Eva Löcherbach
email: locherbach@univ-paris12.fr

Dasha Loukianova
email: dasha.loukianova@univ-evry.fr

The law of iterated logarithm for additive functionals and martingale additive functionals of Harris recurrent Markov processes

Keywords: Harris recurrence, Nummelin splitting, continuous time Markov processes, additive functionals, law of iterated logarithm

We use Nummelin splitting in continuous time in order to prove laws of iterated logarithm for additive functionals of a Harris recurrent Markov process, with deterministic or random renormalization.

Introduction

Let X be a Harris recurrent strong Markov process in continuous time, having invariant measure µ. In [START_REF] Löcherbach | On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions[END_REF], we studied the problem of introducing Nummeling splitting in continuous time. This is a method, firstly introduced independently by Nummelin, [START_REF] Nummelin | A splitting technique for Harris recurrent Markov chains[END_REF], and Athreya and Ney, [START_REF] Athreya | A new approach to the limit theory of recurrent Markov chains[END_REF], in the discrete time case that allows to introduce renewal times for the process on an extended probability space. In [START_REF] Löcherbach | On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions[END_REF], we translate this technique to the situation of processes in continuous time. It is a classical technique in the theory of Markov processes to translate results from discrete time to continuous time by considering what is called the R-chain in the literature. This means that we observe the continuous time process after independent exponential waiting times. We refer the reader to Meyn and Tweedie, [START_REF] Meyn | Proceedings of the Doeblin conference '50 years after Doeblin: development in the theory of Markov chains, Markov processes, and sums of random variables[END_REF] and [START_REF] Meyn | A survey of Foster-Lyapunov techniques for general state space Markov processes[END_REF], for a general survey of the subject. Hence we use the Nummelin splitting technique at random times T n , n ≥ 1, which we get when sampling the process after independent exponential waiting times. This allows to get a sequence of renewal times for the process in the following sense : There exists a sequence of stopping times (S n , R n ) such that 1. For all n, S n < R n < ∞, S n+1 = S n + S 1 • θ Sn , R n = inf{T m : T m > S n }.

2. For every n, X Rn is independent of σ{X s : s ≤ S n } and L(X Rn ) = ν for some fixed probability measure ν.

The details of this construction are recalled in section 4 of this paper.

We apply our technique to the study of the asymptotic behavior of additive functionals, for example A t = t 0 f (X s )ds. In [START_REF] Löcherbach | On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions[END_REF], we have shown the existence of a deterministic equivalent for integrable additive functionals A t . The deterministic equivalent is a deterministic function t → v(t) such that v(t) → ∞ as t → ∞ and such that for any integrable additive functional A t , lim

M →∞ lim inf t→∞ P π (1/M ≤ A t /v(t) ≤ M ) = 1
for any initial measure π. The deterministic equivalent can be defined as follows. Take any fixed positive special function g of the process having µ(g) > 0 (see definition 2.3 below for the exact definition of special functions, for strong Feller processes, any bounded function having compact support is special) and define

v(t) := E η ( t 0 g(X s ds),
where η is an arbitrary initial measure. Then the strong Chacon-Ornstein theorem implies that for any other special function g ′ and any other initial measure η ′ ,

lim t→∞ E η ( t 0 g(X s ds) E η ′ ( t 0 g ′ (X s ds) = µ(g) µ(g ′ ) .
Hence the deterministic equivalent is unique up to a constant in the sense that for two choices of the deterministic equivalent, v and v ′ , we have that lim t→∞ v(t)/v ′ (t) = c, where c is a positive constant. In regular models, it can be shown that v(t) ∼ t α l(t), where l is a function that varies slowly at infinity. For example, for Brownian motion in dimension one, we have α = 1/2.

In the present paper, we generalize results obtained by Chen in [START_REF] Chen | How often does a Harris recurrent Markov chain recur?[END_REF], [START_REF] Chen | The law of the iterated logarithm for functionals of Harris recurrent Markov chains: Self Normalization[END_REF] and [START_REF] Chen | On the limit laws of the second order for additive functionals of Harris recurrent Markov chains[END_REF] on the almost sure asymptotic behavior of integrable additive functionals to the continuous time case.

In the first case, consider A t = t 0 f (X s )ds an additive functional such that µ(f ) > 0. In this case, it is possible to use the Chacon-Ornstein ratio limit theorem in order to compare A t to additive functionals of the R-chain (X Tn ) n where T n is the sum of n independent exponential waiting times : If we write N t := max{n :

T n ≤ t}, then t 0 f (X s )ds Nt k=1 f (X T k ) → 1 almost surely as t → ∞.
Now, it is possible to translate results obtained by Chen in [START_REF] Chen | How often does a Harris recurrent Markov chain recur?[END_REF] directly to the continuous time case, and we get the following first theorem (cf. theorem 3.1): Let L 2 (λ) := (log log λ) ∨ 1. Then for v(t) the deterministic equivalent of the process : there exists a constant C ∈ (0, ∞), depending only on the process but not on f, such that lim sup

t→∞ t 0 f (X s )ds v( t L 2 (v(t)) )L 2 (v(t)) = Cµ(f ) a.s.
Let a(n) be the deterministic equivalent of the R-chain, i.e. lim

M →∞ lim inf n→∞ P π (1/M ≤ n k=1 f (X T k ) a(n) ≤ M ) = 1
for any initial measure π and any positive function f such that µ(f ) > 0. The existence of a(n) has been proven in Chen [START_REF] Chen | How often does a Harris recurrent Markov chain recur?[END_REF]. Then the only difficulty in this first situation consists in the fact that we have to prove equivalence of the deterministic equivalent a(n) of the R-chain and of v(n). This is done in section 4.2 and in section 5 thanks to Nummelin splitting in continuous time.

Secondly, we are interested in strong limit theorems for additive functionals in the case where A t = t 0 f (X s )ds having µ(f ) = 0. This case is much more difficult since now a direct comparison with additive functionals of the R-chain is no more possible : the ratio-limit theorem is no more valid. It is in this situation that the Nummelin spitting in continuous time shows to be very useful : Taking increments of the additive functional over life cycles, i.e. putting

ξ n := Rn R n-1 f (X s )ds,
the ξ n are not independent, but very strongly mixing, and of the same law. (Actually, ξ n is independent of ξ n+2 .) Hence it is possible to apply the classical law of iterated logarithm to ξ 1 + . . . + ξ n under sufficient moment conditions, and we obtain the following law of iterated logarithm with random renormalization : Let f be a µ-integrable function such that µ(f ) = 0 and such that |f | is a special function (see definition 2.3 below for details), f is also called a charge in this case. Let B t = t 0 g(X s )ds be any additive functional such that µ(g) > 0. Then we have lim sup

t→∞ t 0 f (X s )ds √ 2B t L 2 B t = c
almost surely, where c is a constant that is positive if the asymptotic variance of ξ 1 +. . .+ξ n is positive.

Let us give some comments on our results. Firstly, as already pointed out, our results are a direct generalization of results obtained by Chen in the case of Markov chains, see [START_REF] Chen | How often does a Harris recurrent Markov chain recur?[END_REF]- [START_REF] Chen | On the limit laws of the second order for additive functionals of Harris recurrent Markov chains[END_REF]. It has not been possible to translate all results obtained by Chen to our case since in contrary to the discrete time case, the ξ n are no longer independent (in the case of Markov chains, Nummelin splitting does introduce independent increments of additive functionals over life-cycles). Touati, see [START_REF] Touati | Loi fonctionnelle du logarithme itéré pour les processus de Markov récurrents[END_REF], also gives laws of iterated logarithm, also in the continuous time case, but only under the assumption of regularity of the process, i.e. R 1 is in the domain of attraction of some stable law. However, we do not need the assumption of regularity of the process. Finally, let us also point out the work by Csáki, Földes and Hu, [START_REF] Csáki | Strong approximation of additive functionals of a planar Brownian motion[END_REF], in which the authors prove a law of iterated logarithm for additive functionals of planar Brownian motion, however using deterministic renormalization.

The paper is organized as follows. In section 4, we recall the technique of Nummelin splitting in continuous time, as introduced in [START_REF] Löcherbach | On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions[END_REF]. Moreover, we recall known results related to this construction as well as some new technical results that will be useful in the sequel. Section 5 is devoted to the proof of theorem 3.1, section 6 gives the proof of theorems 3.2 and 3.4.

Notation

Consider a probability space (Ω, A, (P x ) x ), and on (Ω, A, (P x ) x ) a process X = (X t ) t≥0 which is strong Markov, taking values in a locally compact Polish space (E, E), with càdlàg paths, and with X 0 = x P x -almost surely, x ∈ E. We write (P t ) t for the semi group of X and we suppose that X is recurrent in the sense of Harris, with invariant measure µ, unique up to multiplication with a constant. Moreover, we shall write (F t ) t for the filtration generated by the process.

We impose the following regularity condition on the transition semi-group P t of X :

Assumption 2.1 1. The transition semi-group P t of the process X is strongly Feller, i.e. for every A ∈ E, x → P t (x, A) is continuous.

2. There exists a sigma-finite positive measure Λ on (E, E) such that for every t > 0, P t (x, dy) = p t (x, y)Λ(dy), where (t, x, y) → p t (x, y) is jointly measurable.

On the deterministic equivalent of additive functionals

We resume results of [START_REF] Löcherbach | On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions[END_REF] on the deterministic equivalent of additive functionals. First, recall the definition of an additive functional:

Definition 2.2 An additive functional of the process X is a Ī R + -valued, adapted process A = (A t ) t≥0 such that 1. Almost surely, the process is non-decreasing, right-continuous, having A 0 = 0.

2. For any s, t ≥ 0, A s+t = A t +A s •θ t almost surely. Here, θ denotes the shift operator.

Examples for additive functionals are A t = t 0 f (X s )ds where f is a positive measurable function. Such an additive functional is said to be integrable, if µ(f ) < ∞. In [START_REF] Löcherbach | On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions[END_REF], we have constructed a deterministic equivalent of any integrable additive functional. This is a deterministic function v → v(t) such that v(0) = 0, v(.) is non-decreasing and v(t) → ∞ as t → ∞ satisfying that for any integrable additive functional A t , A t /v(t) is bounded and bounded away from zero in probability (see corollary 2.8 of [START_REF] Löcherbach | On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions[END_REF]). Recall the notion of a special function (see also [START_REF] Revuz | Markov chains[END_REF], [START_REF] Brancovan | Fonctionnelles additives spéciales des processus récurrents au sens de Harris[END_REF]): Definition 2.3 A measurable function f : E → IR + is called special if for all bounded and positive measurable functions h such that µ(h) > 0, the function

x → E x ∞ 0 exp - t 0 h(X s )ds f (X t )dt
is bounded. In the same way, an additive functional A t is called special if

x → E x ∞ 0 exp - t 0 h(X s )ds dA t is bounded.
By [START_REF] Löcherbach | On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions[END_REF], any special function g of X with µ(g) > 0 defines a version of the deterministic equivalent via

v(t) = E π t 0 g(X s )ds, (2.1)
for any arbitrary initial measure π. v(t) is unique up to a constant in the following sense : For any other choice v ′ (t) of a deterministic equivalent, we have that lim t→∞ v(t)/v ′ (t) = c, where c is a positive constant. v(t) is called deterministic equivalent due to the following result (see corollary 2.19 of [START_REF] Löcherbach | On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions[END_REF]).

