L I Galtchouk

S M Pergamenshchikov
email: serge.pergamenchtchikov@univ-rouen.fr

Efficient big data analysis for ergodic diffusion models on the basis of discrete data. *

Keywords: Adaptive nonparametric drift estimation, Asymptotic efficiency, Discrete time data, Nonasymptotic estimation, Model selection, Quadratic risk, Sharp oracle inequality AMS 2000 Subject Classifications: primary 62G08, secondary 62G05, 62G20

 we show that the constructed model selection procedures are asymptotically efficient in adaptive setting, i.e. in the case when the regularity of the drift coefficient is unknown. To this end, for the first time for such problem, we found in the explicit form the celebrated Pinsker constant which is the sharp lower bound for the minimax squared accuracy normalized by the optimal convergence rate, i.e. it provides the best potentially possible estimation accuracy. Finally, we show that the accuracy of the model selection procedure asymptotically coincides with this lower bound, i.e. the constructed procedure is efficient.

1 Introduction

Problem

In this paper we deal with the process (y t ) t≥0 defined by the stochastic differential equation

dy t = S(y t ) dt + b(y t ) dW t , 0 ≤ t ≤ T , (1.1)
where (W t ) t≥0 is a standard Wiener process, the initial value y 0 is a fixed constant, S(•) is unknown drift and b(•) is unknown diffusion coefficient.

The problem is to estimate the function S(x), for x ∈ [x 0 , x 1 ], on the basis of the observations (y t j ) 1≤j≤N , t j = jδ ,

where the frequency δ = δ T ∈ (0, 1) and the sample size N = N (T ) are some functions of T that will be specified later. We consider the quadratic risk defined, for any estimator S, as

R ϑ ( S) = E ϑ S -S 2 and f 2 = x 1 x 0 |f (x)| 2 dx , (1.3)
where E ϑ is the expectation with respect to the distribution of the process (1.1) for the functions ϑ = ϑ(•) = (S(•) , b(•)).

Motivations

The main motivation for the nonparametric drift in (1.1) is given in the papers [START_REF] Fujimori | The Danzing selector for a linear model of diffusion processes[END_REF][START_REF] De Gregorio | Adaptive LASSO-type estimation for multivariate diffusion processes[END_REF][START_REF] Galtchouk | Non asymptotic sharp oracle inequalities for high dimensional ergodic diffusion models[END_REF] in which Big Data problems are studied in the framework of the high dimension linear diffusion models, i.e. the model (1.1) with

S(x) = q j=1 θ j ψ j (x) , (1.4)
where (ψ j ) 1≤j≤q are known linear independent functions and the dimension is greater than the number of observations, i.e. q > N . Indeed, such problems are important at various applications such that stochastic optimal control, finance, filtration, signal processing and etc (see, for example, [START_REF] Arato | Linear stochastic systems with constant coefficients. A statistical approach[END_REF][START_REF] Ibragimov | Statistical Estimation: Asymptotic Theory[END_REF][START_REF] Lamberton | Introduction to stochastic calculus applied to finance[END_REF][START_REF] Karatzas | Methods of Mathematical Finance[END_REF][START_REF] Kabanov | Two Scale Stochastic Systems: Asymptotic Analysis and Control[END_REF][START_REF] Bayisa | Adaptive algorithm for sparse signal recovery[END_REF] and the references therein). Nonparametric estimation problems for S were studied in a number of papers in the case of complete observations, that is when a whole trajectory (y t ) 0≤t≤T was observed. A sufficiently complete survey one can find in [START_REF] Kutoyants | Statistical Inferences for Ergodic Diffusion Processes[END_REF]. It should be noted that, for the first time, the famous Pinsker constant representing the asymptotic efficiency property for nonparametric diffusion models was found by Dalayan and Kutoyants in [START_REF] Dalalyan | Sharp Adaptive Estimation of the Drift Function for Ergodic Diffusion[END_REF][START_REF] Dalalyan | Asymptotically Efficient Trend Coefficient Estimation for Ergodic Diffusion[END_REF] for a special weighted integral risk using very nice local time tool. In non asymptotic setting Galtchouk and Pergamenshchikov in [START_REF] Galtchouk | Sequential nonparametric adaptive estimation of the drift coefficient in diffusion processes[END_REF][START_REF] Galtchouk | Nonparametric sequential estimation of the drift in diffusion via model selection[END_REF][START_REF] Galtchouk | Nonparametric sequential minimax estimation of the drift coefficient in diffusion processes[END_REF][START_REF] Galtchouk | Asymptotically efficient sequential kernel estimates of the drift coefficient in ergodic diffusion processes[END_REF] developed nonparametric sequential estimation methods for the models (1.1) on the basis of which in [START_REF] Galtchouk | Adaptive sequential estimation for ergodic diffusion processes in quadratic metric[END_REF] they calculated the Pinsker constant for the risk (1.3). It should be noted that in all the cited papers estimation problems were studied in the case of complete observations (y t ) 0≤t≤T .

In practice, usually for the models (1.1) the observations are accessible only at the discrete time moments (1.2). A natural question arises about properties of estimators based on discrete observations. The nonparametric estimation based on discrete time observations for models of the form (1.1) was considered firstly for estimating the unknown diffusion coefficient b(•) on a fixed interval [0, T ], when the observation frequency goes to zero, (see, for example, [START_REF] Florens-Zmirou | On estimating the diffusion coefficient from discret observations[END_REF], [START_REF] Genon-Catalot | Nonparametric estimation of the diffusion coefficient by wavelets methods[END_REF], [START_REF] Hoffmann | Adaptive estimation in diffusion processes[END_REF], [START_REF] Jacod | Non-parametric kernel estimation of the coefficient of a diffusion[END_REF] and the references therein). Later, in [START_REF] Gobet | Nonparametric estimation of scalar diffusions based on low frequency data[END_REF] kernel estimates of the drift and diffusion coefficients were studied for the reflected processes (1.1) with the values in the interval [0, 1]. Minimax optimal convergence rates are found as the sample size goes to infinity. As to the estimation in the ergodic case, it should be noted that firstly a sequential procedure was proposed in [START_REF] Hoffmann | Adaptive estimation in diffusion processes[END_REF] for nonparametric estimating the drift coefficient of the process (1.1) in the integral metric. Some upper and lower asymptotic bounds were found for the L p -risks. Later, in the paper [START_REF] Comte | Nonparametric estimation for a discretely observed integrated diffusion model[END_REF] a non-asymptotic oracle inequality was obtained for a special empiric quadratic risk defined as a function of the observations at the discrete time moments. In the asymptotic setting, when the observation frequency goes to zero and the length of the observation time interval tends to infinity, the constructed estimators reach the minimax optimal convergence rates. But, unfortunately, neither the optimal convergence rate and nor efficiency property were established in [START_REF] Comte | Nonparametric estimation for a discretely observed integrated diffusion model[END_REF].

Key ideas

Our approach is based on the sequential analysis methods developed in the papers [START_REF] Galtchouk | Sequential nonparametric adaptive estimation of the drift coefficient in diffusion processes[END_REF][START_REF] Galtchouk | Nonparametric sequential minimax estimation of the drift coefficient in diffusion processes[END_REF][START_REF] Galtchouk | Asymptotically efficient sequential kernel estimates of the drift coefficient in ergodic diffusion processes[END_REF] for nonparametric estimation problems. This approach makes possible to replace the random denominator by a conditional constant in a sequential Nadaraya-Watson estimator. Let us recall that in the case of complete observations (that is, when a whole trajectory is observed) the sequential estimator efficiency was proved by making use of a uniform concentration inequality (see [START_REF] Galtchouk | Uniform concentration inequality for ergodic diffusion processes[END_REF]), besides an indicator kernel estimator. As it turns out later in [START_REF] Galtchouk | Adaptive sequential estimation for ergodic diffusion processes in quadratic metric[END_REF], the efficient kernel estimate in the above given sense provides constructing a selection model adaptive procedure that appears ef-ficient in the quadratic metric. Therefore, in order to realize this program (i.e. from efficient pointwise estimators to an efficient L 2 -estimator) in the case of discrete time observations, one needs to use a suitable concentration inequalities. Such concentration inequalities are obrained in [START_REF] Galtchouk | Uniform concentration inequality for ergodic diffusion processes observed at discrete times[END_REF] through the uniform geometric ergodicity method for the process (1.1) developed in [START_REF] Galtchouk | Geometric ergodicity for classes of homogeneous Markov chains[END_REF] which provides non asymptotic uppers bounds uniformly over functions S(•) and b(•) for the convergence rate in the ergodic theorem. Using this tool we can show that the corresponding weight least square estimator for S in (1.1) setting on the regularity parameters is efficient for the risk (1.3) in the Pinsker sense [START_REF] Pinsker | Optimal filtration of square integrable signals in gaussian noise[END_REF]. Finally, using the sharp oracle inequalities obtained in [START_REF] Galtchouk | Non asymptotic sharp oracle inequalities for high dimensional ergodic diffusion models[END_REF] we can estimate asymptotically from above the risk of the model selection procedure with the risk efficient weight least square estimator and we obtain the efficiency property in the adaptive sense, i.e. don't using the regularity of the function S.

Plan of the paper

The paper is organized as follows. In Section 2 we describe the functional classes. In Section 3 a sequential estimator of the drift coefficient is constructed. In Section 4 we introduce a regression model based on the above sequential estimator. In Section 5 we construct the model selection procedure and announce the corresponding oracle inequalities. The main results on asymptotic efficiency are formulated in Section 6. In Section 7 we study main asymptotic properties of the basic regression model. In Section 8 we study the efficiency property for non adaptive setting, i.e. in the case when the regularity of S and the Pinsker variance are known. In Section 9 we give the proofs for main results. In the Appendix we give all necessary technical results.

Main Conditions

In the paper we consider the estimation problem for the drift S on the interval [x 0 , x 1 ], where x 0 < x 1 are some arbitrary fixed points. In order to obtain a reliable estimator of S, it is necessary to impose some conditions on this function which are similar to the periodicity of the deterministic signal in the white noise model. One of conditions which is sufficient for this purpose is the assumption that the process (y t ) t≥0 in (1.1) returns to any vicinity of each point x ∈ [x 0 , x 1 ] infinite times. The ergodicity provides this property, when the coefficients of equation are known. In the case of unknown coefficients, one needs to impose a uniform ergodicity property. To obtain the uniform ergodicity property for the process (1.1) we use the functional class introduced in [START_REF] Galtchouk | Efficient pointwise estimation based on discrete data in ergodic nonparametric diffusions[END_REF], i.e. for any fixed L ≥ 1, M > 0 and

x * > |x 0 | + |x 1 | we set Σ L,M = S ∈ C 1 (R) : sup |x|≤x * |S(x)| + | Ṡ(x)| ≤ M , -L ≤ inf |x|≥x * Ṡ(x) ≤ sup |x|≥x * Ṡ(x) ≤ -1/L . (2.1)
Here and in the sequel we denote by ḟ and f the correspoding derivatives. Moreover, for some fixed parameters 0

< b min ≤ b max < ∞ we denote by B the class of functions b from C 2 (R) such that b min ≤ inf x∈R |b(x)| ≤ sup x∈R max |b(x)| , | ḃ(x)| , | b(x)| ≤ b max . (2.2)
Now we set

Θ = Σ L,M × B = (S, b) : S ∈ Σ L,M and b ∈ B . (2.3)
It is easy to see that the functions from Σ L,M are uniformly bounded on [x 0 , x 1 ], i.e. s * = sup

x 0 ≤x≤x 1 sup S∈Σ L,M S 2 (x) < ∞ . (2.4)
It should be noted that, for any ϑ ∈ Θ, (see, for example, [START_REF] Galtchouk | Existence and uniqueness of a solution for stochastic equations with respect to semimartingales[END_REF]) the equation (1.1) has a unique strong solution and, moreover, there exists an invariant density for the process (1.1) which is defined as

q ϑ (x) = R b -2 (z) e S(z) dz -1 b -2 (x) e S(x) , (2.5)
where S(x) = 2

x 0 b -2 (v)S(v)dv (see,e.g., [START_REF] Gihman | Stochastic differential equations[END_REF], Ch.4, 18, Th2). It is easy to see that this density is uniformly bounded on the class (2.3) and bounded away from zero on the interval [-x * , x * ] :

0 < q * = inf |x|≤x * inf ϑ∈Θ q ϑ (x) ≤ sup x∈R sup ϑ∈Θ q ϑ (x) = q * < +∞ (2.6)
We need the following condition for the observation frequency.

