the Semitic languages of Ethiopia: A problem of substratum[END_REF][START_REF] Leslau | The influence of Sidamo on the Ethiopic languages of Gurage[END_REF][START_REF] Leslau | Sidamo features in South Ethiopic phonology[END_REF] and Moreno (1948) about the influence of Cushitic languages upon Semitic languages of the area. But as summarized in CRASS & MEYER Four recent papers, namely Bender (2003), Crass (2002), Crass and Bisang (2004), and [START_REF] Zaborski | Ethiopian language macroarea[END_REF] favor the existence of a linguistic area. Bender (2003) argues against Tosco (2000b) and tries to extend the ELA [Ethiopian Language Area] by testing a number of Nilo-Saharan languages [also present in this zone] using a selection of Ferguson's features. Crass (2002) discusses two phonological features in detail; in Crass and Bisang (2004) the discussion is extended to features such as word order, converbs, and ideophones verbalized by the verb 'to say'. Zaborski presents the most extended list, including twenty-eight features which he considers to be valid for a macro-area including Ethiopia, Eritrea, Djibouti, Somalia, and parts of Sudan, Kenya, and even Tanzania and Uganda. (CRASS & MEYER 2007: 230) A further study by [START_REF] Simeone-Senelle | Is there a Red Sea linguistic area?[END_REF], enlarged to the Red Sea area, claimed that in North-Eastern Africa, influences between Cushitic and Semitic languages were to some extent bidirectional, on the basis of some additional features hardly ever taken into consideration in the previous literature. They showed possible cases of Semitic influence on Cushitic, such as the absence of labiovelars, the position of the head vs the relative clause, or the root and pattern morphological system, which will be the bulk of this paper. However, their study concluded that mutual influences probably do not reflect a long term bilateral symmetric contact situation, a process which is sometimes included in the characterization of a Sprachbund (see e.g. [START_REF] Thomason | Language contact, creolization, and genetic linguistics[END_REF]. These influences could "mirror successive stages due to the changing sociolinguistic situations and levels of bi- [lingualism] or multi-lingualism, as well as to the prestigious role of Arabic for religious, economic and political reasons" (SIMEONE-SENELLE & VANHOVE 2006: 62), as is specially the case for Beja.

On the other hand, [START_REF] Sasse | A southwest Ethiopian language area and its cultural background[END_REF] and [START_REF] Zaborski | The morphology of nominal plural in the Cushitic languages. Wien: Beiträge zur Afrikanistik[END_REF], who were also critical of Ferguson's approach, considered that linguistic factors do not exclude the possibility of smaller linguistic areas in the zone, such as an Eritrean-North Ethiopian area (Zaborski) or a Southwest Ethiopian area (Sasse). This paper is also an attempt to suggest that the Eastern Sudan-Eritrea zone could be another one of these smaller linguistic sub-areas. Moreover, on the basis of several Afro-Semitic and East-Cushitic languages spoken in the Highland East Cushitic/Gurage subarea of Ethiopia, CRASS & MEYER (2007) themselves proposed a new series of morphological and syntactic features to identify this linguistic subarea, and consider that they could be possible candidates for establishing the existence of a North-East African language macro-area.

Given the particular socio-linguistic context in which Beja has evolved (see section 2 below), the aim of this paper is to try and show that language contact may lead to the preservation of typologically rare and endangered morphological strategies, namely the root and pattern schema.

The Beja language: classification and sociolinguistics

Beja (named beɖawije-t by the Beja people) is an unwritten language, traditionally classified as the sole member of the Northern branch of Cushitic of the Afroasiatic phylum. It is mainly spoken in the Red Sea Hills in Eastern Sudan by approximately 1,100,000 speakers, as well as by a much smaller number of speakers in Northern Eritrea and Southern Egypt (where it is endangered).