Theorem 2.4 For any additive functional A of the process having E µ (A 1 ) ∈]0, ∞[, we have lim

M →∞ lim inf t→∞ P π 1 M ≤ 1 v(t) A t ≤ M = 1.
3 The law of iterated logarithm for additive functionals

We put L 2 (λ) := (log log λ) ∨ 1 for every λ ≥ 0.

Theorem 3.1 Let A be a µ-integrable additive functional of X. There exists a constant 0 < c < ∞ that depends only on the process, but not on A, such that

lim sup t→∞ A t v( t L 2 (v(t)) )L 2 (v(t)) = c µ(dx)E x (A 1 ) a.s. (3.2)
Moreover, let f be a measurable µ-integrable function satisfying the following assumptions:

(i) µ(f ) = 0. (ii) |f | is a bounded special function.
Then we have the following: Theorem 3.2 Suppose f satisfies assumptions (i) and (ii) above. Let A = (A t ) t be any µ-integrable additive functional of the process. Then there exists a constant λ f ≥ 0, such that

lim sup t→∞ t 0 f (X s )ds 2A t L 2 (A t ) = (E µ (A 1 )) -1/2 λ f a.s.
Here, λ f > 0 if (6.20) below holds.

Remark 3.3 Under the additional hypothesis

∞ 0 f (x)P t f (x)dt converges in L 1 (E, µ),
we have that

(λ f ) 2 = 2 ∞ 0 f (x)P t f (x)dtµ(dx). (3.3)
Finally, let M be a locally square integrable local (P x , (F t ) t )-martingale, with M 0 = 0, having continuous paths, for any initial value x. We suppose moreover that ∀y, ∀s, t :

M t+s -M t = M s • θ t P y -a.s., (3.4)
and the process < M > is an additive functional of X satisfying

E µ (< M > 1 ) < ∞. (3.5)
Then we have the following theorem:

Theorem 3.4 Under the assumptions (3.4) and (3.5), we have

lim sup t→∞ M t 2A t L 2 (A t ) = (E µ (A 1 )) -1/2 λ M a.s.
Moreover, we have also that

lim sup t→∞ M t v( t L 2 (v(t)) )L 2 (v(t)) = c (E µ (A 1 )) -1/2 λ M a.s.
Here,

λ 2 M = E µ (< M > 1 )
, and c is a constant that depends only on the process.

Remark 3.5 The events lim sup t→∞ Mt √ 2AtL 2 (At) = (E µ (A 1 )) -1/2 λ M etc of theorem 3.1
to 3.4 belong to the sigma-field J of invariant sets, and since X is Harris, by Revuz and Yor, [START_REF] Revuz | Continuous martingales and Brownian motion, 3rd edition[END_REF], chapter X, proposition 3.6 and 3.10, its probability is either zero or one. Hence the convergence holds almost surely, independently of the choice of the initial distribution.

Remark 3.6 Touati, see [START_REF] Touati | Loi fonctionnelle du logarithme itéré pour les processus de Markov récurrents[END_REF], has shown the statement of theorem 3.4 for regular models, i.e. models where v(t) ∼ t α l(t) as t → ∞, for some 0 < α ≤ 1, and where l varies slowly at infinity. However, our result holds always.

Example 3.7 We consider the following statistical problem that has been studied by Höpfner and Kutoyants, see [START_REF] Höpfner | On a problem in statistical inference in null recurrent diffusions[END_REF]. Observe the trajectory of a one-dimensional diffusion

dX t = ϑb(X t )dt + σdW t , X 0 = 0,
where σ > 0 is known, where ϑ is some unknown parameter and b continuous such that the diffusion is recurrent. The model is not necessarily regular.

In this model, the maximum likelihood estimator is given by

θt = t 0 b(X s )dX s / t 0 b 2 (X s )ds. Let I t := t 0 b 2 (X s )ds and M t := t 0 b(X s )σdW s . Hence, I t ( θt -ϑ) = M t .
Now, applying theorem 3.4, we get

lim sup t→∞ √ I t √ L 2 I t ( θt -ϑ) < ∞ a.s.,
which means that we have to consider random rates of convergence.

Preliminaries on Nummelin splitting in continuous time

We use the Nummelin splitting in continuous time, as developed in [START_REF] Löcherbach | On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions[END_REF], in order to introduce a recurrent atom for the process. We recall briefly the construction:

Introduce a sequence (σ n ) n≥1 of i.i.d. exp(1)-waiting times, independent of the process X itself. Let T 0 := 0, T n := σ 1 + . . . + σ n and Xn := X Tn . Then the chain X = ( Xn ) n is recurrent in the sense of Harris and its one-step transition kernel U 1 (x, dy) := ∞ 0 e -t P t (x, dy)dt satisfies a minorization condition:

U 1 (x, dy) ≥ α1 C (x)ν(dy), (4.6)
where 0 < α < 1, µ(C) > 0 and ν a probability measure equivalent to µ(• ∩ C) (cf [START_REF] Revuz | Markov chains[END_REF], [START_REF] Höpfner | Limit theorems for null recurrent Markov processes[END_REF], proposition 6.7). The set C can be chosen to be compact.

Then it is possible to define on an extension of the original space (Ω, A, (P x )) a Markov process Z = (Z t ) t≥0 , taking values in E × [0, 1] × E such that the T n are jump times of the process and such that under P x , ((Z 1 t ) t , (T n ) n ) has the same distribution as ((X t ) t , (T n ) n ). We give the details as introduced in [START_REF] Löcherbach | On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions[END_REF] : First of all, define the following transition kernel Q((x, u), dy) from E × [0, 1] to E :

Q((x, u), dy) =      ν(dy) if (x, u) ∈ C × [0, α] 1 1-α U 1 (x, dy) -αν(dy) if (x, u) ∈ C×]α, 1] U 1 (x, dy) if x / ∈ C . (4.7)
We now recall the construction of

Z t = (Z 1 t , Z 2 t , Z 3 t ) taking values in E × [0, 1] × E as given in [9]. Write u 1 (x, x ′ ) := ∞ 0 e t p t (x, x ′ )dt. Let Z 1 0 = X 0 = x. Choose Z 2 0 according to the uniform distribution U on [0, 1]. On {Z 2 0 = u}, choose Z 3 0 ∼ Q((x, u), dx ′ ). Then inductively in n ≥ 0, on Z Tn = (x, u, x ′ ) : 1. Choose a new jump time σ n+1 according to e -t p t (x, x ′ ) u 1 (x, x ′ ) dt on IR + ,
where we define 0/0 := a/∞ := 1, for any a ≥ 0, and put T n+1 := T n + σ n+1 .

2. On {σ n+1 = t}, put Z 2 Tn+s := u, Z 3 Tn+s := x ′ for all 0 ≤ s < t.

3. For every s < t, choose

Z 1 Tn+s ∼ p s (x, y)p t-s (y, x ′ ) p t (x, x ′ ) Λ(dy).
Choose Z 1 Tn+s := x 0 for some fixed point x 0 ∈ E on {p t (x, x ′ ) = 0}. Moreover, given Z 1 Tn+s = y, on s + u < t, choose

Z 1 Tn+s+u ∼ p u (y, y ′ )p t-s-u (y ′ , x ′ ) p t-s (y, x ′ ) Λ(dy ′ ). Again, on {p t-s (y, x ′ ) = 0}, choose Z 1 Tn+s+u = x 0 . 4. At the jump time T n+1 , choose Z 1 T n+1 := Z 3 Tn = x ′ . Choose Z 2 T n+1 independently of Z s , s < T n+1 , according to the uniform law U. Finally, on {Z 2 T n+1 = u ′ }, choose Z 3 T n+1 ∼ Q((x ′ , u ′ ), dx ′′ ).
Note that by construction, given the initial value of Z at time T n , the evolution of the process during [T n , T n+1 [ does not depend on the chosen value of Z 2 Tn . We will write P π for the measure related to X, under which X starts from the initial measure π(dx), and IP π for the measure related to Z, under which Z starts from the initial measure π(dx) ⊗ U (du) ⊗ Q((x, u), dy). In the same spirit we denote E π the expectation with respect to P π and IE π the expectation with respect to IP π . Moreover, we shall write IF for the filtration generated by Z, C G for the filtration generated by the first two coordinates Z 1 and Z 2 of the process, and IF X for the sub-filtration generated by X interpreted as first coordinate of Z.

4.1 Some auxiliary results and remarks on the construction of Z Proposition 4.1 The law of (Z 1 t ) t under IP π equals the law of (X t ) t under P π . Moreover, for any given fixed time t,

L(Z 3 t |Z 1 t )(dx ′ ) = u 1 (Z 1 t , x ′ )Λ(dx ′ ).
Proof The first assertion is proposition 2.8 c) of [START_REF] Löcherbach | On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions[END_REF]. We show the second assertion : Let f, g : E → IR + be bounded measurable positive functions . Put Φ(x) := u 1 (x, x ′ )g(x ′ )Λ(dx ′ ). We define

A n := σ{F T n-1 , T n }. Note that since Z 1 Tn = Z 3 T n-1 , L(Z Tn |A n ) = δ Z 1 Tn (dx)U (du)Q((Z T 1 n , u), dx ′ ). Then IE x [f (Z 1 t )g(Z 3 t )] = n IE x [f (Z 1 t )g(Z 3 t )1 {Tn≤t<T n+1 } ] = n IE x [IE[f (Z 1 t )g(Z 3 t )1 {t<T n+1 } |A n ]1 {Tn≤t} ] = n IE x 1 0 du Q((Z 1 Tn , u), dx ′ )g(x ′ ) ∞ 0 e -s 1 {t-Tn≤s} ds p t-Tn (Z 1 Tn , y) p s-(t-Tn) (y, x ′ ) u 1 (Z 1 Tn , x ′ ) f (y)Λ(dy) 1 {Tn≤t} = n IE x p t-Tn (Z 1 Tn , y)f (y)e -(t-Tn) ( ∞ 0 e -s p s (y, x ′ )ds)g(x ′ )Λ(dx ′ ) Λ(dy) 1 {Tn≤t} = n IE x p t-Tn (Z 1 Tn , y)f (y)Φ(y)e -(t-Tn) Λ(dy) = n IE x E Z 1 Tn (f • Φ(X t-Tn ); t -T n ≤ T 1 ) 1 {Tn≤t} = E x [f (X t )Φ(X t )],
since Z 1 Tn ∼ X Tn . This yields the assertion.

• Write A := C × [0, α] × E.

Now we put

S 0 := 0, R 0 := 0, S n+1 := inf{T m > R n : Z Tm ∈ A}, R n+1 := inf{T m : T m > S n+1 }.
Then the sequence of IF -stopping times R n generalizes the notion of life-cycle decomposition in the case of existence of a recurrent atom in the sense which is precised in [START_REF] Löcherbach | On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions[END_REF].

