E H El

A Messaoudi
email: messaoud@ibilce.unesp.br

ON THE SPECTRUM OF STOCHASTIC PERTURBATIONS OF THE SHIFT AND JULIA SETS

Keywords: AMS Subject Classifications (2000): 37A30, 37F50, 47A10, 47A35 Markov operator, Markov process, transition operator, stochastic perturbations of the shift, stochastic adding machine, Julia sets, residual spectrum, continuous spectrum

 by investigating in different Banach spaces (ℓ α (N), c0(N), cc(N)) the point, continuous and residual spectra of stochastic perturbations of the shift operator associated to the stochastic adding machine in base 2 and in Fibonacci base. For the base 2, the spectra are connected to the Julia set of a quadratic map. In the Fibonacci case, the spectra involve the Julia set of an endomorphism of C 2 .

Introduction

In this paper, we study in detail the spectrum of some stochastic perturbations of the shift operator introduced by Killeen and Taylor in [START_REF] Peter | A stochastic adding machine and complex dynamics[END_REF]. We focus our study on large Banach spaces for which we complete the Killeen-Taylor study. We investigate also the case of Fibonacci base, but in this case, we are not able to compute the residual and continuous spectra exactly. We recall that in [START_REF] Peter | A stochastic adding machine and complex dynamics[END_REF], Killeen and Taylor defined the stochastic adding machine as a stochastic perturbation of the shift in the following way: let N be a nonnegative integer number written in base 2 as N = ∑ k(N ) i=0 ε i (N )2 i where ε i (N ) = 0 or 1 for all i. It is known that there exists an algorithm that computes the digits of N + 1. This algorithm can be described by introducing an auxiliary binary "carry" variable c i (N ) for each digit ε i (N ) by the following manner:

Put c -1 (N + 1) = 1 and

ε i (N + 1) = ε i (N ) + c i-1 (N + 1) mod ( 2
)
c i (N + 1) = [ ε i (N ) + c i-1 (N + 1) 2
]
where i ≥ 0 and [z] denote the integer part of z ∈ R + .

Let {e i (n) : i ≥ 0, n ∈ N} be an independent, identically distributed family of random variables which take the value 0 with probability 1 -p and the value 1 with probability p. Let N be an integer. Given a sequence (r i (N )) i≥0 of 0 and 1 such that r i (N ) = 1 for finitely many indices i, we consider the sequences (r i (N + 1)) i≥0 and (c ′ i (N + 1)) i≥-1 defined by c ′ -1 (N + 1) = 1 and for all i ≥ 0 r i (N + 1) = r i (N ) + e i (N )c ′ i-1 (N + 1) mod ( 2)

c ′ i (N + 1) = [ r i (N ) + e i (N )c ′ i-1 (N + 1) 2
]
,

With this we have that a number ∑ +∞ i=0 r i (N )2 i transitions to a number ∑ +∞ i=0 r i (N + 1)2 i . In particular, an integer N having a binary representation of the form ε n . . . ε k+1 0 11 . . . 11 k transitions to ε n . . . ε k+1 1 00 . . . 00 k with probability p k+1 and a number having binary representation of the form ε n . . . ε k 11 . . . 11 k transitions to ε n . . . ε k 00 . . . 00 k with probability p k (1 -p). Equivalently, we obtain a Markov process ψ(N ) with state space N by ψ(N ) = ∑ +∞ i=0 r i (N )2 i . The corresponding transition operator is denoted by S p and given in Figure 2.

For p = 1 the transition operator equals the shift operator (cf. Figure 2), hence the stochastic adding machine can be seen as a stochastic perturbation of the shift operator. It is also a model of Weber law in the context of counter and pacemarker errors. This law is used in biology and psychophysiology [START_REF] Peter | How the Propagation of Error Through Stochastic Counters Affects Time Discrimination and Other Psychophysical Judgments[END_REF].