Beja society is organized into clans and tribes, and its islamisation began as early as the tenth century. All Bejas are now Muslims. Until the first half of the 20 th century, their main occupations were camel trading and pastoralism; only the latter survives today to a lesser extent since the terrible droughts of the mid-1980s. Throughout their history, Bejas have had intensive contacts with Tigré (South-Semitic), Nubian (Nilo-Saharan), and Arabic (Central Semitic) speakers and today, in Sudan, most Beja males are bilingual with Sudanese colloquial Arabic. Bilingualism is less frequent among women. Only a minority of the Bejas have knowledge of Tigré, but a few clans at the border with Eritrea have switched to Tigré (with some influence from Beja). In Sudan, although the illiteracy rate is still high, formal education is increasing among young villagers and city dwellers, but girls still seldom go further than primary level. The sole language of education is Classical Arabic, with a mixture of colloquial Sudanese Arabic. In addition Quranic Arabic is taught in Quranic schools to young boys and girls alike.

Roots and patterns in the North-East Africa Macro-area

Within the Afroasiatic phylum, the Semitic and Berber stocks (the latter to a different extent) are renowned for having a conspicuous and robust consonantal root system and a morphological system which is not predominantly organized in a concatenative way1 but on the basis of patterns which determine semantic and functional classes of words. Patterns constitute a closed inventory consisting of qualitative or quantitative vocalic alternation in the stem, syllabic structure changes, and different types of reduplication, to which affixes can also be added. Below is a well-known example from Arabic.

(1) Roots and patterns in Arabic Root KTB > KaTaBa 'to write', KaTTaBa 'to write a lot' mi-KTuːB 'written', KaːTiB 'writer', KiTaːB 'book', KuTuB 'books', etc. This root and pattern morphological structure also exists in some languages of North-East Africa. In the South-Semitic branch to which most of the Semitic languages of this area belong, as well as in Arabic, the structure of the lexicon, for both verbs and nouns, is organized along these lines. But whereas this is true for the South-Semitic languages spoken in the northern part of the area, such as Tigré [START_REF] Raz | Tigré grammar and texts[END_REF] with which some varieties of Beja are in contact, in the South-Semitic languages spoken further South, the root and pattern morphological system is archaic, as in Amharic for instance (LESLAU 2000: 40), and sometimes very marginal as in Gurage (HETZRON 1977: 53). TOSCO (2000: 344) considered that in this area the root and pattern schema is 'Semitic-biased', i.e. almost exclusively limited to the Semitic languages. But this morphological structure is to a lesser degree also characteristic of the three Cushitic languages of the northern part of North-East Africa, Beja, Afar and Saho (which belong to the Eastern branch of Cushitic), as opposed to the other Cushitic languages spoken in the rest the area, where stems are usually invariable and the root and pattern schema at best highly marginal, and most often absent (cf. COHEN 1988: 256).

Following Sasse and Zaborski's argumentation about smaller subareas, this paper claims that the preservation of the root and pattern schema, although genetic, could be an areal feature differentiating the extreme northern part of the North-East African zone from the rest of the Cushitic languages of the area.

The following sections provide a general overview of the root and pattern schemas in Beja, with some limited comparisons (for lack of space) with Arabic, Afar and Saho.

Roots and patterns in Beja

In Beja the root and pattern morphological system concerns a sizeable portion of the lexicon: a large part of the verbal morphology (conjugations, verbal derivations, verb-noun derivations), and part of the nominal morphology (adjectives, nouns, 'broken' plurals, and to a lesser extent place and instrument names).

Verb morphology

Verb classes

There are two different verb classes in Beja. The first class (V1) is made of an invariable stem conjugated with suffixes for all TAMs. Historically, this is the most recent type which in most other Cushitic languages became the sole conjugation system. This verb class is not concerned by the root and pattern schema.