Proposition 4.2 a) Z Rn+• is independent of F R n-1 for all n ≥ 1. b) Z Rn ∼ ν(dx)U (du)Q((x, u), dx ′ ) for all n ≥ 1. c) The sequence of (Z Rn ) n≥1 is i.i.d.
Proof The assertion is true by construction, see proposition 2.13 of [START_REF] Löcherbach | On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions[END_REF]. • Proposition 4.3 Let A t be any integrable additive functional of X. Then, up to multiplication by a constant, for any initial measure π and any n ≥ 1,

IE π (A R n+1 -A Rn ) = IE ν (A R 1 ) = E µ (A 1 ).
Proof This is proposition 2.20 of [START_REF] Löcherbach | On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions[END_REF].

• Moreover, we have the following:

Proposition 4.4 Let f be a measurable µ-integrable function. Put R 0 := 0,

ξ n := Rn R n-1 f (X s )ds, n ≥ 1.
Then the sequence (ξ n ) n is a stationary ergodic sequence under

IP ν . Moreover, for n ≥ 2, ξ n is independent of F R n-2 .
Proof Fix some n ≥ 1 and some positive measurable bounded function Φ : IR n → IR.

Then, for any k ≥ 1, using the Markov property with respect to

F R k-1 , IE ν (Φ(ξ k , . . . , ξ n+k )) = IE ν (IE Z R k-1 (Φ(ξ 1 , . . . , ξ n+1 ))) = IE ν (Φ(ξ 1 , . . . , ξ n+1 )),
which yields the stationarity. For the ergodicity, note that the sigma field of invariant sets of (ξ n ) n is contained in the sigma-field of invariant sets J of the process, which is trivial, as indicated earlier. The last assertion is an immediate consequence the construction of Z.

• Remark 4.5 The last assertion of proposition 4.4 implies in particular that the sequence (ξ n ) n is a strongly mixing sequence with mixing coefficients α n = 0 for n ≥ 2; n ∈ IN.

In the sequel we shall also need the following technical result.

Proposition 4.6 Let f satisfy the assumptions (i) and (ii) of theorem 3.2. Let ξ n be as in proposition 4.4.

Then IE π (|ξ k | 3 ) < ∞.
Proof We interpret X as first coordinate of Z. Note that

ξ 3 k = R k R k-1 R k R k-1 R k R k-1 f (X s )f (X u )f (X v )dsdudv = 6 R k R k-1 ( R k s ( R k u f (X v )dv)f (X u )du)f (X s )ds ≤ 6 R k R k-1 ( R k s ( R k u |f |(X v )dv)|f |(X u )du)|f |(X s )ds.
Now, using Markov's property with respect to F u , we get

IE π (|ξ k | 3 ) ≤ 6IE ν R 1 0 R 1 s |f |(X u )IE Zu ( R 1 0 |f |(X t )dt)du |f |(X s )ds ≤ CIE ν R 1 0 |f |(X s )ds R 1 s |f |(X u )IE Zu (T 1 )du +CIE ν R 1 0 R 1 s |f |(X u )du |f |(X s )ds. (4.8)
In this last step, we have used that

IE Zu ( R 1 T 1 |f |(X t )dt) ≤ IE Z 3 u ( R 1 0 |f |(X t )dt) ≤ C
by proposition 2.16 of [START_REF] Löcherbach | On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions[END_REF]. Recall that IE Z 3 u designs expectation when the first component starts from x = Z 3 u , the second is chosen according to the uniform distribution on [0, 1] and the third component is chosen according to Q.

Have a look at the second expression in (4.8): Using Markov's property with respect to F s , we get that

IE ν R 1 0 R 1 s |f |(X u )du |f |(X s )ds = IE ν R 1 0 |f |(X s )IE Zs R 1 0 |f |(X u )du ds ≤ CIE ν R 1 0 |f |(X s )IE Zs (T 1 )ds + IE ν R 1 0 |f |(X s )IE Zs R 1 T 1 |f |(X u )du ds ≤ CIE ν R 1 0 |f |(X s )IE Zs (T 1 )ds + IE ν R 1 0 |f |(X s )IE Z 3 s R 1 0 |f |(X u )du ds ≤ CIE ν R 1 0 |f |(X s )IE Zs (T 1 )ds + CIE ν R 1 0 |f |(X s )ds = CIE ν R 1 0 |f |(X s )IE Zs (T 1 )ds + Cµ(|f |).
Here we have used proposition 5.16 of [START_REF] Löcherbach | On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions[END_REF]:

IE Z 3 s R 1 0 |f |(X u )du ≤ C.
Hence, writing Ψ(Z s ) := IE Zs (T 1 )|f |(X s ), we have to study

IE ν R 1 0 Ψ(Z s )ds. But IE ν R 1 0 Ψ(Z s )ds = IE ν T 2 0 Ψ(Z s )ds + n≥1 IE ν 1 {Z T k / ∈A ∀1≤k≤n} T n+2 T n+1 Ψ(Z s )ds . (4.9)
Have a look at the first term:

IE ν T 2 0 Ψ(Z s )ds = IE ν T 1 0 Ψ(Z s )ds + IE ν T 2 T 1 Ψ(Z s )ds = IE ν T 1 0 Ψ(Z s )ds + IE ν IE Z 3 0 T 1 0 Ψ(Z s )ds = IE ν T 1 0 Ψ(Z s )ds + ν(dx) u 1 (x, x ′ )Λ(dx ′ )IE x ′ T 1 0 Ψ(Z s )ds. (4.10) But IE ν T 1 0 Ψ(Z s )ds = ∞ 0 IE ν 1 {s≤T 1 } Ψ(Z s ) ds,
and since Ψ(Z s ) does not depend on Z 2 s ,

IE ν 1 {s≤T 1 } Ψ(Z s ) = = ν(dx) u 1 (x, x ′ )Λ(dx ′ ) ∞ s e -t p t (x, x ′ ) u 1 (x, x ′ ) dt p s (x, y)p t-s (y, x ′ ) p t (x, x ′ ) Ψ(y, x ′ )Λ(dy)
= ν(dx) e -s p s (x, y)Λ(dy) u 1 (y, x ′ )Ψ(y, x ′ )Λ(dx ′ ).

A simple calculus shows that

u 1 (y, x ′ )Ψ(y, x ′ )Λ(dx ′ ) = |f |(y).
Hence,

IE ν 1 {s≤T 1 } Ψ(Z s ) = IE ν 1 {s≤T 1 } |f |(X s ) .
The same argument applies to the second expression in (4.10), and we get

IE x ′ T 1 0 Ψ(Z s )ds = IE x ′ T 1 0 |f |(X s )ds.
Finally, for all the other terms in (4.9), we first use Markov's property with respect to F Tn , noticing that {Z T k / ∈ A ∀1 ≤ k ≤ n} ∈ F Tn . Hence we have to investigate the following expression

IE Z Tn T 2 T 1 Ψ(Z s )ds = IE Z 3 Tn T 1 0 Ψ(Z s )ds,
and the same argument as above shows that this equals

IE Z 3 Tn T 1 0 |f |(X s )ds.
We deduce that

IE ν R 1 0 Ψ(Z s )ds = IE ν R 1 0 |f |(X s )ds = µ(|f |).
We are now going to treat the first term in (4.8): Using once again Markov's property, we get

IE ν R 1 0 |f |(X s )ds R 1 s |f |(X u )IE Zu (T 1 )du = IE ν R 1 0 |f |(X s )ds IE Zs R 1 0 |f |(X u )IE Zu (T 1 )du ≤ CIE ν R 1 0 |f |(X s )ds IE Zs T 1 0 IE Zu (T 1 )du (4.11) +IE ν R 1 0 |f |(X s )ds IE Zs R 1 T 1 |f |(X u )IE Zu (T 1 )du
Have a look at the second term in (4.11):

IE Zs R 1 T 1 |f |(X u )IE Zu (T 1 )du = IE Z 3 s R 1 0 |f |(X u )IE Zu (T 1 )du,
and an argument similar to that in (4.9) leads to

IE Z 3 s R 1 0 |f |(X u )IE Zu (T 1 )du ≤ C.
Hence by proposition 5.16 of [START_REF] Löcherbach | On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions[END_REF], the second term in (4.11) is bounded.

Write Φ(Z s ) := IE Zs T 1 0 IE Zu (T 1 )du, hence it remains to investigate

IE ν R 1 0 |f |(X s )Φ(Z s )ds = IE ν T 2 0 |f |(X s )Φ(Z s )ds + n≥1 IE ν 1 {Z T k / ∈A ∀1≤k≤n} T n+2 T n+1 |f |(X s )Φ(Z s )ds .
In each of these terms, we take conditional expectation of Φ(Z s ) with respect to X s = Z 1 s , in the same way as in (4.9). On {X s = x}, we thus have to calculate

IE(Φ(Z s )|X s = x) = u 1 (x, x ′ )Λ(dx ′ ) ∞ 0 e -t p t (x, x ′ ) u 1 (x, x ′ ) dt t 0 du p u (x, y)p t-u (y, x ′ ) p t (x, x ′ ) Λ(dy) ∞ 0 e -s p s (y, x ′ ) u 1 (y, x ′ ) sds = Λ(dx ′ ) ∞ 0 e -t dt t 0 du p u (x, y)p t-u (y, x ′ )Λ(dy) ∞ 0 e -s p s (y, x ′ ) u 1 (y, x ′ ) sds = Λ(dx ′ ) ∞ 0 e -u du p u (x, y)Λ(dy) ∞ 0 e -t p t (y, x ′ )dt ∞ 0 e -s p s (y, x ′ ) u 1 (y, x ′ ) sds = Λ(dx ′ ) ∞ 0 e -u du p u (x, y)Λ(dy)u 1 (y, x ′ ) ∞ 0 e -s p s (y, x ′ ) u 1 (y, x ′ ) sds = ∞ 0 e -u du p u (x, y)Λ(dy) ∞ 0 e -s Λ(dx ′ )p s (y, x ′ ) sds = 1.
This concludes our proof. •

Some results related to the deterministic equivalent

Proposition 4.7 Let f be a bounded, positive special function of X, let C be a constant such that

sup x IE x R 1 0 f (X s )ds ≤ C (see proposition 2.16 of [9]). Put v * (t) := C + E ν t 0 f (X s )ds.
Then we have for any c ≥ 1, v * (ct) ≤ (c + 1)v * (t).

Proof Using Markov's property for X we have:

v * (2t) -v * (t) = E ν (E Xt t 0 f (X s )ds).
Now we identify X with the first coordinate of Z:

E x t 0 f (X s )ds = IE x t 0 f (Z 1 s )ds ≤ ≤ IE x ( R 1 0 f (Z 1 s )ds + t+R 1 R 1 f (Z 1 s )ds) ≤ C + IE x (IE Z R 1 ( t 0 f (X s )ds)) = C + IE ν t 0 f (Z 1 s )ds.
Hence, v * (nt) ≤ nv * (t) for all n. Then, for any c ≥ 1,

v * (ct) ≤ v * (([c] + 1)t) ≤ ([c] + 1)v * (t) ≤ (c + 1)v * (t).
• 5 Proof of theorem 3.1

In this section we give the proof of the law of iterated logarithm for additive functionals as stated in theorem 3.1.

Proof Let f be some special function of X. We suppose f ≥ 0, bounded, with µ(f ) > 0.

It is enough to prove that lim sup

t→∞ t 0 f (X s )ds v( t L 2 (v(t)) )L 2 (v(t))
= cµ(f ) a.s.