In [KT], P.R. Killeen and J. Taylor studied the spectrum of the transition operator S p (of ψ(N )) on ℓ ∞ . They proved that the spectrum σ(S p ) is equal to the filled Julia set of the quadratic map f : C → C defined by: f (z) = (z -(1 -p)) 2 /p 2 , i.e: σ(S p ) = {z ∈ C, (f n (z)) n≥0 is bounded } where f n is the n-th iteration of f . In [5], Messaoudi and Smania defined the stochastic adding machine in the Fibonacci base. The corresponding transition operator is given in Figure 6. Their procedure can be extended to a large class of adding machine and is given by the following manner. Consider the Fibonacci sequence (F n ) n≥0 given by the relation

F 0 = 1, F 1 = 2, F n = F n-1 + F n-2 ∀n ≥ 2.
Using the greedy algorithm, we can write every nonnegative integer N in a unique way as [START_REF] Zeckendorff | Représentation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas[END_REF]). It is known that the addition of 1 in the Fibonacci base (adding machine) is recognized by a finite state automaton (transductor). In [5], the authors defined the stochastic adding machine by introducing a " probabilistic transductor ". They also computed the point spectrum of the transition operator acting in ℓ ∞ associated to the stochastic adding machine with respect to the base (F n ) n≥0 . In particular, they showed that the point spectrum σ pt (S p ) in ℓ ∞ is connected to the filled Julia set J(g) of the function g : C 2 → C 2 defined by:

N = k(N ) ∑ i=0 ε i (N )F i where ε i (N ) = 0 or 1 and ε i (N )ε i+1 (N ) ̸ = 11, for all i ∈ {0, • • • , k(N ) -1} (see
g(x, y) = ( 1 p 2 (x -1 + p)(y -1 + p), x).
Precisely, they proved that

σ pt (S p ) = K p = {λ ∈ C (q n (λ)) n≥1 is bounded }, where q F 0 (z) = z, q F 1 (z) = z 2 , q F k (z) = 1 p q F k-1 (z)q F k-2 (z) - 1 -p p , for all
k ≥ 2 and for all nonnegative integers n, we have q n (z) = q F k 1 . . . q F km where

F k 1 + • • • + F km is the Fibonacci representation of n.
In particular, σ pt (S p ) is contained in the set

E p = {λ ∈ C | (q Fn (λ)) n≥1 is bounded } = {λ ∈ C | (λ 1 , λ) ∈ J(g)}
where Here we investigate the spectrum of the stochastic adding machines in base 2 and in the Fibonacci base in different Banach spaces. In particular, we compute exactly the point, continuous and residual spectra of the stochastic adding machine in base 2 for the Banach spaces c 0 , c, ℓ α , α ≥ 1.

λ 1 = 1 -p + (1-λ-p) 2 p .
For the Fibonacci base, we improve the result in [5] by proving that the spectrum of S p acting on ℓ ∞ contain E p . The same result will be proven for the Banach spaces c 0 , c and ℓ α , α ≥ 1.

The paper is organized as follows. In section 2, we give some basic facts on spectral theory. In section 3, we state our main results (Theorems 1, 2 and 3). Section 4 contains the proof in the case of the base 2 and finally, in section 5, we present the proof in the case of the Fibonacci base.

2. Basic facts from the spectral theory of operators (see for instance [START_REF] Halmos | A Hilbert space problem book[END_REF], [START_REF] Rudin | Functional analysis[END_REF], [START_REF] Schechter | Principles of functional analysis[END_REF], [START_REF] Yoshida | functional analysis[END_REF])

Let E be a complex Banach space and T a bounded operator on it. The spectrum of T , denoted by σ(T ), is the subset of complex numbers λ for which T -λId E is not an isomorphism (Id E is the identity maps).

As usual we point out that if λ is in σ(T ) then one of the following assertions hold:

(1) T -λId E is not injective. In this case we say that λ is in the point spectrum denoted by σ pt (T ). ( 2) T -λId E is injective, not onto and has dense range. We say that λ is in the continuous spectrum denoted by σ c (T ). ( 3) T -λId E is injective and does not have dense range. We say that λ is in residual spectrum of T denoted by σ r (T ). It follows that σ(T ) is the disjoint union

σ(T ) = σ pt (T ) ∪ σ c (T ) ∪ σ r (T ).
It is well known and it is an easy consequence of Liouville Theorem that the spectrum of any bounded operator is a non empty compact set of C. There is a connection between the spectrum of T and the spectrum of the dual operator T ′ acting on the dual space E ′ by T ′ : ϕ → ϕ • T. In particular, we have Proposition 2.1 (Phillips Theorem). Let E be a Banach space and T a bounded operator on it, then σ(T ) = σ(T ′ ).