The second verb class (V2) is made of a stem with ablaut which varies according to TAM 2 , person and number, to which prefixed personal indexes for all TAMs are added. This is diachronically the oldest pattern which survives only in a few Cushitic languages. In Beja this verb class contains the majority of the verbs and they form up to 60% of the total number of verbs, as against approximately 30% in Afar and Saho, and only five verbs in Somali and South Agaw (COHEN 1988: 256). Thus the hypothesis is that contact with Arabic (and possibly also with Afro-Semitic Tigré in Eritrea, but further study is needed) helped to preserve this verb class in the Cushitic languages of the zone. Below is an example of the two basic conjugations (Perfective and Imperfective) for biconsonantal and tri-consonantal roots:

Bi-consonantal DF 'go' 3 Tri-consonantal KTM 'arrive' PFV i-dif 'he went' i-ktim 'he arrived' i-dif-na 'they went' i-ktim-na 'they arrived' IPFV i-n-diːf 'he goes' k<an>tiːm 'he arrives' eː-dif-na 'they go' eː-katim-na 'they arrive' Table 1: Perfective & Imperfective patterns
Prefix conjugations are used in Arabic and South-Semitic languages but their functions and origin are different from Beja. In the Semitic languages the prefix conjugation has an aspectual value of Imperfective while in Cushitic it marks a particular morphological class of verbs. As for their respective origins, the Cushitic prefix conjugation (in the singular) originates in a periphrastic verbal construction with an auxiliary verb meaning 'say, be', while the prefix conjugation of Semitic has various origins, depending on the branches, none of them including a verb 'say' or 'be' (see e.g. [START_REF] Cohen | La phrase nominale et l'évolution du système verbal en sémitique[END_REF]. The grammaticalization pattern of Cushitic occurred also later on for all the personal indexes of the suffix conjugation [START_REF] Cohen | Alternances vocaliques dans le système verbal couchitique et chamito-sémitique[END_REF](COHEN -70 & 1984)). Although different grammaticalization processes were used in the two branches of Afroasiatic, all this suggests that the root and pattern system might have already been very robust in Beja (and possibly also in proto-Cushitic) at an ancient stage of the language 4 , which is unfortunately impossible to date for lack of old documents. It is noteworthy that there are at least traces of vocalic alternation between the Perfective and the Imperfective in all the 2 For details about vocalic alternation in Cushitic and Afroasiatic see [START_REF] Cohen | Alternances vocaliques dans le système verbal couchitique et chamito-sémitique[END_REF][START_REF] Cohen | La phrase nominale et l'évolution du système verbal en sémitique[END_REF]. 3 All examples are taken from the lexicon in ROPER (1928). His transcription has been changed into IPA (except for geminated consonants which are doubled instead of using the semi-colon). 4 COHEN (1984: 94) states (without mentioning on which study his statement is based) that tri-consonantal verbs with prefixed paradigms are often of Semitic origin.

branches of Cushitic (see COHEN 1984: 88-102), thus reinforcing the hypothesis of an ancient root and pattern schema in Cushitic.

Converbs

Among the four non-finite Beja verb forms, only one converb, the negative simultaneous converb, belongs to the root and pattern schema (the others are built with suffixes added to the stem of the Imperative form). It consists of the negative particle baː=, proclitic to a variable stem (with ablaut). Nothing similar exist in Arabic, Afar or Saho.

(2) Negative simultaneous converb5 gid 'throw' > gid-eː, neg. baː=giːd 'while I/you/he... was not throwing' kitim 'arrive' > kitim-eː, neg. baː=katiːm 'while I/you/he... was not arriving' Similar stems do not have the same function in Arabic, and they do not exist in Afar and Saho.

