Antoine Lejay
email: antoine.lejay@univ-lorraine.fr

Paolo Pigato
email: paolo.pigato@inria.fr

Statistical estimation of the Oscillating Brownian Motion

Keywords: Oscillating Brownian Motion, Gaussian mixture, local time, occupation time, Arcsine distribution, Skew Brownian Motion

We study the asymptotic behavior of estimators of a two-valued, discontinuous diffusion coefficient in a Stochastic Differential Equation, called an Oscillating Brownian Motion. Using the relation of the latter process with the Skew Brownian Motion, we propose two natural consistent estimators, which are variants of the integrated volatility estimator and take the occupation times into account. We show the stable convergence of the renormalized errors' estimations toward some Gaussian mixture, possibly corrected by a term that depends on the local time. These limits stem from the lack of ergodicity as well as the behavior of the local time at zero of the process. We test both estimators on simulated processes, finding a complete agreement with the theoretical predictions.

Introduction

Diffusion processes with discontinuous coefficients attract more and more attention for simulation and modelling purposes (see references in [START_REF] Lejay | Simulating diffusion processes in discontinuous media: a numerical scheme with constant time steps[END_REF]). Many domains are actually concerned, such as geophysics [START_REF] Ramirez | Advection-dispersion across interfaces[END_REF], population ecology [START_REF] Cantrell | Diffusion Models for Population Dynamics Incorporating Individual Behavior at Boundaries: Applications to Refuge Design[END_REF][START_REF] Cantrell | Skew Brownian motion: a model for diffusion with interfaces[END_REF], finance [START_REF] Decamps | Self exciting threshold interest rates models[END_REF][START_REF] Lipton | Filling the gaps[END_REF], . . . among others. From a theoretical point of view, diffusions with discontinuous coefficients are an instance of Stochastic Differential Equations (SDE) with local time -also called skew diffusions -for which many results are contained in the work of J.-F. Le Gall [START_REF] Gall | One-dimensional stochastic differential equations involving the local times of the unknown process[END_REF].

Estimation comes together with simulation and modelling, as models need to be calibrated. This article deals with the parametric estimation of the coefficients of a one-dimensional SDE of type

Y t = x + t 0 σ(Y s ) dW s + t 0 b(Y s ) ds, (1.1)
where W is a Brownian motion and x → σ(x) takes two values {σ + , σ -} according to the sign of x, when the process Y is observed at discrete times iT /n, i = 0, . . . , n up to a time T .

In a first part, we consider that (1.1) contains no drift (b = 0). The solution Y to (1.1) is called an Oscillating Brownian Motion (OBM). This process was studied by J. Keilson and J.A. Wellner in [START_REF] Keilson | Oscillating Brownian motion[END_REF], who give some of its main properties and closed form expressions for its density and occupation time. We provide two simple estimators which generalize the integrated volatility (or averaged squared) estimator (in finance, σ is called the volatility when Y represents the logarithm of the price of an asset). For two processes Z, Z , we set

[Z, Z ] n T := n i=1 (Z i,n -Z i-1,n )(Z i,n -Z i-1,n ) and Qn T (Z, +) := 1 n n i=1 1(Z i,n ≥ 0)
with Z i,n = Z iT /n . The quantity Qn T (Z, +) is an approximation of the occupation time of Z on the positive axis up to time T . Our estimator (σ n + ) 2 of σ 2 + is then [Y + , Y + ]/ Qn T (Y, +), where Y + is the positive part of Y . A similar estimator is defined for σ 2 -. Although an analytic form is known for the density, this estimator is simpler to implement than the Maximum Likelihood Estimator based on discrete observations. Besides, it also applies when b = 0, while explicit expressions for the density become cumbersome, at best [START_REF] Lejay | One-dimensional skew diffusions: explicit expressions of densities and resolvent kernel[END_REF][START_REF] Mazzonetto | On the Exact Simulation of (Skew) Brownian Diffusion with Discontinuous Drift[END_REF].

We show first that σn + is a consistent estimator of σ + . Yet it is asymptotically biased. We also prove that √ n((σ n + ) 2σ 2 + ) converges stably to a mixture of Gaussian distributions (in which, unsurprisingly, the occupation time of the positive side appears) plus an explicit term giving the bias.

When estimating σ + , the actual size of the "useful" sample is proportional to the occupation time of R + . Therefore, a dependence on the occupation time is to be expected in any reasonable estimator. The law of the occupation time for the OBM follows an arcsine type distribution, which generalizes the one of the Brownian motion. Since these laws carry much mass close to the extremes, this amounts to say that many trajectories of this process spend most of the time on the same side of 0. Therefore, with a high probability, either σ + or σ -is only based on few observations. This affects our central limit theorem as well as the quality of the estimation of (σ + , σ -), meaning that the limit law will not be a Gaussian one, as one would expect from the approximation of quadratic variation, but a Gaussian mixture displaying heavy tails.

Another tool of importance in this framework, strictly connected with the occupation time, is the local time. Given a stochastic process Y , its local time at a point x, denoted by {L x t (Y )} t≥0 , represents the time spent by Y at x, properly re-scaled. It has a fundamental role in the study of SDEs with discontinuous coefficients. Intuitively, the local time appears when dealing with discontinuous coefficients because it helps to quantify what happens locally at the discontinuity. A Lamperti's type transform applied with the help of the Itô-Tanaka formula shows that Y is intimately related to the Skew Brownian Motion (SBM, see [START_REF] Harrison | On skew Brownian motion[END_REF][START_REF] Lejay | On the constructions of the skew Brownian motion[END_REF]) X, the solution to the SDE X t = x + W t + θL 0 t (X), -1 < θ < 1, through a simple deterministic transform X = Φ(Y ) [START_REF] Étoré | On random walk simulation of one-dimensional diffusion processes with discontinuous coefficients[END_REF][START_REF] Lejay | A scheme for simulating one-dimensional diffusion processes with discontinuous coefficients[END_REF].

In the present paper, the local time plays an important role for two reasons. First, because we use the transform X = Φ(Y ) to apply some convergence results which extend to the SBM some results of J. Jacod on the asymptotic behavior of quantities of type n -1/2 n i=1 f (X i-1,n , X i,n ). Second, because the local time itself appears in the limit of the above quantities. Actually, the asymptotic bias is related to the local time.

We then provide a second simple estimator of σ 2 ± , defined as

(m n ± ) 2 = [Y ± , Y ] n T / Qn T (Y, ±) which is also consistent. We show that √ n((m n ± ) 2 -σ 2 ± ) converges stably to a Gaussian mixture. The asymptotic bias observed in √ n((σ n ± ) 2 -σ 2 ±
) is removed through the introduction of the quadratic term [Y + , Y -] which is related to the local time. The variance of the former limit is not larger than the one of the latter.

In Corollary 3.8, we also generalize these convergence results in presence of a bounded drift term. We prove that the estimators mentioned above converge to analogous limit random variables, depending on the occupation time of the SDE under study. Unlike for the OBM, the limit law is not explicit for SDEs with general drift, since the law of the occupation time is not know, if not in special cases (see e.g. [START_REF] Kasahara | On a generalized arc-sine law for one-dimensional diffusion processes[END_REF][START_REF] Watanabe | Generalized arc-sine laws for one-dimensional diffusion processes and random walks[END_REF][START_REF] Watanabe | A density formula for the law of time spent on the positive side of one-dimensional diffusion processes[END_REF]).

The novelty of the paper lies in the treatment of a discontinuous diffusion coefficient. This implies a drastic difference with the case of regular coefficients, as the situation cannot be reduced to a Brownian one (the measure of the SBM being singular with respect to the one of the Brownian motion [START_REF] Gall | One-dimensional stochastic differential equations involving the local times of the unknown process[END_REF]). This explains the presence of an asymptotic bias for σn ± , which is removed by a correction (leading to m n ± ) involving only a fraction of order n -1/2 of the observations.

Unlike for many estimators, the framework is not the one of ergodic processes, but of null recurrent ones. On the last point, our study does not fit the situations considered e.g., in [START_REF] Altmeyer | Estimation error for occupation time functionals of stationary Markov processes[END_REF][START_REF] Höpfner | Asymptotic statistics[END_REF][START_REF] Höpfner | Limit theorems for null recurrent Markov processes[END_REF].

Differently from many estimators constructed for diffusion processes, we find a mixed normal distribution in the limit. This is due to the lack of ergodicity of the process. For diffusions, asymptotic convergence involving a mixture of normal distributions (with different type of limits) is already observed in the works of F. Florens-Zmirou [START_REF] Florens-Zmirou | On estimating the diffusion coefficient from discrete observations[END_REF] for non-parametric estimation, and of J. Jacod [START_REF] Jacod | Rates of convergence to the local time of a diffusion[END_REF][START_REF] Jacod | Discretization of processes[END_REF], from which we borrow and adapt the general treatment. The core of our proof requires the adaptation to the SBM of some results on the convergence toward the local time given in [START_REF] Jacod | Rates of convergence to the local time of a diffusion[END_REF].

There are several extensions of the present results which we consider with in subsequent articles [START_REF] Lejay | Estimation of the drift parameters of a Drifted Oscillating Brownian motion[END_REF][START_REF] Lejay | The geometric Oscillating Brownian motion: estimation and application to finance[END_REF].

First, the asymptotic behavior of the estimators allows one to construct confidence regions for (σ -, σ + ). In particular, we could set up a statistical hypothesis test to decide whether or not σ -= σ + . This is done in [START_REF] Lejay | The geometric Oscillating Brownian motion: estimation and application to finance[END_REF] to detect leverage effects in stock prices.

Second, there are some situations in which there is no reason to assume a priori knowledge of the threshold for the volatility. Since the transition density function of the Oscillating Brownian motion is explicitly known [START_REF] Keilson | Oscillating Brownian motion[END_REF], our estimator is easily coupled with a model selection procedure such as the Akaike Information Criterion (AIC) [START_REF] Akaike | Information theory and an extension of the maximum likelihood principle[END_REF]. We select possible thresholds that spans the values of the observations. For each threshold r, we estimate (σ -, σ + ) after having shifted the process by r. We then compute the likelihood Λ(r) using the estimated values of (σ -, σ + ). Then, we select the one for which Λ(r) is the highest (all the models have the same number of parameters).

Such a procedure was applied in [START_REF] Lejay | The geometric Oscillating Brownian motion: estimation and application to finance[END_REF] to 21 stocks prices. It gives estimations that are similar with the ones of [START_REF] Mota | On a continuous time stock price model with regime switching, delay, and threshold[END_REF] where a methodology coming from the study of the Self-Exciting Threshold AutoRegressive (SETAR [START_REF] Tong | Threshold models in nonlinear time series analysis[END_REF]) times series were used.

Third, we may want to couple our estimators with ones for the drift. With a two valued drift, this could be useful to detect mean-reverting effects in financial prices. One difficulty is that estimators for the drift necessarily rely on long time. In presence of a drift, our argument relying on the Girsanov theorem no longer applies as the Girsanov weight degenerates as the time horizon increases. Hence, a suitable construction that encompass both the diffusivity and the drift has to be found.

Content of the paper

In Section 2 we define the Oscillating Brownian Motion (OBM) and recall some useful properties. In Section 3 we define our estimators and state precisely the convergence theorems. These results are then proved is Section 4. In Section 5 we consider the Oscillating Random Walk, a discrete process that can be used to construct the OBM, and study an estimator on this discrete process. Section 6 is devoted to the implementation of the estimators of the OBM, and contains numerical experiments showing the good behavior of the estimators in practice.

Notations

For notational convenience, we work on the time interval [0, 1]. Our results can be extended to a general time interval via a space-time re-scaling (see Remark 3.7). Throughout the paper, we use the following notation for convergence of random variables: When we do not specify y, we mean the local time at 0: L t (M ) = L 0 t (M ). For fixed n ∈ N, we consider the discretization of [0, 1] given by 0, 1/n, . . . , 1. For any process (M t ) t∈[0,1] , we write M k,n = M k/n . For any processes M, M we also set the "discrete bracket"

[M, M ] n 1 = n k=1 (M k,n -M k-1,n )( Mk,n -Mk-1,n ).
We also write

[M ] n 1 = [M, M ] n 1 .

Oscillating Brownian Motion

For two parameters σ + , σ -> 0, we define the diffusion coefficient σ as follows:

σ(y) = σ + 1(y ≥ 0) + σ -1(y < 0), ∀y ∈ R. (2.1)
Let W be a Brownian motion with its (completed) natural filtration (G t ) t≥0 on a probability space (Ω, F, P). From now on, we denote by Y the unique strong solution to

Y t = Y 0 + t 0 σ(Y s ) dW s . (2.2)
Strong existence and uniqueness of Y is granted by the results of [START_REF] Gall | One-dimensional stochastic differential equations involving the local times of the unknown process[END_REF]. Following the terminology of [START_REF] Keilson | Oscillating Brownian motion[END_REF], we call Y an Oscillating Brownian Motion (OBM) We recall some known properties of Y , proved in [START_REF] Keilson | Oscillating Brownian motion[END_REF]. For (t, x, y) ∈ (0, 1]×R 2 , let p(t, x, y) be the density of Y t in y with respect to the Lebesgue measure, with initial condition Y 0 = x. In [START_REF] Keilson | Oscillating Brownian motion[END_REF] explicit formulas are given for the transition density, which is based on linear combination of Gaussian ones. The map x → p(t, x, y) (for fixed (t, y)) and y → σ(y) 2 p(t, x, y) (for fixed (t, x)) are continuous. In particular, p(t, x, •) is discontinuous and has a jump at 0.