We also have a classical relation between the point and residual spectra of T and the point spectrum of T ′ . Proposition 2.2. For a bounded operator T we have

σ r (T ) ⊂ σ pt (T ′ ) ⊂ σ r (T ) ∪ σ pt (T ).
In particular, if σ pt (T ) is an empty set then σ r (T ) = σ pt (T ′ ).

Main results.

Our main results are stated in the following three theorems. Theorem 1. The spectrum of the operator S p acting on c 0 , c and ℓ α , α ≥ 1 is equal to the filled Julia set J(f ) of the quadratic map f (z) = (z -(1-p)) 2 /p 2 . Precisely, in c 0 (resp. ℓ α , α > 1), the continuous spectrum of S p is equal to J(f ) and the point and residual spectra are empty. In c, the point spectrum is equal {1}, the residual spectrum is empty and the continuous spectrum equals J(f )\{1}. Theorem 2. In ℓ 1 , the point spectrum of S p is empty. The residual spectrum of S p is not empty and contains a dense and countable subset of the Julia set ∂(J f ), i.e.

∪ +∞ n=0 f -n {1} ⊂ σ r (S p ). The continuous spectrum is equal to the relative complement of the residual spectrum with respect to the filled Julia set J f . Theorem 3. The spectra of S p acting respectively in ℓ ∞ , c 0 , c and ℓ α , α ≥ 1, associated to the stochastic Fibonacci adding machines contain the set E p = {λ ∈ C | (λ 1 , λ) ∈ J(g)} where J(g) is the filled Julia set of the function g and

λ 1 = 1 -p + (1-λ-p) 2 p .
Conjecture: We conjecture that in the case of ℓ 1 , the residual spectrum of the transition operator associated to the stochastic adding machine in base 2 is σ r (S p ) = ∪ +∞ n=0 f -n {1}. For Fibonacci stochastic adding machine, we conjecture that the spectra of S p in the Banach spaces cited in Theorem 3 are equals to the set E p .

Remark 3.1. The methods used for the proof of our results can be adapted for a large class of stochastic adding machine given by transductors. Remark 3.2. We point out that from Killeen and Taylor method one may deduce in the case of ℓ ∞ that the residual and continuous spectrum is empty. On the contrary here we compute directly the residual and continuous spectrum in ℓ α , c 0 and c. We are interested in the spectrum of S p on three Banach spaces connected by duality. The space c 0 is the space of complex sequences which converge to zero, in other words, the continuous functions on N vanishing at infinity. The dual space of c 0 is by Riesz Theorem the space of bounded Borel measures on N with total variation norm. This space can be identified with ℓ 1 , the space of summable row vectors. Finally, the dual space of ℓ 1 is ℓ ∞ the space of bounded complex sequences.

We are also interested in the spectrum of S p as operator on the space ℓ α with α > 1 and also in the space c of complex convergent sequences. Proposition 4.1. The operator S p (acting on the right) is well defined on the space X where X ∈ {c 0 , c, l α , α ≥ 1}, moreover ||S p || ≤ 1.

Since the operator S p is bi-stochastic, the proof of this proposition is a straightforward consequence of the following more general lemma. Lemma 4.2. Let A = (a i,j ) i,j∈N be an infinite matrix with nonnegative coefficients. Assume that there exists a positive constant M such that

(1) sup i∈N   ∞ ∑ j=0 a i,j   ≤ M, (2) sup j∈N ( ∞ ∑ i=0 a i,j ) ≤ M.
Then A defines a bounded operator on the spaces c 0 , c, l ∞ and ℓ α (N) with α ≥ 1. In addition the norm of A is less than M .

