R Z Khasminskii

Yu A Kutoyants

On Parameter Estimation of Hidden Telegraph Process

Keywords: MSC 2000 Classification: 62M05, 62F12, 62F10 Telegraph process, estimator of method of moments, one-step MLE, asymptotic efficiency

The problem of parameter estimation is considered for the twostate telegraph process, observed in white Gaussian observation noise. An online one-step Maximum Likelihood Estimator (MLE) process is constructed, using a preliminary Method of Moments (MM) estimator. The obtained estimation procedure is shown to be asymptotically normal and efficient in the large sample regime.

Introduction

In this work we address the problem of estimating the unknown transition rates λ and µ of a continuous time stationary Markov chain Y (t) , 0 ≤ t ≤ T with two states, y 1 and y 2 , and transition matrix ( -λ λ µ -µ

) .

Trajectory of the chain is observed in a white noise and the unknown twodimensional parameter ϑ = (λ, µ) is to be estimated from the observations 1 X T = (X (t) , 0 ≤ t ≤ T ), generated by the equation dX(t) = Y (t) dt + dW (t), subject to X 0 , [START_REF] Bickel | Asymptotic normality of the maximum likelihood estimator for general hidden Markov models[END_REF] where the initial condition X (0) = X 0 is independent of the standard Wiener process W (t), 0 ≤ t ≤ T . We will assume that ϑ ∈ Θ = (c 0 , c 1 ) × (c 0 , c 1 ) for some known constants 0 < c 0 < c 1 .

Our goal is to construct an online estimator process ϑ ⋆ T = ( ϑ ⋆ t,T , 0 < t ≤ T

) which, on one hand, is easy to compute, and on the other hand, is asymptotically optimal in an appropriate sense, as T → ∞. The construction is carried out in two steps. First we introduce a learning interval [ 0, T δ ] with δ ∈ ( 1 2 , 1) and propose a T δ 2 -consistent preliminary estimator using the Method of Moments. Then we improve it up to asymptotic efficiency with the help of a slightly modified one-step MLE procedure.

Observation models, such as the one studied in this paper, are called Hidden Markov Models (HMM) or partially observed systems. HMMs have been extensively studied in the discrete time setup; see, for example, [START_REF] Elliott | Hidden Markov Models[END_REF], [START_REF] Bickel | Asymptotic normality of the maximum likelihood estimator for general hidden Markov models[END_REF], [START_REF] Cappé | Inference in Hidden Markov Models[END_REF] and the references therein. In particular, asymptotic normality of various estimators in the different settings have been established. Asymptotic theory for continuous time models seems to be less explored. Nonlinear filtering and parameter estimation problems for partially observed continuous time Markov chains are considered, for example, in [START_REF] Elliott | Hidden Markov Models[END_REF]. Identification problems of partially observed linear processes were studied in, e.g., [START_REF] Kutoyants | Parameter Estimation for Stochastic Processes[END_REF], [START_REF] Dembo | Parameter estimation for partially observed continuous time processes via the EM algorithm[END_REF], [START_REF] Zeitouni | Exact filters for the estimation of the number of transitions of finite-state continuous time Marcov processes[END_REF], [START_REF] Kutoyants | Identification of Dynamical Systems with Small Noise[END_REF]. The problem of asymptotically efficient estimation for continuous time telegraph process observed on a discrete grid was studied in [START_REF] Iacus | Estimation for discretly observed telegraph process[END_REF].

The problem, closest in spirit to ours, is addressed in [START_REF] Chigansky | Maximum likelihood estimation for hidden Markov models in continuous time[END_REF], where parameter estimation is considered in the case of hidden finite-state Markov process with continuous time observations. It proves consistency, asymptotic normality and asymptotic efficiency of the MLE under appropriate conditions, which may not be easy to check in general. The case of two-state hidden telegraph process under consideration appears in [START_REF] Chigansky | Maximum likelihood estimation for hidden Markov models in continuous time[END_REF] as an example with onedimensional parameter (ϑ = λ = µ).

The one-step MLE-process proposed here is based on the so-called Fisher scoring method. Let us briefly recall its main elements in the case of i.i.d. observations X n = (X 1 , . . . , X n ), sampled from the density f (ϑ, x). Suppose that we have a √ n-consistent estimator θn , i.e., √ n ( θn -ϑ ) is bounded in probability. Then the estimator ϑ ⋆ n ϑ ⋆ n = θn + I ( θn

) -1 n -1 n ∑ j=1 ḟ ( θn , X j ) f ( θn , X j )
is asymptotically efficient. Here dot stands for the derivative w.r.t. ϑ and I (ϑ) is the Fisher information. This estimator was proposed by Fisher in [START_REF] Fisher | Theory of statistical estimation[END_REF] and studied by Le Cam [START_REF] Cam | On the asymptotic theory of estimation and testing hypotheses[END_REF]. It is also known that the preliminary estimator can have convergence rate slower than √ n, see [START_REF] Robinson | The stochastic difference between econometric statistics[END_REF] and references there in. Newton-Raphson multi-step estimators were introduced in [START_REF] Kamatani | Hybrid multi-step estimators for stochastic differential equations based on sampled data[END_REF] in parameter estimation problems for diffusion processes with discrete time observations. Note that the Newton-Raphson procedure is very close to one-step estimation device. In particular, it was shown in [START_REF] Kamatani | Hybrid multi-step estimators for stochastic differential equations based on sampled data[END_REF] that multi-step Newton-Raphson procedure allows to improve convergence rate of the preliminary estimator up to asymptotic efficiency. The preliminary estimator there is constructed using all available observations. Let us now illustrate the basic idea behind our MLE-process in this classical discrete time setup. Suppose that θN is a preliminary estimator constructed by the first N =

[ n δ ] observations X N = (X 1 , . . . , X N ) with δ ∈ ( 1 2 , 1
)

. Then the one-step MLE-process ϑ ⋆ k,n , N ≤ k ≤ n is given by the following expression

ϑ ⋆ k,n = θN + I k ( θN ) -1 k -1 k ∑ j=N +1 ḟ ( θN , X j ) f ( θN , X j ), N ≤ k ≤ n,
where I k (ϑ) is the empirical Fisher information matrix

I k (ϑ) = 1 k k ∑ j=N +1 ḟ (ϑ, X j ) ḟ (ϑ, X j ) * f (ϑ, X j ) 2 .
Here a * denotes transpose of a vector (or a matrix) a. For any τ ∈ (0, 1) and

k = [nτ ] we have √ k ( ϑ ⋆ k,n -ϑ ) =⇒ N ( 0, I (ϑ) -1 )
as n → ∞. Therefore this estimator-process is asymptotically efficient for all τ ∈ (0, 1]. Remark that its calculation is much simpler than the calculation of the exact MLE.

The MLE-process for the partially observed system (1) with observations X T = (X t , 0 ≤ t ≤ T ), proposed in this paper, uses the preliminary estimator θT δ based on the observations X

T δ = ( X t , 0 ≤ t ≤ T δ ) over the initial learn- ing interval [ 0, T δ ] with δ ∈ ( 1 2 , 1)
and is carried out following the similar construction in [START_REF] Kutoyants | On multi-step MLE-processes for ergodic diffusion[END_REF] (see also [START_REF] Kutoyants | On approximation of the backward stochastic differential equation. Small noise, large samples and high frequency cases[END_REF]).

Problem statement and auxiliary results

Let us start with description of the MLE for the model under consideration. By the innovation theorem (see [START_REF] Liptser | Statistics of Random Processes[END_REF], Theorem 7.12), the stochastic process (1) admits the representation

dX t = m (t, ϑ) dt + d Wt , subject to X 0 , 0 ≤ t ≤ T, where m (t, ϑ) is the conditional expectation m (t, ϑ) = E ϑ [ Y (t) |F X t ] = y 1 P ϑ ( Y (t) = y 1 |F X t ) + y 2 P ϑ ( Y (t) = y 2 |F X t ) .
Here F X t is the σ-algebra of events generated by the observations up to time t, i.e., F X t := σ (X t , 0 ≤ s ≤ t) and Wt , 0 ≤ t ≤ T is the innovation Wiener process. Let us denote

π (t, ϑ) = P ϑ ( Y (t) = y 1 |F X t ) , P ϑ ( Y (t) = y 2 |F X t ) = 1 -π (t, ϑ) .
Then

m (t, ϑ) = y 2 + (y 1 -y 2 ) π (t, ϑ) .
The random process π (t, ϑ) , 0 ≤ t ≤ T satisfies the following equation (see [START_REF] Liptser | Statistics of Random Processes[END_REF], Theorem 9.1 and equation (9.23) therein)

dπ (t, ϑ) = [µ -(λ + µ) π (t, ϑ) +π (t, ϑ) (1 -π (t, ϑ)) (y 2 -y 1 ) (y 2 + (y 1 -y 2 ) π (t, ϑ))] dt + π (t, ϑ) (1 -π (t, ϑ)) (y 1 -y 2 ) dX t . ( 2
)
Denote by

{ P (t)
ϑ , ϑ ∈ Θ } the family of measures, induced by the process 1) on the space of realizations C [0, t] (continuous functions on [0, t]) for different values of the parameter. These measures are equivalent and the likelihood ratio function

X t = (X s , 0 ≤ s ≤ t) from (
L ( ϑ, X t ) = dP (t) ϑ dP (t) 0 ( X t ) , ϑ ∈ Θ, 0 < t ≤ T can be written as follows L ( ϑ, X t ) = exp {∫ t 0 m (s, ϑ) dX s - 1 2 ∫ t 0 m (s, ϑ) 2 ds } .
Here P (t) 0 is the measure corresponding to X t with Y (s) ≡ 0.

4

The MLE-process θt , 0 < t ≤ T is any root of the equation

L ( θt , X t ) = sup ϑ∈Θ L ( ϑ, X t ) , 0 < t ≤ T. ( 3
)
Asymptotic behavior of the MLE θT was studied in [START_REF] Chigansky | Maximum likelihood estimation for hidden Markov models in continuous time[END_REF]. It was shown that in the one-dimensional case

(d = 1, λ = µ = ϑ) the MLE θT is consistent, asymptotically normal √ T ( θT -ϑ ) =⇒ N ( 0, I (ϑ) -1 )
and asymptotically efficient. Here I (ϑ) is the Fisher information.