The transition density function with respect to the Lebesgue measure is

p(t, x, y) =                                  2σ + σ -(σ + + σ -) 1 √ 2πt e -( x σ + -y σ - ) 2 1 2t for x ≥ 0, y < 0, 2σ - σ + (σ + + σ -) 1 √ 2πt e -( y σ + -x σ - ) 2 1 2t for x < 0, y ≥ 0, 1 σ + √ 2πt e -( x σ + -y σ + ) 2 1 2t + σ --σ + σ -+ σ + e -( x σ + + y σ + ) 2 1 2t for x ≥ 0, y ≥ 0, 1 σ + √ 2πt σ - σ + e -( x σ - -y σ - ) 2 1 2t + σ + -σ + σ + + σ - σ - σ + e -( x σ - + y σ - ) 2 1 2t
for x < 0, y < 0.

(2.3) Integrating (2.3),

P(Y t < 0|Y 0 > 0) = 2σ + σ + + σ - Φ -Y 0 /(σ + √ t) , P[Y t > 0 | Y 0 < 0] = 2σ - σ -+ σ + Φ Y 0 /(σ - √ t) , (2.4)
where Φ is the cumulative density function of the standard Gaussian. The law of the occupation time of R + is also computed in [START_REF] Keilson | Oscillating Brownian motion[END_REF]. Let the occupation time of R + be defined as

Q + t = Leb{s ≤ t : Y s ≥ 0}. (2.5)
The distribution of Q + t , with Y 0 = 0 a.s., is explicit. The scaling Q + 1 law = Q + t /t holds, and

P(Q + 1 ∈ du) = 1 π 1 u(1 -u) σ + /σ - 1 -(1 -(σ + /σ -) 2 )u du for 0 < u < 1. (2.6)
This generalizes the arcsine law for the occupation time of the Brownian Motion. The occupation time

Q -on R -is easily computed from Q + since obviously, Q - t + Q + t = t for any t ≥ 0.
We introduce the process (X t ) t≥0 , setting X t = Y t /σ(Y t ) for t ≥ 0. It follows from the Itô-Tanaka formula that X is a Skew Brownian Motion (SBM, see [START_REF] Lejay | On the constructions of the skew Brownian motion[END_REF]), meaning that X satisfies the following SDE:

X t = X 0 + B t + θL t (X), (2.7)
where B is a Brownian Motion, L t (X) is the symmetric local time of X at 0, X 0 = Y 0 /σ(Y 0 ) and the coefficient θ is given by θ

= σ --σ + σ -+ σ + . (2.8)
We write from now on BM for Brownian Motion, SBM for Skew Brownian Motion, OBM for Oscillating Brownian Motion. The local times of X and Y are related by

L t (X) = σ + + σ - 2σ + σ - L t (Y ) (2.9)
(see [START_REF] Lejay | On the constructions of the skew Brownian motion[END_REF] for a special case from which we easily recover this formula).

3 Main results

The stable convergence

Before stating our results, we need to recall the notion of stable convergence, which was introduced by A. Rényi [START_REF] Rényi | On stable sequences of events[END_REF]. We refer to [START_REF] Jacod | Limit theorems for stochastic processes[END_REF] or [START_REF] Jacod | Discretization of processes[END_REF] for a detailed exposition. Let Z n a sequence of E-valued random variables defined on the same probability space (Ω, F, P). Let Z be an E-valued random variable defined on an extension, ( Ω, F, P). We then say that Z n converges stably to Z (and write

Z n sl ---→ n→∞ Z) if: E[Y f (Z n )] ---→ n→∞ Ẽ[Y f (Z)]
for all bounded continuous functions f on E and all bounded random variables Y on (Ω, F) (or, equivalently, for all Y as above and all functions f which are bounded and Lipschitz). This notion of convergence is stronger than convergence in law, but weaker than convergence in probability. We use in this paper the following crucial result: for random variables

Y n , Z n (n ≥ 1), Y and Z, if Z n sl ---→ n→∞ Z and Y n p ---→ n→∞ Y then (Y n , Z n ) sl ---→ n→∞ (Y, Z). (3.1)

Estimators for the parameters of the Oscillating Brownian motion

Let us assume that we observe the process Y solution to (2.2) at the discrete times 0, 1/n, . . . , 1.

We want to estimate σ + , σ -from these observations. A natural estimator for the occupation time Q + 1 defined in (2.5) is given by the Riemann sums (see Section 4.4):

Qn 1 (Y, +) = n k=1 1(Y k,n ≥ 0) n . (3.2)
We define now σn + as

σn + = [Y + ] n 1 Qn 1 (Y, +) . (3.3)
We will show that σn + is a consistent estimator for σ + . Similarly, we set

Qn 1 (Y, -) = n k=1 1(Y k,n < 0) n = 1 -Qn 1 (Y, +) and σn -= [Y -] n 1 Qn 1 (Y, -)
.

Finally, we define our estimator of the vector (σ + , σ -) as

σn = σn + σn - . Theorem 3.1. Let Y be solution of (2.
2) with Y 0 = 0 a.s., and σn defined in (3.3). Then (i) The estimator is consistent:

σn p ---→ n→∞ σ + σ - .
(ii) There exists an extension ( Ω, F , P) of (Ω, F, P) carrying a Brownian motion B independent from W such that

√ n (σ n + ) 2 -σ 2 + (σ n -) 2 -σ 2 - sl ---→ n→∞      √ 2σ 2 + Q + 1 1 0 1(Y s > 0) d Bs √ 2σ 2 - 1 -Q + 1 1 0 1(Y s < 0) d Bs      - 1 Q + 1 1 1-Q + 1 2 √ 2 3 √ π σ -σ + σ + + σ - L 1 (Y ).
(3.4) The stable limit in the above result depends on the path of Y through its local time L 1 (Y ) and its occupation time Q ± 1 . By identifying the distribution of the limit, we can rewrite (3.4) as a convergence in distribution involving elementary random variables as

√ n (σ n + ) 2 -σ 2 + (σ n -) 2 -σ 2 - law ---→ n→∞   √ 2σ 2 + √ Λ N 1 √ 2σ 2 - √ 1-Λ N 2   - 8 √ 2 3 √ π (σ -σ + ) 2 σ + + σ - ξ (1 -Λ)σ 2 -+ Λσ 2 +   1-Λ Λ Λ 1-Λ   =     √ 2σ 2 + √ Λ N 1 -8 3 √ π 1 r+1 ξ √ 1-Λ √ (1-Λ)+Λr 2 √ 2σ 2 - √ 1-Λ N 2 -8 3 √ π 1 1/r+1 ξ √ Λ √ Λ+(1-Λ)/r 2     , (3.5)
where

r = σ + /σ -, ξ, N 1 , N 2 , Λ are mutually independent, ξ ∼ exp(1), N 1 , N 2 ∼ N (0, 1)
and Λ follows the modified arcsine law (2.6) with density

p Λ (τ ) = 1 πτ 1/2 (1 -τ ) 1/2 r 1 -(1 -r 2 )τ . Remark 3.2. The Brownian case is σ =: σ + = σ -, r = 1. The limit law is √ 2σ 2   1 √ Λ N 1 -4 3 √ π ξ √ 1 -Λ 1 √ 1-Λ N 2 -4 3 √ π ξ √ Λ   ,
where Λ follows the classical arcsine law (see [START_REF] Lévy | Sur certains processus stochastiques homogènes[END_REF][START_REF] Revuz | Continuous martingales and Brownian motion[END_REF]).

Remark 3.3. In (4.23) we prove

√ n[Y + , Y -] n 1 sl ---→ n→∞ 2 √ 2 3 √ π σ + σ - σ + + σ - L 1 (Y ).
Actually, each term of type (Y

+ t i -Y + t i-1 )(Y - t i -Y - t i-1 ) vanishes unless sgn(Y t i ) = sgn(Y t i-1 ). Thus, √ n[Y + , Y -] n 1
provides us with as an estimator of the local time since it somehow counts the number of crossings of zero (cf. [START_REF] Jacod | Rates of convergence to the local time of a diffusion[END_REF][START_REF] Lejay | Two consistent estimators for the Skew Brownian motion[END_REF]). Remark 3.4. We assume Y 0 = 0 a.s. because we need Y 0 to visit both R + and R -. This happens a.s. in any right neighborhood of 0 if the diffusion starts from 0. If the initial condition is not 0, we shall wait for the first time at which the process reaches 0, say τ 0 , and consider the (random) interval [τ 0 , T ].

We define now a different estimator for σ ± by

m n + := [Y + , Y ] n 1 Qn 1 (Y, +) , m n -:= [Y -, Y ] n 1 Qn 1 (Y, -)
and

m n := m n + m n - . (3.6)
Theorem 3.5. Let Y be solution of (2.2) with Y 0 = 0 a.s., and m n defined in (3.6). The following convergence holds for n → ∞:

√ n (m n + ) 2 -σ 2 + (m n -) 2 -σ 2 - sl ---→ n→∞    √ 2σ 2 + Q + 1 1 0 1(Y s > 0) d Bs √ 2σ 2 - 1-Q + 1 1 0 1(Y s < 0) d Bs    ,
where B is a BM independent of Y on an extension ( Ω, F, P) of (Ω, F, P). We can rewrite such a convergence as follows:

√ n (m n + ) 2 -σ 2 + (m n -) 2 -σ 2 - law ---→ n→∞   √ 2σ 2 + √ Λ N 1 √ 2σ 2 - √ 1-Λ N 2   , (3.7)
where N 1 , N 2 , Λ are mutually independent, N 1 , N 2 ∼ N (0, 1) and Λ follows the modified arcsine law with density given by (2.6), with r = σ + /σ -.

Remark 3.6. Comparing Theorems 3.1 and 3.5, we see that an asymptotic bias is present in σn , but not in m n . This bias has the same order (∼ 1/ √ n) as the "natural fluctuations" of the estimator. Because the local time is positive, it is more likely that σn ± underestimates σ ± . For a more quantitative comparison between the convergence of the two estimators, see Remark 4.18. In Section 6, we compare the two estimators in practice.

Remark 3.7. Theorem 3.5 gives the asymptotic behavior for an estimator of (σ + , σ -) in the presence of high frequency data, yet with fixed time horizon T = 1. The OBM enjoys a scaling property: if Y is an OBM issued from 0, then ( √ cY t/c ) t∈R + is an OBM issued form 0, for any c > 0 constant (see [38, Exercise IX.1.17, p. 374]). Using this fact, we can easily generalize Theorem 3.5 to the case of data on a time interval [0, T ] for some fixed T > 0. We set

m n,T + := [Y + , Y ] n T Qn T (Y, +) , m n,T -:= [Y -, Y ] n T Qn T (Y, -) . (3.8)
The estimator is consistent and we have the following convergence:

√ n (m n,T + ) 2 -σ 2 + (m n,T -) 2 -σ 2 - sl ---→ n→∞    √ 2T σ 2 + Q + T T 0 1(Y s > 0) d Bs √ 2T σ 2 - T -Q + T T 0 1(Y s < 0) d Bs    ,
where B is a BM independent of W on an extension of the underlying probability space. The limiting random variable follows the law given in (3.7), which actually does not depend on T .

A slightly different approach is to imagine that our data are not in high frequency, but that we observe the process at regular time intervals, for a long time. In this case it is more reasonable to consider an OBM (Y t ) t∈R + , construct an estimator depending on (Y i ) i=0,1,...,T -1,T , T ∈ N, and then take the limit in long time. We set

µ T + := [Y + , Y ] T T QT T (Y, +) , µ T -:= [Y -, Y ] T T QT T (Y, -)
.

Using again Theorem 3.5 and the diffusive scaling of the OBM, we have the following convergence:

√ T (µ T + ) 2 -σ 2 + (µ T -) 2 -σ 2 - law ----→ T →∞   √ 2σ 2 + √ Λ N 1 √ 2σ 2 - √ 1-Λ N 2   .
The limit distribution is again the law given in (3.7). Theorem (3.1) can also be generalized to high frequency data on an interval [0, T ] and to equally spaced data in long time, using the diffusive scaling and (4.21)-(4.22). For example, analogously to (3.8), we define σn,T

+ := [Y + , Y + ] n T Qn T (Y, +) , σn,T -:= [Y -, Y -] n T Qn T (Y, -)
.