Proof. By the assumption (1) it is easy to get that A is well defined on ℓ ∞ (N) and its norms is less than

M . Now, let v = (v n ) n≥0 , v ̸ = 0 such that lim n-→+∞ v n = l ∈ C,
then for any ε > 0 there exists a positive integer j 0 such that for any j ≥ j 0 , we have

|v j -l| ≤ ε 2M . Let d = +∞ ∑ j=0
a n,j , then from the assumption (1), we have that for any n ∈ N,

|(Av) n -d • l| = +∞ ∑ j=0 a n,j (v j -l) ≤ j 0 -1 ∑ j=0 a n,j |v j -l| + ε 2 . (1)
But by the assumption (2), for any j ∈ {0, . . . , j 0 -1}, we have

+∞ ∑ n=0 a n,j < ∞.
Then there exists n 0 ∈ N such that for any n ≥ n 0 and for any j ∈ {0, • • • , j 0 -1}, we have

|a n,j | ≤ ε 2j 0 (δ + 1)
where

δ = sup{|v j -l|, j ∈ N} (2)
Combined ( 1) with (2) we get that

|(Av) n -d • l| ≤ ε, ∀n ≥ n 0 . Hence AX ⊂ X if X = c 0 or c.
Now take α > 1 and v ∈ l α . For any integer integer i ∈ N, we have

|(Av) i | α ≤   +∞ ∑ j=0 a i,j |v j |   α . Let α ′ be a conjugate of α, i.e, 1 α + 1 α ′ = 1.
Then, by Hölder inequality we get

  +∞ ∑ j=0 a i,j |v j |   α ≤   +∞ ∑ j=0 a i,j   α α ′   +∞ ∑ j=0 a i,j |v j | α   . Hence   +∞ ∑ j=0 a i,j |v j |   α ≤   sup l∈N +∞ ∑ j=0 a l,j   α α ′   +∞ ∑ j=0 a i,j |v j | α   . (3) Thus ||Av|| α α ≤ M α α ′ ∞ ∑ i=0   +∞ ∑ j=0 a i,j |v j | α   = M α α ′ ∞ ∑ j=0 ( ∞ ∑ i=0 a i,j
)

|v j | α ≤ M α α ′ sup j∈N ( ∞ ∑ i=0 a i,j
)

||v|| α α ≤ M 1+ α α ′ ||v|| α α . Then ||Av|| α ≤ M ||v|| α .
Hence A is a continuous operator and ||A|| ≤ M .

The case α = 1 is an easy exercise and it is left to the reader. □ From Proposition 4.1, we deduce that S p is a Markov operator and its spectrum is contained in the unit disc of complex numbers.

Consider the map

f : z ∈ C -→ ( z-(1-p) p
) 2 and denote by J(f ) the associated filled Julia set defined by:

J(f ) = { z ∈ C, |f (n) (z)| ̸ -→ ∞ } .
Killeen and Taylor investigated the spectrum of S p acting on ℓ ∞ . They proved that the point spectrum of S p is equal to the filled Julia set of f . In addition, they showed that the spectrum is invariant under the action of f . As a consequence, one may deduce that the continuous and residual spectra in this case are empty.

Here we will compute exactly the residual part and the continuous part of the spectrum of S p acting on the spaces c 0 , c and ℓ α , α ≥ 1.

Theorem 4.3. The spectrum of the operator S p acting on X where X ∈ {c 0 , c, l α , α ≥ 1} is equal to the filled Julia set of f, J(f ). Precisely, in c 0 (resp. ℓ α , α > 1), the continuous spectrum of S p is equal to J(f ) and the point and residual spectra are empty. In c, the point spectrum is the singleton {1}, the residual spectrum is empty and the continuous spectrum is J(f )\{1}.

For the proof of Theorem 4.3 we shall need the following proposition.

E(h 0 , h 1 , . . .) = (h 0 , 0, h 1 , 0, h 2 , . . .),
and

O(h 0 , h 1 , . . .) = (0, h 0 , 0, h 1 , 0, h 2 , . . .), for any h = (h 0 , h 1 , . . .) in X. Precisely, for all v = (v i ) i≥0 ∈ X, we have S p 2 (v) = ES p (v 0 , v 2 , . . . v 2n , . . .) + OS p (v 1 , v 3 , . . . v 2n+1 , . . .).
As a consequence we deduce from the mapping spectral theorem [START_REF] Schechter | Principles of functional analysis[END_REF] that the spectrum of S p is invariant under f . Let us start the proof of Theorem 4.3 by proving the following result.