A 1 ) The frequency δ in the observations (1.2) has the following form

δ = δ T = 1 (T + 1)l T , (2.7)
where the function l T is such that,

lim T →∞ l T ln T = +∞ . (2.8)
For example, one can take l T = (ln T ) 1+ι for some ι > 0.

Remark 2.1. Note that we consider the efficient estimation problem only for the drift function S, i.e. the diffusion coefficient b is considered as a nuisance parameter.

Point-wise estimators

In order to obtain a reliable estimator of the function S on the interval [x 0 , x 1 ] we need some efficient point-wise estimators of this function. To give such estimators, we begin with the partition of the interval [x 0 , x 1 ] by points (z k ) 0≤k≤n defined as

z k = x 0 + k(x 1 -x 0 ) n , (3.1)
where n = n(T ) is an integer-valued function of T such that lim

T →∞ n(T ) √ T = 1 . (3.2)
We can take, for example,

n = n(T ) = 2 √ T 2 + 1 , (3.3)
where [x] is the integer part of the number x. To construct an efficient estimation procedure for S(z k ), at any point z k , we need to use some estimators of the invariant density q ϑ (•) and the squared diffusion coefficient b 2 (•). To this end, we will estimate these both functions by making use of the first N 0 observations, i.e. (y t j ) 1≤j≤N 0 . We set

N 0 = [N γ (T )] (3.4)
where 5/6 < γ < 1.

Estimation of the invariant density

To estimate the density q ϑ we will make use of the following kernel estimator

q(z k ) = 1 2 N 0 h 0 N 0 j=1 χ j,k (h 0 ) , (3.5)
where

h 0 = h 0 (T ) = T -1/2 0 , T 0 = δN 0 , χ j,k (h 0 ) = χ y t j-1 -z k h 0 and χ(y) = 1 {|y|≤1} .
Furthermore, we define a truncated estimator

q(z k ) =          (υ T ) 1/2 , if q(z k ) < (υ T ) 1/2 ; q(z k ) , if (υ T ) 1/2 ≤ q(z k ) ≤ (υ T ) -1/2 ; (υ T ) -1/2 , if q(z k ) > (υ T ) -1/2 , (3.6)
where υ T is a strictly positive function of T verifying the following condition.

A 2 ) Assume, that lim

T →∞ υ T + ln T T (υ T ) 2 + ln T l T (υ T ) 5 = 0 . (3.7)
For example, one can take υ T = ln -ι (T + 1) and l T ≥ ln 1+6ι T , for some ι > 0.

Proposition 3.1. Assume that the conditions A 1 )-A 2 ) hold. Then, for any x 0 < x 1 and for any a > 0 lim

T →∞ T a max x 0 ≤x≤x 1 sup ϑ∈Θ k,r P ϑ (| q T (x) -q ϑ (x)| > υ T ) = 0 .
This proposition is shown in [START_REF] Galtchouk | Efficient pointwise estimation based on discrete data in ergodic nonparametric diffusions[END_REF].

Estimation of the squared diffusion coefficient

To estimate the squared diffusion coefficient b 2 (z k ) we use the following sequential procedure. First we define the sample size for this procedure :

τ * ,k = inf{j ≥ 1 : j l=1 χ l,k (h 0 ) ≥ H * } ∧ N 0 , (3.8)
where H * = h 0 N 0 / ln(T + 1) . Then we set

b k = τ * ,k j=1 χ j,k (h 0 )(y t j -y t j-1 ) 2 δ H * 1 Γ * ,k , (3.9)
where Γ * ,k = { N 0 j=1 χ j,k (h 0 ) ≥ H * }. This estimator satisfies the following property. Proposition 3.2. For any a > 0, lim

T →∞ sup ϑ∈Θ max 1≤k≤n T γ-1/2-a E ϑ | b k -b 2 (z k )| = 0 . (3.10)
The proof is given in [START_REF] Galtchouk | Non asymptotic sharp oracle inequalities for high dimensional ergodic diffusion models[END_REF].

Remark 3.1. Note that in the case of known diffusion coefficient b(•), we can take b k = b 2 (z k ) .

Point-wise estimation of the drift coefficient

Now we estimate S(z k ), at every point z k , by making use of the observations (y t j ) N 0 +1≤j≤N . To this end we use the sequential kernel estimator from [START_REF] Galtchouk | Asymptotically efficient sequential kernel estimates of the drift coefficient in ergodic diffusion processes[END_REF] with the indicator kernel function χ(y) = 1 {|y|≤1} and the duration of observations given by the stopping time

τ k = inf{l ≥ N 0 + 1 : l j=N 0 +1 χ j,k (h) ≥ H k } , (3.11)
where the bandwidth h = (x 1 -x 0 )/2n,

χ j,k (h) = χ y t j-1 -z k h 1 {1≤j≤N } + 1 {j>N }
and H k is some positive random threshold which will be specified later. It is clear that τ k < ∞ a.s., for any H k > 0. Now, we define the correction coefficient 0 < κ k ≤ 1 as

τ k -1 j=N 0 +1 χ j,k (h) + κ k χ τ k ,k (h) = H k . (3.12)
Moreover, using this coefficient we introduce the following weight sequence

κ j,k = 1 {j<τ k } + κ k 1 {j=τ k } . (3.13)
It should be noted, that for any j ≥ 1 the random variables κ k,j are G j-1measurable, where G j = σ y t l , 0 ≤ l ≤ j . Now we define the sequential estimator for S(z k ) as

S * k = 1 δH k τ k j=N 0 +1 κ j,k χ j,k (h)∆y t j 1 Γ k , (3.14)
where ∆y t j = y t j -y t j-1 and

Γ k = {τ k ≤ N } . (3.15)
By the same way used in the paper [START_REF] Galtchouk | Efficient pointwise estimation based on discrete data in ergodic nonparametric diffusions[END_REF] to choose the threshold H k , we use the truncated estimator q(z k ) given in (3.6) and we set

H k = h(N -N 0 )(2 q(z k ) -υ T ) . (3.16)
It should be noted that, as established in [START_REF] Galtchouk | Sequential nonparametric adaptive estimation of the drift coefficient in diffusion processes[END_REF], this form for H k provides the optimal convergence rate.

Regression model

In order to obtain an oracle inequality for discrete time data, we shall pass to a regression model by the same way as in [START_REF] Galtchouk | Adaptive sequential estimation for ergodic diffusion processes in quadratic metric[END_REF].

Denote G * = ∩ n k=1 Γ k and Y k = S * k 1 G * . (4.1)
By making use of (1.1) and (3.14) we define a nonparametric regression model on the set G * :

Y k = S(z k ) + g k + σ k ξ k , σ k = b(z k ) δH k , (4.2)
where

g k = 1 δH k τ k j=N 0 +1 κ j,k χ j,k (h) t j t j-1 S(y u ) du -S(z k ) + 1 δH k τ k j=N 0 +1 κ j,k χ j,k (h) t j t j-1 (b(y s ) -b(z k ))dW s (4.3)
and

ξ k = 1 δH k τ k j=N 0 +1 κ j,k χ j,k (h)∆W t j . (4.4)
Note that the coefficients (σ l ) 1≤l≤n are random variables and using their definitions one can obtain the following bounds

σ 0, * ≤ min 1≤l≤n σ 2 l ≤ max 1≤l≤n σ 2 l ≤ σ 1, * , (4.5)
where

σ 0, * = υ T b min δN h and σ 1, * = b max υ T δ(N -N 0 )h .
We estimate the parameter σ 2 l as follows:

σ l = b l δH l , (4.6)
where b l is the estimator of the squared diffusion coefficient b 2 (z l ) defined in (3.9). In order to obtain the oracle inequality, we need to study the properties of the last estimator. To this end we set *

T = n max 1≤l≤n E ϑ | σ l -σ 2 l | . (4.7)
Proposition 4.1. Assume that the conditions A 1 ) -A 2 ) hold. Then, for any a > 0, lim

T →∞ T γ-1/2-a * T = 0 . (4.8)
This proposition is proved in [START_REF] Galtchouk | Non asymptotic sharp oracle inequalities for high dimensional ergodic diffusion models[END_REF].

Model selection

First we choose a basis (φ j ) j≥1 in L 2 ([x 0 , x 1 ]) such that, for any 1 ≤ i, j ≤ n,

(φ i , φ j ) n = x 1 -x 0 n n l=1 φ i (z l )φ j (z l ) = 1 {i=j} . (5.1)
For example, one can take the trigonometric basis defined as φ 1 (x) ≡ 1/ √ x 1 -x 0 and, for j ≥ 2,

φ j (x) = 2 x 1 -x 0    cos(2π[j/2] l 0 (x)) for even j ; sin(2π[j/2]l 0 (x)) for odd j , (5.2)
where [a] denotes the integer part of a and l 0 (x) = (x -x 0 )/(x 1 -x 0 ). Note that if n is odd, then this basis is orthonormal for the empirical inner product, i.e. satisfies the property (5.1). In the sequel, we assume that the n is odd, i.e. as in (3.3). By making use of this property we define the discrete Fourier representation for S on the sieve (3.1), i.e.,

S(z

k ) = n j=1 θ j,n φ j (z k ) , 1 ≤ k ≤ n , (5.3)
where

θ j,n = (S, φ j ) n = x 1 -x 0 n n l=1 S(z l )φ j (z l ) .
Moreover, using the regression model (4.2) we estimate these coefficients as

θ j,n = (Y, φ j ) n = x 1 -x 0 n n l=1 Y l φ j (z l ) . (5.4)
By the model (4.2), we obtain on the set G *

θ j,n = θ j,n + ζ j,n , ζ j,n = g j,n + x 1 -x 0 n ξ j,n , (5.5)
where

ξ j,n = x 1 -x 0 n n l=1 σ l ξ l φ j (z l ) , g j,n = x 1 -x 0 n n l=1 g l φ j (z l ) .
We estimate the values S(z k ), 1 ≤ k ≤ n, by the weighted least squares estimators

S λ (z k ) = n j=1 λ(j) θ j,n φ j (z k ) , 1 ≤ k ≤ n , (5.6)
where the weight vector λ = (λ(1), . . . , λ(n)) belongs to some finite set Λ from [0, 1] n . In the sequel, we denote by ν the cardinal number of the set Λ, ν = card(Λ), which is a function of T , i.e. ν = ν T . Moreover, we set the norm for Λ as

Λ * = max λ∈Λ n j=1 λ(j) (5.7)
which can be a function of T , i.e. Λ * = Λ * (T ). We assume that the basis functions and the weight set Λ satisfy the following condition.