Again, the limit law does not change and is the one given in (3.5).

A generalization to OBM with drift

We consider now a wider class of processes, adding a drift term to equation (2.2). Formally, let now ξ be the strong solution to

dξ t = b(ξ t ) dt + σ(ξ t ) dW t , (3.9)
with ξ 0 = 0, σ defined in (2.1) and b measurable and bounded. Again, strong existence and uniqueness of the solution to (3.9) is ensured by the results of [START_REF] Gall | One-dimensional stochastic differential equations involving the local times of the unknown process[END_REF]. Let m n (ξ) be defined as in (3.6):

m n + (ξ) = [ξ + , ξ] n 1 Qn 1 (ξ, +) , m n -(ξ) = [ξ -, ξ] n 1 Qn 1 (ξ, -) and m n (ξ) := m n + (ξ) m n - (ξ)
.

Let us also denote

Q + 1 (ξ) = 1 0 1(ξ s > 0) ds. Corollary 3.8. The following convergence holds for n → ∞: √ n (m n + (ξ)) 2 -σ 2 + (m n -(ξ)) 2 -σ 2 - sl ---→ n→∞    √ 2σ 2 + Q + 1 (ξ) 1 0 1(ξ s > 0) d Bs √ 2σ 2 - 1-Q + 1 (ξ) 1 0 1(ξ s < 0) d Bs    ,
where B is a BM independent of W on an extension of the underlying probability space. We can rewrite such convergence as follows:

√ n (m n + (ξ)) 2 -σ 2 + (m n -(ξ)) 2 -σ 2 - law ---→ n→∞   √ 2σ 2 + √ Θ N 1 √ 2σ 2 - √ 1-Θ N 2   ,
where N 1 , N 2 , Θ are mutually independent, N 1 , N 2 ∼ N (0, 1) and

Θ law = Q + 1 (ξ).
Remark 3.9. Unlike for the OBM, the limit law is not explicit in Corollary 3.8, since the law of the occupation time of the positive axes is not know in general (See e.g., [START_REF] Kasahara | On a generalized arc-sine law for one-dimensional diffusion processes[END_REF][START_REF] Lamperti | An occupation time theorem for a class of stochastic processes[END_REF][START_REF] Watanabe | Generalized arc-sine laws for one-dimensional diffusion processes and random walks[END_REF][START_REF] Watanabe | A density formula for the law of time spent on the positive side of one-dimensional diffusion processes[END_REF]).

On the other hand, some information on the law of Θ can be obtained, at least in some special cases, via Laplace transform.

We also stress that this dependence on the occupation time is due the actual sample size of the data giving us useful information. Indeed, when estimating σ + , the number of intervals that we can use is proportional to the occupation time of R + . Analogously for the negative part.

Remark 3.10. Corollary 3.8 holds if the Girsanov weight (G t ) t defined in (4.24) below is uniformly integrable, a condition ensured by the boundedness of b. An example of model with non-bounded b fitting into this framework is the SET-Vasicek [START_REF] Decamps | Self exciting threshold interest rates models[END_REF], a generalization of the Vasicek interest rate model to a non-constant volatility, given exactly by (2.1):

dξ t = -α(ξ t -β) dt + σ(ξ t ) dW t .
Remark 3.11. The scaling property described in Remark 3.7 no longer holds in this situation, so that the estimator can only be used in the "high frequency" setting.

Proofs of the convergence theorem

This section is devoted to the proof Theorems 3.1 and 3.5. We first deal with some general approximation results which are well known for diffusions with regular coefficients (see [START_REF] Jacod | Rates of convergence to the local time of a diffusion[END_REF][START_REF] Jacod | Discretization of processes[END_REF][START_REF] Jacod | Limit theorems for stochastic processes[END_REF]), but not in the framework considered here with discontinuous coefficients (when θ = 0, the law of the SBM is singular with respect to the one of the BM [START_REF] Gall | One-dimensional stochastic differential equations involving the local times of the unknown process[END_REF]).

Following [START_REF] Étoré | On random walk simulation of one-dimensional diffusion processes with discontinuous coefficients[END_REF][START_REF] Lejay | A scheme for simulating one-dimensional diffusion processes with discontinuous coefficients[END_REF], we use the connection between the OBM and the SBM through a Lamperti-type transform. Hence, we apply the results of [START_REF] Lejay | Two consistent estimators for the Skew Brownian motion[END_REF] to the convergence of estimators of quadratic variation, covariation and occupation time for these processes. Finally, we use all these results to prove the main Theorems 3.1 and 3.5.

Approximation results

Let us write

Leb(φ) = ∞ -∞ φ(α) dα
for the Lebesgue integral of a function. In [START_REF] Lejay | Two consistent estimators for the Skew Brownian motion[END_REF], the following approximation result, adapted from [START_REF] Jacod | Rates of convergence to the local time of a diffusion[END_REF], is proved for the SBM solution of (2.7).

Lemma 4.1. Let f be a bounded function such that |x| k |f (x)| dx < ∞ for k = 0, 1, 2 and X be a SBM of parameter θ ∈ [-1, 1] (i.e., the solution to (2.7)). Then for any a > 0,

P 1 √ n n-1 i=1 f (X i,n √ n) -λθ (f )L 1 (X) > a ---→ n→∞ 0,
where (L t (X)) t≥0 is the symmetric local time at 0 of the SBM X and

λθ (f ) = (1 + θ) Leb(f ⊕ ) + (1 -θ) Leb(f ) (4.1) with f ⊕ (x) = f (x)1(x ≥ 0) and f (x) = f (x)1(x ≤ 0).
Remark 4.2. In particular, when θ = 0, X is a BM and the coefficient in front of the local time is simply Leb(f ). We recover there a special case of Theorem 4.1 in [START_REF] Jacod | Rates of convergence to the local time of a diffusion[END_REF] by J. Jacod.

We prove now an approximation result for the OBM.

Lemma 4.3. Let Y be the OBM in (2.2). Let f be a bounded function such that

|x| k |f (x)| dx < ∞ for k = 0, 1, 2.
Then for any a > 0,

P 1 √ n n-1 i=1 f (Y i,n √ n) -λ σ (f )L 1 (Y ) > a ---→ n→∞ 0,
where

(L t (Y )) t≥0 is the local time of Y and λ σ (f ) = Leb(f ⊕ ) σ 2 + + Leb(f ) σ 2 - . (4.2)
Proof. Recall Y t /σ(Y t ) = X t , (2.9) and (2.8). Let f (x) = f (σ(x)x). We have

λθ ( f ) = (1 + θ) Leb( f ⊕ ) + (1 -θ) Leb( f ) = 2σ - σ + (σ -+ σ + ) Leb(f ⊕ ) + 2σ + σ -(σ -+ σ + ) Leb(f -) = 2σ + σ - (σ -+ σ + ) λ σ (f ), so λ σ (f )L t (Y ) = λθ ( f )L t (X)
. Therefore, from Lemma 4.1, for any a > 0,

P 1 √ n n-1 i=1 f (Y i,n √ n) -λ σ (f )L t (Y ) > a = P 1 √ n n-1 i=1 f (X i,n √ n) -λθ ( f )L 1 (X) > a ---→ n→∞ 0.
This concludes the proof.

We state now a very special case of Theorem 3.2 in [START_REF] Jacod | On continuous conditional Gaussian martingales and stable convergence in law[END_REF], that we apply several times in this work. The version in [START_REF] Jacod | On continuous conditional Gaussian martingales and stable convergence in law[END_REF] holds for semimartingales, not only martingales, the processes involved can be multi-dimensional, and the limit process is not necessarily 0. Anyway, we do not need this general framework here. Stating the theorem only for one-dimensional martingales converging to 0 allows us to keep a simpler notation, which we introduce now: for each càdlàg process J we write ∆ n i J = J i/n -J (i-1)/n . Consider a filtered probability space (Ω, F, F , P) carrying a Brownian motion B. The filtration F = (F t ) t∈[0,1] is the natural (completed) one for the Brownian motion. We define the filtration F n as the "discretization" defined by F n t = F [nt]/n . Theorem 4.4 (Simplified form of Lemma 3.2 in [START_REF] Genon-Catalot | On the estimation of the diffusion coefficient for multidimensional diffusion processes[END_REF]). We consider a triangular array {χ n i } i=1,...,n of random variables where each

χ n i is F i/n -measurable, square-integrable and satisfies E[χ n i | F i-1 n ] = 0. If n i=1 E |χ n i | 2 F i-1 n p ---→ n→∞ 0, (4.3)
then Z n 1 = n i=1 χ n i converges to 0 in probability as n → ∞.

Remark 4.5. In [START_REF] Jacod | On continuous conditional Gaussian martingales and stable convergence in law[END_REF] some kind of uniform integrability is assumed in the limit, whereas here we do not ask explicitly for such a condition. The reason is that the uniform integrability assumption is implied by the fact that the limit in (4.3) is 0.

√ n[L(β)] n 1 = √ n n i=1 (L i,n (β) -L i-1,n (β)) 2 p ---→ n→∞ 4 √ 2 3 √ π L 1 (β).
We split the proof of this result in the next tree lemmas. We start with the explicit computations of the moments of the Brownian local time.

Lemma 4.7. For p ≥ 1, we set φ p (α

) := E[L 1 (β) p | β 0 = α]. We have Leb(φ p ) = 2 p + 1 E|N | p+1 , (4.4)
where N denotes a standard Gaussian random variable. Besides, the following tail estimates hold for p = 2, 4 and α = 0.

φ 2 (α) ≤ e -α 2 /2 |α| √ 2π and φ 4 (α) ≤ 16 √ 2 √ π e -α 2 /4 |α| . (4.5)
Remark 4.8. These functions φ p will be useful when applying Lemma 4.1 and Remark 4.2, taking, for fixed p, f = φ p . Inequalities (4.5) imply that the integrability condition for f is satisfied and the theorem can be applied.

Proof. Formula (6) in [START_REF] Takács | On the local time of the Brownian motion[END_REF] gives the following expression for the moments of the Brownian local time

φ p (α) = 2p ∞ 0 x p-1 P(N ≥ |α| + x) dx.
To apply Remark 4.2 we need to compute the following integral

Leb(φ p ) = ∞ -∞ φ p (α) dα = 2 ∞ 0 φ p (α) dα = 2 ∞ 0 2p ∞ 0 x p-1 ∞ 0 e -t 2 /2 √ 2π 1 [t≥|α|+x] dt dx dα.
Changing the order of integration by Fubini-Tonelli's theorem,

Leb(φ p ) = 4p ∞ 0 e -t 2 /2 √ 2π dt ∞ 0 x p-1 dx ∞ 0 1 [t≥|α|+x] dα = 2 p + 1 E|N | p+1 , so (4.4
) is proved. We now use the following bound for Gaussian tails:

∞ x e -t 2 /2 √ 2π dt ≤ e -x 2 /2
x √ 2π . We apply it twice and find the upper bound for p = 2:

φ 2 (α) = 4 ∞ 0 x ∞ 0 e -t 2 /2 √ 2π 1 [t≥|α|+x] dt dx ≤ e -α 2 /2 |α| √ 2π .
For p = 4 we apply the same inequality:

φ 4 (α) = 8 ∞ 0 x 3 ∞ 0 e -t 2 /2 √ 2π 1 [t≥|α|+x] dt dx ≤ 8 ∞ 0 x 3 e -(α+x) 2 /2 (|α| + x) √ 2π dx ≤ 8 ∞ 0 x 2 e -(α+x) 2 /2 √ 2π dx.
Now, since xe -x ≤ e -1 for all x ≥ 0,

φ 4 (α) ≤ 16 ∞ 0 e -(|α|+x) 2 /4 √ 2π dx ≤ 32 e -α 2 /4 |α| √ 2π .
Hence the result.

We consider now the quadratic sum in Lemma 4.6, and write

√ n n i=1 (L i,n (β) -L i-1,n (β)) 2 = √ n n i=1 E[(L i,n (β) -L i-1,n (β)) 2 |H i-1,n ] + √ n n i=1 (L i,n (β) -L i-1,n (β)) 2 -E[(L i,n (β) -L i-1,n (β)) 2 |H i-1,n ] .
In the next two lemmas we prove the convergence of the two summands. Lemma 4.6 follows directly.

Lemma 4.9. Let L(β) be the Brownian local time at 0. The following convergence holds:

√ n n i=1 E[(L i,n (β) -L i-1,n (β)) 2 |H i-1,n ] p ---→ n→∞ 4 √ 2 3 √ π L 1 (β).
Proof. The diffusive scaling property of (β, L(β)) implies that for any p ≥ 1,

E[(L i,n (β) -L i-1,n (β)) p | H i-1,n ] = 1 n p/2 E L 1 (β) p β 0 = √ nβ i-1,n . (4.6)
Setting p = 2 and since Leb(φ 2 ) = 2 3 E|N | 3 from (4.4), Remark 4.2 below Lemma 4.1 implies that

√ n n i=1 E[(L i,n (β) -L i-1,n (β)) 2 |H i-1,n ] = n i=1 1 √ n φ 2 (β i-1,n √ n) p ---→ n→∞ 4 √ 2 3 √ π L 1 (β).
Hence the result.