Proposition 4.5. The point spectrum of S p acting on X where X ∈ {c 0 , l α , α ≥ 1} is empty, and the point spectrum of S p on c is equal to {1}.

For the proof, we need the following lemma from [START_REF] Peter | A stochastic adding machine and complex dynamics[END_REF] . Lemma 4.6. [START_REF] Peter | A stochastic adding machine and complex dynamics[END_REF]. Let n be a nonnegative integer and X n = {m ∈ N : (S p ) n,m ̸ = 0}, then the following properties are valid.

(1) For all nonnegative integers n, we have n ∈ X n and

(S p ) n,n = 1 -p. (2) If n = ε k . . . ε 1 0, k ≥ 2, is an even integer then X n = {n, n + 1} and (S p ) n,n+1 = p. (3) If n = ε k . . . ε t 0 1 . . . 1 s is an odd integer with s ≥ 1 and k ≥ t ≥ s + 1, then X n = {n, n+1, n-2 m +1, 1 ≤ m ≤ s} and n transitions to n+1 = ε k . . . ε t 1 0 . . . 00 s with probability (S p ) n,n+1 = p s+1 , and n transitions to n -2 m + 1 = ε k . . . ε t 0 1 . . . 1 s-m 0 . . . 0 m , 1 ≤ m ≤ s with probability (S p ) n,n-2 m +1 = p m (1 -p).
Proof of Proposition 4.5. Let λ be an eigenvalue of S p associated to the eigenvector v = (v n ) n≥0 in X where where X ∈ {c 0 , c, l α , α ≥ 1} Let λ be an eigenvalue of S p associated to the eigenvector v = (v i ) i≥0 in X. By Lemma 4.6, we see that the operator S p satisfies (S p ) i,i+k = 0 for all i, k ∈ N with k ≥ 2. Therefore, for all integers k ≥ 1, we have

k ∑ i=0 (S p ) k-1,i v i = λv k-1 . (4)
Then, one can prove by induction on k that for all integers k ≥ 1, there exists a complex number q k = q k (p, λ) such that

v k = q k v 0 (5)
By Lemma 4.6 and the fact that (S p -λI)v) 2 n = 0 for all nonnegative integers n, we get

p n+1 v 2 n + (1 -p -λ)v 2 n -1 + n ∑ i=1 p i (1 -p)v 2 n -2 i = 0, ∀n ≥ 0. (6) Hence v 2 n = 1 p A -( 1 p -1)v 0 , where A = - 1 p n ((1 -p -λ)v 2 n-1 +(2 n-1 -1) + n-1 ∑ i=1 p i (1 -p)v 2 n-1 +(2 n-1 -2 i ) .
On the other hand, by the self similarity structure of the transition matrix S p , one can prove that if i and j are two integers such that for some positive integer n we have 2 n-1 ≤ i, j < 2 n , then the transition probability from i to j is equal to the transition probability from i -2 n-1 to j -2 n-1 . Using this last fact and (6), it follows that

v 2 n = 1 p q 2 n-1 v 2 n-1 - ( 1 p -1 ) v 0 .
This gives

q 2 n = 1 p q 2 2 n-1 - ( 1 p -1 ) , ( 7
)
where

q 2 0 = q 1 = - 1 -p -λ p . Case 1: v ∈ c 0 or ℓ α , α ≥ 1.
We have lim n→∞ q 2 n = 0. Thus by [START_REF] Rudin | Functional analysis[END_REF], we get p = 1, which is absurd, then the point spectrum is empty.