A 3 ) For any a > 0, lim

T →∞ ν T T a = 0 and lim T →∞ Λ * (T ) T 1/3+a = 0 . (5.8)
To estimate the function S on the interval [x 0 , x 1 ], we use the step-function approximation, i.e.,

S λ (x) = n l=1 S λ (z l )1 {z l-1 <x≤z l } , x ∈ [x 0 , x 1 ] . (5.9)
Now one needs to choose a cost function in order to define an optimal weight λ ∈ Λ. A best candidate for the cost function should be the empirical squared error given by the relation

Err n (λ) = S λ -S 2 n → min . In our case, the empirical squared error is equal to

Err n (λ) = n j=1 λ 2 (j) θ 2 j,n -2 n j=1 λ(j) θ j,n θ j,n + n j=1 θ 2 j,n . (5.10)
Since coefficients θ j,n are unknown, we need to replace the term θ j,n θ j,n by some estimator which we choose as

θ j,n = θ 2 j,n - x 1 -x 0 n σ j,n and σ j,n = x 1 -x 0 n n l=1 σ l φ 2 j (z l ) , (5.11)
where σ l is the estimator for σ 2 l defined in (4.6). Note that if the diffusion is known, then we take in (5.11) σ j,n = σ j,n and

σ j,n = x 1 -x 0 n n l=1 σ 2 l φ 2 j (z l ) .
(5.12)

It is clear that the inequalities (4.5) imply

σ 0, * ≤ min 1≤l≤n σ l,n ≤ max 1≤l≤n σ l,n ≤ σ 1, * . (5.13)
Now, for using the estimator (5.11) instead of θ j,n θ j,n one needs to add to the cost function a suitable penalty term that we take as

P n (λ) = x 1 -x 0 n n j=1 λ 2 (j) σ j,n (5.14)
if the diffusion is unknown and as

P n (λ) = x 1 -x 0 n n j=1 λ 2 (j) σ j,n (5.15)
when the diffusion is known. Finally, we use the following cost function

J n (λ) = n j=1 λ 2 (j) θ 2 j,n -2 n j=1 λ(j) θ j,n + ρ P n (λ) , (5.16)
where the positive coefficient 0 < ρ < 1 will be specified later.

We define the model selection procedure as λ = argmin λ∈Λ J n (λ) and S * = S λ .

(5.17 For the risk (1.3), we have shown the following result in [START_REF] Galtchouk | Non asymptotic sharp oracle inequalities for high dimensional ergodic diffusion models[END_REF].

Theorem 5.1. Assume that the conditions A 1 ) -A 3 ) hold. Then, for any T ≥ 1, 0 < ρ ≤ 1/8 and ϑ ∈ Θ, the estimation procedure S * defined in (5.17) satisfies the inequality

R ϑ ( S * ) ≤ (1 + ρ) 2 (1 + 5ρ) 1 -6ρ min λ∈Λ R ϑ ( S λ ) + U ϑ,T ρ T , (5.18)
where the remainder term U ϑ,T is such that for any a > 0, lim

T →∞ T -a sup ϑ∈Θ U ϑ,T = 0 . (5.19)
In the sequel to obtain the efficient properties for this procedure we will use the special weight coefficients introduced in [START_REF] Galtchouk | Sharp non-asymptotic oracle inequalities for nonparametric heteroscedastic regression models[END_REF][START_REF] Galtchouk | Adaptive asymptotically efficient estimation in heteroscedastic nonparametric regression[END_REF]. To this end we consider a 2-dimensional numerical grid of the form

A = {1, . . . , k * } × {r 1 , . . . , r m * } , (5.20)
where r i = iε and m * = [1/ε 2 ]. The both parameters k * ≥ 1 and 0 < ε ≤ 1 are some functions of T , i.e. k * = k * T and ε = ε T , such that, for any γ > 0, lim

T →∞ ε T + 1 T γ ε T + 1 k * T + k * T ln T = 0 . (5.21)
One can take, for example,

ε T = 1 ln(T + 1)
and k * = k + ln(T + 1), for some fixed k ≥ 1. For any α = (k, l) ∈ A, we introduce the weight sequence λ α = (λ α (j)) j≥1 as

λ α (j) = 1 {1≤j≤j 0 } + 1 -(j/ω α ) k 1 {j 0 <j≤ωα} , (5.22)
where j 0 = j 0 (α) = [ω α / ln(T + 1)], ω α = ωk (lT ) 1/(2k+1) and ωk = (x 1 -x 0 ) (k + 1)(2k + 1)

π 2k k 1/(2k+1)
.

We set Λ = (λ α ) α∈A .

(5.23)

Note that, in this case, the cardinal ν of the set Λ is the function of T , i.e. ν = ν T = k * m * and the conditions (5.21) imply that, for any a > 0, lim

T →∞ ν T T a = 0 . (5.24)
Moreover, from the definition (5.22) we can obtain that, for any α ∈ A,

n j=1 λ α (j) ≤ ω α ≤ ωk T ε T 1/3 .
Therefore, for any a > 0, lim T →∞ Λ * T 1/3+a = 0 .

(5.25)

Hence, the condition A 3 ) holds and we obtain the following result.

Theorem 5.2. Assume that the conditions A 1 ) -A 2 ) hold. Then, the model selection procedure (5.17) with the weights (5.23) satisfies the oracle inequality (5.18) with the remainder term satisfying the property (5.19), for any a > 0.

Remark 5.2. It should be noted that similarly to [START_REF] Galtchouk | Adaptive sequential estimation for ergodic diffusion processes in quadratic metric[END_REF], we will use the inequality (5.18) to provide the efficiency property in the adaptive setting. This means that without using the regularity properties of the unknown function S we can estimate from above the risk for the model selection procedure by the risk for the efficient estimation procedure constructed on the regularity properties of the function S. The upper bound for the risk of the model selection procedure follows from the sharp oracle inequality.

Main results

In order to study the asymptotic efficiency of estimators we use a suitable class of drift coefficients S. To this end we define the following functional Sobolev ball

W 0 k,r = {f ∈ C k 0 ([x 0 , x 1 ]) : k j=0 f (j) 2 ≤ r} , (6.1)
where r > 0 and the integer k ≥ 1 are some parameters, C k 0 ([x 0 , x 1 ]) is the space of k times differentiable functions f : R → R such that

f (i) (x) = 0 , for 0 ≤ i ≤ k -1 and x / ∈ [x 0 , x 1 ] .
Moreover, let S 0 be a fixed continuously differentiable function from Σ L,M such that S 0 (x) = 0 for x 0 ≤ x ≤ x 1 . We set

W k,r = S 0 + W 0 k,r and Θ k,r = W k,r × B . (6.2)
Note that one can represent the functional class W 0 k,r as an ellipse in L 2 ([x 0 , x 1 ]) with trigonometric basis (φ j ) j≥0 :

W 0 k,r = {f ∈ C k 0 ([x 0 , x 1 ]) : ∞ j=0 a j θ 2 j ≤ r} , (6.3)
where θ j = (f, φ j ) =

x 1 x 0 f (x)φ j (x)d x and

a j = k i=0 2π[j/2] x 1 -x 0 2i .
In order to formulate our asymptotic results, we define, for any ϑ ∈ Θ k,r , the following normalizing coefficient γ(ϑ)

γ(ϑ) = l * J k * ϑ , k * = 2k 2k + 1 , (6.4)
where

l * = (2k + 1)r 1-k * k π(k + 1)(2k + 1) k * and J ϑ = x 1 x 0 b 2 (x) q ϑ (x) d x . (6.5)
It is well known that, for any S ∈ Θ k,r , the optimal rate of convergence of estimators is T -2k/(2k+1) (see, for example, [START_REF] Galtchouk | Nonparametric sequential estimation of the drift in diffusion via model selection[END_REF]). Now we denote by Ξ T the set of all possible estimators of S which are measurable with respect to the σ-field σ{y t , 0 ≤ t ≤ T }, i.e. based on the observations (y t ) 0≤t≤T . Theorem 6.1. For any integer k ≥ 1 and r > 0, the quadratic risk R ϑ ( S) admits the following asymptotic lower bound lim inf

T →∞ T k * inf S∈Ξ T sup ϑ∈Θ k,r υ(ϑ)R ϑ ( S) ≥ l * , where υ(ϑ) = J -k * ϑ .
Assume now, that the penalty parameter ρ in (5.16) is a function of T , i.e. ρ = ρ T such that, for any a > 0, lim T →∞ ρ T = 0 and lim

T →∞ T a ρ T = ∞ . (6.6)
We can take, for example, ρ T = (6 + ln(T + 1)) -1 .

Theorem 6.2. Assume that the conditions A 1 ) -A 2 ) hold. Then, for any integer k ≥ 2 and r > 0, the quadratic risk of the estimation procedure S * defined through the trigonometric basis (5.2) in (5.17) and (5.23) with the parameter ρ of the form (6.6) satisfies the following upper bound lim sup

T →∞ T k * sup ϑ∈Θ k,r υ(ϑ)R ϑ ( S * ) ≤ l * .
Theorems 6.1 -6.2 imply immediately the efficiency property : Theorem 6.3. Assume that the conditions A 1 ) -A 2 ) hold. Then the estimation procedure S * defined through the trigonometric basis (5.2) in (5.17) and (5.23) with the parameter ρ of the form (6.6) is asymptotically efficient, i.e.

lim T →∞ inf S∈Ξ T sup ϑ∈Θ k,r υ(ϑ)R ϑ ( S) sup ϑ∈Θ k,r υ(ϑ)R ϑ ( S * ) = 1 .
Remark 6.1. It should be noted that from Theorems 6.1 -6.2 it follows that the function (6.4) is the Pinsker constant, which is calculated for the first time for the model (1.1). For the model (1.1) with the diffusion coefficient b(•) ≡ 1 the Pinsker constant was calculated in [START_REF] Galtchouk | Adaptive sequential estimation for ergodic diffusion processes in quadratic metric[END_REF] which in this case coincides with (6.4). Note also that the parameter l * is the well -known Pinsker constant for the "signal plus white noise" model obtained in [START_REF] Pinsker | Optimal filtration of square integrable signals in gaussian noise[END_REF]. Therefore, the Pinsker constant for the model (1.1) is obtained by multiplying the constant from [START_REF] Pinsker | Optimal filtration of square integrable signals in gaussian noise[END_REF] by the Pinsker variance (6.5) in the power k * .

7 Properties of the regression model (4.2)

First we recall the main properties of the model (4.2) from [START_REF] Galtchouk | Non asymptotic sharp oracle inequalities for high dimensional ergodic diffusion models[END_REF].

The random variables (ξ k ) 1≤k≤n from (5.5), satisfy the following property.

Proposition 7.1. The random variables (ξ k ) 1≤k≤n are N (0, 1) i.i.d. conditionally to the σ-field G N 0 .

Proposition 7.2. Assume that the conditions A 1 ) -A 2 ) hold. Then, for any a > 0, lim

T →∞ T a sup ϑ∈Θ P ϑ (G c * ) = 0 . (7.1)
These propositions are proved in [START_REF] Galtchouk | Non asymptotic sharp oracle inequalities for high dimensional ergodic diffusion models[END_REF]. Now, we study the heteroscedastic property in the model (4.2). To this end we set

s n = (x 1 -x 0 ) n l=1 σ 2 l , (7.2)
where the variances (σ l ) 1≤l≤n are defined in (4.2).