We consider now the martingale part.

Lemma 4.10. With

H i,n := (L i,n (β) -L i-1,n (β)) 2 -E[(L i,n (β) -L i-1,n (β)) 2 |H i-1,n ], it holds that √ n n i=1 H i,n p ---→ n→∞ 0.
Proof. The statement is proved using Theorem 4.4 by setting χ n i := √ nH i,n . We prove (4.3).

From (4.6) with p = 4,

E H 2 i,n H i-1,n ≤ E (L i,n (β) -L i-1,n (β)) 4 H i-1,n = 1 n 2 φ 4 (β i-1,n √ n). With Remark 4.2 below Lemma 4.1, n -1/2 n i=1 φ 4 (β i-1,n √ n) converges in probability to
Leb(φ 4 )L 1 (β) because of (4.5). Thus,

n n i=1 E H 2 i,n H i-1 ≤ 1 √ n 1 √ n n i=1 φ 4 (β i-1,n √ n) p ---→ n→∞ 0.
The proof is then complete.

Scaled quadratic covariation of skew Brownian motion and its local time

We now give some results on the scaled quadratic covariation between the SBM and its local time. For the Brownian motion W with the filtration G = (G t ) t≥0 of Section 2, we consider X the strong solution to X t = x + W t + θL t (X) for θ ∈ [-1, 1] and L(X) its local time (apart from the results in [START_REF] Gall | One-dimensional stochastic differential equations involving the local times of the unknown process[END_REF], strong existence for the SBM has been proved first in [START_REF] Harrison | On skew Brownian motion[END_REF]).

Lemma 4.11. For X and L(X) as above, the following convergence holds:

√ n[X, L(X)] n 1 = √ n n i=1 (X i,n -X i-1,n )(L i,n (X) -L i-1,n (X)) p ---→ n→∞ 0, (4.7)
√ n[|X|, L(X)] n 1 = √ n n i=1 (|X i,n | -|X i-1,n |)(L i,n (X) -L i-1,n (X)) p ---→ n→∞ 0, (4.8) √ n[X + , L(X)] n 1 = √ n n i=1 (X + i,n -X + i-1,n )(L i,n (X) -L i-1,n (X)) p ---→ n→∞ 0. (4.9)
We set

Z i,n := (X i,n -X i-1,n )(L i,n (X) -L i-1,n (X)) (4.10)
and write

√ n n i=1 Z i,n = √ n n i=1 E[Z i,n |G i-1,n ] + √ n n i=1 Z i,n -E[Z i,n |G i-1,n ] .
We prove (4.7) in the next two lemmas. Once (4.7) is proved, (4.8) follows since |X| is a SBM with parameter θ = 1 (this is immediate from the Itô-Tanaka formula applied to (2.7) by noting that for θ = 1, the SDE (2.7) is the one of a reflected Brownian motion) while (4.9) follows from a combination of (4.7) and (4.8) since X + = |X|+X 2 . Lemma 4.12. With Z i,n defined in (4.10), the following convergence holds:

√ n n i=1 E[Z i,n | G i-1,n ] p ---→ n→∞ 0. (4.11)
Proof. We express first E(X tx) 2 as a function of x using the law of the SBM. The density transition function of the SBM is [START_REF] Lejay | On the constructions of the skew Brownian motion[END_REF][START_REF] Walsh | A diffusion with discontinuous local time[END_REF] p θ (t, x, y) := p(t, xy) + sgn(y)θp(t, |x| + |y|)

where p(t, x) = (2πt) -1/2 e -x 2 /(2t) , the Gaussian density. Therefore

E(X t -x) 2 = EB 2 t + θtψ(x/ √ t) with ψ(x) := ∞ -∞ (x -y) 2 sgn(y) e -(|x|+|y|) 2 /2 √ 2π dy.
We compute ψ for x > 0:

ψ(x) = - 0 -∞ (x -y) 2 e -(x-y) 2 /2 √ 2π dy + ∞ 0 (x -y) 2 e -(x+y) 2 /2 √ 2π dy = - ∞ x z 2 e -z 2 /2 √ 2π dz + ∞ 0 (x -y) 2 e -(x+y) 2 /2 √ 2π dy and ∞ 0 (x -y) 2 e -(x+y) 2 /2 √ 2π dy = ∞ 0 (x + y) 2 e -(x+y) 2 /2 √ 2π dy -4x ∞ 0 y e -(x+y) 2 /2 √ 2π dy = ∞ x z 2 e -z 2 /2 √ 2π dz -4x ∞ 0 (y + x) e -(x+y) 2 /2 √ 2π dy + 4x 2 ∞ 0 e -(x+y) 2 /2 √ 2π dy = ∞ x z 2 e -z 2 /2 √ 2π dz -4x ∞ x z e -z 2 /2 √ 2π dz + 4x 2 ∞ x e -z 2 /2 √ 2π dz.
So for x > 0

ψ(x) = -4x ∞ x z e -z 2 /2 √ 2π dz + 4x 2 ∞ x e -z 2 /2 √ 2π dz = 4x(x(1 -Φ(x)) -p(1, x))
and

∞ 0 ψ(x) dx = 2 E[|N | 3 ] 3 -E[|N |] = - 2 √ 2 3 √ π .
With the change of variable y → -y, we see that ψ is an odd function. Thus,

∞ -∞ ψ(x) dx = 0. Recall now (2.7). Writing (X t -x) -θL t (X) = B t , (X t -x) 2 + θ 2 L t (X) 2 -2θ(X t -x)L t (X) = B 2 t . Recall that (|X|, L(X)) law = (|β|, L(β)), where β is a BM. Moreover, φ 2 defined in Lemma 4.7 is symmetric. So EL t (X) 2 = EL t (β) 2 = tφ 2 (β 0 / √ t) = tφ 2 (X 0 / √ t). Therefore E(X t -x)L t (X) = θ 2 EL t (X) 2 + 1 2θ (E(X t -x) 2 -EB 2 t ) = tθ 2 φ 2 (x/ √ t) + t 2 ψ(x/ √ t) and E[(X i,n -X i-1,n )(L i,n (X) -L i-1,n (X)) | G i-1,n ] = 1 2n (θφ 2 (X i-1,n √ n) + ψ(X i-1,n √ n)).
Since φ 2 is symmetric and applying (4.1),

λθ (φ 2 ) = Leb(φ 2 ) = 2 E[|N | 3 ] 3 .
Since ψ is anti-symmetric and (4.1)

λθ (ψ) = (1 + θ) Leb(ψ + ) + (1 -θ) Leb(ψ -) = 2θ Leb(ψ + ) = 4θ E[|N | 3 ] 3 -E[|N |] so λθ θφ 2 2 + ψ 2 = θ E[|N | 3 ] -2E[|N |] = 0. It is straightforward to check that |x| k ( θ 2 φ 2 (x) + 1 2 ψ(x)) dx < ∞ for k = 0,
√ n n i=1 (Z i,n -E[Z i,n | G i-1,n ]) p ---→ n→∞ 0.
Proof. To apply Theorem 4.4, we prove (4.3):

E (Z i,n -E[Z i,n | G i-1,n ]) 2 G i-1,n ≤ E (X i,n -X i-1,n ) 2 (L i,n (X) -L i-1,n (X)) 2 G i-1,n ≤ E (X i,n -X i-1,n ) 4 G i-1,n 1/2 E (L i,n (X) -L i-1,n (X)) 4 G i-1,n 1/2
and we upper bound the two factors. We know

E (X i,n -X i-1,n ) 4 |G i-1,n 1/2 ≤ C n . Recall again that (|X|, L(X)) law = (|β|, L(β))
, where β is a BM, and that φ 4 is symmetric. From (4.6),

E (L i,n (X) -L i-1,n (X)) 4 G i-1,n 1/2 ≤ 1 n φ 4 (X i-1,n √ n) 1/2 .
Because of (4.5), we apply Lemma 4.1 so that

nE (Z i,n -E[Z i,n | G i-1,n ]) 2 G i-1,n ≤ C √ n n i=1 1 √ n φ 4 (X i-1 √ n) 1/2 p ---→ n→∞ 0.
Hence the result.

Approximation of occupation time

In this section we extend the result in [START_REF] Ngo | On the discrete approximation of occupation time of diffusion processes[END_REF], which is proved for diffusions with smooth coefficients, to the OBM. We consider approximating the occupation time of [0, ∞) up to time 1:

Q + 1 = Leb(s ∈ [0, 1] : Y s ≥ 0) = 1 0 1 {Ys≥0} ds.
As previously, we suppose that we know the values Y i,n of Y on a grid of time lag 1/n.

Theorem 4.14. Let Y be given in (2.2) and Qn 1 (Y, +) be given by (3.2). The following convergence holds:

√ n Qn 1 (Y, +) - t 0 1 {Ys≥0} p ---→ n→∞ 0.
For i = 1, . . . , n, we consider

J i,n = 1 n 1 {Y i-1,n ≥0} - i n i-1 n 1 {Ys≥0} ds = sgn(Y i-1,n ) i n i-1 n 1 {Y i-1,n Ys<0} ds, U i,n = J i,n -E[J i,n | G i-1,n ]. (4.12)
The U i,n are martingale increments. We write

√ n Qn 1 (Y, +) - 1 0 1 {Ys≥0} = √ n n i=1 E[J i,n | G i-1,n ] + √ n n i=1 U i,n .
In the following lemmas we prove the convergence of the two summands.

Remark 4.15. In [START_REF] Ngo | On the discrete approximation of occupation time of diffusion processes[END_REF] it is proved that the estimator times n 3/4 is tight, so the speed of convergence proved there, holding only for smooth coefficients, is faster than the speed proved here. We are actually able to prove that n 3/4 is the speed of convergence for the martingale part n i=1 U i,n also for the OBM (and other diffusions with discontinuous coefficients), but not for the drift part

n i=1 E[J i,n | G i-1,n ].
We would need the central limit theorem giving the speed of convergence corresponding to the law of large numbers proved in Lemma (4.3), but this looks quite hard to get in the case of discontinuous diffusion coefficients. Anyway, for our goal of estimating the parameters of the OBM, the fact that our estimator multiplied with the "diffusive scaling"

√ n converges to 0 is enough. Actually, this result depends on a compensation between two terms in Lemma 4.16 which holds for this particular diffusion but for which we do not have results holding for a wider class of SDEs with discontinuous coefficients.

Lemma 4.16. With J i,n defined by (4.12), the following convergence holds:

√ n n i=1 E[J i,n | G i-1,n ] p ---→ n→∞ 0.
Proof. From (4.12),

√ nE i n i-1 n sgn(Y i-1,n )1 {Y i-1,n Ys<0} ds G i-1,n = √ n i n i-1 n sgn(Y i-1,n )E 1 {Y i-1,n Ys<0} G i-1,n ds = √ n i n i-1 n sgn(Y i-1,n )P[Y i-1,n Y s < 0 | G i-1,n ] ds.
The process Y is time-homogenenous. Using the Markov property and (2.4) we compute this quantity by reducing the distribution of

(Y s , Y i-1,n ) s∈[(i-1)/n,i/n] given G i-1,n to the one of (Y s , Y 0 ) s∈[0,1/n] given Y 0 . When Y 0 > 0, √ n 1 n 0 sgn(Y 0 )P[Y 0 Y s < 0 | Y 0 ] ds = √ n 1 n 0 2σ + σ -+ σ + Φ(-Y 0 /(σ + √ s)) ds = 1 √ n 2σ + σ -+ σ + 1 0 Φ(-Y 0 √ n/(σ + √ t)) dt = 1 √ n f (Y 0 √ n),
where

f (x) := 2σ + σ -+ σ + 1 0 Φ(-x/(σ + √ t)) dt for x > 0.
Now, for Y 0 < 0, we find similary,

√ n 1 n 0 sgn(Y 0 )P[Y 0 Y s < 0 | Y 0 ] ds = 1 √ n f (Y 0 √ n),
where

f (x) := -2σ - σ -+ σ + 1 0 Φ(x/(σ - √ t)) dt for x < 0.
We can compute

∞ 0 f (x) dx = ∞ 0 1 0 2σ + σ -+ σ + Φ(-x/(σ + √ s)) ds dx = 2σ 2 + σ -+ σ + ∞ 0 1 0 Φ(-x/ √ s) ds dx and 0 -∞ f (x) dx = -2σ 2 - σ -+ σ + 0 -∞ 1 0 Φ(x/ √ s) ds dx.
Therefore computing the coefficient in (4.2),

λ σ (f ) = 1 σ 2 + f ⊕ + 1 σ 2 - f = 0. Moreover, |f (x)| ≤ C 1 0 Φ(Cx/ √ t)) dt ≤ Ce -Cx 2 for some constant C, so |x| k f (x) dx < ∞ for all k ≥ 0. Applying Lemma 4.3 with n -1/2 f ( √ nY i-1,n ) = √ nE[J i,n | G i-1,n ],
we prove the statement.