Case 2: v ∈ c. Assume that lim q n = l ∈ C, then by [START_REF] Rudin | Functional analysis[END_REF], we deduce that l = 1 or l = p -1. On the other hand, for any n ∈ N, there exist k nonnegative integers

n 1 < n 2 . . . < n k such that n = 2 n 1 + • • • 2 n k .
We can prove (see [START_REF] Peter | A stochastic adding machine and complex dynamics[END_REF]) that

q n = q 2 n 1 . . . q 2 n k . ( 8
)
Then lim q 2 n-2 +2 n = l 2 = l, thus l = p -1 is excluded. Since S p is stochastic, we conclude that l = 1 and σ pt,c (S p ) = {1}. □ Remark 4.7. By the same arguments as above, Killeen and Taylor in [START_REF] Peter | A stochastic adding machine and complex dynamics[END_REF] proved that the point spectrum of S p acting on ℓ ∞ is equal to the filled Julia set of the quadratic map f . In fact, it is easy to see from the arguments above that σ pt,l ∞ (S p ) = {λ ∈ C, q n (λ) bounded }. Indeed, [START_REF] Rudin | Functional analysis[END_REF] implies that if (q 2 n ) n≥0 is bounded, then for all n ≥ 0, |q 2 n | ≤ 1. This clearly forces σ pt,l ∞ (S p ) = {λ ∈ C, q 2 n (λ) bounded } by [START_REF] Schechter | Principles of functional analysis[END_REF]. Now, since

q 2 n = h • f n-1 • h -1 (q 1 ) = h • f n-1 (λ), ∀n ∈ N, where h(x) = x p - 1 -p p , we conclude that σ pt,l ∞ (S p ) = J(f ). It follows from Proposition 4.4 that σ l ∞ (S p ) = J(f )
and the residual and continuous spectra are empty.

Proposition 4.8. The residual spectrum of S p acting on X ∈ {c 0 , c, ℓ α , α > 1} is empty.

Proof. Let λ be an element of the residual spectrum of S p acting on c 0 (resp. c).

Then, by Proposition 2.2, we deduce that there exists a sequence u = (u k ) k≥0 ∈ l 1 (N) such that u(S p -λId) = 0.

Claim. u k = 1 q k u 0 , ∀k ∈ N.
We have

∀k ∈ 2N, (u(S p -λId)) k+1 = pu k + (1 -p -λ)u k+1 = 0. Hence ∀k ∈ 2N, u k = q 1 u k+1 . ( 9
)
If k is odd, then k = 2 n -1 + t where t = 0 or t = ∑ s j=2 2 n j where 1 ≤ n < n 2 < n 3 , . . . < n s . Since (u(S p -λId)) k+1 = 0, then we have

p n+1 u k + (1 -p -λ)u k+1 + n ∑ i=1 p i (1 -p)u k+2 i = 0. ( 10
)
Observe that the relation [START_REF] Zeckendorff | Représentation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas[END_REF] between u k and u k+2 n is similar to the relation (6) between v 2 n and v 0 . Hence, by induction on n, we obtain

u k = q 2 n u k+2 n . (11)
Indeed, if n = 1 then by ( 10) and ( 9), we get

p 2 u k + (q 1 (1 -p -λ) + p(1 -p))u k+2 = 0. Therefore u k = ( q 2 1 p - 1 -p p ) u k+2 = q 2 u k+2 .
Then (11) is proved for n = 1. Now, assume that (11) holds for the numbers 1, 2,

• • • , m -1.
Take n = m and 1 ≤ i < m, then k

+ 2 i = 1 + 2 + • • • + 2 m-1 + 2 i + t = 2 i -1 + t ′
where t ′ = 2 m + t. Applying the induction hypothesis, we get

u k+2 i = q 2 i u k+2 i+1 = q 2 i q 2 i+1 . . . q 2 m-1 u k+2 m .
On the other hand, since

2 i + • • • + 2 m-1 = 2 m -2 i , we have u k+2 i = q 2 m -2 i u k+2 m . (12)
Considering [START_REF] Zeckendorff | Représentation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas[END_REF] with n = m and (12) yields

u k = - 1 p m+1 ( (1 -p -λ) q 2 m -1 + m ∑ i=1 p i (1 -p)q 2 m -2 i u k+2 m ) .
Combined ( 5) and ( 6), we obtain (11) for n = m. Then (11) holds for all integers n ≥ 1.