In our model the observations are given by ( 1) and the unknown parameter ϑ = (λ, µ) is to be estimated from the sample path X T = (X t , 0 ≤ t ≤ T ). We will construct an MLE-process ϑ ⋆ t,T , 0 < t ≤ T , such that for any fixed τ ∈ (0, 1] and t = τ T we have

√ t ( ϑ ⋆ t,T -ϑ ) =⇒ N ( 0, I (ϑ) -1 ) , T → ∞.
Here I (ϑ) is the corresponding Fisher information matrix. Recall that an estimator is asymptotically efficient if it is asymptotically normal with the limit covariance matrix I (ϑ) -1 . The family of measures in the statistical problem under consideration is locally asymptotically normal and therefore it can be shown that the latter implies asymptotic efficiency in the usual minimax sense (see [START_REF] Ibragimov | Statistical Estimation -Asymptotic Theory[END_REF]).

Note that the construction of the MLE-process θt , 0 < t ≤ T according to (3) and ( 2) is a computationally hard problem, because it requires solving the family of equations (2) for all ϑ ∈ Θ and equations (3) for all t ∈ (0, T ]. Computational complexity can be reduced as follows. First we construct an estimator θT = ( λT , μT ) by means of the Method of Moments (see (17) below) and show that it is √ T -consistent, that is,

E ϑ √ T ( θT -ϑ ) 2 ≤ C,
where the constant C > 0 does not depend on T . Then applying this estimator θT δ to the observations on the learning interval [ 0, T δ ] with 1 2 < δ < 1, we introduce the one-step MLE-process

ϑ ⋆ t,T = θT δ + t -1/2 I t ( θT δ ) -1 ∆ t ( θT δ , X t ) , T δ ≤ t ≤ T. ( 4
)
Here the empirical Fisher information matrix satisfies

I t (ϑ) = 1 t ∫ t T δ ṁ (ϑ, s) ṁ (ϑ, s) * ds -→ I (ϑ) , as t → ∞, T δ = o (t) and ∆ t ( ϑ, X t ) = 1 √ t ∫ t T δ ṁ (ϑ, s) [dX s -m (ϑ, s) ds]
is the vector score-function process.

Computation of this MLE-process uses the solutions m (ϑ, t) and ṁ (ϑ, t) of the equation ( 2) and of the corresponding equation for ṁ (ϑ, t) only for one value of the parameter, namely for ϑ = θT δ . Solving the maximum likelihood equation ( 3) is avoided and the estimator ϑ ⋆ t,T can be easily calculated using (4). This is the main computational advantage of the proposed estimation procedure. On the other hand, as mentioned above, the estimator ϑ ⋆ t,T is asymptotically efficient and, in particular,

ϑ ⋆ T,T = ϑ ⋆ T satisfies √ T (ϑ ⋆ T -ϑ) =⇒ N ( 0, I (ϑ) -1
) , being therefore asymptotically equivalent to the exact MLE.

Hereafter we will use the following notations: the derivative w.r.t. parameter ϑ is denoted by a dot, ṁ (ϑ, t) stands for the column vector of the derivatives ṁλ (ϑ, t), ṁµ (ϑ, t).

Let us recall some relevant well known properties of the (stationary) telegraph process Y (t) , t ≥ 0.

1. The stationary distribution of the process Y (t) is given by

P ϑ {Y (t) = y 1 } = µ λ + µ , P ϑ {Y (t) = y 2 } = λ λ + µ (5)
2. Let us denote P ij (t) = P ϑ {Y (t) = y j |Y (0) = y i }, then solving the Kolmogorov equation we obtain

P 11 (t) = µ λ + µ + λ λ + µ e -(λ+µ)t , P 12 (t) = λ λ + µ - λ λ + µ e -(λ+µ)t , P 21 (t) = µ λ + µ - µ λ + µ e -(λ+µ)t , P 22 (t) = λ λ + µ + µ λ + µ e -(λ+µ)t (6)
It follows from ( 5) and ( 6) that

K (s) = E ϑ [Y (t) Y (t + s)] = ( y 1 µ + y 2 λ λ + µ ) 2 + (y 2 -y 1 ) 2 λµ (λ + µ) 2 e -(λ+µ)s = ( Ȳ ) 2 + De -(λ+µ)s , (7)
where

Ȳ = E ϑ Y (t) = y 1 µ + y 2 λ λ + µ , D = (y 2 -y 1 ) 2 λµ (λ + µ) 2 . ( 8
)
3. Let F Y t ⊂ F be the family of σ-algebras of events

{Y (s) = y i , 0 ≤ s ≤ t, i = 1, 2} .
It follows from ( 6) that for some constant K > 0 and A < T and for all s > A, t > 0 the inequality

E ϑ { Y (s + T ) Y (t + T ) |F Y A } -E ϑ [Y (s) Y (t)] < Ke -(λ+µ)(T -A) (9) holds.

Method of moments estimator

Let us first address the problem of constructing a √ T -consistent estimator of the unknown parameter ϑ by the Method of Moments. Recall that we observe the stochastic process

dX t = Y (t) dt + dW t , X 0 , 0 ≤ t ≤ T, ( 10
)
in continuous time, where W t , 0 ≤ t ≤ T is a standard Wiener process, X 0 is an independent random initial condition, and Y (t), t ≥ 0 is the stationary Markov process with two states y 1 and y 2 and infinitesimal rate matrix

( -λ λ µ -µ
) .

The processes Y (t) , t ≥ 0 and W t , t ≥ 0 are independent.

For simplicity we assume that T is an integer and impose the condition

λ ∈ [c 0 , c 1 ] , µ ∈ [c 0 , c 1 ] (11)
where c 0 and c 1 are some positive constants. To introduce our preliminary estimator we need the following objects:

• The function

Φ (x) = 1 x - 1 x 2 ( 1 -e -x ) . ( 12
)
• The statistics

ζ T = 1 T T -1 ∑ i=0 [X i+1 -X i ] 2 -1. (13)
• The random variable α T is defined as a solution of the equation

ζ T = ( X T T ) 2 + 2η T Φ (α T ) , ( 14
)
where

η T = ( X T T -y 1 ) ( y 2 - X T T ) . ( 15
)
• The event A T ="the equation ( 14) has a solution α T ∈ [2c 0 , 2c 1 ]".

• The random variable

β T = α T 1I {A T } + (c 0 + c 1 ) 1I {A c T } ( 16
)
Define the estimator θT = ( λT , μT ) where

λT = X(T ) T -y 1 y 2 -y 1 β T ; μT = y 2 -X(T ) T y 2 -y 1 β T . ( 17
)
Properties of these estimators are given in the following theorem.

Theorem 1 Assume the condition [START_REF] Kutoyants | Parameter Estimation for Stochastic Processes[END_REF] holds; then the estimators in ( 17) satisfy

E ϑ [ √ T ( λT -λ ) ] 2 < C, E ϑ [ √ T (μ T -µ) ] 2 < C. ( 18
)
with a constant C > 0, independent of T .

The proof is split below into several steps.

The next lemma gives √ T -consistent estimator for Ȳ (see [START_REF] Ibragimov | Statistical Estimation -Asymptotic Theory[END_REF]).

Lemma 1 Under the assumption [START_REF] Kutoyants | Parameter Estimation for Stochastic Processes[END_REF], the estimator X T /T is uniformly consistent for Ȳ :

E ϑ ( X T T - Ȳ ) 2 ≤ C T , T > 0 ( 19
)
with a constant C > 0 independent of ϑ and T .

Proof. Using [START_REF] Iacus | Estimation for discretly observed telegraph process[END_REF] we obtain the following relations

E ϑ ( X T T - Ȳ ) 2 = E ϑ 1 T ∫ T 0 [ Y (t) - Ȳ ] dt + W T T 2 = 1 T + 1 T 2 E ϑ ∫ T 0 [ Y (t) - Ȳ ] dt 2 ≤ 1 T ( 1 + 2λµ (λ + µ) 3 (y 2 -y 1 ) 2 ) ≤ 1 T ( 1 + c 2 1 4c 3 0 (y 2 -y 1 ) 2
) .

Corollary. Existence of consistent estimators for λ λ+µ and µ λ+µ follows from (5) and Lemma 1. Indeed, from the equality

Ȳ = λ λ + µ y 2 + µ λ + µ y 1
and Lemma 1 we obtain

E ϑ [ √ T ( T -1 X T -y 1 y 2 -y 1 - λ λ + µ )] 2 < C, E ϑ [ √ T ( y 2 -T -1 X T y 2 -y 1 - µ λ + µ )] 2 < C. ( 20
)
The statistic

X T T = 1 T ∫ T 0 Y (t) dt + W T T
is a sum of a bounded random variable and an independent normal random variable N (0, T -1 ). Hence η T defined in [START_REF] Kutoyants | On multi-step MLE-processes for ergodic diffusion[END_REF] satisfies

E ϑ [ √ T (η T -D) ] 2 < C, ( 21
)
where the constant C > 0 does not depend on T and ϑ. The constant D is defined in [START_REF] Ibragimov | Statistical Estimation -Asymptotic Theory[END_REF].