Proposition 7.3. Assume that the condition A 1 ) holds. Then

lim T →∞ sup ϑ∈Θ k,r E ϑ s n - J ϑ x 1 -x 0 = 0 . (7.3)
Proof. Using the definition of σ l in (4.2) and taking account of the form of h given in (3.11), we can represent the quantity s n as

s n = n l=1 nb 2 (z l ) δ h (N -N 0 )(2 q T (z l ) -υ T ) (z l -z l-1 ) = 1 x 1 -x 0 n l=1 b ϑ (z l )(z l -z l-1 ) + R 1 (ϑ) + R 2 (ϑ) , (7.4)
where

b ϑ (x) = b 2 (x)/q ϑ (x), R 1 (ϑ) = 1 x 1 -x 0 n 2 δ (N -N 0 ) -1 n l=1 b ϑ (z l ) (z l -z l-1 )
and

R 2 (ϑ) = n l=1 nb 2 (x l ) δ h (N -N 0 ) 1 2 q T (z l ) -υ T - 1 2q ϑ (z l ) (z l -z l-1 ) .
First, note, that the function b ϑ (•) and its derivative are uniformly bounded, i.e. sup

ϑ∈Θ k,r max x 0 ≤z≤x 1 b ϑ (z) + | b ϑ (z)| < ∞ . (7.5)
Therefore, lim

T →∞ sup ϑ∈Θ k,r n l=1 b ϑ (z l )(z l -z l-1 ) -J ϑ = 0 .
As the second term (7.4), note that, in view of the condition (3.2), lim

T →∞ n 2 δ (N -N 0 ) = 1 .
Therefore, the property (7.5) provides that lim

T →∞ sup ϑ∈Θ k,r |R 1 (ϑ)| = 0 .
Moreover, taking into account that 2 q T (z l ) -υ T > υ

1/2
T , we obtain, for sufficiently large T , that for some C > 0

|R 2 (ϑ)| ≤ C n l=1 υ -1/2 T (| q T (z l ) -q ϑ (z l )|) (z l -z l-1 ) + √ υ T .
Note here, that for any 1 ≤ l ≤ n and for sufficiently large T ,

υ -1/2 T E ϑ | q T (z l ) -q ϑ (z l )| ≤ 2υ -1 T P ϑ (| q T (z l ) -q ϑ (z l )| > υ T ) + √ υ T .
Therefore, due to Proposition 3.1, lim

T →∞ sup ϑ∈Θ k,r |R 2 (ϑ)| = 0 .
Hence Proposition 7.3.

Non adaptive estimation

We start with the non adaptive case, i.e. we assume that the parameters k, r and the function J ϑ are known. In this case, we use the weight estimator (5.9) from the family (5.23)

S = S λ and λ = λ α , (8.1)
where α = (k, l), l = l T = [r(ϑ)/ε] ε and r(ϑ) = r/J ϑ . We remind that ε = ε T = 1/ ln(T + 1). Firstly, we need to study the asymptotic properties of the weights λ. To this end we set

Υ T (ϑ) = n j=1 (1 -λ(j)) 2 θ 2 j,n + J ϑ (x 1 -x 0 )n 2 n j=1 λ 2 (j) . (8.2)
Proposition 8.1. For any k ≥ 2 and r > 0, lim sup

T →∞ T k * sup ϑ∈Θ k,r υ(ϑ) Υ T (ϑ) ≤ l * . (8.3)
Proof. First of all, note that 0 < inf

ϑ∈Θ k,r J ϑ ≤ sup ϑ∈Θ k,r J ϑ < ∞ . (8.4)
This implies directly that lim

T →∞ sup ϑ∈Θ k,r l T r(ϑ) -1 = 0 , (8.5)
where r(ϑ) = r/J ϑ . Moreover, note that

T k * υ(ϑ) Υ T (ϑ) ≤ T k * υ(ϑ) S T + (J ϑ ) 1-k * T 1-k * (x 1 -x 0 ) n j=1 λ 2 (j) , (8.6)
where S T = n j=1 (1 -λ(j)) 2 θ 2 j,n . We decompose S T as

S T = S 1,T + S 2,T with S 1,T = [ ω] j=j 0 +1 (1 -λ(j)) 2 θ 2 j,n and S 2,T = n j=[ ω]+1 θ 2 j,n .
Let us recall that ω = ω α = ωk T l T

1/(2k+1)

.

Lemma A.2 yields

S 1,T ≤ (1 + ε) [ ω] j=j 0 (1 -λ(j)) 2 θ 2 j + 2r(1 + ε -1 ) ω n 2k ,
and Lemma A.3 yields

S 2,T ≤ (1 + ε) j> ω θ 2 j + (1 + ε -1 ) r n 2 ω 2(k-1) .
These inequalities imply that

S T ≤ (1 + ε)S * T + 2r(1 + ε -1 ) ωT , (8.7)
where

S * T = j≥1 (1 -λ(j)) 2 θ 2 j = j≤ ω (1 -λ(j)) 2 θ 2 j + j> ω θ 2 j := S * 1,T + S * 2,T and ωT = ω n 2k + 1 n 2 ω 2(k-1) . Note, that T k * υ(ϑ)S * 1,T = T k * υ(ϑ) [ ω] j=j 0 (1 -λ(j)) 2 θ 2 j = υ(ϑ) ω2k k ( l T ) k * [ ω] j=j 0 j 2k θ 2 j ≤ υ(ϑ) ω2k k ( l T ) k * j 0 [ ω] j=j 0 a j θ 2 j ,
where n = sup j≥n j 2k /a j . It is clear that

lim n→∞ n = (x 1 -x 0 ) 2k π 2k .
Therefore, from (8.5) we obtain that lim sup

n→∞ sup ϑ∈Θ k,r T k * υ(ϑ) S * 1,T [ ω] j=j 0 a j θ 2 j ≤ (x 1 -x 0 ) 2k π 2k ω2k k r k * = k (2k + 1)(k + 1)rπ k * .
Futher note, that for any 0 < ε < 1 and for sufficiently large T ,

S * 2,T = j> ω θ 2 j ≤ (x 1 -x 0 ) 2k π 2k ω 2k j> ω a j θ 2 j = (x 1 -x 0 ) 2k π 2k (T l T ) k * ω2k k j> ω a j θ 2 j ≤ (1 + ε) J k * ϑ (x 1 -x 0 ) 2k π 2k (rT ) k * ω2k k j> ω a j θ 2 j .
This inequality implies lim sup

n→∞ sup ϑ∈Θ k,r T k * υ(ϑ) S * 2,T j> ω a j θ 2 j ≤ (x 1 -x 0 ) 2k π 2k ω2k k r k * = k (2k + 1)(k + 1)rπ k * .
So, taking into account in (8.7) that for k ≥ 2 lim

T →∞ sup S∈W k,r T k * ωT = 0 , we obtain that lim sup T →∞ sup ϑ∈Θ k,r T k * υ(ϑ) S T ≤ r 1-k * k (2k + 1)(k + 1)π k * . ( 8.8)
Moreover, we can check directly that lim sup

T →∞ sup ϑ∈Θ k,r J 1-k * ϑ T 1-k * (x 1 -x 0 ) n j=1 λ 2 (j) ≤ u * k , (8.9)
where

u * k = ωk r 1-k * 2k 2 (x 1 -x 0 )(2k + 1)(k + 1) = 2kr 1-k * k (2k + 1)(k + 1)π k * .
Using now (8.8) and (8.9) in (8.6) we obtain the limit equality (8.3) (5.23). Therefore, through the oracle inequality we can estimate the risk of the model selection procedure with the risk of the estimator (5.23) and, therefore, using the property (8.11) we can provide the efficiency property for the procedure (5.17).

Proofs

9.1 Proof of Theorem 6.1

First, we introduce the following auxiliary class

Θ 0 k,r = ϑ = (S, b 0 ) : S ∈ W k,r , b 0 ≡ 1 . It is clear that Θ 0 k,r ⊂ Θ k,r and sup ϑ∈Θ k,r υ(ϑ) R( S, S) ≥ sup ϑ∈Θ 0 k,r υ(ϑ) R( S, S) .
Using here Theorem 5.2 from [START_REF] Galtchouk | Adaptive sequential estimation for ergodic diffusion processes in quadratic metric[END_REF] we obtain Theorem 6.1.

Proof of Theorem 8.2

First note that, in view of Proposition 7.2, to prove this theorem it suffices to show that lim sup

T →∞ T k * sup ϑ∈Θ k,r υ(ϑ) E ϑ χ G * S -S 2 n ≤ l * . (9.1)
Indeed, note that due to (5.3)-(5.6) on the set G * we obtain that

S -S 2 = n j=1 λ 2 (j)(θ j,n + ζ j,n ) 2 -2 λ(j)(θ j,n + ζ j,n )θ j,n + θ 2 j,n = n j=1 ( λ(j) -1) 2 θ 2 j,n + 2 λ(j)( λ(j) -1)ζ j,n θ j,n + λ 2 (j)ζ 2 j,n , (9.2)
where θ j,n = (S , φ j ) n and λ is defined in (8.1). Taking into account that

ζ j,n = g j,n + x 1 -x 0 n ξ j,n , using the inequality 2xy ≤ x 2 + -1 y 2 , we have from (9.2) on the set G * S -S 2 n ≤ (1 + ) n j=1 (1 -λ(j)) 2 θ 2 j,n + B 1 + 1 + 2 B 2 + 2 x 1 -x 0 n n j=1 λ(j)( λ(j) -1)θ j,n ξ j,n , (9.3)
where

B 1 = x 1 -x 0 n n j=1 λ 2 (j)ξ 2 j,n and B 2 = n j=1 λ 2 (j)g 2 j,n . (9.4)
Moreover, taking into account the property E ξ j,n = 0, one gets

E ϑ S -S 2 n χ G * ≤ (1 + ) n j=1 (1 -λ(j)) 2 θ 2 j,n + E ϑ B 1 + 1 + 2 E ϑ B 2 χ G * -2E ϑ B 3 χ G c * , (9.5)
where

B 3 = x 1 -x 0 n n j=1 λ(j)( λ(j) -1)ξ j,n θ j,n . Note that (x 1 -x 0 )E ϑ ξ 2 j,n |G N 0 = s n n + x 1 -x 0 n n l=1 σ 2 l φ j (z l ) (9.6)
where s n is defined in (7.2) and φ j (z l ) = (x 1 -x 0 )φ 2 j (z l ) -1. Setting now

s n = s n - J ϑ x 1 -x 0 ,
we represent the terme E ϑ B 1 as

E ϑ B 1 = 1 n 2 E ϑ n j=1 λ 2 (j) s n + (x 1 -x 0 ) n l=1 σ 2 l φ j (x l ) = J ϑ (x 1 -x 0 )n 2 n j=1 λ 2 (j) + B 11 + B 12 , (9.7)
where

B 11 = 1 n 2 n j=1 λ 2 (j) E ϑ s n , B 12 = x 1 -x 0 n 2 E ϑ n j=1 λ 2 (j) n l=1 σ 2 l φ j (x l ) .
Now from (9.5) and (9.7) we obtain that

E ϑ S -S 2 n χ G * ≤ (1 + ) Υ T + B 11 + B 12 + 1 + 2 E ϑ B 2 χ G * -2E ϑ B 3 χ G c * . ( 9.8)
The property (8.9) and Proposition 7.3 yield for the terme B 11 lim sup

T →∞ T k * sup ϑ∈Θ k,r | B 11 | = 0 .
As to the term B 12 , due to Lemma A.4, one has

|B 12 | ≤ x 1 -x 0 n 2 E ϑ n l=1 σ 2 l n j=1 λ 2 (j)φ j (x l ) ≤ 2 k+1 x 1 -x 0 n 2 E ϑ n l=1 σ 2 l =, 2 k+1 E ϑ s n n 2 . (9.9)
Therefore, in view of Proposition 7.3, we obtain that lim sup

T →∞ T k * sup ϑ∈Θ k,r |B 12 | = 0 .
Moreover, by the definition,

B 2 = n j=1 λ 2 (j)g 2 j,n
and g j,n = (g, φ j ) n .