Lemma 4.17. With U i,n defined by (4.12), the following convergence holds:

√ n n i=1 U i,n p ---→ n→∞ 0.
Proof. We consider (4.3). We have

E U 2 i,n G i-1,n = E J 2 i,n G i-1,n -E[J i,n | G i-1,n ] 2 ≤ E J 2 i,n G i-1,n .
From (4.12),

E J 2 i,n G i-1,n ≤ 1 n E i n i-1 n 1 {(Y i-1,n Ys)<0} ds G i-1,n = 1 n i n i-1 n P[Y i-1,n Y s < 0 | G i-1,n ] ds = 1 n 2 |f (Y i-1,n √ n)|.
Therefore, with (4.2),

λ σ (|f |) = 1 σ 2 + f ⊕ - 1 σ 2 - f = 2 σ -+ σ + ∞ -∞ 1 0 Φ(x/ √ s) ds dx. Lemma 4.3 implies n n i=1 E J 2 i,n G i-1,n = n i=1 1 n f (Y i-1,n √ n) p ---→ n→∞ 0.
This proves the result.

Proof of the main results

In this section we use all the results proved so far to prove Theorems 3.1 and 3.5.

Proof of Theorem 3.1. We set

ξ t = σ + t 0 1(Y s > 0) dW s and η t = σ - t 0 1(Y s < 0) dW s .
Itô-Tanaka's formula (see [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF]) gives the following equation for the positive and negative part of Y :

Y + t = ξ t + 1 2 L t (Y ), and
Y - t = -η t + 1 2 L t (Y ). (4.13)
Moreover, ξ is a martingale with quadratic variation

ξ t = t 0 σ(Y s ) 2 1(Y s > 0) ds = σ 2 + t 0 1(Y s > 0) ds.
It is well known that the quadratic variation of a martingale can be approximated with the sum of squared increments over shrinking partitions. Thus,

[ξ] n 1 p ---→ n→∞ ξ 1 = 1 0 σ 2 + 1(Y s > 0) ds = σ 2 + Q + 1 (Y ). (4.14)
We write now

(σ n + ) 2 -σ 2 + (σ n -) 2 -σ 2 - =     [Y + ] n 1 -σ 2 + Qn 1 (Y, +) Qn 1 (Y, +) [Y -] n 1 -σ 2 - Qn 1 (Y, -) Qn 1 (Y, -)     =     [Y + ] n 1 -ξ 1 Qn 1 (Y, +) [Y -] n 1 -η 1 Qn 1 (Y, -)     +     σ 2 + (Q + 1 -Qn 1 (Y, +)) Qn 1 (Y, +) σ 2 -(1 -Q + 1 -Qn 1 (Y, -)) Qn 1 (Y, -)     .
Recall that Qn 1 (Y, +) and Qn 1 (Y, -) converge almost surely to

Q + 1 (Y ) and Q - 1 (Y ) = 1 -Q + 1 . Besides, 0 < Q + 1 < 1 a.s.
, because Y 0 = 0. Therefore, from Theorem 4.14,

√ n     σ 2 + (Q + 1 -Qn 1 (Y, +)) Qn 1 (Y, +) σ 2 -(1 -Q + 1 -Qn 1 (Y, -)) Qn 1 (Y, -)     p ---→ n→∞ 0 0 .
Using again (3.1) and (4. [START_REF] Kasahara | On a generalized arc-sine law for one-dimensional diffusion processes[END_REF],

√ n σn ) 2 -σ 2 + σn ) 2 -σ 2 - sl ---→ n→∞      √ 2σ 2 + Q + 1 1 0 1(Y s > 0) d Bs √ 2σ 2 - 1 -Q + 1 1 0 1(Y s < 0) d Bs      -     1 Q + 1 1 1 -Q + 1     √ 2 3 √ π 2σ + σ - σ + + σ - L 1 (Y ).
(4.20)

The statement is now proved, but we would like to get a more explicit expression for the law of the limit random variable. Recall Q + t (Y ) = Q + t (X). From Corollary 1.2 in [START_REF] Appuhamillage | Occupation and local times for skew Brownian motion with applications to dispersion across an interface[END_REF], standard computations give that the joint density of (L t (X),

Q + t ) is, for b > 0, τ ∈ [0, t]: p Lt(X),Q + t (b, τ ) = (1 -θ 2 )b 4πτ 3/2 (t -τ ) 3/2 exp - (1 + θ) 2 b 2 8τ - (1 -θ) 2 b 2 8(t -τ ) .
We set now

Z t = L t (X) 4 (1 + θ) 2 Q + t + (1 -θ) 2 t -Q + t .
Changing variable in the integration, the joint density of

(Z t , Q + t ) is p Zt,Q + t (x, τ ) = x exp - x 2 2 1 πτ 1/2 (t -τ ) 1/2 1 -θ 2 (1 + θ) 2 (t -τ ) + (1 -θ) 2 τ . (4.21)
We also find the joint density of (Z t , t -Q + t ) as

p Zt,t-Q + t (x, τ ) = x exp - x 2 2 1 πτ 1/2 (t -τ ) 1/2 1 -θ 2 (1 + θ) 2 τ + (1 -θ) 2 (t -τ ) . (4.22)
As we can factorize p Zt,Q + t (x, τ ) = p Zt (x)p Q + t (τ ), Z t and Q + t are independent and their laws are explicit. In particular from (2.6), for t = 1, p Z 1 (x) = x exp(-x 2 2 ),

p Q + 1 (τ ) = 1 πτ 1/2 (1 -τ ) 1/2 × 1 -θ 2 (1 + θ) 2 (t -τ ) + (1 -θ) 2 τ = 1 πτ 1/2 (1 -τ ) 1/2 × σ + /σ - 1 -(1 -(σ + /σ -) 2 )τ ,
and

p 1-Q + 1 (τ ) = 1 πτ 1/2 (1 -τ ) 1/2 × 1 -θ 2 (1 + θ) 2 τ + (1 -θ) 2 (1 -τ ) = 1 πτ 1/2 (1 -τ ) 1/2 × σ -/σ + 1 -(1 -(σ -/σ + ) 2 )τ
.

Let now Λ be a random variable with the same law of Q + 1 , and let ξ be an independent exponential random variable of parameter 1. From (4.21) and (4.22),

1 Q + 1 1 1-Q + 1 L 1 (X) law = 4ξ (1 -Λ)(1 + θ) 2 + Λ(1 -θ) 2   1-Λ Λ Λ 1-Λ   = 2(σ + + σ -)ξ (1 -Λ)σ 2 -+ Λσ 2 +   1-Λ Λ Λ 1-Λ   . Moreover,    √ 2σ 2 + Q + 1 1 0 1(Y s > 0) d Bs √ 2σ 2 - 1-Q + 1 1 0 1(Y s < 0) d Bs    law =   √ 2σ 2 + √ Λ N 1 √ 2σ 2 - √ 1-Λ N 2   ,
where N 1 , N 2 are standard Gaussian random variables independent of ξ and Λ. Therefore the limit law has the expression given in the statement.

Proof of Theorem 3.5. Using (4.13), we obtain

[Y + , Y ] n 1 = ξ + 1 2 L(Y ), Y n 1 = [ξ, Y ] n 1 + 1 2 [L(Y ), Y ] n 1 .
From the Central Limit Theorem for martingales [START_REF] Jacod | Discretization of processes[END_REF]Theorem 5.4.2] and since

ξ t + η t = Y t , √ n [ξ, Y ] n 1 [η, Y ] n 1 - ξ 1 η 1 = √ n [ξ, ξ] n 1 + [ξ, η] n 1 [η, ξ] n 1 + [η, η] n 1 - ξ 1 η 1 sl ---→ n→∞ √ 2 1 0 σ 2 + 1(Y s > 0) σ 2 -1(Y s < 0) d Bs ,
where B is a BM independent of the filtration of W . Both W and B are defined on an extension ( Ω, F, P) of (Ω, F, P) with P = P ⊗ P where P carries the BM B. Moreover,

[L(Y ), Y ] n 1 = [L(Y ), Y + ] n 1 -[L(Y ), Y -] n 1 = 2σ 2 + σ - σ + + σ - [X + , L(X)] n 1 - 2σ + σ 2 - σ + + σ - [X -, L(X)] n Proof. Let us write Q = Q ⊗ P . The Girsanov weight G T is F T -measurable
f , E[GY f (Z n )] converges to E P[GY f (Z)].
Under Q, W , W = 0 as W and W are independent and the bracket does not change under a Girsanov transform. This implies that W is still a Brownian motion under Q. Hence the result.

Proof of Corollary 3.8. Let ξ be solution to dξ t = σ(ξ t ) dW t with an underlying Brownian motion W on (Ω, F, P). We denote by (G t ) t≥0 the filtration of W . Thus, ξ is an OBM.

The Girsanov theorem is still valid for discontinuous coefficients [START_REF] Gall | One-dimensional stochastic differential equations involving the local times of the unknown process[END_REF]. Let us set

G t = exp t 0 b(ξ s ) σ(ξ s ) dW s - 1 2 t 0 b(ξ s ) σ(ξ s ) 2 ds . ( 4
√ n (m n + (ξ)) 2 -σ 2 + (m n -(ξ)) 2 -σ 2 - Q-sl ---→ n→∞    √ 2σ 2 + Q + 1 (ξ) 1 0 1(ξ s > 0) d Bs √ 2σ 2 - 1-Q + 1 (ξ) 1 0 1(ξ s < 0) d Bs    ,
where B is a BM independent of W and W also under Q.

Oscillating Random Walk

In [START_REF] Keilson | Oscillating Brownian motion[END_REF] the OBM is constructed also as a limit of discrete processes, called Oscillating Random Walks (ORW), analogously to how the BM is constructed as a limit of Random Walks. The aim of this section is to exemplify the phenomena of dependence of the limit law on the occupation time, in a simpler framework and with a non-technical proof. We define the ORW as the following process. Fix 0 < p, q ≤ 1. For k ∈ N, we introduce the following random variables:

U k iid, P(U k = 1) = P(U k = -1) = q 2 , P(U k = 0) = 1 -q, V k iid, P(V k = 1) = P(V k = -1) = p 2 , P(V k = 0) = 1 -p, Z k iid, P(Z k = 1) = P(Z k = -1) = 1 2 . Now we set Y * 0 = 0 and Y * k+1 =      Y * k + U k+1 if Y * k > 0, Y * k + V k+1 if Y * k < 0, Y * k + Z k+1 if Y * k = 0.
We consider the re-normalized process

Y n t = n -1/2 Y * [nt] .
Let Y be an OBM of parameters σ 2 + = q, σ 2 -= p driven by a Brownian motion W . As shown in [START_REF] Stone | Limit theorems for random walks, birth and death processes, and diffusion processes[END_REF], it is possible to construct a birth and death process from the Brownian motion W whose distribution is that of the normalized random walk Y n . Hence, for this particular choice of Y n and Y with the same underlying Brownian motion, as shown in [START_REF] Keilson | Oscillating Brownian motion[END_REF],

sup 0≤t≤K |Y n t -Y t | p ---→ n→∞ 0 for all K > 0.
In this setting, 0 < σ -, σ + ≤ 1. Yet we do not loose in generality because we can always rescale time and space so that the random walk Y n with p = σ 2 + /(σ 2 + +σ 2 -) and q = σ 2 -/(σ 2 + +σ 2 -) converges in distribution to the OBM with diffusivity σ 2 + and σ 2 -on the space of continuous functions with the uniform norm.

In this appendix, we recover from the observations of Y n for some large n an estimator for the parameters of the OBM.

We set

β n = #{k ∈ N, k ≤ n : Y n k/n > 0}, α n = #{k ∈ N, k ≤ n : Y n k/n > 0, Y n (k+1)/n = Y n
k/n }, and introduce the following estimator of q = σ 2 + : qn = α n β n .

(5.1)

Theorem 5.1. Let qn be the estimator defined above. The following convergence holds:

√ n (q n -q) law ---→ n→∞ N q(1 -q) Λ ,
where Λ follows the law in (2.6), N is a standard Gaussian and they are independent.

Proof. When Y n k/n > 0, Y n (k+1)/n = Y n k/n with probability q, and Y n (k+1)/n = Y n k/n with probability 1q. We can compute the log-likelihood and maximize it as in the statistics of Binomial variables, finding that the maximum likelihood estimator for q is qn in (5.1). In [START_REF] Keilson | Oscillating Brownian motion[END_REF] it is proved that

#{k ≤ n : Y k ≥ 0} n law ---→ n→∞ Λ,
where Λ follows the law in (2.6). This easily implies

β n n law ---→ n→∞ Λ. (5.2)
Conditioning to β n , we have that α n follows is a binomial distribution with parameters q, β n . We write the event √ n (q nq) ≤ x = β n q(1q) (q nq) ≤ x β n nq(1q) .