Note that from the condition [START_REF] Kutoyants | Parameter Estimation for Stochastic Processes[END_REF] we have

λµ (λ + µ) 2 > c 2 0 4c 2 1
and we easily obtain the bound [START_REF] Zeitouni | Exact filters for the estimation of the number of transitions of finite-state continuous time Marcov processes[END_REF] for the estimator

ηT = max { η T , (y 2 -y 1 ) 2 c 2 0 8c 2 1 } (22) Lemma 2
The following equality holds

E ϑ ζ T = Ȳ 2 + 2DΦ (λ + µ) (23)
and under condition [START_REF] Kutoyants | Parameter Estimation for Stochastic Processes[END_REF] we also have

E ϑ [ √ T (ζ T -E ϑ ζ T ) ] 2 < C. ( 24
)
Proof. By [START_REF] Iacus | Estimation for discretly observed telegraph process[END_REF] and stationarity of the process Y (t) we obtain

E ϑ ζ T = E ϑ [X 1 -X 0 ] 2 -1 = E ϑ ∫ 1 0 ∫ 1 0 Y (s) Y (t) dsdt + 1 -1 = Ȳ 2 + 2DΦ (λ + µ) . ( 25
)
Denote

γ i = ∫ i+1 i Y (t) dt; ∆W (i) = W i+1 -W i .
Further, the equality

ζ T -E ϑ ζ T = 1 T T -1 ∑ i=0 ( γ 2 i -E ϑ γ 2 i ) + 2 T T -1 ∑ i=0 γ i ∆W (i) + 1 T T -1 ∑ i=0 ( ∆W (i) 2 -1 ) implies E ϑ (ζ T -E ϑ ζ T ) 2 ≤ 3 T 2 E ϑ ( T -1 ∑ i=0 ( γ 2 i -E ϑ γ 2 i ) ) 2 + 12 T 2 E ϑ ( T -1 ∑ i=0 γ i ∆W (i) ) 2 + 3 T 2 E ϑ ( T -1 ∑ i=0 ( ∆W (i) 2 -1 ) ) 2 := 3J 1 + 12J 2 + 3J 3 . ( 26
)
By stationarity of Y (t)

J 1 = 1 T 2 E ϑ ( T -1 ∑ i=0 ( γ 2 i -E ϑ γ 2 i ) ) 2 = 1 T 2 T -1 ∑ i=0 T -1 ∑ j=0 E ϑ ( γ 2 i -E ϑ γ 2 i ) ( γ 2 j -E ϑ γ 2 j ) = 1 T 2 T -1 ∑ i,j=0 {∫ 1 0 ∫ 1 0 ∫ 1 0 ∫ 1 0 E ϑ {Y (s) Y (t) E ϑ [ Y (|i -j| + s 1 ) Y (|i -j| + t 1 ) |F Y 1 ]} dsdtds 1 dt 1 - ( E ϑ γ 2 0 ) 2 } . ( 27
)
The bound [START_REF] Kamatani | Hybrid multi-step estimators for stochastic differential equations based on sampled data[END_REF] gives

E ϑ [ Y (|i -j| + s 1 ) Y (|i -j| + t 1 ) |F Y 1 ] -E ϑ Y (s 1 ) Y (t 1 ) ≤ K e -(λ+µ)|j-i| .
and, using ( 27) and ( 11), we obtain

J 1 ≤ K T 2 T -1 ∑ i,j=0 e -(λ+µ)|j-i| ≤ K 1 T .
The obvious bounds

J 2 = 1 T 2 E ϑ ( T -1 ∑ i=0 γ i ∆W (i) ) 2 = 1 T 2 T -1 ∑ i=0 E ϑ γ 2 i = E ϑ γ 2 0 T ≤ K T , J 3 = 1 T 2 E ϑ ( T -1 ∑ i=0 [ ∆W (i) 2 -1 ] ) 2 = 1 T 2 T -1 ∑ i=0 E ϑ [ ∆W (i) 2 -1 ] 2 ≤ K T
and (26) imply the second assertion of the lemma.

Lemma 3 The function Φ (x) (see [START_REF] Kutoyants | Identification of Dynamical Systems with Small Noise[END_REF]) has the following properties

lim x→0+ Φ (x) = 1 2 , ( 28
)
lim x→∞ Φ (x) = 0, (29)
Φ ′ (x) < 0, for x > 0. ( 30
)
Proof. Expanding (12) into power series, we get

Φ (x) = 1 2 - x 3! + x 2 4! - x 3 5! + . . . Φ ′ (x) = - ( 1 3! - 2x 4!
) -

( 3x 2 5! - 4x 3 6! ) -. . .
which give the limits (28) and ( 29) and the bound (30) for x < 2. For x ≥ 2 it follows from the explicit expression for this derivative

Φ ′ (x) = 1 x 2 ( 2 x -1
) -

( 2 x 3 + 1 x 2 ) e -x .
Let us consider the equation ( 14) for α T , where ζ T and η T are defined in ( 13) and ( 15) respectively. By Lemma 3 this equation has at most one solution. Recall that on the event A T , the equation ( 14) does have a solution and consider the statistic β T defined in ( 16) (here c 0 , c 1 are the constants from the condition ( 11)).

Lemma 4 Under the condition [START_REF] Kutoyants | Parameter Estimation for Stochastic Processes[END_REF], the estimate

β T is √ T -consistent for λ + µ: E ϑ [ √ T (β T -(λ + µ)) ] 2 < C, ( 31
)
with a constant C > 0, independent of ϑ and T .

Proof. It follows from Lemmas 1 and 2 that

ζ T = Ȳ 2 + 2DΦ (λ + µ) + ε 1 (T ) . ( 32
)
Here and below ε i (T ) , i = 1, 2 . . . satisfy

E ϑ ( √ T ε i (T ) ) 2 < C.
By Lemma 1, estimates ( 21), ( 22) and boundedness of Φ (x) we also obtain

ζ T = Ȳ 2 + 2η T Φ (λ + µ) + ε 2 (T ) . ( 33
)
It follows from ( 14) and (33) that on the event

A T 2η T Φ (α T ) = 2η T Φ (λ + µ) + ε 3 (T ) .
Since ηT is bounded away from zero by a positive constant (see Corollary to Lemma 1), the latter equality and Lemma 3 imply that on

A T Φ (α T ) -Φ (λ + µ) = ε 4 (T ) .
Therefore by Lemma 3 we obtain

E ϑ { 1I {A T } √ T (β T -(λ + µ)) } 2 < C. ( 34
)
If ω ∈ A c T then the equation

Ȳ 2 + 2DΦ (λ + µ) + ε 3 (T ) = ( X (T ) T ) 2 + 2η T Φ (x) (35)
has no solution x ∈ [2c 0 , 2c 1 ].

It follows from (35), Lemma 1 and the Corollary that the equation

Φ (x) = Φ (λ + µ) + ε 4 (T ) has no solution for x ∈ [2c 0 , 2c 1 ]. This means that [ Ȳ 2 + 2DΦ (λ + µ) + ε 3 - ( Ȳ + ε 0 ) 2 ] (2D + 2ε 5 ) -1 ̸ ∈ [Φ (2c 0 ) , Φ (2c 1 )] ,
where we used the notations

ε 0 = T -1 X T -Ȳ and ε 5 = ηT -D.
Hence we can write

A c T ⊂ {|ε 0 (T )| > α 1 } ∪ {|ε 3 (T )| > α 2 } ∪ {|ε 5 (T )| > α 3 }
for some positive constants α 1 , α 2 and α 3 , which do not depend on T . This inclusion implies

P (A c T ) ≤ P {|ε 0 (T )| > α 1 } + P {|ε 3 (T )| > α 2 } + P {|ε 5 (T )| > α 3 } < C T
and the claim of Lemma 4 holds by (34).

Proof of Theorem 1. Let us now apply obtained results to prove √ Tconsistency of the estimators defined in [START_REF] Robinson | The stochastic difference between econometric statistics[END_REF]. To this end, the obvious equality

λT -λ = β T X(T ) T -y 1 y 2 -y 1 -β T λ λ + µ + λ λ + µ (β T -(λ + µ)) ,
the bounds [START_REF] Wonham | Some applications of stochastic differential equations to optimal non-linear filtering[END_REF] and Lemma 4 give

E ϑ ( √ T ( λT -λ ) ) 2 ≤ 2E ϑ [ √ T β T ( X(T ) T -y 1 y 2 -y 1 - λ λ + µ )] 2 + 2 ( λ λ + µ ) 2 E ϑ [ √ T (β T -(λ + µ)) ] 2 < C.
The second inequality in [START_REF] Robinson | The stochastic difference between econometric statistics[END_REF] is shown similarly and thus the estimator θT = ( λT , μT

) is √ T -consistent.

One-step MLE

Our next goal is to construct the asymptotically efficient MLE-process for the parameter ϑ = (λ, µ) ∈ Θ. We do it in two steps. First we define the preliminary estimator θT δ = ( λT δ , μT δ ), obtained by applying the estimator from the previous section to the observations

X T δ = ( X t , 0 ≤ t ≤ T δ ) on the learning interval [ 0, T δ ] , with δ ∈ ( 1 2 , 1
)

. By Theorem 1 this estimator satisfies the condition:

sup ϑ∈K T δ E ϑ θT δ -ϑ 2 ≤ C,
with a constant C > 0 independent of T and ϑ ∈ Θ. Let us introduce the additional condition (recall that Θ = (c 0 , c 1 ) × (c 0 , c 1 )):

M (N )

. There exists N ≥ 2, such that

c 0 (y 1 -y 2 ) 2 > 2N + 9 8 . ( 36
)
With the preliminary estimator θT δ at hand, we propose one-step MLE:

ϑ ⋆ t,T = θT δ + t -1 I t ( θT δ ) -1 ∫ t T δ ṁ( θT δ , s) [ dX s -m( θT δ , s)ds ] , ( 37
)
based on the slightly modified score-function

∆ t ( ϑ, X t ) = 1 √ t ∫ t 0 ṁ(ϑ, s) [dX s -m(ϑ, s)ds] , T δ ≤ t ≤ T.
Here we defined the vector

ṁ(ϑ, s) = (y 1 -y 2 ) ∂π (s, ϑ) ∂ϑ = (y 1 -y 2 ) ( ∂π (t, ϑ) ∂λ , ∂π (t, ϑ) ∂µ ) *
and the empirical Fisher information matrix I t (ϑ)

I t (ϑ) = 1 t ∫ t T δ ṁ(ϑ, s) ṁ(ϑ, s) * ds -→ I(ϑ), t → ∞
where the limit holds by the Law of Large Numbers; I(ϑ) is the Fisher information matrix

I(ϑ) = (y 1 -y 2 ) 2 E ϑ ∂π (s, ϑ) ∂ϑ ∂π (s, ϑ) * ∂ϑ .
The stochastic process ∂π(s,ϑ) ∂ϑ has relevant ergodic properties (see the proof of Lemma 6 below) and we assume here that ∂π(s,ϑ) ∂ϑ has invariant distribution. Therefore the expectation above does not depend on s.

Let us change the variable τ = tT -1 ∈ [0, 1] and introduce the stochastic process ϑ ⋆ T (τ ) , τ δ ≤ τ ≤ 1, where ϑ ⋆ T (τ ) = ϑ ⋆ τ T,T and τ δ = T δ-1 → 0. Below ϑ 0 denotes the true value of the parameter.