In particular, we have

u 2 n-1 -1 = q 2 n-1 u 2 n -1
, for all integers n ≥ 1. Thus

u 2 n -1 = 1 q 2 0 q 2 . . . q 2 n-1 u 0 = 1 q 2 n -1 u 0 , ∀n ≥ 1. ( 13
)
On the other hand, for all integers n ≥ 1, by [START_REF] Yoshida | functional analysis[END_REF] we have u 2 n = q 2 0 u 2 n +2 0 . and from (11), we see that

u 2 n = q 2 0 q 2 1 u 2 2 -1+2 n = . . . = q 2 0 q 2 1 . . . q 2 n-1 u 2 n+1 -1 . ( 14
)
Consequently from ( 13) and ( 14), we obtain

u 2 n = 1 q 2 n u 0 , ∀n ≥ 1. ( 15
)
Now fix an integer k ∈ N and assume that k = s ∑ i=1 2 n i where 0 ≤ n 1 < n 2 < . . . < n s . We will prove by induction on s that the following statement holds

u k = 1 q 2 n 1 q 2 n 2 . . . q 2 ns u 0 = 1 q k u 0 . (16)
Indeed, it follows from (15), that ( 16) is true for s = 1. Now assume that( 16) is true for all integers 1 ≤ i < s. Case 1. k is odd.

In this case k = s ∑ i=1 2 n i = 2 n -1 + l = n-1 ∑ j=0 2 j + l where l = 0 if n = s + 1 and l = s ∑ i=n 2 n i if n ≤ s.
If n ≥ 2, we use (11) to get u k-2 n-1 = q 2 n-1 u k and by induction hypothesis, we have

u k = 1 q 2 n-1 q k-2 n-1 u 0 = 1 q k u 0 .
If n = 1, we consider (9) to write u k = 1 q 1 u k-1 . Thus, we deduce, by induction hypothesis, that

u k = 1 q 1 q k-1 u 0 = 1 q k u 0 .
Case 2. k is even. In this case n 1 > 0. and by ( 9), we deduce that

u k = q 2 0 u k+2 0 = q 2 0 u k+2 1 -1 .
Applying (11), it follows that

u k = q 2 0 q 2 1 u k+2 2 -1 = . . . = q 2 0 q 2 1 . . . q 2 n 1 -1 u k+2 n 1 -1 = q 2 0 q 2 1 . . . q 2 n 1 -1 u (k-2 n 1 )+2 n 1 +1 -1 .
Hence

u k = q 2 0 . . . q 2 n 1 -1 q 2 n 1 +1 u (k-2 n 1 )+2 n 1 +2 -1 = q 2 0 . . . q 2 n 1 -1 q 2 n 1 +1 . . . q 2 n 2 -1 u (k-2 n 1 -2 n 2 )+2 n 2 +1 -1 .
Thus

u k = ns ∏ i=0 q 2 i s ∏ i=1 q 2 n i u 2 ns+1 -1 . (17)
By ( 17) and (13) we get

u k = 1 s ∏ i=1 q 2 n i u 0 = 1 q k u 0 .
Therefore we have proved that for all nonnegative integers

u k = 1 q k u 0 . (18)
We conclude that u is in ℓ 1 (N) if and only if

+∞ ∑ k=1 1 q k (λ) < ∞.
But this gives that the residual spectrum of S p acting on c 0 or c satisfy

σ r,C 0 (S p ) ⊂ { λ ∈ D(0, 1) : +∞ ∑ k=1 1 q k (λ) < ∞ } . (19) We claim that +∞ ∑ k=1 1 q k (λ) < ∞ implies |q 2 n -1 | ≥ 1, for all integers n ≥ 1.
Indeed, by D'Alembert's Theorem, we have

lim sup |q n | |q n+1 | ≤ 1. (20) Now assume that n is even. Then n = 2 k 0 + • • • + 2 km where 1 ≤ k 0 < k 1 < . . . < k m (representation in base 2). In this case n + 1 = 2 0 + 2 k 0 + • • • + 2 km .
Using [START_REF] Schechter | Principles of functional analysis[END_REF], we obtain

|q n | |q n+1 | = 1 |q 1 |
and by (20), we get

|q 1 | ≥ 1. (21)
Since for all integers n ≥ 0, we have q

2 n = 1 p q 2 2 n-1 - ( 1 p -1
)