Therefore,

B 2 ≤ n j=1 g 2 j,n = g 2 n and E ϑ 1 G * B 2 ≤ E ϑ 1 G * g 2 n = x 1 -x 0 n n k=1 E ϑ 1 G * g 2 k .
It is easy to verify (see [START_REF] Galtchouk | Non asymptotic sharp oracle inequalities for high dimensional ergodic diffusion models[END_REF]) that

sup ϑ∈Θ max 1≤ k≤ n E ϑ 1 G * g 2 k ≤ M 2 h 2 + o(1/T ) .
These bounds yield that lim

T →∞ T 2k 2k+1 sup ϑ∈Θ k,r E ϑ 1 G * B 2 = 0 . (9.10)
To estimate the last term in (9.5) note that through Lemma A.5 we can estimate the square of the expectation of this term as

E ϑ B 2 3 ≤ σ 1, * x 1 -x 0 n n j=1 θ 2 j,n = σ 1, * (x 1 -x 0 ) n S 2 n .
Therefore, 9.3 Proof of Theorem 6.2

E ϑ 1 G c *
Taking into account that for sufficiently large T the estimator (8.1) belongs to the family ( S λ ) λ∈Λ indexed by the set (5.23), we obtain that lim sup

T →∞ T k * E ϑ S * -S 2 ≤ lim sup T →∞ T k * E ϑ S λ -S 2 .
So, Theorem 8.2 implies immediately Theorem 6.2.

Acknowledgments. The work of the last author was partially supported by the Russian Federal Professor program, project no. 1.472.2016/1.4 (the Ministry of Science and Higher Education of the Russian Federation).

A Appendix

A.1 Properties of the trigonometric basis. where c j are some constants and the sequence (z j ) 0≤j≤n is given in (3.1).

Then, for any ε > 0, the function ∆ = f -g satisfies the following inequalities

∆ 2 ≤ (1 + ε) ∆ 2 n + 1 + 1 ε ḟ 2 n 2 (x 1 -x 0 ) 2 ,
and

∆ 2 n ≤ (1 + ε) ∆ 2 + 1 + 1 ε ḟ 2 n 2 (x 1 -x 0 ) 2 .
The proof is given in Lemma A.2 from [START_REF] Konev | Robust model selection for a semimartingale continuous time regression from discrete data[END_REF].

Lemma A.2. For any 1 ≤ j ≤ n, the trigonometric Fourier coefficients (θ j,n ) 1≤j≤n of the functions S from the class W k,r satisfy, for any ε > 0, the following inequality By applying Lemma A.1 with f = ∆ n , g = 0, we obtain that

∆ n 2 n ≤ 2 ∆ n 2 + 2 ∆n 2 (x 1 -x 0 ) 2 n 2 = 2 l>n θ 2 l + 2 ∆n 2 (x 1 -x 0 ) 2 n 2 ≤ 2ra -1 n+1 + 2 ∆n 2 (x 1 -x 0 ) 2 n 2 .
Taking account of (6.3) and that 2[l/2] ≥ l -1 for l ≥ 2, we have

∆ n 2 n ≤ 2r(x 1 -x 0 ) 2k π 2k n 2k + 2 ∆n 2 (x 1 -x 0 ) 2 n 2 .

A.2 Technical lemma

Lemma A.5. Let (u j ) 1≤j≤n be nonrandom sequence. Then Proof. First we denote D n = n j=1 u j ξ j,n . Now, using the definition of ξ j,n in (5.5) we obtain that

D n = x 1 -x 0 n n l=1
σ l ξ l,n n j=1 u j φ j (z l ) .

This implies

E ϑ D 2 (u) = x 1 -x 0 n E ϑ n l=1 σ 2 l ( n j=1 u j φ j (z l )) 2 ≤ σ 1, * n j=1 u 2 (j) = σ 1, * u 2 .
Hence Lemma 3.1.

) Remark 5 . 1 .

 51 It should be emphasized that if in the model (1.1) the diffusion coefficient b(•) in known, then the model selection procedure (5.17) is defined through minimazing the cost function J n (λ) with the penalty term (5.15).

|B 3 | ≤ σ 1 ,E ϑ 1

 11 * (x 1 -x 0 ) n S n P ϑ G c * and taking into account Proposition 7.2, we obtain that lim G c * |B 3 | = 0 . Therefore, using Proposition 8.1 in (9.8) we obtain Theorem 8.2.

Lemma A. 1 .

 1 Let f be an absolutely continuous [x 0 , x 1 ] → R function with ḟ < ∞ and g be [x 0 , x 1 ] → R a step-wise function of the formg(z) = n j=1 c j χ (z j-1 ,z j ] (z),

θ 2 j

 2 ,n ≤ (1 + ε) θ 2 j + (1 + ε -1 ) řk n 2k , (A.1)whereřk = 2r(π 2 + 1)(x 1 -x 0 ) 2k π 2k .Proof. First we represent the function S as the Fourier series in L 2 ([x 0 , x 1 ]) :S(x) = n l=1 θ l φ l (x) + ∆ n (x) , where ∆ n (x) = l>n θ l φ l (x) . (A.2)Making now the Fourier decomposition of S on the grid (z k ) 1≤ k≤ n with an odd n, one gets the Fourier coefficients :θ j,n = (S, φ j ) n = θ j + (∆ n , φ j ) nand, for any 0 < ε < 1,θ 2 j,n ≤ (1 + ε)θ 2 j + (1 + ε -1 ) ∆ n 2 n .

  and, hence Proposition 8.1. Now, through Proposition 8.1 we can obtain the following upper bound. It should be noted that the inequality (8.11) together with Theorem 6.1 mean that the estimator S defined by (8.1) is efficient. Unfortunately, we can't calculate this estimator, since it depends on unknown parameters k, r and J ϑ . But, this estimator belongs to the family ( S λ ) λ∈Λ with the weight vectors defined in

	Theorem 8.2. For any k ≥ 2 and r > 0, the estimator S from (8.1) satisfies
	the inequality
	lim sup T →∞	ϑ∈Θ k,r T k * sup	υ(ϑ) E ϑ S -S 2 n ≤ l * .	(8.10)
	Note that Lemma A.1 implies the following upper bound.
	Theorem 8.3. The quadratic risk for the estimating procedure S from (8.1)
	has the following asymptotic upper bound
	lim sup T →∞	ϑ∈Θ k,r T k * sup	υ(ϑ)R( S, S) ≤ l * .	(8.11)
	Remark 8.1.

Note here, that for any n ≥ 1,

Lemma A.3. For any n ≥ 2, 1 ≤ m < n and r > 0, the coefficients (θ j,n ) 1≤j≤n of functions S from the class W k,r satisfy, for any ε > 0, the following inequality

where

.

Proof. First we note that n j=m+1 θ 2 j,n = min

where the function ∆ m (•) is defined in (A.2). By applying Lemma A.1 with f = ∆ m , g = 0, and taking account of the inequality (A.3), we obtain the bound (A.4). Hence Lemma A.3

where φ l (x) = (x 1 -x 0 )φ 2 l (x) -1. Proof of this result is given in Lemma A.2 from [START_REF] Galtchouk | Sharp non-asymptotic oracle inequalities for nonparametric heteroscedastic regression models[END_REF].


Lucrecia López Romina

Steinmetz

Bing Shao

Emile Fong

Lorena Boyer

López Cecilia

Norberto Elio Steinmetz

Vincent Tejerina

Meuric

López Steinmetz

Shao Bing Fong

Emile Boyer

Lorena Cecilia López Steinmetz
email: lucrecialopezsteinmetz@hotmail.com

Norberto Elio Tejerina

Vincent Meuric

Lithium and Boron in Calcified Tissues of Vicuna and Their Relation to Chronic Exposure by Water Ingestion in The Andean Lithium Triangle

Keywords: lithium, vicuna, bones, teeth, salt pan, Andes

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

The lithium triangle defines a zone within the Andean plateau that is characterized by the presence of lithium (Li)-rich salt pans (Fig. 1). This region involves a major part of the known global Li resources [START_REF] Kesler | Global lithium resources: relative importance of pegmatites, brine and other deposits[END_REF][START_REF] Munk | Lithium brines: A Global Perspective[END_REF]. Li is a relatively abundant element on Earth and its concentration in the upper continental crust is about 35 ± 11 ppm [START_REF] Teng | Lithium isotopic composition and concentration of the upper continental crust[END_REF]).

As Li is a very reactive and geochemical incompatible element, it is often present in waters (i.e., in solution) in its ionic form Li + . The mean Li concentration in sea water is 0.18 mg L -1 [START_REF] Riley | The lithium content of sea water[END_REF]. Continental waters are characterized by containing large variable Li grades. In the Andean plateau, river waters involve relatively high Li contents that are typically up to 10 mg L -1 (López [START_REF] Steinmetz | Lithium-and boron-bearing brines in the Central Andes: exploring hydrofacies on the eastern Puna plateau between 23° and 23°30[END_REF]. These rivers systems are frequently fed by hydrothermal sources that contain even higher Li grades, which is conducive to increasing Li concentrations in river waters often exceeding 15 mg L -1 [START_REF] Giordano | Structural control on geotermal circulation in the Cerro Tuzgle-Tocomar geotermal volcanic area (Puna plateau, Argentina)[END_REF][START_REF] Giordano | Structural control on geotermal circulation in the Cerro Tuzgle-Tocomar geotermal volcanic area (Puna plateau, Argentina)[END_REF][START_REF] Arnold | Fluid geochemistry of a deep-seated geothermal resource in the Puna plateau (Jujuy Province, Argentina)[END_REF]. Moreover, in the lithium triangle, brines within salt pans typically contain mean Li concentrations ranging between 200 and 400 mg L -1 (Fig. 2 A), and that can even exceed 1,000 mg L -1 [START_REF] Ericksen | Geology and resources of salars in the central Andes[END_REF][START_REF] Risacher | Quaternary geochemical evolution of the Salar of Uyuni and Coipasa, Central Altiplano, Bolivia[END_REF]Garret 2004;[START_REF] Kesler | Global lithium resources: relative importance of pegmatites, brine and other deposits[END_REF][START_REF] Godfrey | The role of climate in the accumulation of lithium-rich brine in the Central Andes[END_REF][START_REF] Steinmetz | Lithium-and boron-bearing brines in the Central Andes: exploring hydrofacies on the eastern Puna plateau between 23° and 23°30[END_REF][START_REF] Steinmetz | Northern Puna plateau-scale survey of Li brine-type deposits in the Andes of NW Argentina[END_REF].

Drinking water supply in Andean villages is obtained from such natural sources.

The presence of Li in drinking waters can be traced back in time. Li was found in calcified, keratinized, and soft tissues of ancient mummies from the Puna de Atacama in northern Chile [START_REF] Farell | Bioimaging of trace metals in ancient Chilean mummies and contemporary Egyptian teeth by laser ablation-inductively coupled plasma-mass spectrometry (LA-ICP-MS)[END_REF][START_REF] Figueroa | Environmental lithium exposure in the northern of Chile -Tissue exposure indices[END_REF]; Table 1). In sampled teeth the .

Accepted Article

concentrations in the crown were high in enamel as well as in dentin, suggesting a prenatal exposure to Li-bearing drinking waters [START_REF] Farell | Bioimaging of trace metals in ancient Chilean mummies and contemporary Egyptian teeth by laser ablation-inductively coupled plasma-mass spectrometry (LA-ICP-MS)[END_REF].

The passage of time has not changed these situations, and in modern times, the water potabilization process does not include the regulation of Li contents prior to human ingestion (Figueroa et al. 2012). For example, in the town of San Antonio de los Cobres in northwest Argentina (for location see Fig. 1), besides Li concentrations reaching 1 mg L -1 , up to 0.21 mg L -1 of arsenic (As), and 5.90 mg L -1 of boron (B) have been reported in drinking waters [START_REF] Concha | High-level exposure to lithium, boron, cesium, and arsenic via drinking water in the Andes of northwestern Argentina[END_REF]. The chronic ingestion of drinking water of such quality is observed to induce an increase of up to one order of magnitude in the concentration of these chemical elements in the urine, as well as in diverse body tissues including hair, blood and breast milk, relative to people unexposed [START_REF] Concha | High-level exposure to lithium, boron, cesium, and arsenic via drinking water in the Andes of northwestern Argentina[END_REF][START_REF] Figueroa | Environmental lithium exposure in the northern of Chile -Tissue exposure indices[END_REF][START_REF] Harari | Environmental exposure to lithium during pregnancy and fetal size: a longitudinal study in the Argentinean Andes[END_REF].