Theorem 2 Suppose ϑ 0 ∈ Θ, δ ∈ ( 1 2 , 1)
and the condition M (2) holds. Then the one-step MLE-process is consistent: for any ν > 0 and any τ ∈ (0, 1]

P ϑ 0 {|ϑ ⋆ T (τ ) -ϑ 0 | > ν} → 0 (38) as T → ∞. Moreover, it is asymptotically normal √ τ T (ϑ ⋆ T (τ ) -ϑ 0 ) =⇒ N ( 0, I(ϑ 0 ) -1 ) . ( 39
)
Proof. Let us denote the partial derivatives

πλ (t, ϑ) = ∂π (t, ϑ) ∂λ , πµ (t, ϑ) = ∂π (t, ϑ) ∂µ , πλ,λ (t, ϑ) = ∂ 2 π (t, ϑ) ∂λ 2 ,
and so on.

Lemma 5 Suppose ϑ 0 ∈ Θ and N > 1. If the condition

c 0 (y 1 -y 2 ) 2 > N + 3 4 (40)
holds, then

sup ϑ∈Θ E ϑ 0 ( | πλ (t, ϑ)| N + | πµ (t, ϑ)| N ) < C 1 , ( 41
)
and, if the condition

c 0 (y 1 -y 2 ) 2 > 2N + 9 8 (42)
holds, then

sup ϑ∈Θ E ϑ 0 ( |π λ,λ (t, ϑ)| N + |π λ,µ (t, ϑ)| N + |π µ,µ (t, ϑ)| N ) < C 2 . ( 43
)
Here the constants C 1 > 0, C 2 > 0 do not depend on t.

Proof. For simplicity we write

πλ (t, ϑ) = πλ , πµ (t, ϑ) = πµ , π (t, ϑ) = π.
Taking the derivative of

dπ = [µ -(λ + µ) π -π (1 -π) (y 1 -y 2 ) (y 2 + (y 1 -y 2 ) π)] dt + π (1 -π) (y 1 -y 2 ) dX t . ( 44
)
we obtain the equations

d πλ = -π dt -πλ [λ + µ + (1 -2π) (y 1 -y 2 ) [y 2 + (y 1 -y 2 ) π] +π (1 -π) (y 1 -y 2 ) 2 ] dt + πλ (1 -2π) (y 1 -y 2 ) dX (t) , ( 45
) d πµ = [1 -π] dt -πµ [λ + µ + (1 -2π) (y 1 -y 2 ) [y 2 + (y 1 -y 2 ) π] +π (1 -π) (y 1 -y 2 ) 2 ] dt + πµ (1 -2π) (y 1 -y 2 ) dX (t) . ( 46
)
Using the standard arguments it can be shown that the stochastic process π (t, ϑ) has continuous derivatives w.r.t. λ and µ with probability 1. If we denote the true value of the parameters by ϑ 0 and π (t, ϑ 0 ) = π o etc., then these equations with ϑ = ϑ 0 read

d πo λ = -π o dt -πo λ [ λ 0 + µ 0 + π o (1 -π o ) (y 1 -y 2 ) 2 ] dt + πo λ (1 -2π o ) (y 1 -y 2 ) d W (t) , ( 47
)
d πo µ = [1 -π o ] dt -πo µ [ λ 0 + µ 0 + π o (1 -π o ) (y 1 -y 2 ) 2 ] dt + πo µ (1 -2π) (y 1 -y 2 ) d W (t) , ( 48
)
where

dπ o = [µ 0 -(λ 0 + µ 0 ) π o ] dt + π o (1 -π o ) (y 1 -y 2 ) d W (t) . ( 49
)
This system is linear with respect to πλ and πµ and it can be written as follows (

x t = π o , y t = πo λ , z t = πo µ , a = λ 0 + µ 0 , b = y 1 -y 2 ) dx t = [µ 0 -ax t ] dt + bx t (1 -x t ) d Wt , ( 50
)
dy t = -x t dt - [ a + b 2 x t (1 -x t ) ] y t dt + b (1 -2x t ) y t d Wt , ( 51
)
dz t = [1 -x t ] dt - [ a + b 2 x t (1 -x t ) ] z t dt + b (1 -2x t ) z t d Wt . ( 52
)
Note that as λ 0 > 0 and µ 0 > 0, the process π (t, ϑ 0 ) = x t ∈ (0, 1) is ergodic with two reflecting boundaries at 0 and 1. Therefore the process π o (t, ϑ 0 ) is ergodic with the invariant density

f (ϑ 0 , x) = [x (1 -x)] 2(µ 0 -λ 0 ) (y 1 -y 2 ) 2 -2 G (ϑ 0 ) exp { - 2µ 0 + 2 (λ 0 -µ 0 ) x (y 1 -y 2 ) 2 x (1 -x) } = [x (1 -x)] γ(µ 0 -λ 0 )-2 G (ϑ 0 ) exp { - γµ 0 x - γλ 0 1 -x }
where we denote γ = 2 (y 1 -y 2 ) -2 and define the normalizing constant

G (ϑ 0 ) = ∫ 1 0 [x (1 -x)] γ(µ 0 -λ 0 )-2 exp { - γµ 0 x - γλ 0 1 -x } dx.
The processes y t and z t admit the following expressions

y t = - ∫ t 0 exp { - ∫ t v [ a + b 2 x s (1 -x s ) - b 2 2 (1 -2x s ) 2 ] ds +b ∫ t v (1 -2x s ) d Ws } x v dv, (53)
z t = ∫ t 0 exp { - ∫ t v [ a + b 2 x s (1 -x s ) - b 2 2 (1 -2x s ) 2 ] ds +b ∫ t v (1 -2x s ) d Ws } [1 -x v ] dv. ( 54
)
Let us put x s = 1 2 -x s . Then we have

x s (1 -x s ) - 1 2 (1 -2x s ) 2 = -3x 2 s + 1 4
and

y t = ∫ t 0 ( x v - 1 2 ) e - ( a+ b 2 4 ) (t-v) exp { 3b 2 ∫ t v x 2 s ds + 2b ∫ t v x s d Ws } dv.
To estimate the moments E ϑ 0 |y t | N we note that x v -1 2 ≤ 1 2 and use the Hölder inequality

(∫ t 0 |f (v) g (v)| dv ) N ≤ (∫ t 0 |f (v)| N N -1 dv ) N -1 ∫ t 0 |g (v)| N dv with f (v) = exp {-a (t -v) ε} and g (v) = exp { - ( a (1 -ε) + b 2 4
) (t -v) +3b 2 ∫ t v x 2 s ds + 2b ∫ t v x s d Ws } ,
where ε > 0. This yields the estimate

E ϑ 0 |y t | N ≤ C (N, ε) ∫ t 0 e -N ( a(1-ε)+ b 2 4 ) (t-v) E ϑ 0 e 3N b 2 ∫ t v x 2 s ds+2N b ∫ t v xsd Ws dv,
where the constant C (N, ε) > 0 does not depend on t. Further, we can write

E ϑ 0 exp { 3N b 2 ∫ t v x 2 s ds + 2N b ∫ t v x s d Ws } = E ϑ 0 ( exp { 2N b ∫ t v x s d Ws -2N 2 b 2 ∫ t v x 2 s ds } exp { N b 2 (2N + 3) ∫ t v x 2 s ds }) ≤ exp { N b 2 4 (2N + 3) (t -v) } because x 2 s ≤ 1/4, and
E ϑ 0 exp { 2N b ∫ t v x s d Ws -2N 2 b 2 ∫ t v x 2 s ds } = 1.
Therefore,

E ϑ 0 |y t | N ≤ C (N, ε) ∫ t 0 e -N ( a(1-ε)+ b 2 4 -b 2 4 (2N +3
)

) (t-v) dv = C (N, ε) ∫ t 0 e -N ( a(1-ε)-b 2 2 (N +1) ) (t-v) dv.
We see that if

λ 0 + µ 0 (y 1 -y 2 ) 2 > 1 2 + N 2 , then E ϑ 0 |y t | N ≤ C.
In particular, if we set N = 2 in the condition (40) and choose a sufficiently small ε > 0, we get the bound

sup ϑ 0 ∈Θ E ϑ 0 ∂π (t, ϑ 0 ) ∂λ 2 ≤ C, ( 55
)
where the constant C > 0 does not depend on t.

We also need to estimate the derivatives (45), ( 46) for the values ϑ ̸ = ϑ 0 . The equation for πλ becomes

d πλ = -π dt -πλ [ λ + µ + (1 -2π) (y 1 -y 2 ) 2 ( π -π 0 ) +π (1 -π) (y 1 -y 2 ) 2 ] dt + πλ (1 -2π) (y 1 -y 2 ) d W (t) . ( 56
)
Hence if we put a = λ + µ, y t = πλ and b = y 1 -y 2 , we obtain the equation

dy t = -x t dt - [ a + b 2 (1 -2x t ) ( x t -x 0 t ) + b 2 x t (1 -x t ) ] y t dt + b (1 -2x t ) y t d Wt .
The solution of this equation can be written explicitly, similarly to (53), but with additional term b 2 (1 -2x t ) (x t -x 0 t ) in the exponent. This term satisfies the inequality

(1 -2x t ) ( x t -x 0 t ) ≥ -1.
Hence by calculations as above, E ϑ 0 |y t | 2 will be bounded, if

λ + µ (y 1 -y 2 ) 2 > 3 2 + N 2 .
For the second derivative π = πλ,λ (t, ϑ) we obtain similar bounds for the moments as follows. The equation for π is

dπ = -y t [ 2 -2b 2 y t ( x t -x 0 t ) + 2b 2 y t (1 -2x t ) ] dt -2by 2 t d Wt - π [ a + b 2 (1 -2x t ) ( x t -x 0 t ) + b 2 x t (1 -x t ) ] dt + bπ (1 -2x t ) d Wt .
Let us write it as

dπ = A (t) dt + B (t) d Wt -π [a + C (t)] dt + πt D (t) d Wt
using the obvious notations. Its solution satisfies

∂ 2 π (t, ϑ) ∂λ 2 = ∫ t 0 e - ∫ t v [a+C(s)-1 2 D(s) 2 ]ds+ ∫ t v D(s)d Ws [ A (v) dv + B (v) d Wv ] .
The bound

C (s) - D (s) 2 2 = b 2 (1 -2x s ) ( x s -x 0 s ) + b 2 2 [ 2x s (1 -x s ) -(1 -2x s ) 2 ] ≥ -b 2 -3b 2 ( x - 1 2
) 2 + b 2 4 = - 3b 2 4 -3b 2 ( x - 1 2
) 2 ≥ - 3b 2 2 holds since ( x -1 2 ) 2 ≤ 1 4 . Therefore, a - 3 2 b 2 - 2N + 3 4 b 2 = a - 9 4 b 2 - N 2 b 2 and if λ + µ (y 1 -y 2 ) 2 > 9 4 + N 2 ,
then we obtain

E ϑ 0 ∂ 2 π (t, ϑ) ∂λ 2 N < C.
Hence under the condition (42) we have

sup ϑ∈Θ E ϑ 0 ∂ 2 π (t, ϑ) ∂λ 2 N < C.
Similar estimates can be obtained for other derivatives and the claimed assertion follows.