(5.12) 1) random variables independent of the process X and T 0 := 0, T n = σ 1 + . . . + σ n , as introduced in the section 5 for splitting construction. Define Xn := X Tn . Then the chain ( Xn ) is recurrent in the sense of Harris, with the same invariant measure µ. The special functions of the chain ( Xn ) and those of the process X are the same.

with v(t) = E ν t 0 f (X s )ds. Let σ n , n ∈ IN , be i.i.d. exp(

Now, let

a(n) := IE ν n k=1 f ( Xk ).
This is a version of deterministic equivalent of the chain ( Xn ), see Chen ([3]).

In the sequel, c denotes a positive constant that depends only on the process, not on f. Chen's law of iterated logarithm for discrete Harris chains, see [START_REF] Chen | How often does a Harris recurrent Markov chain recur?[END_REF], theorem 2.2, applied to the chain ( Xn ) gives :

lim sup n→∞ n k=0 f ( Xk ) a( n L 2 (a(n)) )L 2 (a(n))
= Lµ(f ) a.s.

(5.13)

Here the constant L depends only on the recurrence behavior of the chain ( Xn ) : Chen shows (5.13) first for a particular additive functional which is the number of visits to an atom before time n. The constant L is the limit obtained for this chain. Then, (5.13) is obtained by passing to a general additive functional, using Chacon-Ornstein theorem.

Let N t = n≥1 1 {Tn≤t} . N t is a Poisson process with intensity 1, lim t→∞ N t = ∞ a.s., and the substitution of N t in (5.13) gives:

lim sup t→∞ Nt k=0 f ( Xk ) a( Nt L 2 (a(Nt)) )L 2 (a(N t )) = Lµ(f ) a.s. ( 5.14)
Using the Chacon-Ornstein limit-quotient theorem for additive functionals or the law of large numbers for Poisson process we get:

lim t→∞ Nt k=0 f ( Xk ) t 0 f (X s )ds = 1 a.s., (5.15)
and thus lim sup

t→∞ t 0 f (X s )ds a( Nt L 2 (a(Nt)) )L 2 (a(N t )) = Lµ(f ) a.s. (5.16)
We now want to compare the normalization in (5.12) with that of (5.16).

Remark that

M t = Nt k=0 f ( Xk ) - t 0 f (X s )ds = t 0 f (X s )dN s - t 0 f (X s )ds
is a martingale. For all k > 0, T n ∧ k is a bounded stopping time. Using the stopping-rule,

IE ν N Tn∧k k=0 f ( Xk ) = IE ν Tn∧k 0 f (X s )ds.
(5.17)

By monotone convergence, we can replace in the previous equation T n ∧k by T n and obtain

a(n) = IE ν Tn 0 f (X s )ds.
But under IP ν , (T n ) n and X are independent and T n is the sum of n independent random variables that are exponentially exp(1) distributed. So if we firstly integrate with respect to X, we obtain

a(n) = E(v(T n )), (5.18)
where expectation is taken only with respect to T n which is the sum of n independent exp(1)-variables.

Now denote as previously

v * (t) = C + v(t), where C > 0 is a constant such that sup x IE x R 1 0 f (X s )ds ≤ C. We write v * (T n ) ≤ v * (n)1 {Tn≤n} + v * ( T n n n)1 {Tn≥n} . But v * ( T n n n)1 {Tn≥n} ≤ ( T n n + 1)v * (n)1 {Tn≥n} .
Hence we get

v * (T n ) ≤ v * (n) + T n n v * (n)1 {Tn≥n} ≤ v * (n)(1 + T n n ),
and taking expectations with respect to T n yields

a * (n) ≤ 2v * (n), where a * (n) = C + a(n).
In the same way,

v * (T n ) v * (n) ≥ v * ( Tn n n)1 { Tn n >1-ε} v * (n(1 -ε) 1 1-ε} ) ≥ v * ((1 -ε)n)1 { Tn n >1-ε} (1 + 1 1-ε )v * (n(1 -ε)) hence, v * (T n ) v * (n) ≥ 1 { Tn n >1-ε} (1 + 1 1-ε )
, and taking expectation yields

a * (n) ≥ v * (n) 1 2 + ε ′ p n ,
where p n = P (T n ≥ n(1 -ε)) → 1 as n → ∞. Hence we have shown:

1 2 = lim inf a * (n) v * (n) ≤ lim sup a * (n) v * (n) ≤ 2.
And thus almost surely

1 2 ≤ lim inf a * (N t ) v * (N t ) ≤ lim sup a * (N t ) v * (N t ) = 2.
In the same way, using lim t→∞ Nt t = 1, one shows that almost surely

1 2 ≤ lim inf v * (N t ) v * (t) ≤ lim sup v * (N t ) v * (t) = 2,
and finally, since

v(t)/v * (t) → 1, 1 2 ≤ lim inf a(N t ) v(t) ≤ lim sup a(N t ) v(t) = 2.
In the same way we show that with some constants

C 1 > 0 and C 2 < ∞ C 1 = lim inf a( Nt L 2 a(Nt) )L 2 a(N t ) v( t L 2 v(t) )L 2 v(t) ≤ lim sup a( Nt L 2 a(Nt) )L 2 a(N t ) v( t L 2 v(t) )L 2 v(t) = C 2 . (5.19)
Hence,

lim sup t 0 f (X s )ds v( t L 2 v(t) )L 2 v(t) ≤ lim sup t 0 f (X s )ds a( Nt L 2 a(Nt) )L 2 a(N t ) lim sup a( Nt L 2 a(Nt) )L 2 a(N t ) v( t L 2 v(t) )L 2 v(t) = LC 2 µ(f ),
and this yields the desired result.

• Corollary 5.1 As a consequence of the above proof, in particular of (5.18), it is evident that in the regular case, i.e. if

v(t) ∼ t α l(t) as t → ∞
for some 0 < α ≤ 1 and for some function l that varies slowly at infinity,

a(n) = v(n).
6 Proof of theorem 3.2 and theorem 3.4

Let f be a measurable µ-integrable function such that µ(f ) = 0 and such that |f | is a special function. Put ξ n := Rn R n-1 f (X s )ds. Then the sequence of ξ 1 , ξ 3 , ξ 5 , . . . is a sequence of i.i.d. variables having IE(ξ i ) = 0 and IE(|ξ i | 3 ) < ∞, see proposition 4.6. The same is true of the sequence ξ 2 , ξ 4 , . . . . As a consequence, (ξ n ) n is a strictly stationary sequence that is uniformly strong mixing in a very strong sense, since for any n, σ{ξ k , k ≤ n} is independent of σ{ξ n+l , l ≥ 2}. Put

Σ n = ξ 1 + . . . + ξ n , then IE ν (Σ 2 n ) = nIE ν (ξ 2 1 ) + 2(n -1)IE ν (ξ 1 ξ 2 ) ∼ nσ 2 where σ 2 = IE ν (ξ 2 1 ) + 2IE ν (ξ 1 ξ 2 )
. We suppose that σ 2 > 0. (6.20)

We apply the Hartman-Wintner law of iterated logarithm for strongly mixing variables which is a consequence of theorem 2 of Rio, [START_REF] Rio | The functional law of the iterated logarithm for stationary strongly mixing sequences[END_REF]. Note that the condition (1.5) of [START_REF] Rio | The functional law of the iterated logarithm for stationary strongly mixing sequences[END_REF] on exponential mixing rates is satisfied due to proposition 4.6. We thus get, putting

λ f := σ, (6.21) lim sup Σ n √ 2nL 2 n = lim sup Rn 0 f (X s )ds √ 2nL 2 n = λ f almost surely as n → ∞. (6.22)
We are now able to give the proof of theorem 3.2 :

Proof of theorem 3.2 Let N t := n≥1 1 {Rn≤t} .
Using (6.22), we get immediately that lim sup

R N t 0 f (X s )ds 2N t L 2 (N t ) = λ f . (6.23)
Here λ f depends only on the function f and on the transition of the process, but not on the specific choice of R n . Now, note that, using Markov's property,

IP ν sup Rn≤t≤R n+1 | t Rn f (X s )ds| ≥ ε nL 2 (n) = IP ν sup 0≤t≤R 1 | t 0 f (X s )ds| ≥ ε nL 2 (n) ≤ IP ν 1 ε 2 ( sup 0≤t≤R 1 | t 0 f (X s )ds|) 2 ≥ n .
Summing over n yields

n IP ν sup Rn≤t≤R n+1 | t Rn f (X s )ds| ≥ ε nL 2 (n) ≤ 1 ε 2 IE ν ( sup 0≤t≤R 1 | t 0 f (X s )ds|) 2 ≤ 1 ε 2 IE ν R 1 0 |f |(X s )ds 2 < ∞
by proposition 4.6. Using Borel-Cantelli's lemma, we thus conclude that lim sup

t 0 f (X s )ds 2N t L 2 (N t ) = λ f , since lim sup sup R N t ≤s≤t | s R N t f (X u )du| 2N t L 2 (N t ) = 0.
Note that the additive functional N t can be replaced by any µ-integrable additive functional A, using the ratio limit theorem. This concludes the proof.

•

Proof of remark 3.3 The proof is given in several steps and follows the ideas of the proof of lemma 2.3 of Chen [START_REF] Chen | The law of the iterated logarithm for functionals of Harris recurrent Markov chains: Self Normalization[END_REF].

1. We start by showing the following result. Let

N t := n≥1 1 {Sn≤t} . Let v(t) = IE ν (N t + 1). Put R t = inf{R n : R n > S Nt+1 }. Then we have lim t→∞ 1 v(t) IE ν ( Rt 0 f (X s )ds) 2 = λ 2 f (6.24)
and in particular also

lim n→∞ 1 n 0 v(t)dt n 0 IE ν ( Rt 0 f (X s )ds) 2 dt = λ 2 f . (6.25)
Equation ( 6.24) is shown as follows. As in the proof of proposition 3.8 of [START_REF] Löcherbach | On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions[END_REF], writing S 0 := R 0 = 0, we have

Rt 0 f (X s )ds 2 =   n≥0 1 {Sn≤t} R n+1 Rn f (X s )ds   2 = n≥0 m≥0 1 {Sn≤t} 1 {Sm≤t} ( R n+1 Rn f (X s )ds)( R m+1 Rm f (X s )ds) = n≥0 m≥0
1 {Sn≤t} 1 {Sm≤t} ξ n+1 ξ m+1 . Now, using the independence of ξ n+1 of {S n ≤ t} and the independence properties of the sequence (ξ n ) n itself, we obtain immediately, as in [START_REF] Löcherbach | On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions[END_REF],