Studies on the effect of ingesting Li have been, for a long period of time, mainly focused in its psychiatric use, which is the most common source of exposure to Li. Li salts, especially the carbonate (Li 2 CO 3 ) and acetate (LiCH 3 COO) have psychotropic properties, and are employed since the mid-1800s in the treatment of bipolar disorders (Cade 1949;[START_REF] Schou | Lithium in psychiatric therapy and prophylaxis[END_REF][START_REF] Aral | Toxicity of lithium to humans and the environment -A literature review[END_REF]. Treatment of these disorders require the surveillance of the Li concentration due to its toxicity (Price and Heninger 1964). Some major differences exist between these two kinds of Li exposures. First, although psychiatric Li exposures are conducted under long-term durations, the people living in Andean villages are exposed to Li during their entire life. Furthermore, inhabitants of the Andean plateau where drinkable water contains high Li grades are not .

Accepted Article

systematically subjected to serum surveys, contrarily to persons following a psychiatric treatment. For example, according to pharmacological Li carbonate leaflets, a standard posology during a psychiatric treatment for a young adult person weighing ~70 kg is from 500 to 1200 mg day -1 . This implies a daily absorption of 93 to 225 mg of Li. The resulting therapeutic serum concentrations are expected remain below 8.40 mg L -1 , as the toxic stage begins at about 10.50 -17.50 mg L -1 [START_REF] Litovitz | 1993 annual report of the American Association of Poison Control Centers Toxic Exposure Surveillance System[END_REF][START_REF] Jaeger | Lithium Medicine[END_REF]). In populations residing in the Puna de Atacama, [START_REF] Figueroa | Environmental lithium exposure in the northern of Chile -Tissue exposure indices[END_REF] have reported average serum concentrations of 5.30 mg L -1 resulting from daily ingestions of high Ligraded drinkable waters.

Absorbed Li reaches its maximum concentration in serum about 30 min after oral ingestion. Li is distributed throughout body tissues, and is excreted primarily through the kidneys after approximately 24 h [START_REF] Freeman | Lithium: clinical considerations in internal medicine[END_REF]. Additionally, a long term metabolization of Li occurs through ossification. Li accumulates in mammal bones, and seems to be especially incorporated into the minerals of rapidly growing bones [START_REF] Birch | Lithium accumulation in bone after oral administration in rat and human[END_REF]. Naturally, this incorporation ends when the subject is no longer exposed to Li. However, only a small fraction is rapidly removed from the bone tissue. This slow decrease suggests a reduced bone turnover state [START_REF] Peppersack | Effects of lithium on bone resorption in cultured foetal rat long-bones[END_REF][START_REF] Mak | Effects of lithium therapy on bone mineral metabolism: a two-year perspective longitudinal study[END_REF][START_REF] Zamani | Lithium's effect on bone mineral density[END_REF].

It is important to note that there is no Li in human bones of persons that are not exposed to Li [START_REF] Alexander | Boron and lithium content of human bones[END_REF]. Bone and tooth tissues are composed of water, inorganic, and organic constituents. The proportion of constituting components in bone and tooth are variable and depends on several factors such as the tissue type, the considered species, the age, etc. In human tooth, enamel is the most mineralized tissue .

Accepted Article

with 96%, and in dentin, the inorganic material, represents 70% (Gutiérrez-Salazar and Reyes-Gasga 2003). The mineral fraction in bone is also the most abundant constituent, where it is majorly composed of phosphate (PO 4 3-) and calcium (Ca 2+ ), and represents about 65% of normal human bone, and up to ~76% in bovines [START_REF] Quelch | Chemical composition of human bone[END_REF].

Living beings in the Andean plateau are exposed to altitudes of more than 3,000 m above sea level (m asl), high daily thermic amplitudes, and regional aridity. Vicuna (Vicugna vicugna) is a wild Andean camelid living under these extreme conditions in the high plateau areas (Tirado et al. 2012, and references therein;Fig. 2 B). During periods of enhanced aridity (from April to November), the adaptation strategy of vicuna consists of drinking from brine pools within the salt pans (Fig. 2 C). The salinity of these brines is >100,000 mg L -1 , and dissolved elements include high concentrations of Li, As, and B, among other ions.

In this article we present results of chemical analysis on the bone and tooth composition of two specimens of vicuna we have found at the salt pan of Salinas Grandes in the Andean plateau of northwestern Argentina. We analyzed the presence of Li, B, and As in bones, teeth, waters and brines, and discuss its environmental significance in the context of the lithium triangle.

THE STUDIED SPECIMEN AND ITS ENVIRONMENT

The vicuna sp.

Wild South American camelids represent one of the main groups of vertebrates in the Andean plateau [START_REF] Franklin | Biology, ecology, and relationship to man of the South American camelids[END_REF]. Andean camelids live in arid and semiarid environments, and are distributed according to the altitudinal range. Vicuna (Vicugna vicugna), an endangered species protected by laws, is found in the high plateau areas, .

Accepted Article

between 3,000 and 4,600 m asl [START_REF] Franklin | Family Camelidae (camels)[END_REF]Fig. 2). Adult specimens of vicuna measure between 1.45 to 1.60 m, and usually weigh 35 to 65 kg [START_REF] Cajal | Uso de Hábitat por Vicuñas y Guanacos en la Reserva San Guillermo, Argentina[END_REF][START_REF] Aguilar | Estimación de la composición botánica en la dieta de la vicuña (Vicugna vicugna) en la puna jujeña[END_REF][START_REF] Borgnia | Interaction between wild camelids and livestock in an andean semidesert[END_REF][START_REF] Cassini | Spatial ecology and behavior of vicuñas[END_REF][START_REF] Franklin | Family Camelidae (camels)[END_REF]Tirado et al. 2012).

The vicuna diet is based on grasses and shrubs (~80%), complemented by pseudograsses and herbs (Tirado et al. 2012). These proportions are however variable, and change following sedentary and migratory tendencies of different populations according to seasonal and environmental restrictions [START_REF] Franklin | Biology, ecology, and relationship to man of the South American camelids[END_REF][START_REF] Contreras | Patrón de migración altitudinal y rango de hogar de guanacos en un ambiente andino del centro norte de Chile[END_REF]). An interesting aspect of the vicuna is that, together with guanaco (Llama guanicoe), they are one of the rare Andean mammals capable of drinking saline waters. This adaptation strategy is mainly employed during the driest periods, when caravans of vicunas can be observed traversing salt pans.

The Salinas Grandes salt pan

Salinas Grandes is a large salt pan of 280 km 2 placed at 3,410 m asl within the largest (>17,000 km 2 ) endorheic basin of the Argentinean plateau (Fig. 1). The regional climate is arid and ruled by scarce rainfalls averaging 300 to 400 mm. Under these conditions, soils are skeletal, and vegetation is mostly represented by halophytes, grasses and resinous shrubs. The relative humidity barely reaches 47%, and the annual average temperature is 8°C, with day-night amplitudes of >18°C [START_REF] Bianchi | Las precipitaciones en el noroeste argentino[END_REF][START_REF] Buitrago | El clima de la Provincia de Jujuy[END_REF].

Water is scarce and contains high saline contents. Fluvial systems mainly consist of ephemeral streams that infiltrate before reaching the salt pan. Some major fluvial collectors feeding the Salinas Grandes systems over the year are the San Antonio de los .

Accepted Article

Cobres and Las Burras rivers (Fig. 1). Springs are abundant in the headwaters of the basin catchment, and supply large amounts of solubilized salts (López Steinmetz 2017).

The Salinas Grandes salt pan is composed of halite (NaCl, Fig. 2), which holds brines of the Na-K/Cl-SO 4 type, nearly neutral pH values (ranging from 7.10 to 8.00), and with salinities reaching 120,539 mg L -1 (López [START_REF] Steinmetz | Northern Puna plateau-scale survey of Li brine-type deposits in the Andes of NW Argentina[END_REF]). These brines are stored beneath the salt crust, and often emerge through natural brine pools due to dissolution at certain sites of the salt pan (Fig. 2 C). Brines contain a large variety of dissolved elements, including B and Li. In the Salinas Grandes, the concentration of B is typically between 300 and 540 mg L -1 , while Li + grades are largely variable ranging between 25 and 1,018 mg L -1 (López [START_REF] Steinmetz | Northern Puna plateau-scale survey of Li brine-type deposits in the Andes of NW Argentina[END_REF]). The highest saline concentrations in the Salinas Grandes, including B and Li, are found in the western margin of the salt pan.

METHODS

Water and brine sampling

We have collected water samples from the San Antonio de los Cobres (SAC1) and Cobres (CO) Rivers, and from a spring source (SAC2) in the headwaters of the SAC River. A brine sample (SG) was collected from a brine pool within the salt pan, in the western Salinas Grandes. Two other samples from previous studies were included in the analysis: a river water sample (LB; López Steinmetz 2017) and a brine sample (SG7; López [START_REF] Steinmetz | Northern Puna plateau-scale survey of Li brine-type deposits in the Andes of NW Argentina[END_REF]. The locations of samples are summarized in Table 2, and shown in Fig. 1. All the water and brine sampling were conducted by employing the same sampling methodology: samples were collected from natural sources in 1.50 L .

Accepted Article

polyethylene bottles pre-cleaned with the water/brine being sampled, and were kept at ~0 °C until they were transported to the laboratory for analysis.

Electrical conductivity of samples were determined with a Hanna HI2314 multirange conductivity meter. Total dissolved solids (TDS), representing the salinity of samples in mg L -1 , have been obtained by employing a conversion factor of 0.50 on the measured electrical conductivity. The concentrations of Li + , B, and As were conducted at the Agua de los Andes S.A. laboratory of water chemistry (Jujuy, Argentina) and were determined following standard recommendations [START_REF] Clesceri | Standard Methods for the Examination of Water and Waste Water[END_REF]. Li + was determined by Atomic Emission Spectroscopy, colorimetry using carmine reactive was employed for B, and Arsenic was measured as total As by atomic absorption spectrometry. Analytical results are summarized in Table 3.

Bone and tooth sampling

We have collected the remains of two young vicuna specimens lying in the western border of the Salinas Grandes (locations are shown in Fig. 1). Partial skeletons were cleaned up and bones were classified following [START_REF] Sato | Atlas de osteología de camélidos sudamericanos y de la alpaca adulta (Vicugna pacos)[END_REF], and Sierpe González (2015) (Fig. 3 A).

Parts of the mandible with teeth of one specimen (HSG1) were sampled. Sample HSG1 is a composite sample that includes the left side of the mandible with the mandibular angle, the lower premolar and molars; the right mandible with the lower premolar and molars; and the mandibular incisive part with the lower incisive (Fig. 3 B andC). Sample HSG1b only involves teeth (lower premolar, molars, and incisive).

Sample HSG2 is a composite sample including partial femur, tibia, and metatarsus (hind leg) together that were selected from the second specimen (HSG2). For sample HSG2, .

Accepted Article

only the body of these bones were retained, and heads and bases were removed by cutting them (Fig. 3 D,E,F).