Lemma 6 The solutions (x t , y t , z t ) of the equations (50)-( 52) have the following ergodic properties

1 T ∫ T 0 ṁλ (t, ϑ 0 ) 2 dt = b 2 T ∫ T 0 y 2 t dt -→ I 11 (ϑ 0 ) , 1 T ∫ T 0 ṁλ (t, ϑ 0 ) ṁµ (t, ϑ 0 ) dt = b 2 T ∫ T 0 y t z t dt -→ I 12 (ϑ 0 ) , 1 T ∫ T 0 ṁµ (t, ϑ 0 ) 2 dt = b 2 T ∫ T 0 z 2 t dt -→ I 22 (ϑ 0 ) ,
where convergence is in the mean square.

Proof. The proof of existence of the invariant measure can be found in [START_REF] Chigansky | Maximum likelihood estimation for hidden Markov models in continuous time[END_REF], section 4.2. Note that the equations (50)-(52) do not coincide with those in [START_REF] Chigansky | Maximum likelihood estimation for hidden Markov models in continuous time[END_REF], where the case λ = µ and y 1 = 1, y 2 = 0 is considered; however the same arguments apply to the system of equations ( 50)-(52).

The strong mixing coefficient α (t) for ergodic diffusion process (50) satisfies the estimate α (t) < e -c|t| , as shown in [START_REF] Veretennikov | Bounds for the mixing rate in the theory of stochastic equations[END_REF]. The conditions, under which this bound holds, can be checked by applying the following transformation to the equation (50):

ξ t = g (x t ) , g (x) = ∫ x 1/2 dv bv (1 -v) , x ∈ (0, 1) .
The obtained process solves the stochastic differential equation

dξ t = A (ξ t ) dt + dW t , ξ 0 = g (x 0 ) , 0 ≤ t ≤ T,
whose drift coefficient A (•) satisfies the assumptions needed in [START_REF] Veretennikov | Bounds for the mixing rate in the theory of stochastic equations[END_REF].

Now to verify the convergence

E ϑ 0 ( 1 T ∫ T 0 y 2 t dt - 1 T ∫ T 0 E ϑ 0 y 2 t dt ) 2 = E ϑ 0 ( 1 T ∫ T 0 [ y 2 t -E ϑ 0 y 2 t ] dt ) 2 -→ 0 (57)
we can apply the result of the following lemma.

Lemma 7 Let {Y t , t > 0} be a stochastic process with zero mean, satisfying

E |Y t | m(2k-1) < C 1 , ∫ ∞ 0 t k-1 [α (t)] (m-2)/m dt < C 2 ,
for some m > 2 and k ≥ 1, where α (t) is the strong mixing the coefficient. Then

E ∫ T 0 Y t dt 2k ≤ C 3 T k .
Proof. For the proof see Lemma 2.1 in [START_REF] Khasminskii | On stochastic processes defined by differential equations with small parameter[END_REF].

The limit (57) is obtained by setting Y t = y 2 t -E ϑ 0 y 2 t , m = 3 and k = 1. Let us now verify consistency of the one-step MLE-process. To this end, we have

P ϑ 0 {|ϑ ⋆ T (τ ) -ϑ 0 | > ν} ≤ P ϑ 0 { θT δ -ϑ 0 > ν 2 } + P ϑ 0 { I τ T ( θT δ ) -1 τ T ∫ τ T T δ ṁ( θT δ , s) [ dX s -m( θT δ , s)ds ] > ν 2 } .
By the Theorem 1, the first term on the right satisfies

P ϑ 0 { θT δ -ϑ 0 > ν 2 } ≤ 4 ν 2 E ϑ 0 θT δ -ϑ 0 2 ≤ C ν 2 T δ → 0.
The second term can be bounded as follows

P ϑ 0 { I τ T ( θT δ ) -1 τ T ∫ τ T T δ ṁ( θT δ , s) [ dX s -m( θT δ , s)ds ] > ν 2 } ≤ P ϑ 0 { I τ T ( θT δ ) -1 τ T ∫ τ T T δ ṁ( θT δ , s) d Ws > ν 4 } + P ϑ 0 { I τ T ( θT δ ) -1 τ T ∫ τ T T δ ṁ( θT δ , s) ∆m ( θT δ , s ) ds > ν 4 } ,
where ∆m ( θT δ , s ) = m(ϑ 0 , s) -m( θT δ , s). We can write

I τ T ( θT δ ) -1 τ T ∫ τ T T δ ṁ( θT δ , s) d Ws ≤ I τ T ( θT δ ) -1 T γ 1 T δ-γ ∫ τ T T δ ṁ( θT δ , s) d Ws ,
where 0 < γ < δ - 1 2 . Hence

P ϑ 0 { I τ T ( θT δ ) -1 τ T ∫ τ T T δ ṁ( θT δ , s) d Ws > ν 4 } ≤ P ϑ 0 { 1 T δ-γ ∫ τ T T δ ṁ( θT δ , s) d Ws > √ ν 2 } + P ϑ 0 { I τ T ( θT δ ) -1 T γ > √ ν 2 } -→ 0,
as T → ∞, because

P ϑ 0 { 1 T δ-γ ∫ τ T T δ ṁ( θT δ , s) d Ws > √ ν 2 } ≤ 1 νT 2δ-2γ E ϑ 0 ∫ T T δ ṁ( θT δ , s) 2 ds ≤ C νT 2δ-2γ-1 → 0.
Recall that 2δ -2γ -1 > 0; then

P ϑ 0 { I τ T ( θT δ ) -1 τ T ∫ τ T T δ ṁ( θT δ , s) ∆m ( θT δ , s ) ds > ν 4 } ≤ P ϑ 0 { 1 τ T 1-γ ∫ τ T T δ ṁ( θT δ , s) ∫ 1 0 ṁ(ϑ v , s) * dvds ( θT δ -ϑ 0 ) > √ ν 2 } + P ϑ 0 { I τ T ( θT δ ) -1 T γ > √ ν 2 } -→ 0 as T → ∞, because θT δ -ϑ 0 = O ( T -δ/2
) and other terms are bounded in probability (here

ϑ v = ϑ 0 + v( θT δ -ϑ 0 )).
To prove (39) let us write

√ τ T (ϑ ⋆ T (τ ) -ϑ 0 ) = √ τ T ( θT δ -ϑ 0 ) + I τ T ( θT δ ) -1 √ τ T ∫ τ T T δ ṁ( θT δ , s) d Ws + I τ T ( θT δ ) -1 √ τ T ∫ τ T T δ ṁ( θT δ , s) [ m(ϑ 0 , s) -m( θT δ , s) ] ds,
and note that

E ϑ 0 1 √ τ T ∫ τ T T δ [ ṁ( θT δ , s) -ṁ(ϑ 0 , s) ] d Ws 2 ≤ 1 τ T ∫ τ T T δ E ϑ 0 ṁ( θT δ , s) -ṁ(ϑ 0 , s) 2 ds -→ 0
as T → ∞. By the Central Limit Theorem (see, e.g., Theorem 1.19 in [START_REF] Kutoyants | Statistical Inference for Ergodic Diffusion Processes[END_REF]) the convergence in distribution

1 √ τ T ∫ τ T T δ ṁ(ϑ 0 , s) d Ws =⇒ N (0, I(ϑ 0 ))
holds, since by the Law of Large Numbers

1 τ T ∫ τ T T δ ṁ(ϑ 0 , s) ṁ(ϑ 0 , s) * dt -→ I(ϑ 0 ).
Further, let us denote

vT δ = √ τ T ( θT δ -ϑ 0 )
, then we can write

vT δ + I τ T ( θT δ ) -1 √ τ T ∫ τ T T δ ṁ( θT δ , s) [ m(ϑ 0 , s) -m( θT δ , s) ] ds = I τ T ( θT δ ) -1 ( I τ T ( θT δ ) - 1 τ T ∫ 1 0 ∫ τ T T δ
ṁ( θT δ , s) ṁ(ϑ r , s) * dr ds

) vT δ ,
where ϑ r = θT δ + r ( θT δ -ϑ 0 ) . The expression

ṁ(ϑ r , s) = ṁ( θT δ , s) + ∫ 1 0 m(ϑ q , s)dq ( θT δ -ϑ 0 )
and the equality

I τ T ( θT δ ) = 1 τ T ∫ τ T T δ ṁ( θT δ , s) ṁ( θT δ , s) * ds give vT δ + I τ T ( θT δ ) -1 √ τ T ∫ τ T T δ ṁ( θT δ , s) [ m(ϑ 0 , s) -m( θT δ , s) ] ds = √ τ T θT δ -ϑ 0 2 O (1) = T 1 2 -δ O (1) -→ 0, as T → ∞,
Let us verify that the Fisher information matrix is nondegenerate. To this end, it suffices to show invertibility of the matrix

J (ϑ 0 ) = ( E ϑ 0 ỹ2 t , E ϑ 0 ỹt zt E ϑ 0 ỹt zt , E ϑ 0 z2 t ) ,
where ỹt , zt are stationary solutions of (51) and ( 52) respectively. If this matrix is degenerate, then

E ϑ 0 ỹ2 t E ϑ z2 t = (E ϑ 0 ỹt zt ) 2 . ( 58
)
By the Cauchy-Schwartz inequality

(E ϑ 0 ỹt zt ) 2 ≤ E ϑ 0 ỹ2 t E ϑ 0 z2
t with equality if and only if zt = cỹ t with some constant c ̸ = 0. Therefore in the case of equality we have E ϑ 0 (cỹ t -zt ) 2 = 0.