From Berry-Essen inequality [START_REF] Berry | The accuracy of the Gaussian approximation to the sum of independent variates[END_REF][START_REF] Esseen | On the Liapounoff limit of error in the theory of probability[END_REF], we have P √ n(q nq) ≤ x β n -Φ x β n nq(1q) ≤ C q / β n , for some constant C q . Now, from (5.2) and Portmanteau Lemma,

E Φ x β n nq(1 -q) ---→ n→∞ E Φ x Λ q(1 -q) .
Moreover, E[C q / √ β n ] → 0. Recalling P √ n(q nq) ≤ x = E P √ n (q nq) ≤ x β n , we obtain the following convergence

P √ n (q n -q) ≤ x ---→ n→∞ E Φ x Λ q(1 -q) ,
which implies the statement.

Empirical evidence

In this section we implement estimators σn , m n and use them on simulated data. For doing so, we reduce the OBM (2.2) to a SBM (2.7), and we simulate it through the simulation method given in [START_REF] Lejay | Simulating diffusion processes in discontinuous media: a numerical scheme with constant time steps[END_REF]. This method gives the successive positions {X k∆t } k≥0 of the SBM, hence the OBM, on a time grid of size ∆t. Recall Remark 3.7, in particular estimators m n,T ± , σn,T ± , for which we have central limit theorems with the same limit laws of (3.5), (3.7). We use the parameters: T = 5, ∆t = 0.01 (thus n = 500), σ -= 0.5, σ + = 2 (so that θ = -0.48 in (2.7), pushing the process to the negative side). In Figure 1, we plot the density of M n ± := √ n((m n,T ± ) 2σ 2 ± ) and S n ± := √ n((σ n,T ± ) 2σ 2 ± )

for N realizations of these estimators (meaning the simulation of N = 10 000 paths of the OBM). Their empirical densities are compared with the ones of

M ± := √ 2σ 2 ± N √ Λ and S + S - :=     √ 2σ 2 + √ Λ N 1 -8 3 √ π 1 r+1 ξ √ 1-Λ √ (1-Λ)+Λr 2 √ 2σ 2 - √ 1-Λ N 2 -8 3 √ π 1 1/r+1 ξ √ Λ √ Λ+(1-Λ)/r 2     ,
given in (3.7) and (3.5), with N ∼ N (0, 1) and ξ ∼ exp(1). The densities of M and S (which do not depend on T ) are obtained by simulation. The occupation time Λ is simulated by inverting its distribution function [START_REF] Lamperti | An occupation time theorem for a class of stochastic processes[END_REF][START_REF] Lejay | Simulation of a stochastic process in a discontinuous layered medium[END_REF]:

Λ law = σ 2 -V σ 2 -V + σ 2 + (1 -V )
with V = sin 2 U π 2 , U uniform on [0, 1).

We see that the limit distribution on the positive side has a larger variance than the one in the negative side. This is due to the sample size, proportional to the occupation time, which is on average larger on the side where the diffusion coefficient is smaller. We also obtain a good agreement of the normalized empirical error with the prediction of the central limit theorem. On the pictures on the right, we observe the difference between biased and non-biased estimator; the grey area is the effect given by the shift to the left of the distribution, caused by the local time term. This shift is more visible when the diffusion coefficient is smaller.

We have also checked that the distribution of √ n

[Y +,Y -] n T T is close to the distribution of 2 √ 2 3 √ π σ + σ - σ + +σ - L T (Y ) √ T law = √ 2 3 √ π 2σ + σ - σ + +σ - 2
L 1 (X), which is straightforward since the density of L 1 (X)

is known (for this, we use P. Lévy's identity that relates the local time to the supremum of the Brownian motion whose density is explicitly known). The agreement is good. Finally, the same simulation work can be done using the random walks defined in Section 5 using the simple approach. Again, the numerical results are in good agreement with the theory, although some instabilities appear due to the fact that the occupation time may take small values with high probability.

p-→1

  in probability; law --→ in law; sl -→ stable in law; law = denotes equality in law. The Lebesgue measure is denoted by Leb. The positive and negative parts of x ∈ R are denoted by x + = x ∨ 0, x -= (-x) ∨ 0. For any continuous semimartingale M , we write M for its quadratic variation process. For y ∈ R we define the (symmetric) local time of M at y as the process (L y t (M )) t∈[0,1] , with (See [38, Corollary VI.1.9, p. 227]) {y-ε≤Ms≤y+ε} d M s almost surely.

Figure 1 :

 1 Figure 1: Densities of: (a), (d) normalized error of the estimators M n ± (solid line) and theoretical limits M ± (dashed lines); (b), (e) normalized error of the estimators S n ± (solid line) and theoretical limits S ± (dashed lines); (c), (f) normalized error of the estimators M n ± and normalized error of the estimators S n ± .

  ) t∈[0,1] be the natural (completed) filtration of β. For i = 1, . . . , n, we write H i,n = H i/n . Lemma 4.6. Let L(β) be the Brownian local time at 0. The following convergence holds:

	4.2 Scaled quadratic variation of Brownian local time
	Let (β t ) (t∈[0,1]) be a BM and L(β) its local time at 0. Let us recall the diffusive scaling property (β ct , L t (β)) t>0 law = ( √ cβ, √ cL(β)) for any c > 0 (see e.g. [38, Exercise 2.11, p. 244]).
	Let H = (H t

  1, 2. With Lemma 4.1, this proves (4.11). 4.13. With Z i,n defined by (4.10), the following convergence holds:

	Lemma

  and integrable with respect to P. Hence, it is easily shown that for any bounded, F T -measurable random variable Y and any bounded, continuous function

Acknowledgement. We want to thank the referees for their careful reading. We also thank Prof. Dr. Stefan Ankirchner who showed us the article of Keilson & Wellner on the Oscillating Brownian motion which was the starting point of this article.

From (4.13),

The local time L(Y ) is of finite variation, Y + is continuous. Thus [L(Y )] n 1 as well as [Y + , L(Y )] n 1 converge to 0 almost surely. From (4.14),

Recall the definition of Qn

Then

We consider separately the tree summands. From the central limit theorem for martingales (see for example [START_REF] Jacod | Discretization of processes[END_REF], (5.4.3) or Theorem 5.4.2), since

where B is a Brownian motion independent of the filtration of W . Therefore it is also independent of L(Y ). Consider now the second summand. The OBM Y is linked to a SBM X solution to (2.7) through Y t = X t σ(X t ). With (2.9) and (4.9) in Lemma 4.11,

Clearly this also holds for [Y -, L(Y )] n 1 , and we obtain the convergence in probability of

to (0, 0). We use Lemma 4.6 for dealing with the third summand:

We obtain, using (3.1),

This is the analogous of (4.19) in the proof of Theorem 3.1. From now on the proof follows as in Theorem 3.1, but without the local time part. Remark 4.18. We look for the origin of the asymptotic bias present in σn , but not in m n . Consider first the difference between the approximation of quadratic variation used in the two different estimators:

From (4.13),

From the central limit theorem for martingales [17, Theorem 5.4.2],

We then see that the asymptotic bias in σ2 ± is related to the bracket [Y + , Y -] n 1 .

Proof of Corollary 3.8: adding a drift term via Girsanov Theorem

Let us start with a remark on the stability of the stable convergence under a Girsanov transform.

Lemma 4.19. For two probability spaces (Ω, F, P) and (Ω , F , P ), let us define an extension ( Ω, F, P) by of (Ω, F, P) of the form Ω = Ω × Ω , F = F ⊗ F and P = P ⊗ P .

Assume that (Ω, F, P) and (Ω , F , P ) carry respectively Brownian motions W and W with natural (completed) filtrations F = (F t ) t>0 and F = (F t ) t>0 . Assume also that W and W are independent. On (Ω, F, P), let G be an exponential F -martingale which is uniformly integrable. Let Q be the measure such that dQ dP Ft = G t . Suppose now that a sequence Z n on (Ω, F, P) of F T -measurable random variables converges stably to a random variable Z = AW B on the extension ( Ω, F, P) of (Ω, F, P) where A and B are F T -random variables on (Ω, F, P) and B is a positive random variable.

Then Z n converges stably to Z = AW B on ( Ω, F, Q ⊗ P ) where W is a Brownian motion independent from A and B (the laws of A and B are of course changed).


Éric De

La Clergerie

Benoît Sagot

Djamé Seddah

The ParisNLP entry at the ConLL UD Shared Task 2017: A Tale of a #ParsingTragedy

We present the ParisNLP entry at the UD CoNLL 2017 parsing shared task. In addition to the UDpipe models provided, we built our own data-driven tokenization models, sentence segmenter and lexiconbased morphological analyzers. All of these were used with a range of different parsing models (neural or not, feature-rich or not, transition or graph-based, etc.) and the best combination for each language was selected. Unfortunately, a glitch in the shared task's Matrix led our model selector to run generic, weakly lexicalized models, tailored for surprise languages, instead of our dataset-specific models. Because of this #ParsingTragedy, we officially ranked 27th, whereas our real models finally unofficially ranked 6th.

Introduction

The Universal Dependency parsing shared task (Zeman et al., 2017) was arguably the hardest shared task the field has seen since the CoNLL 2006 shared task (Buchholz and Marsi, 2006) where 13 languages had to be parsed in gold token, gold morphology mode, while its follow up in 2007 introduced an out-of-domain track for a subset of the 2006 languages (Nivre et al., 2007). The SANCL "parsing the web" shared task (Petrov and McDonald, 2012) introduced the parsing of English non-canonical data in gold token, predicted morphology mode and saw a large decrease of performance compared to what was usually reported in English parsing of the Penn Treebank . As far as we know, the SPMRL shared tasks (Seddah et al., 2013, 2014) were first to introduce a non gold tokenization, predicted morphology, scenario for two morphologically rich languages, Arabic and Hebrew while, for other languages, complex source tokens were left untouched (Korean, German, French. . . ). Here, the Universal Dependency (hereafter "UD") shared task introduced an endto-end parsing evaluation protocol where none of the usual stratification layers were to be evaluated in gold mode: tokenization, sentence segmentation, morphology prediction and of course syntactic structures had to be produced1 for 46 languages covering 81 datasets. Some of them are low-resource languages, with training sets containing as few as 22 sentences. In addition, an out-of-domain scenario was de facto included via a new 14-language parallel test set. Because of the very nature of the UD initiative, some languages are covered by several treebanks (English, French, Russian, Finnish. . . ) built by different teams, who interpreted the annotation guidelines with a certain degree of freedom when it comes to rare, or simply not covered, phenomena. 2 Let us add that our systems had to be deployed on a virtual machine and evaluated in a total blind mode with different metadata between the trial and the test runs.

All those parameters led to a multi-dimension shared task which can loosely be summarized by the following "equation":

Lang.T ok.W S.Seg.M orph.DS.OOD.AS.Exp, where Lang stands for Language, T ok for tokenization, W S for word segmentation, Seg for sentence segmentation, M orph for predicted morphology, DS for data scarcity, OOD for out-ofdomainness, AS for annotation scheme, Exp for experimental environment.

In this shared task, we earnestly tried to cover all of these dimensions, ranking #3 in UPOS tagging and #5 in sentence segmentation. But we were ultimately strongly impacted by the Exp parameter (cf. Section 6.3), a parameter we could not control, resulting in a disappointing rank of #27 out of 33. Once this variable was corrected, we reached rank #6. 3 Our system relies on a strong pre-processing pipeline, which includes lexicon-enhanced statistical taggers as well as data-driven tokenizers and sentence segmenters. The parsing step proper makes use for each dataset of one of 4 parsing models: 2 non-neural ones (transition and graphbased) and extensions of these models with character and word-level neural layers.

Architecture and Strategies

In preparation for the shared task, we have developed and adapted a number of different models for tokenization4 and sentence segmentation, tagging-predicting UPOS and the values for a (manually selected, language-independent) subset of the morphological attributes (hereafter "partial MSTAGs")-and parsing. For each dataset for which training data was available, we combined different pre-processing strategies with different parsing models and selected the best performing ones based on parsing F-scores on the development set in the predicted token scenario. Whenever no development set was available, we achieved this selection based on a 10-fold crossevaluation on the training set.

Our baseline pre-processing strategy consisted in simply using the data annotated using UDPipe (Straka et al., 2016) provided by the shared task organizers. We also developed new tools of our own, namely a tagger as well as a joint tokenizer and sentence segmenter. We chose whether to use the baseline UDPipe annotations or our own annotations for each of the following steps: sentence segmentation, tokenization, and tagging (UPOS and partial MSTAGs). We used UDPipe-based information in all configurations for XPOS, lemma, and word segmentation, based on an a posteriori character-level alignment algorithm.