Verb derivation

Beja is the only Cushitic language that uses qualitative ablaut in the stem as a derivational device, for the expression of various semantic and voice derivation. The ablaut may be combined with reduplication of one syllable or the use of prefixes The intensive pattern of Beja V2 consists of a qualitative and quantitative vocalic alternation, usually oː for bi-consonantal verbs, with a few exceptions (such as dir 'kill' above), and aː in the first syllable of tri-consonantal verbs. Among the Semitic languages of North-East Africa and the Arabian Peninsula, this ablaut pattern for intensive derivation is only known in the modern South Arabian (MSA) languages (South-Semitic) spoken in Eastern Yemen. (It is also used for causative and transitivisation). The MSA languages are direct cognates of the South-Semitic languages of North-East Africa and it is usually admitted that South Semitic languages of Africa were imported by South-Arabian speakers [START_REF] Ullendorf | The Semitic languages of Ethiopia: A comparative phonology[END_REF]. Still this ablaut pattern was not retained in the South-Semitic languages of North-East Africa. It is also unknown in Arabic with an intensive value, a semantic derivation which is made through the reduplication of the second consonant of the root. The Arabic derived form with aː in the first syllable has a goal, or sometimes reciprocal, meaning.

Frequentative

FREQ bis 'bury' > baːbis, baːbabis 'bury in several graves' kitim 'arrive' > kaːtatim 'arrive at many places'

The frequentative derivation pattern, which is generally used for transitive verbs with plural objects, consists for bi-consonantal verbs in the reduplication of the unique syllable + a qualitative and quantitative ablaut, and for tri-consonantal verbs in the reduplication of the beginning of the second syllable + a qualitative and quantitative ablaut.

The semantics, the syntax and the forms are specific to Beja. They are unknown to other Cushitic or Semitic languages, although the reduplication pattern (without ablaut) is widespread in Cushitic and South-Semitic. Syllabic reduplication does not occur in Arabic.

Reflexive

REFL fif 'pour' > faf 'boil over, overrun' rimid 'avenge' > rimad 'avenge oneself' Beja is the sole Cushitic language to differentiate between active and reflexive voice by means of vocalic alternation. The reflexive voice has a characteristic short a in the Perfective (in the second syllable in triconsonantal verbs) as opposed to i for the active base form.

Remnants of this pattern exist in some Semitic languages, among them Arabic, but in a fossilized form. It is worth mentioning that the Imperfective form of the reflexive voice in Beja attests the presence of a productive and historically more recent device in Semitic, very much alive in Arabic, i.e. the shared innovation of a prefix t-added to the stem in Beja: a-t-ramiːd 'I avenge myself'. In Arabic, the t is infixed after the first root consonant for the basic stem or prefixed to two derived stems.

The t characteristic of the reflexive is unknown in other Cushitic languages.

Passive/middle

PASS min 'shave' > aːtoː-maːn 'shave oneself' dibil 'collect' > at-dabaːl 'be collected'

In Cushitic, qualitative ablaut for the passive/middle voice only occurs in Beja. It is characterized by a long vowel aː (in the second syllable for tri-consonantal verbs, in addition to an ablaut to a in the first syllable) in the Perfective. In addition, a prefix with a dental consonant -t-, whose form depends on the syllabic structure of the verb, is added to the passive/middle stem. This pattern is also used with recent borrowings from Arabic: kitib 'write', at-kataːb 'be written'.

In classical Arabic there is also a qualitative ablaut for the expression of the passive voice, but this is unknown in colloquial Arabic varieties (or at best fossilized in some varieties of the Arabian Peninsula). This pattern does not include the addition of a prefix. In Arabic, the passive/reflexive form is now made via a t prefixed to or infixed in the stem, or via a prefix n-, depending on the stem of the base and derived forms, or by a combination of both consonants in some colloquial varieties.

In South-Semitic of North-East Africa, ablaut is not used for voice.

Causative

CAUS dir 'kill' > soː-dir 'make kill' kitim 'arrive' > si-katim 'make arrive'

Although an -s suffix (not a prefix as in Beja) is common in Cushitic, Beja is once more the only Cushitic language which uses ablaut for the causative derived form; it is attested in both Perfective and Imperfective for tri-consonantal verbs, but only in the Imperfective for bi-consonantal verbs. They are characterized by a short a in the first syllable of triconsonantal roots. The prefix with an s, whose vowel quality depends on the syllabic structure of the verb, is added to the stem.