Introduce a new process ṽt = cỹ t -zt as a solution of the equation

dṽ t = [x t (1 -c) -1] dt - [ a + b 2 xt (1 -xt ) ] ṽt dt + b (1 -2x t ) ṽt d Wt ,
where ṽt and xt are stationary. Hence

E ϑ 0 (∫ t 0 e -a(t-s) [x s (1 -c) -1] ds ) 2 ≤ 4 ( 1 + e -2at ) E ϑ 0 ṽ2 t + 4b 4 a ∫ t 0 e -a(t-s) 1 16 E ϑ 0 ṽ2 s ds + 4b 2 ∫ t 0 e -2a(t-s) E ϑ 0 ṽ2 s ds ≤ CE ϑ 0 ṽ2
t with some constant C > 0 which does not depend on t. By stationarity E ϑ 0 ṽ2 t does not depend on t and hence non-degeneracy of the matrix J (ϑ 0 ) follows, if we can show that

lim t→∞ E ϑ 0 (∫ t 0 e -a(t-s) [x s (1 -c) -1] ds ) 2 > 0,
for all c. The stochastic process

ζ t = ∫ t 0 e -a(t-s) [x s (1 -c) -1] ds
is the solution of the equation

dζ t dt = -aζ t + xt (1 -c) -1, ζ 0 = 0.
Elementary calculations show that for all ϑ 0 and c

lim t→∞ E ϑ 0 ζ 2 t = E ϑ 0 [π 0 (1 -c) -1] 2 a 2 > 0,
where π0 is a random variable with the density of the invariant distribution of the stochastic process π (t, ϑ 0 ). Thus the Fisher information matrix I (ϑ 0 ) is non degenerate for all ϑ 0 ∈ Θ.

Note that, by the Theorem 2, the limit covariance matrix of one-step MLE-process coincides with the covariance of the asymptotically efficient MLE (see [START_REF] Chigansky | Maximum likelihood estimation for hidden Markov models in continuous time[END_REF]), therefore ϑ ⋆ T (τ ) is asymptotically efficient as well.

Discussion

In this section we dwell on several possible generalizations of the obtained results.

Using the same arguments as in [START_REF] Kutoyants | On multi-step MLE-processes for ergodic diffusion[END_REF] it is possible to prove the uniform consistency : for any ν > 0 lim

T →∞ P ϑ 0 { sup τ δ ≤τ ≤1 |ϑ ⋆ T (τ ) -ϑ 0 | ≥ ν } = 0.
Moreover it can be shown that the normalized one-step MLE-process

η T (τ ) = τ √ T I(ϑ 0 ) -1/2 (ϑ ⋆ T (τ ) -ϑ 0 ) , τ * ≤ τ ≤ 1,
converges to the two-dimensional standard Wiener process W (τ ) , τ * ≤ τ ≤ 1, where τ * is any value satisfying τ * ∈ (0, 1]. For the details see the proof of a similar result in [START_REF] Kutoyants | On multi-step MLE-processes for ergodic diffusion[END_REF].

The condition M (2) can probably be relaxed, but it needs a special study. The finitness of the Fisher information follows from [START_REF] Chigansky | Maximum likelihood estimation for hidden Markov models in continuous time[END_REF] and the condition M (2) is basically used to control the matrix of the second derivatives.

Markov chain with d > 2 states

In the case of Markov chain Y (t) , t ≥ 0 with d > 2 states, the only essential difficulty in our approach is the construction of the preliminary estimator. For two states construction of a suitable preliminary estimator is relatively easy and it is not immediately clear how to extend it to the more general setup. If we have some √ T consistent estimator θT of the unknown parameter ϑ ∈ Θ ⊂ R m , then the corresponding one-step MLE-process is

ϑ ⋆ t = θT δ + t -1 I t ( θT δ ) -1 ∫ t T δ ṁ ( θT δ , s ) [ dX s -m ( θT δ , s ) ds ] , T δ ≤ t ≤ T, where δ ∈ ( 1 2 , 1 ) . Asymptotic efficiency √ τ T (ϑ ⋆ τ T -ϑ 0 ) =⇒ N ( 0, I (ϑ 0 ) -1
) , can be shown, applying the arguments from this paper. Note that the estimator ϑ ⋆ T is computationally simpler than the MLE θT . For the discrete time hidden Markov models with d ≥ 2 states, the MLE studied in e.g. [START_REF] Bickel | Asymptotic normality of the maximum likelihood estimator for general hidden Markov models[END_REF], can be used as the preliminary estimator for the one-step MLE-process, constructed as in our paper. The obtained online estimator of the unknown parameter will have asymptotically optimal properties.

Two-step MLE-process

The learning interval [0, T δ ] with δ ∈ ( 1 2 , 1) is negligible with respect to the observations time T . It can be made even shorter, if we use two-step MLEprocess approach, as it was proposed in [START_REF] Kutoyants | On multi-step MLE-processes for ergodic diffusion[END_REF]. This modification uses the learning interval [0, T δ ) with δ ∈ ( 1 3 , 1 2 ]. The procedure is as follows. First we obtain the preliminary estimator θT δ by applying our Method of Moments estimator to the observations X

T δ = ( X s , 0 ≤ s ≤ T δ )
. By ( 18)

T δ E ϑ 0 θT δ -ϑ 0 2 ≤ C,
where the constant C > 0 does not depend on T .

Then we introduce the second preliminary estimator-process θt,T = θT δ + t -1/2 I t ( θT δ ) -1 ∆ t ( θT δ , X It can be shown that for all τ ∈ (0, 1] and t = τ T we have the asymptotic normality of the estimator ϑ ⋆⋆ T (τ ) = ϑ ⋆⋆ τ T,T :

√ τ T (ϑ ⋆⋆ T (τ ) -ϑ 0 ) =⇒ N ( 0, I (ϑ 0 ) -1 ) . ( 59
)
The first step of the proof is to study the expression

T γ 2 ( θτT,T -ϑ 0 ) = T γ 2 ( θT δ -ϑ 0 ) + T -1+ γ 2 I τ T ( θT δ ) -1 ∆ τ T,T (ϑ 0 , X t ) + T -1+ γ 2 I τ T ( θT δ ) -1 ∫ τ T T δ ṁ ( θT δ , s ) [ m (ϑ 0 , s) -m ( θT δ , s )] ds,
We can show that if 2 3 < γ < 2δ, then T γ 2

( θτT,T -ϑ 0 ) is bounded in probability and we have (59).

The proof is omitted, being similar in spirit to that in [START_REF] Kutoyants | On multi-step MLE-processes for ergodic diffusion[END_REF]. Note however that such two-step MLE-process lacks the advantage of being computationally efficient: its construction requires generating the family of filters m ( θs,T , s ) for all θs,T , s ∈ [ T δ , t ] , while for one-step MLE-process, m (ϑ, s) is to be computed only for the single value ϑ = θT δ .

Kalman filter

The proposed device is universal and can be applied to construct asymptotically efficient estimators (estimator-processes) in other models. Let us consider the following simple example of the partially observed system dX t = (λ + µY t ) dt + dW t , X 0 = 0, dY (t) = -aY (t) dt + dV t , Y 0 , where W t , t ≥ 0 and V t , t ≥ 0 are two independent Wiener processes and a > 0 is a known constant. The unknown parameter ϑ = (λ, µ) ∈ Θ = (c 0 , c 1 ) × (c 3 , c 4 ) is to be estimated, using the sample X T = (X t , 0 ≤ t ≤ T ). Calculation of the MLE θT requires m (ϑ, t) for all ϑ ∈ Θ, t ∈ [0, T ] (see, e.g., Section 3.1 in [START_REF] Kutoyants | Statistical Inference for Ergodic Diffusion Processes[END_REF]) and hence we are faced with the computational problem of the same complexity as for the model considered in this paper. One-step procedure can reduce the computational load, while attaining the optimal performance in the large sample limit.

A reasonable choice of the preliminary estimator is θT = ( λT , μT ) with

λT = X T T , μT = √ ζ T K .
where we denoted ζ T = max ( ζT , 0 ) and

ζT = 1 T T -1 ∑ i=0 [ X i+1 -X i -λT ] 2 -1, K = E (∫ 1 0 Y * s ds ) 2 .
Here Y * s is the stationary process satisfying the same equation as Y (s). It can be shown that

E ϑ 0 √ T ( θT -ϑ 0 ) 2 < C.
Introduce the learning interval [ 0, T δ ] with δ ∈ ( 1 2 , 1

) and the one-step MLE

ϑ ⋆ T = θT δ + I T ( θT δ ) -1 T -1 ∫ T T δ ṁ ( θT δ , t ) [ dX t -( λT δ + μT δ n ( θT δ , t ) )dt ] .
Using the same arguments as above we can verify the convergence √ T (ϑ ⋆ T -ϑ 0 ) =⇒ N ( 0, I (ϑ 0 ) -1 ) .

Further, following Section 4 2 ∫ t 0 e

 420 of[START_REF] Chigansky | Maximum likelihood estimation for hidden Markov models in continuous time[END_REF], we can writeṽt = ṽ0 e -at + ∫ t 0 e -a(t-s) [x s (1 -c) -1] ds -b -a(t-s) xs (1 -xs ) ṽs ds + b ∫ t 0 e -a(t-s) (1 -2x s ) ṽs d Ws .

  whereF X t = σ {X s , 0 ≤ s ≤ t}. The conditional expectation n (ϑ, t) satisfies the Kalman-Bucy filtering equations dn (ϑ, t) = -an (ϑ, t) dt + µγ t (µ) [dX t -(λ + µn (ϑ, t)) dt] , n (ϑ, 0) , ∂γ t (µ) ∂t = -2aγ t (µ) -γ t (µ) 2 µ 2 + 1, γ 0 , 0 ≤ t ≤ T.

	and the log-likelihood function is given by
	L	(	ϑ, X T )	=	∫ T 0	m (ϑ, t) dX t -	1 2	∫ T 0	m (ϑ, t) 2 dt,	ϑ ∈ Θ.