IE ν ( Rt 0 f (X s )ds) 2 = n≥0 IE ν (1 {Sn≤t} ξ 2 n+1 ) + 2 n≥0 IE ν (1 {S n+1 ≤t} ξ n+1 ξ n+2 ). (6.26)
The first term in (6.26) equals

n≥0 IE ν (1 {Sn≤t} ξ 2 n+1 ) = IE ν (N t + 1)IE ν (ξ 2
IE ν (1 {Sn≤t} ξ 2 n+1 ) v(t) = IE ν (ξ 2 1 ).
Concerning the second term in (6.26), note that

n≥0 IE ν (1 {S n+1 ≤t} ξ n+1 ξ n+2 ) = n≥0 IE ν (1 {Sn≤t} ξ n+1 ξ n+2 ) - n≥0 IE ν (1 {Sn≤t<S n+1 } ξ n+1 ξ n+2 ) = IE ν (N t + 1)IE ν (ξ 1 ξ 2 ) - n≥0 IE ν (1 {Sn≤t<S n+1 } ξ n+1 ξ n+2 ). (6.28)
The first term converges to IE ν (ξ 1 ξ 2 ), once divided by v(t). In order to treat the second term in (6.28), note that

n≥0 IE ν (1 {Sn≤t<S n+1 } ξ n+1 ξ n+2 ) = n≥0 IE ν (1 {Sn≤t<S n+1 } ( S n+1 Rn f (X s )ds)( R n+2 R n+1 f (X s )ds)) + n≥0 IE ν (1 {Sn≤t<S n+1 } ( R n+1 S n+1 f (X s )ds)( R n+2 R n+1 f (X s )ds)). (6.29)
Recall that (G t ) t is the filtration generated by the first two coordinates of Z. In the first term of (6.29), we are going to use conditional expectation with respect to G S n+1 :

IE ν (1 {Sn≤t<S n+1 } ( S n+1 Rn f (X s )ds)( R n+2 R n+1 f (X s )ds)) = IE ν (1 {Sn≤t<S n+1 } ( S n+1 Rn f (X s )ds)IE ν [ R n+2 R n+1 f (X s )ds|G S n+1 ]).
But, since L(Z 3 S n+1 |G S n+1 ) = ν(dx ′ ), and then using Markov's property firstly with respect to F S n+1 and then with respect to

F T 1 , IE ν [ R n+2 R n+1 f (X s )ds|G S n+1 ] = ν(dx ′ )IE (Z 1 S n+1 ,Z 2 S n+1 ,x ′ ) R 1 T 1 f (X s )ds = ν(dx ′ )IE x ′ R 1 0 f (X s )ds = IE ν ξ 1 = cµ(f ) = 0,
where the last equality follows from proposition 4.3 and from µ(f ) = 0. Hence the first term in (6.29) vanishes. In order to treat the second term, we use that |f | is bounded by a constant. Hence, writing

A n := σ{F S n+1 , R n+1 }, IE ν (1 {Sn≤t<S n+1 } ( R n+1 S n+1 |f |(X s )ds)( R n+2 R n+1 |f |(X s )ds)) ≤ C IE ν 1 {Sn≤t<S n+1 } (R n+1 -S n+1 )IE ν R n+2 R n+1 |f |(X s )ds | A n . But IE ν [ R n+2 R n+1 |f |(X s )ds | A n ] does not depend on R n+1 , and since Z 1 R n+1 = Z 3 S n+1 , we get IE ν R n+2 R n+1 |f |(X s )ds | A n = IE Z 3 S n+1 R 1 0 |f |(X s )ds ≤ C,
since |f | is a special function, see proposition 2.19 of [START_REF] Löcherbach | On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions[END_REF]. Thus,

IE ν 1 {Sn≤t<S n+1 } (R n+1 -S n+1 )IE ν R n+2 R n+1 |f |(X s )ds | A n ≤ CIE ν 1 {Sn≤t<S n+1 } (R n+1 -S n+1 ) = CIE ν 1 {Sn≤t<S n+1 } IE ν [R n+1 -S n+1 |G S n+1 ] ≤ C α IE ν 1 {Sn≤t<S n+1 } , since IE ν [R n+1 -S n+1 |G S n+1 ] = ν(dx ′ ) ∞ 0 t e -t p t (Z 1 S n+1 , x ′ ) u 1 (Z 1 S n+1 , x ′ ) dt ≤ ∞ 0 t e -t dt 1 α p t (Z 1 S n+1 , x ′ )Λ(dx ′ ) = 1 α , since by construction, u 1 (Z 1 S n+1 , x ′ ) ≥ αν(x ′ ).
Here ν(dx ′ ) = ν(x ′ )Λ(dx ′ ). Thus, after summation in n, the second term in (6.29) is bounded by

C ′ n IE ν 1 {Sn≤t<S n+1 } = C ′ , since n 1 {Sn≤t<S n+1 } = 1.
Thus, the second term in (6.28), once divided by v(t), converges to 0, as t → ∞. Finally, (6.26), (6.27) and (6.28) give the desired (6.24).

Let

h(x) := f (x) ∞ 0 P t f (x)dt.
Note that Unfortunately, the term on the right hand side of (6.32) is not an additive functional of the process, since the function h t-s depends also on s. That's why we have to integrate once more. Integrating with respect to t over the interval [0, n] and using Fubini gives , where θn is the shift operator of the discrete time chain Xn = X Tn . Let a(n) be the deterministic equivalent of the chain X, then by theorem 6.1 of Chen, [START_REF] Chen | On the limit laws of the second order for additive functionals of Harris recurrent Markov chains[END_REF], On the other hand, by the first assertion of theorem 3.

E ν ( t 0 f (X s )ds) 2 = 2E ν t 0 t s f (X s )f (X r )drds = 2E ν t 0 t-s 0 f (X s )P r f (X s )drds = 2E ν t 0 h(X s )ds -2E ν t 0 f (X s ) ∞ t-s P r f (X s )drds. ( 6
lim

P

  r f (X s )drds|dt ≤ E ν n 0 ds( n-s 0 |h t |(X s )dt)ds t |(x)dt,then as in the proof of proposition 3.8 of[START_REF] Löcherbach | On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions[END_REF], the right hand side of (6.33) can be bounded byt |(X s )dt)ds ≤ IE ν (N n + 1) • H n (x)µ(dx) = IE ν (N n + 1) n 0 µ(|h t |)dt.

(

L 2

 2 sup M Tn a( n L 2 (a(n)) )L 2 (a(n)) = λ M almost surely. Write for short b(n) := a( n L 2 (a(n)) )L 2 (a(n)). Hence we have lim sup M Nt b( Nt ) = λ M almost surely,where Nt := n 1 {Tn≤t} . The same argument as in the proof of theorem 3.1 shows that for c(t) := v( t L 2 (v(t)) )L 2 (v(t)), 0 < lim inf b(v( Nt)) )L 2 (v( Nt )) = cλ Malmost surely, where the constant c depends only on the process. Hence we get the lower bound lim sup M t v( t L 2 (v(t)) )L 2 (v(t)) ≥ cλ M .

  Since t → v(t) is non-decreasing, we have n 0 v(t)dt ≥ v(n/2) • n/2. Recall the definition of v * (t) of proposition 4.7. Then by proposition 4.7 and by the Chacon-Ornstein ratio limit theorem. Moreover, µ(|h t |) → 0 as t → ∞ by our assumption. Hence, This is seen as follows. Write ∆ t := Rt t f (X s )ds. Then IE ν ( Similar arguments to that of proof of proposition 4.6 give IE ν ∆ 2 t ≤ C for some constant independent of t, and thus

												6.34)
	As a consequence,
			E ν		n 0 ds( n 0 |h t |(X s )dt)ds n 0 v(t)dt	≤	IE ν (N n + 1) n 0 µ(|h t |)dt n 0 v(t)dt	.	(6.35)
	lim sup	IE ν (N n + 1) v(n/2)	= lim sup	v(n) v(n/2)	= lim sup	v(n) v * (n)	•	v * (n) v * (n/2)	•	v * (n/2) v(n/2)	< ∞
									1 n	0	n	µ(|h t |)dt → 0.
	As a consequence of (6.35) and (6.30), (6.31), we obtain finally
						lim n→∞	n 0 E ν ( t 0 f (X s )ds) 2 dt n 0 v(t)dt	= 2µ(h).	(6.36)
	3. Finally,
	lim n→∞	1 0 v(t)dt n	0	n	IE ν (						0	t	f (X s )ds) 2 dt = 0.	(6.37)
							0	Rt	f (X s )ds) 2 -IE ν (	0	t	f (X s )ds) 2
						= IE ν ∆ 2 t -2IE ν (	0	t	f (X s ds)∆ t .
								n→∞ lim	n 0 IE ν ∆ 2 t dt
												1/2	0	n	t dt IE ν ∆ 2	1/2

Rt 0 f (X s )ds) 2 -IE ν ( n 0 v(t)dt = 0, since n/( n 0 v(t)dt) → 0. Moreover, n 0 IE ν ( t 0 f (X s ds)∆ t dt ≤ n 0 IE ν ( t 0 f (X s ds)) 2 dt

( n 0 v(t)dt) 1/2 = 0, which gives (6.37). (6.25), (6.36) and (6.37) finally yield the desired result

Since M has continuous paths, it is well-known that M Rn is measurable with respect to F Rn-. Hence we have

Since M is an additive functional, we have that

(ξ 1 ). Now, as in (4.9), we write

in order to conclude that for all x,

Hence the sequence (ξ n ) n is a σ{ξ 1 , . . . , ξ n }-martingale difference sequence, which is stationary and ergodic due to proposition 4.4. The same argument as above applies and allows us to get that

Then the Hartmann-Wintner law of iterated logarithm, see [START_REF] Stout | The Hartman-Wintner law of the iterated logarithm for martingales[END_REF], yields lim sup

almost surely. Now, the proof of the first part of theorem 3.4 is exactly as the proof of theorem 3.2.

In order to prove the second assertion of theorem 3.4, note moreover that Mn := M Tn is a F Tn -martingale such that Mn+k -Mn = Mk • θ Tn = Mk • θn ,


Sylvain Kubler
email: s.kubler@univ-lorraine.fr

William Derigent

André Thomas

Éric Rondeau

Embedding data on "communicating materials" from context-sensitive information analysis

Keywords: Intelligent product, Product life cycle management, Data dissemination, Internet of Things, Radio-frequency identification

Nowadays, intelligent products carrying information or having decision-making abilities are becoming widespread. The idea of using intelligent products to ensure an information continuum all along the product life cycle (PLC) is more and more shared today. However, it is not that easy to identify the information that must be linked to the product. As a result, this paper proposes an information dissemination process for selecting information sensitive to the context of use of the product. This information is then stored on the products themselves using a new type of augmented material, referred to as "communicating material".

Introduction

New challenges and opportunities arise with concepts such as the Internet of Things (IoT) and ubiquitous/pervasive computing [START_REF] Weiser | Hot topics-ubiquitous computing[END_REF]. Through these concepts, real-world objects are linked with the virtual world, thereby enabling connectivity anywhere, anytime, for anything and anyone. The IoT based on RFID (Radio Frequency IDentification) usage is also a substantial topic, dealing with access to information disseminated via any kind of physical object and the development of new smart services and applications [START_REF] Yan | The Internet of Things: From RFID to the Next-Generation Pervasive Networked Systems[END_REF].