Tooth and bone samples HSG1, HSG1b, and HSG2 were prepared, processed and analyzed at the IACA Laboratories (Bahia Blanca, Argentina). For sample preparation, the sampled bone and tooth were freed of soft tissue, and partially defatted with ether for 24 hours. Samples were then crushed with a special stainless steel cutting tool, and ground to obtain a fine powder using a mortar. Samples were homogenized to obtain representative fractions, and, finally, dry matter was determined gravimetrically as the residue remaining after drying the samples.

Sample processing included acid digestion and dilution by first treating the sampled bone and tooth with 9 mL of HNO 3 and 3 mL of HCl by microwave digestion using a Milestone Ethos Easy digester. Following which, 0.5 mg of each sample were considered and diluted to a final volume of 20 mL for HSG1 and HSG2, and to 25 mL for HSG1b.

The determinations of the total Li, B, As, calcium (Ca), magnesium (Mg), sodium (Na), and potassium (K) were performed by inductively coupled plasma optical emission spectroscopy (ICP-OES) using an optic ICP (HORIBA JY 2000). Internal standards and standard solutions (HORIBA Advanced Techno Co., Ltd) were prepared in 20% HNO 3 and employed for calibrations. The limit of detection of measured elements was 3 times the standard deviation of the blank values. Different methods were used to make the determinations: B was determined using a three-point curve method (0, 200 y 500 ppb), four-point curve fitting was employed for Li and As (0, 10, 100, 1000 ppb), and K (0, 0.5, 1, 10 mg L -1 ), and five-point curves for Ca, Mg, and Na (0, 1, 10, 50, 100 mg L -1 ).

.

Accepted Article

Different dilutions were employed for each determination to adjust the concentrations found in samples with the different calibration standard solutions: Li, B, As, and K 1:2, Ca and Mg 1:100, and Na 1:10. The employed analytical wavelengths were: Li 670.784 nm, B 208.959 nm, As 189.042 nm, Ca 317.933 nm, Mg 279,806 nm, Na 558.995 nm, and K 766.490 nm. The determination of phosphates was performed using a spectral photometer by measuring the absorption of a 190-1100 nm (UV-visible) reference beam.

Results are summarized in Table 4.

Statistical analysis

All analyses were conducted using the R Studio 3.1.1 software (2014-07-10) and an alpha value for rejection of null hypotheses was set at p < 0.05 for all analyses. The normality of distributions and homogeneity of variances were evaluated by applying the Shapiro-Wilk (W), and the Bartlett tests (Bartlett's K-squared), respectively. Data are not normally distributed, consequently, further analyses were conducted using nonparametric tests by applying the Kruskal-Wallis test (K test ) and the Spearman (rho) correlation coefficient. Assessment of differences and associations were analyzed first: i) between Li in water, brine, and vicuna calcified tissues (bones, bones plus teeth, and teeth) samples, and ii) between B in water, brine, and vicuna calcified tissues (bones, bones plus teeth, and teeth) samples, according to the data we have obtained in our study (Tables 3 and4), which are summarized in Table A of the Electronic Supporting Information; and then between Li in water and body tissues (of vicuna and human) categorized in calcified, soft, keratinized and fluids, by considering the dataset presented in Table 5. The latter tested .

Accepted Article

data include values reported in this study and in previous studies. Details concerning statistical computations are available in the Electronic Supporting Information.

RESULTS

Hydrochemistry

The salinity of samples ranged from ~1,000 to ~57,000 mg L -1 . According to the limnologic criteria [START_REF] Cole | Manual de limnología[END_REF], water samples collected from rivers (CO and SAC1) are brackish, and the sample from the spring (SAC2) corresponds to brines (Fig. 4).

The Li concentrations in tributaries discharging into the Salinas Grandes are between 1.15 and 3.75 mg L -1 . Brines from pools in the salt pan are characterized by Li grades of 673 and 1,018 mg L -1 (Table 3).

The B concentration in feeding rivers is measured between 18 and 44 mg L -1 , and reaches its maximum in brine pools with 406 mg L -1 . As was detected in stream channels, springs and brines, with concentrations of 0.06 to 1.32 mg L -1 . Concentration ratios (in meq L -1 ) between Li, B, and As are extremely variable.

The B/Li ratio is comprised between 0.23 and 13.74, Li/As ranges between 177.33 and 5,503.34, and B/As grades from 1,807.77 up to 17,642.84 (Table 6). The concentration ratios with respect to the salinity (TDS normalized values) are of 58.72 to 371.65 for [(Li/As)/TDS], and 36.99 to 5,107.68 for [(B/As)/TDS] (meq L -1 per mg L -1 ; Table 6).

Vicuna bone and tooth chemistry

The measured concentrations of all analyzed constituents of sampled bones and teeth are higher in the HSG1 (mandible and teeth together as composite sample) than in HSG2 (femur, tibia, and metatarsus together), except for K (Table 4). PO 4 and Ca represent the majoritarian (98.54 -98.53 %) constituents of the bone and/or tooth mineral .

Accepted Article

fraction, with 42.36 and 36.20 g% of PO 4 and 25.50 and 21.50 g% of Ca in HSG1 and HSG2, respectively. Na, Mg, and K are minor constituents, and their concentrations are in the range of 0.62 -0.53 g% of Na, 0.32 -0.24 g% of Mg, and 0.06 -0.09 g% of K.

Traces of Li and B have been detected in bone composition. Li is 20.80 and 13.50 mg Kg -1 in samples HSG1 and HSG2, respectively, and B is 46.80 and 40.00 mg Kg -1 .

Teeth (HSG1b) contain 40 mg Kg -1 of Li and 41.70 mg Kg -1 of B. The concentrations of As were below the detection limit of the employed analytical method (0.50 mg Kg -1 ). The B/Li concentration ratios (in meq L -1 ) are comprised between 0.67 and 1.90 (Table 6).

Statistical analysis

Li and B.

The analysis of the differences between the concentrations of Li and B in the three groups of samples (water, brine, and vicuna calcified tissues) reveal no significant differences (Li: K test = 5.44, df = 2, p-value = 0.07; B: K test = 4.28, df = 2, p-value = 0.12). The analysis of associations has revealed that there is a significant, and positive association between the concentrations of Li and B (rho = 0.80, p-value = 0.01, alternative hypothesis: two sided). For this association, with a significance level of 0.05 and n = 9, the power of the test is 80% (power = 0.80).

Li in water and body tissues.

The assessment of the dataset shown in Table 5 reveal that there are no significant differences between the concentrations of Li in the four categories of human and vicuna tissues (K test = 3.75, df = 3, p-value = 0.29). Significant differences are found between the concentrations of Li in water and those in body tissues (K test = 6.45, df = 2, p-value = 0.04). The analysis of associations has shown that there is a significant, and positive .

Accepted Article

association between the concentrations of Li in water and body tissues (rho = 0.64, pvalue = 0.02, alternative hypothesis: two sided). For this association, with a significance level of 0.05 and n = 13, the power of the test is 70% (power = 0.70).

DISCUSSION

Available water resources in the Salinas Grandes area are characterized by containing high levels of salinity, Li, B, and As. In rivers and spring waters (samples SAC1, SAC2, CO, and LB), the concentrations of B are ~1 order of magnitude higher than those of Li, whereas concentrations are in the same order of magnitude in samples collected in brine pools (samples SG and SG7; Table 6). However, the analysis of differences between Li and B in the three groups of samples (water, brine, and calcified tissues) has revealed that these differences are not significant.

There is no clear proportionality between As and the concentrations of B and Li (Li/As and B/As ratios, Table 6). The lack of proportionality between As, Li and B even persists when the salinity of samples is taken into consideration (i.e., TDS normalized ratios, Table 6). On the other hand, the analysis of associations has revealed that the concentrations of Li and those of B are positively associated. Based on the association between Li and B and the lack of proportionality with respect to the ionic ratios of As, it can be deduced that natural processes regulating the Li and B grades in waters are similar between them, but are different from those ruling As.

To the best of our knowledge, this is the first study to report chemical data on vicuna bones and teeth, and its relation with the Andean plateau environment. The analyzed calcified tissues contain up to 40 mg Kg -1 of Li and 46.80 mg Kg -1 of B. The B/Li ratios are between 0.67 and 1.90 (Table 6). No As has been found in vicuna bones .

Accepted Article

and teeth while it can accumulate in other animals [START_REF] Rodríguez | A review of metal exposure and its effects on bone health[END_REF].

According to the results, either the amount of As is below the detection limit of the analytical technique used, or it is simply absent in the vicuna calcified tissues. This would imply that, unlike Li and B, As does not seem to be assimilated from drinking water and then incorporated to bones and teeth in vicuna, in contrast, As ingestion has been reported to be related to glucose and lipid metabolism in mice [START_REF] Zuo | Prolonged inorganic arsenic exposure via drinking water impairs brown adipose tissue function in mice[END_REF], and to accumulate in human soft and hard tissues [START_REF] Rodríguez | A review of metal exposure and its effects on bone health[END_REF].

The measured concentrations of Li in persons living in the Andean region (2.82 to 4.55 Kg -1 ; Table 1) are consistent and within the range of the Li concentrations reported in Andean mummies (1.10 to 8.80 Kg -1 ; Table 1) [START_REF] Concha | High-level exposure to lithium, boron, cesium, and arsenic via drinking water in the Andes of northwestern Argentina[END_REF][START_REF] Farell | Bioimaging of trace metals in ancient Chilean mummies and contemporary Egyptian teeth by laser ablation-inductively coupled plasma-mass spectrometry (LA-ICP-MS)[END_REF][START_REF] Figueroa | Environmental lithium exposure in the northern of Chile -Tissue exposure indices[END_REF]. It is important to note that, similar to ancient people living in the Andean plateau, modern settlements are supplied with drinkable water that does not undergo the regulation of Li contents prior to human ingestion. This implies that the Li concentration in mummy tissues can be considered as accurate and likely representative data of modern inhabitants.

The measured concentrations of Li in the vicuna calcified tissues (13.50 to 40 Kg - 1 ; Table 4) are higher than those reported in human tissues (1.10 and 8.80 Kg -1 ; Table 1) [START_REF] Concha | High-level exposure to lithium, boron, cesium, and arsenic via drinking water in the Andes of northwestern Argentina[END_REF][START_REF] Farell | Bioimaging of trace metals in ancient Chilean mummies and contemporary Egyptian teeth by laser ablation-inductively coupled plasma-mass spectrometry (LA-ICP-MS)[END_REF][START_REF] Figueroa | Environmental lithium exposure in the northern of Chile -Tissue exposure indices[END_REF]. The weight of a human being is comparable with that of an adult vicuna (measuring from 35 to 65 kg ;[START_REF] Cajal | Uso de Hábitat por Vicuñas y Guanacos en la Reserva San Guillermo, Argentina[END_REF][START_REF] Aguilar | Estimación de la composición botánica en la dieta de la vicuña (Vicugna vicugna) en la puna jujeña[END_REF][START_REF] Borgnia | Interaction between wild camelids and livestock in an andean semidesert[END_REF][START_REF] Cassini | Spatial ecology and behavior of vicuñas[END_REF][START_REF] Franklin | Family Camelidae (camels)[END_REF]Tirado et al. 2012). The studied vicuna specimens are young and, despite this, the measured Li concentration in bones and teeth are higher than those reported from human in the Andes, especially concerning calcified tissues. Therefore, the comparatively higher .