Let us define n (ϑ, t) = E ϑ ( Y (t) |F X t )

and m (ϑ, t) = λ + µn (ϑ, t),

Acknowledgment. We thank Pavel Chigansky for his valuable comments and we are grateful to the anonymous referees for their comments and remarks. This work was done under partial financial support (second author) of the grant of RSF number 14-49-00079.


Edgardo Civallero
email: edgardocivallero@gmail.com

The archive of the community's voices Libraries and oral tradition in the digital era

Keywords: Oral tradition, library, academic library, research, ICTs

A great part of the current human knowledge is still transmitted through non-written means -oral tradi on, song and music, etc. The collec on, organiza on, research and management of these non-wri"en sources -and par cularly, of oral tradi on-has usually been carried out by scholars from an strictly academic point of view; accordingly, the information collected by these specialists has been kept in archives for academic use only. This behavior stripped oral tradition of much of its practical use (especially for the people providing the original materials) and opened a gap between the Academia and the communities of practice. This paper outlines a brief approach to the nature and the importance of oral tradition in our modern world, and to how to connect traditional knowledge and communities, the Academia and its researchers, and libraries as a common space. It also drafts some ideas about how to deal with it from a LIS (Library and Information Sciences) perspective, and how to use ICTs (Information and Communication Technologies) in a planned, sensitive and responsible manner, in the collection, organization and revitalization of oral tradition.

Introduction

Part of the current human knowledge is still conveyed through non-written means, one of the most important being orality. Oral tradi on -tradi onal knowledge transmi"ed by oral channels-forms the founda ons of many individual or collec ve identities and cultures all around the world, including those of modern, urban-based societies.

The systematic collection of these non-written sources, and their organization, study and management, has been led primarily by scholars in the field of Humanities, and has been carried out from an academic point of view and for academic purposes. The informa on collected -ny fragments of a huge corpus-has been kept in archives for academic use only, and has hence (with the exception of oral history) been devoid of most of its true value. For when oral tradition is isolated from its community of practice (creators and users), it becomes just a snapshot of a bigger picture, a stillphotograph at any fixed time documenting a frozen moment in the past. It turns into a piece of information that may be observed and analyzed by a few curious eyes, and may help future generations work out how a group of people thought or behaved at some point in history. Oral tradition, however, is much more than that. Besides preserving a record of the past, oral tradition continues to be the dominant form of human communication and knowledge transmission in the 21st century. And although it can evidently benefit from being documented and archived, oral tradition is a living tradition maintained by collective memory, created and perpetuated by societies to serve a purpose.

In recent years, there has been much criticism and debate on how the Humanities, i.e. academic disciplines that study human culture, handle traditional knowledge in general, and oral tradition in particular. However, there still remains much to be done to bring researchers and communities together to find better ways to interact and collaborate in such as way as to prevent abuses and misunderstandings. In supporting the search for common ground and points of active cooperation, libraries may play a valuable role not only as spaces for meeting and debate to help bridge the gaps, but as institutions able to make significant contributions. Unlike archives, libraries tend to see information as something with a purpose, aimed at being circulated, disseminated, and (inter)connected; aimed at being useful and contributing something to society. Besides, libraries act upon some of the same categories and mechanisms used by traditional, orally-based networks: both consider knowledge as common human heritage that should belong to and benefit all.

In addition to bring researches and communities closer, there are a number of activities and services that libraries may develop to become the places where community's voices and memories are kept alive and thriving. New technologies and virtual spaces can make a significant contribution towards this goal when all the pros and cons have been weighed up and where feasible and sustainable. This paper outlines a brief approach to the nature and the importance of oral tradition in our modern world, and to how to connect traditional knowledge and communities, the Academia and its researchers, and libraries as a common space. It also drafts some ideas about how to deal with it from a LIS (Library and Information Sciences) perspective, and how to use ICTs (Information and Communication Technologies) in a planned, sensitive and responsible manner, in the collection, organization and revitalization of oral tradition.

The importance of oral tradition

Human knowledge includes several spheres; one of the most interesting is the one labeled as "traditional". It is a corpus of selected knowledge, communally "curated" by many hands over time, and transmitted from generation to generation, usually by oral means and using well-defined forms; it collects what a human group -no ma"er its size, composi on and structure-has deemed necessary or important to preserve, codify and perpetuate.

Generally speaking, this type of knowledge forms the basic culture of an individual or a community, determining and keeping their identity alive, allowing them to know where they come from and to decide where they want or are able to go. Curiously enough, it is not easy to realize the strong presence of this knowledge in our lives: only when we compare our cultural maps to the ones that guide another person, or when we live in a society that is not our own, differences are perceived and we come to understand how deep are the roots of that heritage which, to a great extent, make us who we are.

All the values, teachings, ideas and experiences transmitted during the first stages of socialization belong to the "traditional" corpus: from the very concept of "family" and the different kinds of bonds that occur between its members, to customary treatment and respect given to "strangers" (and the definition of who is and who is not "a stranger"), as well as basic rules of behavior and interaction with other people, games and songs to play and sing, food and drink to be consumed, beliefs, expressions and interjections, ways and moments to laugh and cry... And not just that: the corpus also includes legends and stories, myths of creation and destruction, awareness about the environment and its challenges, general guidelines about which is the individual's position in the (natural and social) universe, historical and pseudo-historical narratives, family/community anecdotes, taboos and non-written laws, ways of navigating through major milestones in life, ways of tackling problems and taking advantage of opportunities, and a very long etcetera.

The above list reveals the close relationship between traditional knowledge and the social fabric. Traditional knowledge encodes a significant share of the information and instructions necessary for the construction of any given community's internal links and fundamental structures, the design of strategies for its conservation, adaptation and transformation, and the development of values, beliefs and behaviors that are consistent with those strategies. It also includes the -o=en unwri"en-history of a human group; a group that can range from a handful of individuals with some interest or trait in common to a nation. In short, that form of knowledge expresses and conveys the paths traveled so far by a number of persons and the strategies that have allowed them to thread these paths together.

In general terms, the traditional knowledge of a community is maintained and transmitted through the spoken word. It is not the only method, though: the same goal may be achieved by using textiles, facial paintings, tattoos and body adornments, baskets, designs on ceramic and gourds, carvings, sculptures, photographs, paintings and drawings. And, of course, by books, magazines, journals, newspapers, letters and other written and printed documents. But the spoken word is, without question, the most popular method.

The oral tradition of a particular community -an urban tribe, the members of a union, the shepherds of several neighboring villages-is made up of spoken knowledge. As other spoken contents, oral tradition is characterized by being grammatically complex (much more than written text), diverse, dynamic, relatively spontaneous (although it uses a number of formulas), immediate and unstable. And very fragile. It can also be misleading, confusing and thorny at times, even for its own users: in oral tradition the tangible and the intangible, the objective and the subjective are so intimately mingled that there are times when they cannot be told apart.

Although some of the pieces of information that form oral tradition are "fossilized", i.e. they have been transmitted without being substantially altered for centuries, it is flexible and changes: on the one hand as the result of losing part of its contents and gaining new ones, and on the other hand by undergoing modifications at different levels. Such changes can be forced, but usually respond to the needs of a particular social group, which is itself changing and evolves over me -sometimes adopting new routes, sometimes adjusting its steps according to the terrain and the circumstances.

Oral tradition is passed on in two direc ons: ver cally -genera on a=er genera onand horizontally, between the members of the same generation. Therefore, in addition to perpetuating a corpus of knowledge, it reinforces the ties among peers and among adults and young people, and strengthens community structures. It also favors the processes of socialization and education, cultural production, and languages' vitality.

What structures and channels oral tradition uses to gather and convey its contents are limited only by imagination. Items of oral tradition can be delivered as a song or a story, an epic declamation or a nursery rhyme. They can be accompanied by gestures or music, be part of a broader cultural expression -such as theater or dance-or not. They can be transmitted in a bar or in a kitchen, in a workshop or in a classroom, in a formal context or in an informal one.

And the contents conveyed... are practically infinite. Simple anecdotes or the genealogy of a royal family, human or divine, from the beginning of time. The mental map of the Andean pastures for lamas, or the sites for collecting mussels on a rugged coastline, with all its detailed micro-geography. The story of a beverage's invention, of the composition of a piece of music, of the creation of a saying. The anecdotes behind a photo. The instructions to properly manage a piece of forest or a river, to care for a crop, to find a medicinal plant. The trajectory of a political party, a company, a factory, a publisher, a shop. The story that goes through a book, a corner in any city, a wall covered with graffiti... All these contents, small and insignificant as they seem, are pieces of our memory, our history, our essence. Parts of us as persons, as members of a society, as components of a group, as inhabitants of a certain place... Oral tradition is not limited to those societies or groups lacking reading/writing skills; it is also present amongst literate societies. Actually, there are few branches of human activity, regardless of their size, not having an oral corpus associated with them. In contemporary urban communities, oral tradition has a strong presence, especially amongst social groups (minority, disadvantaged, alternative or dissident) who have not been able, for whatever reason, to see their opinions and memories fixed in writing. But it also channels certain knowledge among hegemonic literate sectors: it keeps alive family memories, local histories, individual or group experiences related to a number of historical events, traditions, games, crafts... Therefore, besides keeping the threads of the social fabric together, those narratives supplement and balance the "official history", i.e. the tailored account that typify a people, a culture or a country. And they allow the existence of plurality and diversity, two values often pushed to the background to favor prevailing stereotypes and mainstream discourses.

Oral tradition is also a "living repository" of a category of knowledge that cannot usually be found in any other place. In fact, in some cases oral tradition becomes the last refuge of information on the verge of disappearance, encoded in equally endangered languages.

A wide majority of this intangible tradition still circulates from mouth to ear, devoid of any material support. This is the most delicate part, and the one that needs more attention and care, especially when it belongs to social groups suffering cultural pressure, discrimination, or oblivion.