Neither the qualitative ablaut nor the s prefix exist in Arabic (s-occurs in other Semitic languages of the Near East like Aramaic). The prefix s only occurs in Arabic in combination with t with mostly a reflexive/ middle sense (of the derived form with a prefix Ɂa-) and a denominative sense. Arabic has two causative forms: the same as the intensive one, i.e. with a geminated second root consonant, and the Ɂa-C1C2aC3a pattern.

Reciprocal

RECP gid 'throw' > amoː-gaːd 'throw at each other' girim 'be inimical' > am-garaːm 'be inimical to each other' Like the passive/middle voice, the reciprocal is characterized by a qualitative ablaut in aː in the stem (in the second syllable for triconsonantal verbs), but the prefix is different and consists of am-(+ a long vowel -oː for monosyllabic verbs).

This m is not used for verbal derivation in Arabic (but m is used to build passive participles -and active ones). Arabic, marginally though, uses a quantitative ablaut of the first vowel of the stem to aː to express, marginally, the reciprocal of the base form. But most often the reciprocal meaning is expressed by means of the more specifically reflexive forms with the t-prefixed or infixed to the derived form in aː in the first syllable or to the base form.

In some other Cushitic languages m is used as a suffix for a passive/ middle voice (without ablaut in the stem). In Beja m can also marginally be used as a passive marker, together with ablaut, for a few transitive intensive verbs such as ameː-saj 'be flayed', ameː-biɖan 'be forgotten'.

Verb-noun derivation

As with the verbal morphology, the root and pattern morphology only concerns the largest verb class V2, i.e. the one conjugated with prefixes. This morphological structure is limited to two of the three verb-noun forms of Beja: nomen actionis (or masdar) and agent noun (the third one, the so-called 'gerund' is built with a suffix -a added to the stem of the imperative).

Nomen Actionis / masdar

For the base forms of old verb class V2, masdars are generally, and freely, formed through a qualitative ablaut, most often to uː (in the second syllable for tri-consonantal roots), sometimes aː for bi-consonantal roots or defective verbs:

(3) Masdars of base forms of V2 kitim 'arrive' > kituːm 'arrival' ʃɁiʃ 'cough' > ʃɁuːʃ 'coughing' bir 'snatch' > baːr 'act of snatching' riwi 'arrest, cause to stop' > raːwaː 'halting, ceasing' A C1aC2uːC3 pattern with a characteristic second vowel uː (and a different vowel than in Beja in the first syllable) is a masdar pattern in classical Arabic, but it is semantically limited almost exclusively to verbs expressing movements and positions of the body (see BLACHÈRE & GAUDEFROY-DEMOMBYNES 1975: 81).

In Afar and Saho, the masdars of the prefix verb class are regularly characterized by a long or short vowel a(ː) (not uː) alternating with the vowel(s) of the verb stem (see SIMEONE-SENELLE & VANHOVE 2006: 43).

In Beja, the vowel a is also characteristic of the stem of masdars for a few bi-consonantal and defective verbs, which in addition have a prefix mV-(the list below is probably exhaustive):

(4) Masdars of a few bi-consonantal and defective V2

dir 'kill' > ma-dar 'murder, killing', ma-doːr 'murder(ous)' Ɂamit 'grasp' > ma-Ɂmat 'act of grasping' hiw 'give' > mi-jaw 'gift, act of giving' ah 'take' > miː-jaj 'receipt, act of receiving or taking' jiwid 'dress the hair by curling and frizzing' > miː-wad 'curl' di 'say' > miː-jad 'act of saying' firi 'give birth' > mi-freːj 'birth' rik w i 'be afraid' > mi-rk w ay 'fear'

For another handful of masdars with the prefix mV-, there is no ablaut on the stem, just a syllabic change for tri-consonantal roots:

(5) Masdars without ablaut of V2 ɖha 'be fat' > ma-ɖha 'fatness' ɖhan 'live' > mi-ɖhan 'life, health' g w imid 'lengthen' > muː-gmid 'length' rak w uk 'thick' > mi-rk w uk 'thickness'

The absence of ablaut is also the case for the few masdars of the derived reflexive forms:

(6) Masdars without ablaut of Reflexive V2 REFL: lag 'dance' > ma-lag 'dancing'

In Arabic, one class of (tri-consonantal) verbs, those with a or u in the Imperfective, and most of the derived verbs with aː and a goal meaning, form their masdars with a prefix ma-.