At the parsing level, we developed and tried five different parsers, both neural and non-neural, which are variants of the shift-reduce (hereafter "SR") and maximum spanning-tree algorithms (hereafter "MST"). The next two sections describe in more detail our different pre-processing and parsing architectures, give insights into their performance, and show how we selected our final architecture for each dataset.

3 Pre-processing

Tagging

Tagging architecture Taking advantage of the opportunity given by the shared task, we developed a new part-of-speech tagging system inspired by our previous work on MElt (Denis and Sagot, 2012), a left-to-right maximum-entropy tagger relying on features based on both the training corpus and, when available, an external lexicon. The two main advantages of using an external lexicon as a source of additional features are the following: (i) it provides the tagger with information about words unknown to the training corpus; (ii) it allows the tagger to have a better insight into the right context of the current word, for which the tagger has not yet predicted anything.

Whereas MElt uses the megam package to learn tagging models, our new system, named alVW-Tagger, relies on Vowpal Wabbit. 5 One of Vowpal Wabbit's major advantages is its training speed, which allowed us to train many tagger variants for each language, in order to assess, for each language, the relative performance of different types of external lexicons and different ways to use them as a source of features. In all our experiments, we used VW in its default multiclass mode, i.e. using a squared loss and the one-against-all strategy. Our feature set is a slight extension of the one used by MElt (cf. Denis and Sagot, 2012).

The first improvement over MElt concerns how information extracted from the external lexicons is used: instead of only using the categories provided by the lexicon, we also use morphological features. We experimented different modes. In the baseline mode, the category provided by the lexicon is the concatenation of a UPOS and a sequence of morphological features, hereafter the "full category". In the F mode ("ms mode" in Table 1), only the UPOS is used (morphological features are ignored). In the M mode, both the full category and the sequence of morphological features are used separately. Finally, in the FM mode, both the UPOS and the sequence of morphological features are used separately.

The second improvement over MElt is that alVWTagger predicts both a part-of-speech (here, a UPOS) and a set of morphological features. As mentioned earlier, we decided to restrict the set of morphological features we predict, in order to reduce data sparsity. 6 For each word, our tagger first predicts a UPOS. Next, it uses this UPOS as a feature to predict the set of morphological features as a whole, using an auxiliary model. 7 Extraction of morphological lexicons As mentioned above, our tagger is able to use an external lexicon as a source of external information. We therefore created a number of morphological lexicons for as many languages as possible, relying only on data and tools that were allowed by the shared task instructions. We compared the UPOS accuracies on the development sets, or on the training sets in an 10-fold setting when no development data was provided, and retained the best performing lexicon for each dataset (see Table 1). Each lexicon was extracted from one of the following sources or several of them, using an a posteriori merging algorithm:

• The monolingual lexicons from the Apertium project (lexicon type code "AP" in Table 1); • Raw monolingual raw corpora provided by the shared task organizers, after application of a basic rule-based tokenizer and the appropriate Apertium or Giellatekno morphological analyzers (codes "APma" or "GTma"); • The corresponding training dataset (code "T") or another training dataset for the same language (code "Tdataset"); • The UDPipe-annotated corpora provided by the shared task organizers (code "UDP"); • A previously extracted lexicon for another language, which we automatically "translated" using a dedicated algorithm, which we provided, as a seed, with a bilingual lexicon automatically extracted from OPUS sentence-aligned data (code "TRsource language").

All lexical information not directly extracted from UDPipe-annotated data or from training data was converted to the UD morphological categories (UPOS and morphological features).

For a few languages only (for lack of time), we also created expanded versions of our lexicons using word embeddings re-computed on the raw data provided by the organizers, assigning to words unknown to the lexicon the morphological information associated with the closest known word (using a simple euclidian distance on the word embedding space).8 When the best performing lexicon is one of these extended lexicons, it is indicated in Table 1 by the "-e" suffix.

Tokenization and sentence segmentation

Using the same architecture as our tagger, yet without resorting to external lexicons, we developed a data-driven tokenizer and sentence segmenter, which runs as follows. First, a simple rule-based pre-tokenizer is applied to the raw text of the training corpus, after removing all sentence boundaries.9,10 This pre-tokenizer outputs a sequence of "pre-tokens," in which, at each pre-token boundary, we keep trace of whether a whitespace was present in the raw text or not at this position. Next, we use the gold train data to label each pre-token boundary with one of the following labels: not a token boundary (NATB), token boundary (TB), sentence boundary (SB). 11 This model can then be applied on raw text, after the pre-tokenizer has been applied. It labels each pre-token boundary, resulting in the following decisions depending on whether it corresponds to a whitespace in the raw text or not: (i) if it predicts NATB at a non-whitespace boundary, the boundary is removed; (ii) if it predicts NATB at a whitespace boundary, it results in a tokenwith-space; (iii) if it predicts TB (resp. SB) at a non-whitespace boundary, a token (resp. sentence) boundary is created and "SpaceAfter=No" is added to the preceedings token; (iv) if it predicts We compared our tokenization and sentence segmentation results with the UDPipe baseline on development sets. Whenever the UDPipe tokenization and sentence segmentation scores were both better, we decided to use them in all configurations. Other datasets, for which tokenization and sentence segmentation performance is shown in Table 2, were split into two sets: those on which our tokenization was better but sentence segmentation was worse-for those, we forced the UD-Pipe sentence segmentation in all settings-, and those for which both our tokenization and sentence segmentation were better.

Preprocessing model configurations

As mentioned in Section 2, we used parsing-based evaluation to select our pre-processing strategy for each corpus. More precisely, we selected for each dataset one of the following strategies:

1. UDPIPE: the UDPipe baseline is used and provided as such to the parser. 2. TAG: the UDPipe baseline is used, except for the UPOS and MSTAG information, which is provided by our own tagger. 3. TAG+TOK+SEG and TAG+TOK: we apply our own tokenization and POS-tagger to produce UPOS and MSTAG information; sentence segmentation is performed either by us (TAG+TOK+SEG (available for datasets with "yes" in the last column in Whenever we used our own tokenization and not that of the UDPipe baseline, we used a characterlevel alignment algorithm to map this information to our own tokens.

Table 1 shows the configuration retained for each language for which a training set was provided in advance. 12 For surprise language datasets, we always used the UDPipe configuration. 13 For PUD corpora, we used the same configuration as for the basic dataset for the same language (for instance, we used for the fr pud dataset the same configuration as that chosen for the fr dataset). 14 Table 1 indicates for each dataset which configuration was retained.

Parsing Models

We used 4 base parsers, all implemented on top of the DYALOG system (de La Clergerie, 2005), a logic-programming environment (à la Prolog) specially tailored for natural language processing, in particular for tabulation-based dynamic programming algorithms.

Non-neural parsing models The first two parsers are feature-based and use no neural components. The most advanced one is DYALOG-SR, a shift-reduce transition-based parser, using dynamic programming techniques to maintain beams (Villemonte De La Clergerie, 2013). It accepts a large set of transition types, besides the usual shift and reduce transitions of the arcstandard strategy. In particular, to handle nonprojectivity, it can use different instances of swap transitions, to swap 2 stack elements between the 3 topmost ones. A noop transition may also be used at the end of parsing paths to compensate differences in path lengths. Training is done with a structured averaged perceptron, using early aggressive updates, whenever the oracle falls out of the beam, or when a violation occurs, or when a margin becomes too high, etc. 15 Feature templates are used to combine elementary standard features:

• Word features related to the 3 topmost stack elements s i=0 13 For surprise languages, the UDPipe baseline was trained on data not available to the shared task participants.

14 Because of a last-minute bug, we used the TAG configuration for trpud and ptpud although we used the UDPIPE configuration for tr and pt. We also used the TAG setting for fipud rather than the TAG+TOK+SEG setting used for fi. 15 By "violation," we mean for instance adding an edge not present in the gold tree, a first step towards dynamic oracles. We explored this path further for the shared task through dynamic programming exploration of the search space, yet did not observe significant improvements yet. morphosyntactic features, Brown-like clusters (derived from word embeddings), and flags indicating capitalization, numbers, etc.

• Binned distances between some of these elements • Dependency features related to the leftmost/rightmost dependency labels for s i (and dependent [lr]s i ), label set for the dependents of s i and [lr]s i , and number of dependents • Last action (+label) leading to the current parsing state.

The second feature-based parser is DYALOG-MST, a parser developed for the shared task and implementing the Maximum Spanning Tree (MST) algorithm (McDonald et al., 2005). By definition, DYALOG-MST may produce nonprojective trees. Being recent and much less flexible than DYALOG-SR, it also relies on a much smaller set of first-order features and templates, related to the source and target words of a dependency edge, plus its label and binned distance. It also exploits features related to the number of occurrences of a given POS between the source and target of an edge (inside features) or not covered by the edge but in the neighborhood of the nodes (outside features). Similar features are also implemented for punctuation.

Neural parsing models Both feature-based parsers were then extended with a neural-based component, implemented in C++ with DyNet (Neubig et al., 2017). The key idea is that the neural component can provide the best parser action or, if aksed, a ranking of all possible actions. This information is then used as extra features to finally take a decision. The 2 neural-based variants of DYALOG-SR and DYALOG-MST, straightforwardly dubbed DYALOG-SRNN and DYALOG-MSTNN, implement a similar architecture, the one for DYALOG-SRNN being a bit more advanced and stable. Moreover, DYALOG-MSTNN was only found to be the best choice for a very limited number of treebanks. In addition to these models, we also investigated a basic version of DYALOG-SRNN that only uses, in a feature-poor setting, its character-level component and its joint action prediction, and which provides the best performance on 3 languages. The following discussion will focus on DYALOG-SRNN.

The architecture is inspired by Google's PARSEYSAURUS (Alberti et al., 2017), with a first left-to-right char LSTM covering the whole sentence and (artificial) whitespaces introduced to separate tokens. 16 The output vectors of the char LSTM at the token separations are used as (learned) word embeddings that are concatenated (when present) with both the pre-trained ones provided for the task and the UPOS tags predicted by the external tagger. The concatenated vectors serve as input to a word bi-LSTM that is also used to predict UPOS tags as a joint task (training with the gold tags provided as oracle). For a given word w i , its final vector representation is the concatenation of the output of the bi-LSTM layers at position i with the LSTM-predicted UPOS tag.

The deployment of the LSTMs is done once for a given sentence. Then, for any parsing state, characterized by the stack, buffer, and dependency components mentioned above, a query is made to the neural layers to suggest an action. The query fetches the final vectors associated with the stack, buffer, and dependent state words, and completes it with input vectors for 12 (possibly empty) label dependencies and for the last action. The number of considered state words is a hyper-parameter of the system, which can range between 10 and 19, the best and default value being 10, covering the 3 topmost stack elements and 6 dependent children, but only the first buffer lookahead word 17 and no grandchildren. Through a hidden layer and a softmax layer, the neural component returns the best action paction (and plabel) but also the ranking and weights of all possible actions. The best action is used as a feature to guide the decision of the parser in combination with the other features, the final weight of an action being a linear weighted combination of the weights returned by both perceptron and neural layers. 18 A dropout rate of 0.25 was used to introduce some noise. The Dynet AdamTrainer was chosen for gradient updates, with its default parameters. Many hyperparameters are however available as options, such as the number of layers of the char and word LSTMs, the size of input, hidden and output dimensions for the LSTMs and feedforward layers. A partial exploration of these parameters was run on a few languages, but not in a systematic way given the lack of time and the huge 16 A better option would be to add whitespace only when present in the original text. 17 We suppose the information relative to the other lookahead words are encapsulated in the final vector of the first lookahead word. 18 The best way to combine the weights of the neural and feature components remains a point to further investigate. number of possibilities. Clearly, even if we did try 380 distinct parsing configurations through around 16K training runs, 19 we are still far away from language-specific parameter tuning, thus leaving room for improvement.

Results

Because of the greater flexibility of transitionbased parsers, MST-based models were only used for a few languages. However, our results, provided in the Appendix, show the good performance of these models, for instance on Old Church Slavonic (cu), Gothic (got), Ancient Greek (grc), and Kazakh (kk). Already during development, it was surprising to observe, for most languages, a strong preference for either SR-based models or MST-based ones. For instance, for Ancient Greek, the best score in gold token mode for a MST-based model is 62.43 while the best score for a SR-based one is 60.59. On the other hand, for Arabic (ar), we get 74.87 for the best SR model and 71.44 for the best MST model.

Altogether, our real, yet unofficial scores are encouraging (ranking #6 in LAS) while our official UPOS tagging, sentence segmentation and tokenization results ranked respectively #3, #6 and #5. Let us note that our low LAS official results, #27, was the result of a mismatch between the trial and test experimental environments provided by the organizers (cf. Section 6.3). However, we officially ranked #5 on surprise languages, which were not affected by this mismatch.