The other part has been converted, for different reasons and by different methods, into tangible documentation: written, printed, graphic or audiovisual materials. Academic researchers and scholars in the field of Humanities have had a remarkable role in this process -a role that has not been free from issues, and has been subject to criticism and debate.

Academic research: collection and archive

Being practically ubiquitous, orally-transmitted knowledge could not go unnoticed by scholars for too long. Popular storytelling was under the focus of many authors for centuries, although it was not the only element to be targeted -portraying customs and traditions was also the aim of a good number of works, from travel diaries to chronicles and ethnographic descriptions.

However, academic, systematic research in the field of Humanities is relatively recent, especially the one related to oral tradition.

Since the first decades of the 20th century, the development of early recording technologies has boosted this kind of investigations; they allowed researchers to recover a number of sound-based cultural expressions, including oral tradition, in a more systematic, precise way. This was done within a very specific, narrow scope of academic disciplines: namely, anthropology, linguistics and ethnology.

Audio-visual media improvements (and Information technology later on) opened up new possibilities for working with oral sources, widening the range of research activities and surveys and getting more fields of Humanities (politics, sociology, history...) involved in the process of collec ng sounds. Oral history -the methodology of which was already used by Thucydides and Herodotus in classical Greecewitnessed a renaissance after World War II. Adding to this, alternative, personal and collective approaches to specific historical events were provided by Spanish miners and fighters, French anarchists, Latin American guerrillas and trade unionists, Brazilian sem-terras among many others.

Oral archives were then created to preserve, organize, and study these collec onsjust small pieces of a seemingly endless mosaic. But its access and use at the time was -and, in many cases, s ll is-limited to the Academia. Oral tradi on had become a new object to be analyzed and theorized about in the "dissecting rooms" of social science schools, and, more often than not, ended up becoming "dead knowledge". While those contents had been collected and preserved to "save" them from the uncertain fate awaiting most oral knowledge, once "rescued" they were isolated from the living socio-cultural structure that produces, revises, maintains and re-elaborates oral tradition day after day. The very reason why those contents exist and are remembered is to be transmitted, in order to continue living in somebody else's lips and memory. By just storing them in archives for faculty/academic-staff use only, they had been (a) turned into a mere record or a simple piece of writing (stripping them from many of their main features), and (b) isolated from the living chains and the linking agents that made them "oral" and "a tradition".

The management of oral collections has been just one of the many reasons for a notorious disconnec on between the academic world -specifically disciplines, researchers, repositories and publications in the area of Humani es-and the communities, i.e. the (re)producers of orally-transmitted knowledge from which researchers borrow their study materials. For some time, gaps between the work done by academic researchers (mostly serving their own interests, or pursuing their own goals within their closed circles) and society got wider, the former failing to help communities to keep their memories alive, active, circulating from mouth to ear, and also documented, organized and stored to be easily accessed.

Only recently, a few tentative first steps in the right direction have been taken to bring the Academy closer to people. On the one hand, by trying to establish a different type of relationship and, on the other, by looking for ways and spaces to work together for a common good.

Libraries in general and academic libraries in particular may become a key piece in building the relationship between researchers and communities of practice, and getting it to work. And not just by providing physical spaces for meetings and activities, but as institutions capable of making a number of important contributions in terms of know-how, techniques, contents, strategies, services and activities. Not for nothing have libraries worldwide exercised a thoughtful and insightful leadership in organizing human knowledge over the centuries.

Libraries and oral tradition

Libraries are institutions that, within their possibilities, bring together the materials and tools necessary to provide societies with the knowledge they may need.

Libraries make every effort to integrate into and be part of the community they serve. In responding to their users' needs and queries in a planned, sensitive and responsible manner, sometimes they work as lighthouses, while on other occasions operate as harbors. All of them provide access to strategic information; support literacy and education; promote broader and more in-depth understanding; bring distant horizons closer, and much more. Besides, they are memory archives, hosting and storing different manifestations of the cultural heritage of humankind.

However, for a number of reasons "traditional" knowledge in general and oral tradition in particular have not been included in library collections and services as assiduously as they should, despite of the fact that this knowledge is considered to be essential for any human group, and that a number of international recommendations concerning cultural diversity and intangible heritage have encouraged the use of oral tradition as a means for preserving and disseminating different societies' cultural expressions and languages.

With their extensive experience as information managers, libraries could do a brilliant, much needed work with oral tradition. But they have to understand that the object of this work is not to record orality or other "non-conventional" sources, but to promote and making the best possible use of them. As already said, "traditional" knowledge, its creation, acquisition and transmission are part of an ongoing, dynamic process in constant motion. By limiting the work with oral tradition to its collection and storage much of its importance, meaning and value are lost.

Therefore, as a first step LIS professionals should gain a greater knowledge of what oral tradition is, how it works, why it is important, what its flaws and biases are, and what their problems and risks are -something for which, unfortunately, there is no specific, LIS-oriented training, guidelines or recommendations. Librarians willing to deal with oral tradition might have to reframe their own concepts, tools and techniques, or create new ones from an interdisciplinary perspective.

Once this initial step has been given, libraries would be ready to organize, promote and disseminate oral tradition, taking into consideration particular formats, channels, contents, actors and spaces. Memory and knowledge encoded orally or through "nonconventional" means would then become alive and vibrant in libraries; this would make libraries' collections richer and would enhance their role as the memory archives of their communities, developing stronger bonds with one another. This inclusion would also serve as a form of acknowledgement and recognition for certain social sectors -for example, indigenous peoples-that have been systema cally underrepresented or ignored in/by libraries.

By including oral tradition in their collections and services and fostering its use, libraries may reinforce some damaged threads of the social fabric they are part of. They may help make certain groups visible, challenge stereotypes and prejudices and favor exchange, learning and mutual recognition; they may recover local history and local expertise while rediscovering customs and traditions and updating some of them in a conscious manner. Libraries may identify the best practitioners of oral tradition in their communities ("living books") and support and encourage them; they may inform their users (individuals, groups, organizations, institutions) about the body of knowledge they preserve and transmit, explaining why that tradition is important, and what can be done with it and for it. In addition, books, audiovisual materials and any other kind of documents may be provided to inform and support those interested in approaching, understanding and put orality into practice. Libraries may also create spaces for oral practitioners to teach their art and exchange their wisdom, expand storytelling possibilities, and raise cultural awareness through heritage workshops, language classes, and all the rest of it.

Orally-delivered contents such as local geography and history, ethno-botany and ethno-zoology, recipes, crafts, and many others fragments of traditional knowledge, should be added to the tales, legends and myths usually kept in libraries' "oral tradition" sections (when/where they exist). These contents could be collected by the library staff or by any other member of the community through different means, and arranged in such a way that they become available for the rest of the library's users.

In the same vein, libraries can join forces and collaborate with schools and other cultural, social and educational organizations and institutions, to foster the use of oral tradition. Formal and informal courses on geography, history, literature, language, music, natural sciences and crafts may benefit from the information contained in oral tradition; additionally, intercultural, bilingual programs should make full use of them to support endangered languages and strengthen the preservation of cultural identities.

Libraries that have learnt how to support oral tradition and its practitioners provide the best environments for academic researchers and communities to work together in the study, preservation and revitalization of traditional knowledge and traditional means of knowledge transmission. New Information and Communication technologies (ICTs) may be helpful tools in this process.

New technologies

"Digital Humanities" are a set of disciplines belonging to the social sciences branch, benefiting from the use of electronic means for the production, management and dissemination of information -i.e. ICTs.

It should be noticed that ICTs are tools, and as such they should be used according to needs and requirements; the fact that they exist does not imply that they must always be used, or that they are always necessary or effective, much less that they must be imposed. It should neither be forgotten that while technology allows doing a number of things, it also forces its users to make a number of others of which they are not aware on many occasions.

With that in mind, there is no doubt that ICTs can open up new possibilities, especially when it comes to designing and developing library activities related to oral tradition: from the production of ebooks on local history to the creation of online oral collections taking advantage of open software and cloud storage, and the distribution, through certain networks and platforms, of videos, images and recordings, accompanied with explanations, as a way of allowing other people to get to know community cultural expressions.

To achieve certain goals, appropriate tools, techniques and formats should be selected. There will be occasions when new technologies may join old ones: depending on the context, sometimes a handwritten book or fanzine can be much more effective than an ebook; an exhibition of photographs at a community center, sponsored by the library and supplemented with relevant background materials, can be much more attractive than a Flickr gallery; a radio ad, much more useful than a post in WhatsApp or Facebook; an outdoor storytelling session, much more interesting than a video on YouTube. Practice has shown that a balanced combination of "traditional" and "modern" elements is often the wisest and fairest course of action.

New alliances

From what has been exposed so far, it is worth considering the elaboration of common projects taking into account (a) libraries know-how, structures and spaces; (b) the opportunities that technology offers today; (c) the academic research and writing on oral tradition; (d) communities where orally-transmitted teachings consolidate and enhance their traditional knowledge; and (e) communities where oral tradition has been broken.

Libraries can help academic researchers and communities to collect, organize and disseminate oral tradition, and materials related to it (papers, books, audiovisual documents...). They can help communities reconnect with their traditional knowledge (oral tradition collected for academic purposes in the past and stored in archives) and update it -which may also trigger a number of research programs. Together with their community libraries can develop oral tradition activities, involving researchers and other actors in cases where this is advisable or necessary; through some of these activities they can cast light on local problems, challenge stereotypes, debunk fake stories, etc. Libraries could bring into contact communities' oral tradition and written documents and collections, opening doors to numerous opportunities (and to more academic-supported research). They could even help the Academia to reformulate its position about a number of past issues regarding traditional knowledge and oral cultures.

By making these and any other connections possible, and by exploring new paths for improvement, libraries, researchers and communities may gain knowledge and experience and discover mutually beneficial ways of interaction with each other, cultural exchange, collaboration, and learning. Lastly, it is worth noting that a large number of voices, past and present, in rural and urban contexts, have found and continue to find refuge in "traditional" knowledge and all its forms of transmission, including oral tradition. All of them are part of human memory. If libraries are meant to be the managers of that memory, the archive of the community's voices, they should include all these fragile expressions in their collections.

Acknowledgements

I would like to thank Sara Plaza Moreno for her valuable suggestion and contributions to this paper, her careful revision of the text, and the discussion of the ideas included in it.