The masdars of three other derived forms (causative, reciprocal and passive/middle), are all characterized by a suffix -oːj, added to the derived or to the basic forms of the verb.

(7) Masdars of derived V2 CAUS: soː-dir-oːj 'act of murdering'; si-katm-oːj 'act of making arrive' PASS: atoː-min-oːj 'act of being shaved'; at-Ɂabk-oːj 'act of being caught' RECP: am-garim-oːj 'mutual enemity'

The other two remaining verb forms have no masdar or share it with the base form.

Although two patterns are common to some extent with Arabic, the Beja system is different in its general organization.

Agent Nouns

Agent nouns combine qualitative ablaut and the suffixation of a morpheme, -aːna, the same suffix as the one used to form agent nouns for V1 verbs (for some tri-consonantal roots a suffix -i is added instead of -aːna). For disyllabic agent nouns, the stem is in fact that of the derived intensive verb form.

(8) Agent nouns of V2 bir 'snatch' > boːr-aːna 'snatcher' gid 'throw' > geːd-aːna 'thrower, a good shot' bis 'bury' > boːs-aːna 'burier' min 'shave' > moːn-aːna 'barber' tuːl 'point with the finger' > tuːl-aːna 'index' kitim 'arrive' > kaːtm-aːna / kaːtim-i 'newcomer'

The Beja agent noun pattern for tri-consonantal roots is reminiscent of the active participle of the base form of Arabic verbs (but no suffix is added in Arabic), which can also have a value of agent noun.

Noun morphology

Adjectives

Beja has seven different patterns which are used to derive adjectives from verbs. All are based on a qualitative vocalic alternation, in a few cases with the addition of a suffix -a. Arabic has no dedicated adjectival patterns (but the active participle pattern of the verbal base form C1aːC2iC3 may correspond to adjectives), and Afar and Saho do not have an adjectival category. The last pattern with a geminated second consonant is most probably a fossilized form.

Pattern

Nouns

There are eleven different patterns with qualitative ablaut for nouns related to the old prefix verb form V2, of which two (C1aC2i and C1iC2i) are shared with the adjectival patterns. In addition, two patterns can have a suffix -a. Most of the patterns of tri-consonantal roots are common with Arabic, not surprisingly considering the small number of vowels in each language.

Pattern

Instrument nouns

There are a few instrument nouns whose pattern is made with qualitative ablaut and a prefix m-, like in Arabic, where, contrary to Beja, this prefixed pattern is frequent and productive. But note that the Beja instrument nouns do not seem to be borrowings from Arabic. In one instance C2 is reduplicated. Another pattern consists of a suffix -an (which does not belong to the patterns of instrument nouns in Arabic), in addition to the ablaut. They seem to be frozen forms in Beja.

(9) Instrument nouns of V2 himi 'cover' > m-himmeːj 'blanket' Ɂafi 'prevent, secure' > m-Ɂafaj 'nail, rivet, fastener' min 'shave' man-an 'razor'

Afar and Saho also have some instrument nouns following a similar pattern (see SIMEONE-SENELLE & VANHOVE 2006: 44).

Place nouns

A few place nouns are also based on the pattern with the ablaut and the prefix mV-. While this is a frequent and productive category in Arabic, it is a frozen one in Beja.

(10) Place nouns of V2 ginif 'kneel' > mi-gnaf 'camp' ridif 'set on a pillion (sp. on camel)' > mi-rdaf 'rump' moːk 'take shelter' > ma-k w a 'place of shelter'.