It is not uncommon today to use intelligent products for ensuring an information continuum all along the PLC. Indeed, a product moves through numerous companies and technical, semantic and organizational interoperability is not always ensured, thereby contributing to information loss. Considering the product as an information vector (to which information can be linked) should contribute to improved the transfer of information among actors of the PLC. Over the last decade, those involved with intelligent-manufacturing systems have demonstrated that systems that integrate intelligent products can be more efficient, flexible and adaptable [START_REF] Brewer | Intelligent tracking in manufacturing[END_REF][START_REF] Mehrabi | Reconfigurable manufacturing systems: key to future manufacturing[END_REF][START_REF] Wong | The Intelligent Product Driven Supply Chain[END_REF][START_REF] Babiceanu | Development and applications of holonic manufacturing systems: a survey[END_REF]. Främling et al. [START_REF] Främling | Agent-based model for managing composite product information[END_REF] argue that it is a formidable challenge to link product-related information to the products themselves (i.e., making the information easily accessible). However, most of the time, products only provide a network pointer to a linked database (via a RFID tag) and a decision making software agent [START_REF] Meyer | Intelligent products: A survey[END_REF]. It should be noted that these products have limitations in some respects: risk of tag damage, small memory capacity, etc.

After many years of considering a communicating product as a physical product associated with an informational product (realized via Auto-ID technologies such as RFID), a new paradigm has been proposed by Thomas [START_REF] Thomas | RFID et nouvelles technologies de communication; enjeux économiques incontournables et problèmes d'éthique[END_REF] which changes drastically the way to consider the material. This concept aims to give the ability for the material to be intrinsically and wholly communicating, as discussed in [START_REF] Kubler | Problem definition methodology for the "Communicating Material" paradigm[END_REF][START_REF] Kubler | Prototyping of a communicating textile[END_REF][START_REF] Kubler | Storage of information on manufactured products using "communicating materials[END_REF]. The futuristic idea behind this concept is to imagine that the material is by its very nature, communicating; what ever the technical solution is. Although current technologies do not allow building an "intrinsically communicating material", the designing of new solutions providing new services to users may be addressed. For instance, a product made of "communicating material" could have special abilities like data storage, copy/redundancy/backup information solutions. Accordingly, some open questions remain to be addressed, these include for example a specification for the information to be gathered, stored and distributed over the PLC and the linkage of new hardware and software systems with current systems [START_REF] Mcfarlane | Intelligent products in the supply chain -10 years on[END_REF][START_REF] Zbib | Heterarchical production control in manufacturing systems using the potential fields concept[END_REF]. To address some of these matters, this paper develops an information dissemination process which consists of two main steps :

1. selecting relevant information from the external database that should be stored/replicated on the product, at a given moment of the PLC, 2. storing information on the product and, subsequently, retrieving it [START_REF] Kubler | Prototyping of a communicating textile[END_REF][START_REF] Kubler | Storage of information on manufactured products using "communicating materials[END_REF]. This paper mainly focuses on the first process step which is the subject of section 2. A brief summary of previous works carried out on the second process step is nevertheless provided in section 3. Finally, the data dissemination process is applied in a case study in section 4.

Process step 1: relevant data identification

During its life cycle, a product is accessed by many actors and undergoes numerous operations. It is therefore essential to identify context-sensitive information. Accordingly, an identification method that uses the logical data model (LDM) is developed.

Fig. 1 gives insight into part of such an LDM. A given LDM entity corresponds, once implemented, to the relational table shown as MaterialDefini-tion in Fig. 1, where the attributes listed for each entity correspond to the table columns and each table row is referred to as a tuple/instance of the relation. The identification method has two stages: A. Identification of all product-related information throughout all tables: it requires searching all tuples related somehow to the network pointer carried by the communicating product (i.e., the identifier, denoted d p in our study). The identification is achieved via an algorithm detailed in section 2.1. Fig. 1 gives an example in which tuple 3 of MaterialDefinition (hatched background) is a product-related tuple that has been retrieved via the algorithm. B. Assessment of the information relevance 1 : it indicates if data must be stored on the product according to user concerns, environmental details, etc. To achieve this, the quantitative model developed by Chan and Roddick [START_REF] Chan | Context-sensitive mobile database summarisation[END_REF] is implemented and adapted. Their approach is interesting in the sense that they try to match the context with data, to identify contextsensitive information. All tables are split in data items 2 and a relevance value is computed for each of them.

Fig. 1 gives an example of three data items (belonging to the productrelated tuple 3 retrieved in Stage A), for which the relevance values have been calculated and displayed. The higher the relevance value, the higher is the probability that this data item will be stored on the product. The approach developed by Chan and Roddick is detailed in section 2.2.

★

Primary Key (PK):

✩

Foreign Key (FK):

x Relation number:

MaterialLot ID MatLot ★ Description Status Quantity ID MatDef ManufacturingBill ID ManBill ★ Description Quantity ID MatDef ✩ ID ProdDef ✩ ProductDefinition ID ProdDef ★ Description PublishedDate MaterialDefinition ID MatDef ★ Description Value ➠ 2 1 3 1 (★) 2

3600... 4

Data item noted T MD{3,1} whose relevance is equal to 0.2 Fig. 1 View of an LDM and a relational table

Protocol for identifying product-related information

In this section, the goal is to identify all tuples from all tables that are somehow related to the product instance d p . Concretely, it is necessary to define the first query to retrieve d p and then, to use the results of that query to build the next 1 Relevance is only computed for the set of product-related tuples retrieved in stage A. 2 One data item corresponds to a table cell.

queries dynamically to explore neighbouring tables, which themselves give rise to new queries and so on (until the entire LDM is explored).

One possibility consists in using database-automation programming languages such as Java DataBase Connectivity. Based on this approach, the set of tables T are extracted in a matrix format at appropriate times (i.e., when users want to store information on the product) and then, are explored via the RetrievalData algorithm given in Algorithm 1. This algorithm relies on two subfunctions, namely ExplorePK and ExploreFK. All variables used in these algorithms are listed in Table 1. RetrievalData retrieves all tuples which are somehow related to d p by moving between the tables included in T . To achieve this, it is necessary to identify the relations between tables based on the primary keys (PK) and foreign keys (FK) 3 . This functionality is provided by the subfunction ExplorePK (see Algorithm 2) which identifies the FKs in the table currently being explored, and moves to the corresponding tables (i.e., where the FKs are PKs) to identify new product-related tuples. Note that ExplorePK may not be sufficient to explore all tables because some relations may not be reached. Indeed, when a table has no further FK, it is necessary to continue in reverse. This means searching relations where the PKs from tables already explored are FKs in tables yet to be explored. This is achieved by a second subfunction ExploreFK, given in Algorithm 3. The algorithm steps are detailed in the case study in section 4, which should make it easier to understand.

output: I ′ , R 1 begin 2 I ′ ←-∅ forall the j 1 ∈ {1, nt} do 3 Rt{i, j 1 } ←-true if Kt{j 1 } = 2 then 4 forall the t ′ ∈ T , j 2 ∈ {1, n t ′ }|value(Tt{j 1 })=value(T t ′ {j 2 }) & K t ′ {j 2 } = 1 do 5 forall the i 2 ∈ {1, m t ′ }|cont(Tt{i, j 1 })=cont(T t ′ {i 2 , j 2 }) do 6 I ′ ←-I ′ ∪ Tt{i 2 , j 2 } Algorithm 3: ExploreFK(E, T , R) output: I 1 begin 2 I ←-∅ N ←-∅ forall the Tt ∈ E, T t ′ ∈ T |D Tt,T t ′ = 1 & T t ′ ∈ E do 3 N ←-N ∪ T t ′ 4 forall the Tt ∈ N , j 1 ∈ {1, nt}|Kt{j 1 } = 2 do 5 forall the T t ′ ∈ E, j 2 ∈ {1, n t ′ }|value(Tt{j 1 })=value(T t ′ {j 2 }), K t ′ {j 2 } = 1 do 6 forall the i 1 ∈ {1, n t ′ }|R t ′ {i 1 , j 2 } = true do 7 forall the i 2 ∈ {1, mt}|cont(Tt{i 2 , j 1 }) = cont(T t ′ {i 1 , j 2 }) do 8 I ←-I ∪ Tt{i 2 , j 1 }

Data selection method

In literature, the vast majority of approaches do not take into account the context of use of the database (or the product) to disseminate data as argued by Chan and Roddick [START_REF] Chan | Context-sensitive mobile database summarisation[END_REF]. Indeed, their attention is mainly focused on query frequencies to partition and allocate data in database systems [START_REF] Abuelyaman | An optimized scheme for vertical partitioning of a distributed database[END_REF]. Accordingly, Chan and Roddick developed an approach which uses the notion of priorities to select context-sensitive information, with these priorities being computed at the level of the data item [START_REF] Chan | Context-sensitive mobile database summarisation[END_REF]. The priorities are numerical values either supplied or generated via observation and experimentation. They are assigned using a multifaceted evaluation of eight criteria. In our approach, only the most appropriate criteria with regard to our context (i.e., PLC) are implemented, namely three of the eight criteria. For each criterion, the calculation of a relative priority ρ x (where x represents a criterion) and an assigned priority φ x (l) is performed (where l represents a data item). The first (relative) priority indicates the importance of criterion x with respect to the other criteria. The second (assigned) priority corresponds to the priority (between 0 and 1) computed for a data item l with respect to criterion x. Both priorities are combined in a unique formula (equation 1) that provides the relevance value P l for the data item l, where len l is its length (e.g., in bytes). Fig. 1 gives the relevance value P l for each product related data item, as exemplified with T MD {3, 1} which has a relevance of 0.43. Data items can therefore be classified in order of relevance (i.e., according to P l ).

P l = x ρx.φx(l) ln (len l + 1) (1)
The three criteria used in our approach are as follows:

1. Enumeration (C e ): users enumerate data that they consider useful (i.e., data that they recommend be attached to the product), 2. Contextual (C c ): this can be used to include data guided by standards or expert recommendations. Indeed, knowledge of the context of use can be useful in inferring the data that may be needed by users. Let us note that a multitude of information systems exist over the PLC (e.g. ERP, PDM, MES) which are not concerned by the same data (i.e. the same LDM entities). The idea is to identify specific "entity groups" through the LDM according to, for instance, the information systems and, therefore, to evaluate their importance over the PLC. Evaluations are performed by experts who focus on the entire PLC, 3. Model-based (C m ): this is based on the relationships implied by the LDM.

This criterion favors data close to the product table 4 . Indeed, according to Chan and Roddick, the shorter the distance between tables, the higher will be the data correlation.

The following subsections detail when and how adjustments and computations of priorities ρ x and φ x (l) are carried out throughout the PLC.

Adjustment of the relative priority ρ x

Regarding the priority ρ x , it is necessary to specify the importance of each criterion at a given stage of the PLC. In our study, the decision maker performs pairwise comparisons between criteria as in equation 2, with q the number of criteria. The importance of criterion i over criterion j is noted s ij . This evaluation is based on the 1 to 9-point scale designed by Saaty [START_REF] Saaty | The Analytic Hierarchy Process[END_REF]: {1, 3, 5, 7, 9}. s ij = 1 means that critetria i and j are equal in importance and s ij = 9 means that criterion i is strongly favored over criterion j. The relative importance of a criterion x is obtained by computing the eigenvalues of the matrix (eigenvalue noted λ(C x )) as in equation 3 and are synthesized by the vector Λ ρ in equation 4.