Discussion

While developing our parsers, training and evaluation were mostly performed using the UDPipe pre-processing baseline with predicted UPOS and MSTAGs but gold tokenization and gold sentence segmentation. For several (bad) reasons, only in the very last days did we train on files tagged with our preprocessing chain. Even later, evaluation (but no training) was finally performed on dev files with predicted segmentation and tokenization, done by either UDPipe or by our pre-processing chains (TAG, TAG+TOK+SEG or TAG+TOK). Based on the results, we selected, for each language and treebank, the best preprocessing configuration and the best parsing model.

In general, we observed that neural-based models without features often worked worse than pure feature-based parsers (such as srcat), but great when combined with features. We believe that, being quite recent, our neural architecture is not yet up-to-date and that we still have to evaluate several possible options. Between using no features (srnnsimple and srnncharsimple models) and using a rich feature set (srnnpx models), where the predicted actions paction and plabel may be combined with other features, we also tested, for a few languages, a more restricted feature set with no combinations of paction and plabel with other features (srnncharjoin models). These latter combinations are faster to train and reach good scores, as shown in Table 3 For the treebanks without dev files, we simply did a standard training, using a random 80/20-split of the train files. Given more time, we would have tried transfer from other treebanks when available (as described below).

To summarize, a large majority of 47 selected models were based on DYALOG-SRNN with a rich feature set, 29 of them relying on predicted data coming from our processing chains (TAG or TAG+TOK+SEG), the other ones relying on the tags and segmentation predicted by UDPipe. 10 models were based on DYALOG-MSTNN, with 5 of them relying on our preprocessing chain. Finally, 5 (resp. 2) were simply based on DYALOG-SR (resp. DYALOG-MST), none of them using our preprocessing.

OOV Handling

Besides the fact that we did not train on files with predicted segmentation, we are also aware of weaknesses in handling unknown words in test files. Indeed, at some point, we made the choice to filter the large word-embedding files by the vocabulary found in the train and dev files of each dataset. We made the same for the clusters we derived from word embeddings. It means that unknown words have no associated word embed-dings or clusters (besides a default one). The impact of this choice is not yet clear but it should be a relatively significant part of the performance gap between our score of the dev set (with predicted segmentation) and on the final test set.20

Generic Models

We also started exploring the idea of transferring information between close languages, such as Slavic languages. Treebank families were created for some groups of related languages by randomly sampling their respective treebanks as described in Table 4. A fully generic treebank was also created by randomly sampling 41k sentences from almost all languages (1k sentences per primary treebank). The non-neural parsers were trained on these new treebanks, using much less lexicalized information (no clusters, no word embeddings, no lemmas, and keeping only UPOS tags, but keeping forms and lemmas when present). We tested using the resulting models, whose named are prefixed with "ZZ", as base models for further training on some small language treebanks. However, preliminary evaluations did not show any major improvement and, due to lack of time, we put on hold this line of research, while keeping our generic parsers as backup ones.

Some of these generic parsers were used for the 4 surprise languages, with Upper Sorbian using a ZZSSlavic parser21 (LAS=56.22), North Saami using ZZFinnish (LAS=37.33), and the two other ones using the fully generic parser (Kurmanji LAS=34.8; Buryat LAS=28.55).

The Tragedy

Ironically, the back-off mechanism we set up for our model selection was also a cause of failure and salvation. Because of the absence of the name field in the test set metadata, which was nevertheless present in the dev run and crucially also in the trial run metadata, the selection of the best model per language was broken and led to the selection of back-off models, either a family one or in most cases the generic one. The Tira blind test configuration prevented us from discovering this experimental discrepancy before the deadline. Once we adapted our wrapper to the test metadata, the appropriate models were selected, resulting in our real run results. It turned out that our non language-specific, generic models performed surprisingly well, with a macro-average F-score of 60% LAS. Of course, except for Ukrainian, our language-specific models reach much better performance, with a macro-average F-score of 70.3%. But our misadventure is an invitation to further investigation.

However, it is unclear at this stage whether or not mixing languages in a large treebank really has advantages over using several small treebanks. In very preliminary experiments on Greek, Arabic, and French, we extracted the 1000 sentences present in the generic treebank for these languages and trained the best generic configuration (srcat, beam 6) on each of these small treebanks. As shown in Table 5, the scores on the development sets do not exhibit any improvement coming from mixing languages in a large pool and are largely below the scores obtained on a larger language-specific treebank.

Impact of the Lexicon

We also investigated the influence of our tagging strategy with respect to the UDPipe baseline. Figure 1 plots the parsing LAS F-scores with respect to training corpus size. It also show the result of logarithmic regressions performed on datasets for which we used the UDPipe baseline for preprocessing versus those for which we used the TAG configuration. As can be seen, using the UD-Pipe baseline results in a much stronger impact of training corpus size, whereas using our own tagger leads to more stable results. We interpret this

Conclusion

The shared task was a excellent opportunity for us to develop a new generation of NLP components to process a large spectrum of languages, using some of the latest developments in deep learning. However, it was really a challenging task, with an overwhelming number of decisions to take and experiments to run over a short period of time.

We now have many paths for improvement. First, because we have a very flexible but newly developed architecture, we need to stabilize it by carefully selecting the best design choices and parameters. We also plan to explore the potential of a multilingual dataset based on the UD annotation scheme, focusing on cross-language transfer and language-independent models.

Figure 1 :

 1 Figure 1: LAS F-score w.r.t. training corpus size

Table 1 :

 1 UPOS accuracies for the UDPipe baseline and for our best alVWTagger setting.

	Dataset	ours (best setting)	UDPipe Dataset	ours (best setting)	UDPipe Dataset	ours (best setting)	UDPipe
		lexicon	ms	overall overall	lexicon	ms	overall overall	lexicon	ms	overall	overall
		type	mode acc.	acc.	type	mode	acc.	acc.	type	mode acc.	acc.
	ar	AP-e	M	94.71	94.57 fr	AP-e		97.30	97.08 nl	AP	F	94.70	94.07
	bg	AP	F	97.61	97.72 frsequoia AP-e	FM	97.54	96.60 nllassysmall AP+Tnl F	96.74	95.65
	ca	AP-e	FM	98.42	98.15 ga	UDP	M	-	-nobokmaal AP		97.66	97.34
	cs	AP	M	98.83	98.48 gl	AP	FM	97.45	96.77 nonynorsk AP	M	97.23	96.74
	cscac	Tcs		99.24	98.78 got	T		94.53	94.22 pl	AP	M	97.03	95.34
	cscltt	AP+Tcs F	94.34	92.06 grc	Tgrcproiel-e		89.56	81.54 pt	AP	FM	97.21	97.00
	cu	T	F	95.15	94.07 grcproiel UDP-e	FM	96.40	96.01 ptbr	AP+Tpt FM	97.96	97.40
	da	AP		96.30	95.19 he	AP	FM	96.68	95.72 ro	AP		97.34	96.98
	de	AP	M	92.70	91.39 hi	AP	F	96.59	95.79 ru	AP	M	96.62	94.95
	el	AP	F	95.53	94.17 hr	TRsl	M	96.94	96.15 rusyntagrus AP	FM	98.54	98.20
	en	AP	F	94.68	94.43 hu	T		93.90	92.31 sk	TRcs	FM	96.00	93.14
	enlines AP	FM	96.08	94.75 id	AP	M	92.98	93.36 sl	AP	FM	97.82	96.34
	enpartut AP+T	FM	95.90	94.39 it	AP	F	97.55	97.23 sv	AP	FM	96.32	95.17
	es	AP		96.47	96.24 itpartut Trit	M	97.89	95.16 svlines	AP	F	96.01	94.63
	esancora AP	FM	98.39	98.16 ja	no lexicon	96.87	96.72 tr	APma	FM	93.65	92.25
	et	GTms	FM	89.28	87.52 kk	APms		-	-ug	UDP		-	-
	eu	AP	F	94.48	92.80 ko	no lexicon	93.77	93.68 uk	AP	M	-	-
	fa	no lexicon	96.04	96.17 laittb	TRit+T		97.15	96.86 ur	AP		93.01	92.45
	fi	GTms	FM	95.06	94.52 laproiel TRit+T-e FM	95.62	95.43 vi	no lexicon	88.60	88.68
	fiftb	GTms	F	92.50	92.34 lv	AP	FM	93.43	90.81 zh	AP+UDP		91.40	91.21

TB (resp. SB) at a whitespace boundary, a token (resp. sentence) boundary is created.

Table 2

 2

	Dataset	ours		UDPIPE	Use our
		tok.	sent.	tokenis.	sent.	sent.
		F-sc.	F-sc.	F-sc.	F-sc.	seg.?
	ar	99.54	92.75	99.99	77.99	yes
	ca	99.95	99.01	99.97	98.77	yes
	cscac	100.00	99.5	100.00	99.09	yes
	cu	99.98	39.44	100.00	37.09	yes
	da	100.00	84.01	99.68	84.36	no
	el	99.57	92.61	99.87	88.67	yes
	et	99.69	90.82	99.79	84.91	yes
	eu	99.93	99.42	99.99	99.00	yes
	fa	99.95	99.42	100.00	97.14	yes
	fi	99.54	87.76	99.69	86.47	yes
	fiftb	100.00	85.94	99.94	82.52	yes
	gl	99.73	96.8	99.06	92.44	yes
	got	99.99	28.95	100.00	23.51	yes
	hu	99.74	96.83	99.91	94.55	yes
	it	99.73	96.31	99.82	93.20	yes
	ja	92.63	99.61	89.53	99.71	no
	laittb	99.90	78.71	99.88	77.38	yes
	laproiel	100.00	19.92	99.99	19.76	yes
	lv	99.21	91.48	98.91	96.48	no
	nonynorsk	99.82	94.30	99.92	93.05	yes
	pt	99.89	90.23	99.82	89.27	yes
	vi	87.25	92.23	83.99	96.28	no

) or by the UDPipe baseline (TAG+TOK, available for datasets with "no" in Table

2

).

Table 2 :

 2 Tokenization and sentence segmentation accuracies for the UDPipe baseline and our tokenizer (restricted to those datasets for which we experimented the use of our own tokenization).

  •••2 , 4 first buffer elements I j=1•••4 , leftmost/rightmost children [lr]s i /grandchildren of the stack elements [lr]2s i , and governors. These features include the lexical form, lemma, UPOS, XPOS,

Table 3 :

 3 . Neural models & feature impact (Dev)

				srnn
	Lang	srcat charsimple charjoin	px
	sk	76.85	67.85	78.87	80.84
	cscac	84.48	77.07	84.80	84.85
	lv	63.86	59.57	66.95	68.74
	ko	54.43	57.19	70.91	71.45

Table 4 :

 4 Generic models partition

Table 5 :

 5 Generic pool vs. small treebank vs. full treebank (with srcat models (LAS, Dev)

Although baseline prediction for all layers were made available throughStraka et al.'s (2016) pre-trained models or pre-annotated development and test files.

See for example, the discrepancy between frpartut and the other French treebanks regarding the annotation of the not so rare car conjunction, 'for/because', and the associated syntactic structures, cf. https://github.com/ UniversalDependencies/docs/issues/432.

http://universaldependencies.org/ conll17/results-unofficial.html

We follow the shared task terminology in differentiating tokenization and word segmentation. A tokenizer only performs token segmentation (i.e. source tokens), and does not predict word segmentation (i.e. wordforms, or tree tokens).

https://github.com/JohnLangford/ vowpal_wabbit/

The list of features we retained is the following: Case, Gender, Number, PronType, VerbForm, Mood, and Voice.

We also experimented with per-feature prediction, but it resulted in slightly lower accuracy results on average, as measured on development sets.

We did not used the embeddings provided by the organizers because we experimentally found that the 10-token window used to train these embeddings resulted in less accurate results than when using smaller windows, especially when the raw corpus available was of a limited

size. 9 Apart from paragraph boundaries whenever

available.10 On languages such as Japanese and Chinese, each nonlatin character is a pre-

token on its own. 11 Our tokenizer and sentence segmenter relies on the almost same features as the tagger, except for two special features, which encode whether the current pre-token is a strong (resp. weak) punctuation, based on two manually crafted lists.

Note that our parsing-performance-based selection strategy did not always result in the same outcome as what we would have been chosen based solely on the comparison of our own tools with UDPipe's baseline. For instance, our new tagger gets better results than UDPipe in UPOS tagging on all development corpora but one, yet we used UDPipe-based UPOS for 24 non-PUD corpora.

We count as a training run the conjunction of a parser configuration, a treebank, and a beam size. Please note that a synthesis may be found at http://alpage.inria.fr/ ˜clerger/UD17/synthesis.html

An average of 6 points between dev and tests. Dev results available at goo.gl/lyuC8L.

 21 We had planned to use a ZZWSlavic parser but made a mistake in the configuration file.

Acknowledgments

We thank the organizers and the data providers who made this shared task possible within the core Universal Dependencies framework (Nivre et al., 2016), namely the authors of the UD version 2.0 datasets (Nivre et al., 2017), the baseline UDPipe models (Straka et al., 2016), and of course the team behind the TIRA evaluation platform (Potthast et al., 2014) to whom we owe a lot.