Afar and Saho also have some place nouns following a similar pattern (see SIMEONE-SENELLE & VANHOVE 2006: 44) 6 .

Plural patterns

The so-called 'broken plural' patterns are common and frequent in the South-Semitic languages of North-East Africa and in Arabic. Beja (as Afar and Saho) also has a limited set of broken plural patterns, but it developed its own system which is partly predictable. Ablaut patterns for plural formation mainly concern non-derived nouns either containing a long vowel, or ending in a diphthong. Whatever the number of root consonants and the syllabic structure, both long iː and uː in the singular turn into short i in the plural, and long aː, eː and oː turn to short a, sometimes with the addition of the plural suffix -a for the former. Nouns ending in -aj turn to a long vowel -eːj.

(12) Plural patterns ang w iːl, pl. ang w il 'ear' luːl, pl. lil 'rope' asuːl, pl. asil 'blister' hasaːl, pl. hasal / hasal-a 'bridle' meːk, pl. mak 'donkey' boːk, pl. bak 'he-goat' ganaj, pl. ganeːj 'gazelle' 6 The semantics of some Beja words following the patterns with the mV-prefix may be different and not predictable: rifif 'drag an object along the ground' > mi-rfaf 'reptile'; maʃʃa 'hair swept to the sides of the face' > mi-ʃuːʃi 'spinster' (married women wear a fringe on the forehead, or plaits). Some old and recent borrowings from Arabic also conform to these patterns:

(13) Plural patterns of borrowings kaːm, pl. kam 'camel' (Arabic gamal) baːbuːr, pl. baːbir 'steamer' (pl. in Sudanese Arabic is bawaːbiːr)

Reduplication of the beginning of the first syllable together with ablaut is rare and limited to adjectives: (14) Plural patterns of adjectives win, pl. waːwin 'big' dis, pl. daːdis 'small' Prosody is also employed in Beja, with a change of stress assignment to mark plurals in a few nouns:

(15) Plural patterns with stress shift haɖáː-b 'a lion', pl. háɖaː-b.

Even though broken plurals can be considered as a genetic feature, the fact that they are very rare or absent in other Cushitic languages [START_REF] Zaborski | The morphology of nominal plural in the Cushitic languages. Wien: Beiträge zur Afrikanistik[END_REF]) speaks for a possible influence of Semitic upon Beja (and Afar and Saho).

Conclusion

Although Beja and Arabic share a typologically similar morphological system based on the root and pattern schema, the above overview has shown that each language has developed its own system, both in terms of semantics and forms. In this morphological domain, although these languages have been in contact for centuries, neither small scale nor massive borrowing from Arabic morphological patterns can be postulated from the Beja data.

Although it has not been possible within the limits of this paper to make a thorough comparison with Afar and Saho, the two other Cushitic languages in which the root and pattern morphology is not marginal or absent, it is nevertheless important to emphasize that this morphological type is less developed in these two languages. In particular, vocalic alternation is not used in their verbal derivation systems and is restricted in the conjugation paradigms to only 30% of the verbs as opposed to 60% in Beja.

Three characteristics of these languages deserve our attention. Firstly, the morphological systems of other Cushitic languages are predominantly or, most often, exclusively concatenative. Secondly, the root and pattern schema is less developed in neighboring Afar and Saho for which contact with Arabic has been less pervasive. Thirdly, Arabic is the dominant and prestigious language for the Beja speakers, even though very few of them have entirely shifted to Arabic. If one admits the hypothesis that the root and pattern morphological system is inherited in both languages, given these three characteristics, it is tempting to think that contact with Arabic was a strong factor for the preservation of this system in a large part of the Beja morphology and that we have here a case of convergence.