Dρ =      1 . . . q 1 s 11 . . . s 1q . . . . . . . . . . . . q s q1 . . . sqq      s ji = 1 i = j s -1 ij i = j (2) λ(Cx) = q k=1 s xk q k=1 q l=1 s kl ∀ x = [1, 2, .., q] (3)
Λρ = λ(C 1 ) • • • λ(Cq) = ρ 1 • • • ρq (4)
2.2.2 Adjustment of the assigned priority φ x (l)

This section details the priority computations φ x (l) with regard to each criterion: C e , C c , C m . i. C e : as mentioned above, this criterion gives a certain freedom to users to select information they deem relevant to store on the product. Concretely, users select the class attributes they consider useful. Let T t (v) be an attribute of table t. The score of this attribute, noted s(T t (v)) is equal to 1 if the user enumerates T t (v), 0 otherwise, as detailed in equation 5.

If a data item l belongs to the attribute T t (v), its score with respect to C e , noted φ e (l) is therefore equal to s(T t (v)).

s(Tt(v)) = 1 enumerated 0 not enumerated with v an attribute from table t

ii. C c : this criterion aims at moderating and balancing C e . Indeed, users in C e enumerates attributes they deem useful but they may have limited knowledge regarding the entire PLC. As introduced previously, the idea is to identify specific "entity groups" through the LDM according to, for instance, the information systems and, therefore, to evaluate their importance over the PLC. In our study, decision makers perform pairwise comparisons between entity groups as in equation 6, with w the number of entity groups. The importance of group i over group j is noted s ij (based on the Saaty's scale). The relative importance of an entity group i is obtained by computing its eigenvalue, noted λ(G i ), as in equation 7. All relative entity group importances are synthesized by the vector Λ g in equation 8.

If a data item l is contained in a table included in G i , its priority with respect to C c , noted φ c (l), is therefore equal to λ(G i ).

Dg =      G 1 • • • Gw G 1 s 11 • • • s 1z . . . . . . . . . . . . Gw s w1 • • • sww      s ij = 1 i = j s -1 ji i = j (6) λ(G i ) = w k=1 s ik w k=1 w l=1 s kl ∀ i = [1, 2, .., w] (7)
Λg = λ(G 1 ) • • • λ(Gw) (8)
iii. C m : the model-based criterion favors data close to the product table. This criterion is based on the relationships implied by the LDM. First, it is necessary to compute all distances between the product table A and any other table B. The distance corresponds to the shortest path reaching B from A (i.e. the minimal number of relations that separate them). For example, let MaterialLot be the product table A and ManufacturingBill be the table B in Fig. 1. The distance between both tables is 2 (relations ①-②). The product table would be the focus of our interest, with φ m decreasing as the modeled distance increases. Chan and Roddick propose equation 9, with k ∈ [1; ∞] a constant and a ∈ N the distance. The coefficient k must be adjusted. Fig. 2 highlights the fact that, for small values of k, the more remote information will be favored, and conversely (cf. k = 1.01 and 1.08). It is therefore necessary to study the entire LDM to fix values for k. Indeed, maximum distances within the LDM of 10 or 200 will certainly lead to different values for k. It can be noted that it is not so simple for an expert to choose the most suitable value of k. To avoid such a problem, self-learning systems could be further implemented in order to learn about the used data model, the application features and thus, to automatically adjust k. Process step 2 deals with the storage of data items (identified in process step 1) on/from the product. Appropriate materials and architectures have been developed in our previous works. In [START_REF] Kubler | Prototyping of a communicating textile[END_REF], a communicating textile is designed in which a huge quantity of RFID µtags is spread (≈ 1500 tags/m 2 ). Fig. 3 shows the communicating textile being processed. In [START_REF] Kubler | Storage of information on manufactured products using "communicating materials[END_REF], a communication protocol for splitting data items among several RFID tags is developed. Fig. 4 resumes the architectures/tools required to store data items on the product/textile (Fig. 4(a)) and then, to retrieve it (Fig. 4(b)):

1. Storage of information : information is written "on the fly" on the material as depicted in Fig. 4(a). The architecture consists of :

a RFID reader, -a software, named "Java/Matlab" (developed in Java R and Matlab R programming languages), for extracting information from the database and for computing the data item relevances. This software implements the approach proposed in this paper, -a software, named "Java Split" (developed in Java programming language), for splitting data items over the material. This software uses a specific protocol header developed in [START_REF] Kubler | Storage of information on manufactured products using "communicating materials[END_REF], 2. Retrieval of information : information carried by the material is read "on the fly" as depicted in Fig. 4. The architecture consists of :

a RFID reader, -the "Java Split" application to reconstruct the set of data items/tables, -optionally, an access to the database when information retrieved from the product must be updated in the database or when additional information is required (e.g. to answer a query).

RFID tag

Fig. 3 Design of a communicating textile prototype: [START_REF] Kubler | Prototyping of a communicating textile[END_REF] The platform detailed in this section is generic and can be used with any kind of communicating material (wood, cement, . . . ). The only condition is to use a communicating material that integrates RFID tags R/W (Readable & Writable), whatever the memory space available in tags.

Case study

In our case study, only the part of PLC related to communicating textile reels is considered and is considerably simplified, as shown in the case study is to focus on moments of the PLC where users want to store or retrieve information on/from the textile. Let us assume that communicating textiles have already been designed in phase 1 of the PLC (see Fig. 5). In our application, the communicating textile prototype designed in [START_REF] Kubler | Prototyping of a communicating textile[END_REF] is used. Let us consider that a supply chain member wants to store information on the textile at the Writing point A in phase 1 of the PLC. Accordingly, the architecture for writing information on this one is implemented as illustrated in Fig. 5. Then, a second supply chain member in phase 2 of the PLC, who works in cutting the textile in several pieces, wants to retrieve information carried by the textile reels (before cutting it). Accordingly, the architecture for reading information on the textile is implemented. In this study, the database implements a part of B2MML (Business To Manufacturing Markup Language) standard 5 [7] (19 entities exactly).

Section 4.1 details, on the one hand, the specifications/adjustments made by users concerned by the Writing point A and, on the other hand, the stages of computation needed to obtain the data item relevance values and ranking. Finally, section 4.2 briefly focuses on the retrieval of information from the textile in the cutting operation.

Writing point A: phase 1 of the PLC

This section details the specifications made by the different experts with regard to the different criteria. These multiple sources of expertise will be described in section 4.1.1 in the following order: i. Enumeration expertise: users enumerate information they deem important to store on the product,

Writting Point A Cutting Reading point ...

Phase 1 Phase 2

Java Matlab Java Split Java Split Fig. 5 Sequence of manufacturing activities related to a communicating textile application

ii. Contextual expertise: experts define the entity groups and adjust their importance over the PLC, iii. Model-based expertise: experts adjust the coefficient k, iv. Criterion expertise: experts define the criteria importance.

Then, section 4.1.2 details respectively the execution of process step 1 and 2 when the product arrives to the writing point A.

Expert specifications

i. The users concerned by the writing point A enumerate the information (i.e., the table attributes) listed in Table 2. These relate to the Product data group (G 1 ) and the Production data group (G 3 ). ii. The group of experts has convened a meeting before the textile reel begins its processing to adjust the contextual weights. They define four entity groups using the LDM, which are detailed in Table 3. The Equipment and Personal information groups report on equipment and people that are somehow related to the product (e.g., equipment used in its manufacture). The Material and Production data groups report on bills of materials(e.g., raw materials and component parts) and operations (e.g., production rules and production scheduling), respectively. Fig. 1 shows four of the 19 entities that are included in G 1 and G 3 . The experts then carry out pairwise comparisons for the four groups with respect to each phase of the textile-transformation process. They define the pairwise comparison given in equation 10 regarding the first phase of the PLC (see Fig. 5). Note that the experts strongly favor G 1 over G 2 (s 12 = 9) and slightly favor G 1 over G 3 (s 13 = 3). The normalized eigenvalue with regard to G 1 , noted λ(G 1 ), is computed in equation 11. Eigenvalues related to the four entity groups are finally synthesized by Λ g in equation 12 and it can be noted that information related to G 1 is much more important than information from the other groups:

λ(G 2 ) < λ(G 4 ) < λ(G 3 ) < λ(G 1 ). Dg =        G 1 G 2 G 3 G 4 G 1 1 9 3 7 G 2 1 9 1 1 7 1 5 G 3 1 3 7 1 7 G 4 1 7 5 1 7 1        (10)
λ(G 1 ) = 1 + 9 + 3 + 7 1 + 9 + 3 + 7 + 1 9 + 1 + . . . iii. The experts then proceed to the adjustment of coefficient k. Because the model is not very large (in our case study, the maximum distance through the LDM is 15), the experts agree on a value of 1.08 for k (see Fig. 2). iv. Finally, experts carry out pairwise comparisons for all criteria as in equation 13. The normalized eigenvector Λ ρ is then computed and is provided in 14. Note that the experts highly favor C e over both C c and C m (ρ m < ρ c < ρ e ). This means that the experts prefer to store information they deem relevant instead of information contextually important.

+ 1 7 + 1 = 0.55 (11) Λc = λ(G 1 ) λ(G 2 ) λ(G 3 ) λ(G 4 ) 0.
Dρ =     Ce Cc Cm Ce 1 5 7 Cc 1 5 1 5 Cm 1 7 1 5 1     (13)
Λρ = ρe ρc ρm 0.72 0.22 0.06 [START_REF] Meyer | Intelligent products: A survey[END_REF] All specifications needed for the writing point A are now defined and the process steps 1 and 2 are respectively launched when the communicating textile arrives at this point.

Execution of process step 1 and 2

Assume now that a textile arrives at the writing point A (cf. Fig. 5).

First, it is necessary to start the protocol for identifying product-related information from all tables. To achieve this, the set of tables T that compose the LDM are explored using the function RetrievalData (see Algorithm 1). Now, the algorithm steps will be outlined by focusing on three of the relational tables ∈ T , namely {MaterialLot, MaterialDefinition and ManufacturingBill} illustrated in Fig. 6. In the following explanation, these three tables are denoted {ML, MD, MB}, respectively. To run the algorithm, the RFID reader obtains the textile identifier d p , which refers to the data item T ML {2, 1} = LPB61 (i.e., the product-lot instance), as shown in Fig. 6 (cf. "☞"). An exploration of the LDM based on ExplorePK then begins with d p = LPB61. First, the tuple containing d p is retrieved (i.e. row 2 of ML which is a "product-related tuple"). Second, the function discovers if there are FKs with respect to this tuple. This is true for MD06 (i.e., T ML {2, 5}) which is a PK in MD as highlighted in Fig. 6 (blue/solid arrow). The function can therefore continue and row 2 of MD is identified as a product-related tuple. The function then checks if there are FKs in this tuple. This is not true and, consequently, the algorithm cannot explore the LDM further. To sidestep such a break, then exploration based onExploreFK, which searches for other relations where PKs from tables already explored are FKs in tables yet to be explored. In our LDM, MB is a neighbour of MD (see Fig. 1) and has yet to be explored. As a result, ExploreFK checks if there are PKs from MD which are FKs in MB. This is true for MD06 as highlighted in Fig. 6 (red/dashed arrow). Therefore, new productrelated tuples are identified (tuples 1 and 3 of MB). These new tuples serve as inputs for exploring the rest of the LDM by reusing ExplorePK. Second, it is necessary to compute the relevance of the retrieved productrelated information. Fig. 7 details the example of MaterialDefinition. All data items that have no relation with d p are set to 0. Regarding the Enumeration criterion, attributes IDMatDef and Description are set to 1 because they are enumerated in Table 2. Regarding the Contextual criterion, all attributes are set to 0.55 because MaterialDefinition is included in G 1 , which has an importance of 0.55 (cf. equation 12). For the last criterion, Model-based, the weight is set to k -1 because the distance between MaterialDefinition and MaterialLot6 is 1 (see Fig. 1). The relevance computation is then performed for all data items as exemplified with T MD{3,1} with k = 1.08, ρ e , ρ c , ρ m respectively equal to 0.72, 0.22, 0.06 (see equation 14) and len(MD06001) = 7 (one ASCII character uses 1 byte). Finally, the relevance P TMD{3,1} becomes 0.43. In this part, the results about relevances of all data items are discussed. The term "list" is used to refer to all data items having a prioritization P l > 0. Fig. 8(a) gives the ordered list from the highest P l to the lowest. The most relevant data item (with P l = 0.7066) is the one located in the ProductSegment table, for the attribute Duration, related to the tuple whose PK is PSTB0B. Because of the large number of data items included in the list (524 exactly), results are presented in the form of diagrams.