Accepted Article

concentrations of Li in vicuna calcified tissues (20.80, 40.00, and 13.50 mg kg-1) relative to human bones (8.80 mg kg -1 ), and teeth (1.59 mg kg -1 ), could be related to the peculiar adaptation of vicuna to the arid conditions of the plateau by ingesting saline waters rather than to the time of exposure (i.e., the age). Brackish waters and brines are richer in Lifrom 1.15 to >1,000 mg L -1 -than fresh drinkable water consumed by humans, i.e. 1 mg L -1 at San Antonio de los Cobres [START_REF] Concha | High-level exposure to lithium, boron, cesium, and arsenic via drinking water in the Andes of northwestern Argentina[END_REF]), and 5.17 mg L -1 in the Camarones River's waters (Table 1; [START_REF] Zaldívar | High lithium concentrations in drinking water and plasma of exposed Subjects[END_REF][START_REF] Farell | Bioimaging of trace metals in ancient Chilean mummies and contemporary Egyptian teeth by laser ablation-inductively coupled plasma-mass spectrometry (LA-ICP-MS)[END_REF]. Indeed, we have found a positive association between the Li concentration in water with Li in all body tissues (i.e., soft, fluid, keratinized and calcified) in human and vicuna, which is consistent with the reported dependency of the Li concentration in serum and other tissues on the Li exposure in human and other species [START_REF] Aral | Toxicity of lithium to humans and the environment -A literature review[END_REF][START_REF] Concha | High-level exposure to lithium, boron, cesium, and arsenic via drinking water in the Andes of northwestern Argentina[END_REF]Figueroa et al. 2012;[START_REF] Figueroa | Environmental lithium exposure in the northern of Chile -Tissue exposure indices[END_REF][START_REF] Harari | Environmental exposure to lithium during pregnancy and fetal size: a longitudinal study in the Argentinean Andes[END_REF], and in human and rat bones and teeth [START_REF] Birch | Lithium accumulation in bone after oral administration in rat and human[END_REF][START_REF] Mak | Effects of lithium therapy on bone mineral metabolism: a two-year perspective longitudinal study[END_REF]Pepersack et al. 1994;[START_REF] Zamani | Lithium's effect on bone mineral density[END_REF][START_REF] Farell | Bioimaging of trace metals in ancient Chilean mummies and contemporary Egyptian teeth by laser ablation-inductively coupled plasma-mass spectrometry (LA-ICP-MS)[END_REF].

The toxicology of Li is based on a serum-measuring scale, and the toxic stage in humans begins at about 10.50 -17.5 mg L -1 [START_REF] Litovitz | 1993 annual report of the American Association of Poison Control Centers Toxic Exposure Surveillance System[END_REF][START_REF] Jaeger | Lithium Medicine[END_REF]). Andean populations exposed to Li through drinking waters have average serum concentrations of 5.30 mg L -1 [START_REF] Figueroa | Environmental lithium exposure in the northern of Chile -Tissue exposure indices[END_REF]. There is no data concerning toxic ranges in serum for vicunas, nor for humans in a bone nor tooth-Li concentration scale. The Li concentration measured in the vicuna calcified tissues is twice to triple higher than that was reported in humans, which begs the question: what would be the consequence of such levels of Li in bone tissues upon the health of vicunas?

.

Accepted Article

In vicuna, teeth contain more Li than bones (tooth sample HSG1b = 40 mg Kg -1 > bone + tooth sample HSG1 = 20.80 mg Kg -1 > bone sample HSG2 = 13.50 mg Kg -1 ; Table 4). Interestingly, this is consistent with the increasing proportion of the mineral fraction in human teeth (70 to 96%; Gutiérrez-Salazar and Reyes-Gasga 2003) with respect to bones (about 65%; [START_REF] Quelch | Chemical composition of human bone[END_REF]. This Li concentration pattern is likely dependent on the mineralized fraction of the calcified tissue and might be related to mechanisms involved in the metabolization of Li. Such mechanisms might include, for instance, the incorporation of Li + during the substitution of Ca 2+ by Mg 2+ in the mineralized bone and tooth structure Ca 10 (PO 4 ) 6 (OH) 2 as Li + and Mg 2+ have similar ionic ratios (i.e., the size of these ions are quite similar, Li + = 0.60 Å and Mg 2+ = 0.65 Å, and thus they might occupy the same position in the crystalline structure of the mineral fraction in bone and/or tooth). It is also possible that isovalent replacements of Na + and K + sites by Li + might take place. However, involved osteological mechanisms are likely to be more complex than that and, for instance, the substitution of Ca 2+ by Mg 2+ (and/or Na + , K + , Li + ?) results in the decrease of the crystal mesh size, and require the incorporation of HPO 4 2-in order for the charges to be compensated [START_REF] Serre | Influence of magnesium substitution on a collagen-apatite biomaterial on the production of a calcifying matrix by human osteoblasts[END_REF][START_REF] Cabrejos-Azama | Magnesium substitution in brushite cements for enhanced bones tissue regeneration[END_REF]and references therein). Furthermore, the observed pattern of Li concentrations that increase consistently with the tissue's mineralized fraction was not found in Andean human tissues (i.e., there is no tooth > bone > soft and keratinized tissue Li concentration trend; Table 1), which highlights the complexity of processes ruling the metabolization of Li. [START_REF] Alexander | Boron and lithium content of human bones[END_REF] have reported that human bones contain a normal average concentration of 63.70 mg Kg -1 of B, which would not be dependent on the time of .

Accepted Article

exposure. As previously mentioned, the concentrations of Li in waters are positively associated with those in body tissues. In addition, we also found that Li concentrations correlate positively with B in water, brine, and calcified tissues. According to these results, we consider that the concentrations of Li and B observed in vicuna would be reflecting the ingestion of saline, high Li and B-graded waters. Despite the consistency of our reported data, due to the absence of other studies reporting the tissue compositions of vicuna for comparison, it is unclear if the presence of B and Li we have found might correspond to an anomaly of the studied specimens, or to a normal characteristic of the Vicugna vicugna species and the camelidae family, associated with the intrinsic characteristics of the Andean plateau environment.

CONCLUSION

Available water resources in the Salinas Grandes area are characterized by maximum concentrations reaching 1,019 mg L -1 of Li; 406 mg L -1 of B; and 1.32 mg L -1

of As. This study reports the first chemical data on vicuna bones and teeth with up to 40 mg Kg -1 of Li, and 46.80 mg Kg -1 of B. No As has been found in vicuna calcified tissues, which would imply that, unlike Li and B, As does not seems to be assimilated from drinking water and incorporated to bones and teeth. The measured Li concentration in vicuna is higher than that was reported from calcified tissues of other species exposed to

Li through drinkable water. The analysis of associations has shown that there are significant, and positive associations between: i) the concentrations of Li in water and body tissues, and ii) the concentrations of Li and B.

The toxicology of Li in vicuna should be assessed in order to address: i) the consequence of such levels of Li upon the health of vicunas, and ii) investigate the .

Accepted Article

potential existence of metabolic processes of adaptation in vicuna that would eventually reveal useful information for the treatment of human pathologies.

Further studies will be necessary in order to set out whether these compositional values correspond to an anomaly of the studied specimens, or to a normal characteristic of the Vicugna vicugna species and the characteristics of the Andean plateau environment.

Supplemental Data -The Supplemental Data are available on the Wiley Online Library at DOI: 10.1002/etc.xxxx. .

Accepted Article

Accepted Article

Accepted Article

Figure 1 .Figure 2 .

 12 Figure 1. Map showing the location of the study area and sampling sites. The water

Figure 4 .

 4 Figure 4. Water types according to the saline content

Table 1 .

 1 Summary

	Sample ID	Water (mg L -1 )	Li concentration Human (mg Kg -1 ) tissues	Tissue type	Li water/Li tissue
		1.00
	C10	San Antonio de los Cobres	4.55	urine	modern	0.22
		River
	F14	5.17	2.82	urine		1.83
	F13	Camarones	1.59	tooth	Mummies	3.25

of lithium (Li) concentrations in urine, calcified, keratinized, and soft human tissues reported from the Andean plateau. Sample ID correspond to the source of data: C10 is from Concha et al. (2010); F13 is from Farell et al. (2013); F14, and F14a to F14g are from Figueroa et al. (2014).

Table 2 .

 2 Detailed location of collected water, brine and vicuna samples considered in the

	present study.
	sample	Latitude	Longitude
	SAC1	24°08'58.23"	66°17'51.19"
	SAC2	24°17'07.47"	66°22'25.86"
	CO	23°39'01.55"	66°18'08.52"
	LB	23°25'19.10"	6°13'36.80"
	SG	23°43'18.73"	66°13'23.25"
	SG7	23°43'45.23"	66°10'54.41"
	HSG1	23°44'42.56"	66°13'18.32"
	HSG2	23°42'30.58"	66°12'20.82"

Table 3 .

 3 Chemical composition of water and brine samples. TDS: total dissolved solids, Li: Lithium, B: Boron, As: Arsenic. All concentrations are expressed in mg L -1 . Sample LB correspond to data reported by López[START_REF] Steinmetz | Lithium-and boron-bearing brines in the Central Andes: exploring hydrofacies on the eastern Puna plateau between 23° and 23°30[END_REF], and SG7 is from López[START_REF] Steinmetz | Northern Puna plateau-scale survey of Li brine-type deposits in the Andes of NW Argentina[END_REF]. Remaining samples are original data from the present study.

			Water samples		Brine samples
		SAC1	SAC2	CO	LB	SG	SG7
	TDS	1,054	45,370 1,002 1,240	57,640	116,136
	Li	1.15	27.15	2.07	3.75	673.00	1,018.23
	B	18.26	280.04 44.31 20.03	406.15	360.93
	As	0.07	0.11	0.06	-	1.32	-

Table 4 .

 4 Composition of vicuna bone and tooth samples.

		HSG1	HSG1b	HSG2	units
		bone+tooth	tooth	bone
	Li	20.80	40.00	13.50	mg kg -1
	B	46.80	41.70	40.00	mg kg -1
	As	<0.50	-	<0.50	mg kg -1
	Ca	25.50	-	21.50	g%
	Mg	0.32	-	0.24	g%

Table 5 .

 5 Summary of the dataset and the categories of body tissues considered during statistical analysis. a Mean value obtained from the lithium (Li) concentrations measured in water samples SAC1, SAC2, CO, LB, SG, and SG7. Sample ID correspond to the source of data: C10 is from[START_REF] Concha | High-level exposure to lithium, boron, cesium, and arsenic via drinking water in the Andes of northwestern Argentina[END_REF]; F13 is from[START_REF] Farell | Bioimaging of trace metals in ancient Chilean mummies and contemporary Egyptian teeth by laser ablation-inductively coupled plasma-mass spectrometry (LA-ICP-MS)[END_REF]; F14, and F14a to F14g are from[START_REF] Figueroa | Environmental lithium exposure in the northern of Chile -Tissue exposure indices[END_REF]. Further details concerning these samples are given in Table1. Remaining data is original from the present study.

	Sample	Li in water	Li in body tissues	Categories of body tissues	Details
	HSG1	287.56 a	20.80	calcified	bone+tooth
	HSG2	287.56 a	13.50	calcified	bone
	HSG1b	287.56 a	40.00	calcified	tooth
	C10	1.00	4.55	fluid	urine
	F13	5.17	1.59	calcified	tooth
	F14	5.17	2.82	fluid	urine
	F14a	5.17	8.80	calcified	rib
	F14b	5.17	8.00	keratinized	hair (trunk)
	F14c	5.17	1.10	keratinized	hair (not specified)
	F14d	5.17	7.30	keratinized	nail
	F14e	5.17	2.80	soft	liver
	F14f	5.17	5.00	soft	kidney
	F14g	5.17	3.40	soft	muscle (ns)

Acknowledgment -This work was supported by grant 2018-2019 of the SECTER -UNJu. Special thanks to C. Meuric for large field logistical support, and to the Secretary of Environment and Sustainable Development for authorizing wildlife bone collection.

The authors declare that there is no conflict of interest regarding the publication of this article.

Data accessibility -Data, associated metadata, and calculation tools are available from the corresponding author (lucrecialopezsteinmetz@hotmail.com).

Accepted Article