Thus, it seems that intensive language contact does not only lead to morphological borrowings, copying, replication, etc., as is generally discussed in the literature on language contact, but may help to preserve a morphological system in genetically related languages which otherwise would have disappeared as is the case in most other Cushitic languages. Beja morphology would not show a case of contact-induced change, but a case of contact-induced preservation. A situation particularly tricky to disentangle if one keeps in mind that "if genetically related languages are in contact, trying to prove that a shared feature is contact induced and not a 'chance' result of Sapir's drift may be next to impossible" (Aikhenvald 2007: 9), let alone if it is not a matter of change, but a matter of preservation of a proto-system, not in every single detail of course, but as an overall picture. This study is meant as a first step towards a more comprehensive and in-depth research.

  . Below is an overview of the six basic verbal derivation patterns in Beja for V2:

INT bis 'bury' > boːs kitim 'arrive' > kaːtim FREQ bis 'bury' > baːbis, baːbabis kitim 'arrive' > kaːtatim REFL fif 'pour' > faf rimid 'avenge' > rimad PASS min 'shave' > aːtoː-maːn dibil 'collect' > at-dabaːl CAUS dir 'kill' > soː-dir kitim 'arrive' > si-katim RECP gid 'throw' > amoː-gaːd girim 'be inimical' > am-garaːm

Table 2 :

 2 Verb derivation patternsEach of these patterns is discussed in turn in what follows.

	4.1.3.1 Intensive
	INT bis 'bury' > boːs 'bury repeatedly'	dir 'kill' > daːr 'slaughter'	kitim 'arrive' > kaːtim 'arrive repeatedly at the
			same place'

Table 3 :

 3 Adjective patterns

		Adjective	Verb form
	aC1aːC2 C1aC2a C1aC2i(C3) majik w 'right-handed, amaːg 'bad' ataːb 'full' ataːl 'stopped, blocked' marɁa 'wide' downright'	mig 'do evil' tib 'fill' til 'stop' mirɁ 'be wide' mijak w 'be dextrous'
	sagi 'distant' C1aːC2i(C3) naːsi 'advisory' naːk w is 'short' C1aC2aːC3 tak w aːk w 'prepared, ready, finished'	sig 'be distant' nasa 'advice' nik w is 'be short' tik w ik w 'prepare'
	C1iC2aːC3	ʃik w aːn 'aromatic'	ʃik w an 'emit pleasant odour, perfume oneself'
	C1aC2C2iC3 ʃallik 'few'	ʃilik 'be few'

Table 4 :

 4 Noun patterns

		Adjective	Verb form
	C1aC2 C1iC2a C1aC2i C1aC2a C1iC2i(C3)	nak w 'pregnancy' nisa 'advise, counsel' sari 'wakefulness' nada 'dew' sala 'roast meat' mirɁi 'width' riʃid 'wealth'	nik w i 'become pregnant' nasa 'advice' sir 'keep awake' nidaj 'sweat, exude water' asal 'roast, fry' mirɁ 'be 'wide' riʃid 'raise, attend to, care for cultivation or cattle'
	C1aC2iːC3	ʃaɖiːɖ 'strip (of bark)' ʃak w iːn 'fragrance'	ʃiɖiɖ 'strip off' ʃik w an 'emit pleasant odour, perfume oneself'
	C1aC2iːC3-a	raʃiːd-a 'cattle'	riʃid 'raise, attend to, care for cultivation or cattle'
	C1iːC2aːC3	safiːf-a (f) 'splice' tiːlaːl 'stride'	sifif 'splice (us. rope)' tilil 'stride, be, go far away from home'
	C1uC2uC3	sukum 'colostrum'	sik w im 'to draw off thick milk from parturient animal'
	C1aC2oːC3	taboːk 'double-handful' tiboːk 'fill scoop with cupped hands'
	C1iC2uːC3-a	tiluːl-a 'exile'	tilil 'to stride, be, go far away from home'

Suffixed and prefixed morphemes do exist also in Semitic and Berber languages.

The citation form of Beja verbs used throughout this paper is the stem of the imperative form devoid of its inflectional morphemes, as in[START_REF] Roper | Tu Beḍawie". An elementary handbook for the use of Sudan government officials[END_REF].

Abbreviations