IDMatDef Description Value 0 0 0 0 0 0 1 1 0 0 0 0 IDMatDef Description Value 0 0 0 0 0 0 k -1 k -1 k -1 0 0 0 IDMatDef Description
First, consider the whisker diagram in Fig. 8(b). For each table T t ∈ T , the figure shows the minimum, the 1 st and 3 rd quartile, the mean and the maximum P l value(s) related to the set of data items included in the list and belonging to T t . Note that the highest relevant data items come from ProductSegment, but also highly relevant are MaterialDefinition, ProductionOrder and MaterialLot. This is because some attributes of these tables are enumerated (see Table 2) and the experts place high importance on the opinion of the users (ρ m < ρ c < ρ e ). Moreover, note that these four tables are included in the entity groups G 1 and G 3 and that the experts recommended highly that information from both these groups should be selected

(λ(G 2 ) < λ(G 4 ) < λ(G 3 ) < λ(G 1 )).
Consider now the pie chart in Fig. 8(c). The percentages are relative to a given data group G i , indicating the proportion of the data items that are in the first part of the list (i.e., that should be stored on the communicating textile) and that belong to G i . At the writing point A, no more than 159 data items can be stored on the communicating textile because there is insufficient memory space, as highlighted in Fig. 8(a). Fig. 8(c) indicates that 48% of the 159 data items (i.e., ≈ 72) are data items from the production data group and should be stored on the communicating textile. The pie chart clearly accords with the previous explanation, because most of the data items stored on the textile come from G 1 and G 3 , accounting for 37% and 48%, respectively. This is largely due to the choices made for the enumeration and contextual criteria.

In contrast, the personal data group (G 2 ) is not relevant with 2% of data items embedded on the product, because there is no enumeration and no high recommendation from the contextual criterion. At this stage, data items to be stored on the communicating product are selected starting from the top of the list until the storage limit of the communicating material is reached. However, further work should be undertaken to optimize this list and to ensure that the maximum number of queries are answerable.

The process step 2 is then performed and the 159 data items are stored/ spread over the communicating textile reel thanks to the software "Java Split" (see Fig. 4(a)). For information purposes, the software spent ≈ 40s to perform this operation, but let us remind ourselves that the primarily goal of the research presented in this paper is devoted to investigate the new concept of "communicating material" and to propose the first tools to communicate with it. Further work will lead to more effective tools.

Reading phase: cutting operation in Phase 2 of the PLC

The textile reel arrives at the cutting operation in Stage 2 (see Fig. 5), where the machine/user wishes to retrieve information carried by that one. As described above, the communication architecture defined in Fig. 4(a) is implemented on the production line. The textile is then read and the dataretrieving operation is achieved. The software "Java Split" is once again used, but this time, for rebuilding and displaying the 159 data items on a mobile device as shown in Fig. 9 with the table MaterialDefinition (the software spent ≈ 30s this time to perform the operation). It can be seen that the three data items composing MaterialDefinition were included in the first 159 th ranks of the list. However, in some cases, it is impossible to rebuild the entire tuple because of some data items are located in the second part of the list (i.e., > 159 th ) and are not stored on the communicating textile. It is not uncommon today to use intelligent/communicating products to create an information continuum over the product life cycle (PLC) (e.g., for traceability purposes). Indeed, considering the product as an information vector (within which information can be stored) would contribute to improve interoperability throughout its PLC. However, it is not that easy to identify, at any given stage in the PLC, the information that should be stored on the product. To address this issue, a data-dissemination process is developed in this paper for selecting context-sensitive information from a database and storing it on the product. A case study gives insights into appropriate storage/retrieval of data on/from a new kind of material described as a "communicating material". The results of this scenario shows that the selected data (i.e. data that is stored on the product) largely meets the expectations formulated by users at a given stage of the PLC. Further work should investigate new approaches to deal with particularities of the context of use of the product. For instance, ontologies could be used as a complement or a substitute of our approach [START_REF] Chandra | Knowledge management for consumer-focused product design[END_REF][START_REF] Goncalves | Knowledge framework for intelligent manufacturing systems[END_REF] and should be examined.

As regards the "communicating material" concept, several advantages may be emphasized compared to products currently being used (e.g. a product with a unique RFID tag). For example, it is both possible to perform data redundancy on the product and to ensure data sustainability over the PLC. Indeed, a same data can be copied to several parts over the material that is useful in cases where it is relevant that little or no product-related data is lost. Further, new substances could be considered to design "communicating materials" like wood, cement or still paint.

Algorithm 1 : 2 I 3 E

 123 RetrievalData(T , d p ) ←dp; // Initialization of I with the data item referenced by the product ←-∅; // The list of tables already explored is set to empty 4 while I = ∅ do //

Fig. 2

 2 Fig. 2 Adjustment of the coefficient k according to the LDM structure

Fig. 5 .Fig. 4

 54 Fig. 4 Architectures and tools of the data dissemination process

Fig. 6

 6 Fig.6Identification of "product-related tuples" thanks to RetrievalData

PFig. 7

 7 Fig.7Relevance computation of "product-related data items"

Fig. 9

 9 Fig. 9 JAVA software to display the product-related information

Table 1

 1 Variable definition used in Algorithms 1, 2 et 3

	Variables	Description
	T = [T1..Tt..Tz]	set of tables in the database where Tt represents the table t and z indicates
		the number of tables extracted from the LDM
	Tt{i, j}	represents the data item located at row i, column j from Tt, with i ∈
		{1, mt} and j ∈ {1, nt}, where mt is the number of tuples and nt is the
		number of attributes in Tt
	cont(Tt{i, j})	indicates the content of Tt{i, j}. cont(Tt{i1, j1}) = cont(T t ′ {i2, j2})
		means that contents of both data items are identical
	value(Tt{j})	indicates the name of column j from Tt. value(Tt{j1}) = value(T t ′ {j2})
		means that names of attributes j1 and j2 from Tt and T t ′ respectively are
		identical (e.g., when an attribute primary key -PK, is a foreign key -FK)
	K = [K1..Kt..Kz]	set of keys related to the LDM, where Kt is a vector giving the key type of
		each attribute of Tt. An element of Kt is noted Kt{j} ∈ {0, 1, 2}, where 0
		means that the attribute j of Tt is neither a PK nor a FK, 1 means that it
		is a PK and 2 means that it is an FK
	dp = Tp{ip, jp}	starting point of the algorithm such that Kp{jp} = 1. dp is provided by the
		machine/device used when writing/reading the product
	R = [R1..Rt..Rz]	set of matrices, where Rt represents an image of the Tt, indicating if data
		items are product-related information. A data item in Rt is noted Rt{i, j} =
		true if it is a product-related information and Rt{i, j} = f alse otherwise
	D	Matrix giving the distances between tables, where D a,b is the distance in
		term of relations between tables Ta and T b
	E	Set of tables already explored in the LDM
	N	Set of tables, yet to be explored, which are linked to tables already explored
	I	Set of data-items that remain to be explored (I l is the l th element of I)

  Table exploration is carried on as long as I = ∅

5 while I = ∅ do //

Table exploration is carried on as long as I = ∅ 6 [I ′ ; R] ←-ExplorePK(T , R, I 1 ); // Data items concerned by the product are set to true in R and data items not explored yet are identified 7 I ←-I ∪ I ′ ; // data items not explored yet are added to I 8 E ←-E ∪ I 1 ; // We memorize that the table including I1 was explored 9 I ←-I -I 1 ; // We remove the data item used for the exploration: I1 10 I ←-ExploreFK(E,T ) // New relations are searched in not explored tables Algorithm 2: ExplorePK(T , R, T t {i, j})

Table 2

 2 Table attributes enumerated in step 7

Table 3

 3 Definition of four entity groups

		55	0.04	0.31	0.10	(12)
	Material (G1)	Personal (G2)		Production (G3)	Equipment (G4)
	MaterialLot	Person		ProductionOrder	Equipment
	MaterialDefinition	PersonClass		ProductSegment	EquipmentClass
	MaterialClass	ActualPersonSeg.		ProductDefinition	EquipmentSegmentSpecif.
	ManufacturingBill PersonSegmentSpecif.	SegmentRequirement	ActualEquipmentSegment
	ActualMaterialLot			SegmentResponse
	MaterialSegmentSpecif.

  Results of the data item relevance

	Rang 1 st 2 st 3 th • • •	Table Name ProductSegment ProductSegment MaterialDefinition • • •	Attribute Name Duration Unit duration IDMatDef • • •	PK PSTB0B PSTB0B MD06 • • •	Cell Value 0.5 Hours 15C • • •	P l 0.7066 0.6102 0.4311 • • •	High probability that data items are stored on the product
	159 th	SegmentResponse	ActualStartTime	PSR0001	10:15:00	0.0567	Storage limit on
	. . . 523 th 524 th	. . . Equipment PersonClass	. . . Description Description	. . . ET3A01 PCPLW01	. . . Line 3A,... Production...	. . . 0.0051 0.0051	the material Lesser probability of being stored on the product
			(a) List of data items ordered from the highest P l to the lowest
	d Data item relevance: P	0 0.2 0.4 0.6 0.8					Production data group G3	48%	Equipment data group G4
		P e r s o n P e r s o n C l a s s E q u i p m e n t E q u i p m e n t C l a s s M a t e r i a l D e f i n i t i o n M a t e r i a l C l a s s M a t e r i a l L o t P r o d u c t S e g m e n t M a t e r i a l S e g m e n t S p e c i f . E q u i p m e n t S e g m e n t S p e c i f . P e r s o n S e g m e n t S p e c i f . P r o d u c t D e f i n i t i o n M a n u f a c t u r i n g B i l l P r o d u c t i o n O r d e r S e g m e n t R e q u i r e m e n t S e g m e n t R e s p o n s e A c t u a l M a t e r i a l L o t A c t u a l E q u i p m e n t S e g A c t u a l P e r s o n S e g m e n t	37% data group Material G1	3% 11% Personal data group G2
			(b) Whisker diagram			(c) Pie chart
	Fig. 8

The product table is defined as that which includes dp.

B2MML is meant to be a common data format to link business enterprise applications with manufacturing enterprise applications.

MaterialLot is the reference table because it contains dp.