. It follows, from the triangle inequality, that |q 2 n | ≥ 1 for all integers n ≥ 1. Let i be a positive integer. Since

2 i -1 = i-1 ∑ j=0
2 j , we obtain by ( 8) that

q 2 i -1 = q 2 i-1 q 2 i-2 • • • q 1 .
Hence |q 2 i -1 | ≥ 1, for any integer i ≥ 1. On the other hand, consider the first coordinate of the vector µ(S p -λId) = 0. Then we have

(1 -p -λ)µ 0 + +∞ ∑ i=1 p i (1 -p)µ 2 i -1 = 0.
Dividing the two members of the last equality by p, we obtain

q 1 = +∞ ∑ i=1 p i-1 (1 -p)/q 2 i -1 . (22)
We claim that there exists an integer i 0 ∈ N such that |q

2 i 0 -1 | > 1. Indeed, if not the series ∑ i∈N 1 |q 2 i -1 | will diverge. Thus |q 1 | < +∞ ∑ i̸ =i 0 p i (1-p)+p i 0 -1 (1-p) < 1.
Absurd. We conclude that the residual spectrum of S p acting on c 0 (resp. c ) is empty.

The same proof yields that the residual spectrum of S p acting on ℓ α , α > 1, is empty and the proof of the proposition is complete. □ Proof. Assume that X ∈ {c 0 , c}. Then, by Phillips Theorem, we see that the spectrum of S p in X is equal to the the spectrum of S p in ℓ ∞ and from Propositions 4.5 and 4.8, we obtain the result. Now, assume X = l α , α > 1. According to Propositions 4.4, 4.5 and 4.8, it is enough to prove that J(f ) ⊂ σ(S p ). Consider λ ∈ J(f ). We will prove that λ belongs to the approximate point spectrum of S p . For all integers k ≥ 2, put w (k) = (1, q 1 (λ), . . . , q k (λ), 0 . . . 0, . . .) t ∈ l α where (q k (λ)) k≥1 = (q k ) k≥1 is the sequence defined in (5) of the proof of Theorem 4.3 and let u

(k) = w (k) ||w (k) || α ,
then we have the following claim.

Claim: lim n→+∞ ||(S p -λId)u (2 n ) || α = 0.
Indeed, we have ∀i ∈ {0, . . . , k -1},

( (S p -λId)u (k) ) i = 0. Thus +∞ ∑ i=0 ((S p -λId)u (k) ) i α = +∞ ∑ i=k k ∑ j=0 (S p -λId) i,j w (k) j ||w (k) || α α α .
Putting a i,j = |(S p -λId) i,j | for all i, j and using (3), we get

k ∑ j=0 (S p -λId) i,j w (k) j α ≤ C k ∑ j=0 |(S p -λId) i,j ||w (k) j | α where C = sup i∈N   ∞ ∑ j=0 |(S p -λId) i,j |   α α ′
and α ′ is the conjugate of α.

Observe that C is a finite nonnegative constant because S p is a stochastic matrix and λ belongs to J(f ) which is a bounded set.

In this way we have

(S p -λId)u (k) α α ≤ C +∞ ∑ i=k ( ∑ k j=0 |w (k) j | α |(S p -λId) ij | ) ||w (k) || α α = C ||w (k) || α α k ∑ j=0 |w (k) j | α +∞ ∑ i=k |(S p -λId) ij |.
Now, for k = 2 n , we will compute the following terms

A kj = +∞ ∑ i=k |(S p -λId) ij |, 0 ≤ j ≤ k.
Assume that 0 ≤ j < k = 2 n . Then (S p -λId) ij = (S p ) ij for all i ≥ k. Case 1: j is odd. Then by Lemma 4.6, (S p ) ij ̸ = 0 if and only i = j -1 or i = j. Hence (S p ) ij = 0 for all i ≥ k. Thus

A kj = 0. ( 23
)
Case 2: j = 0 . Then by Lemma 4.6, we have

A kj = +∞ ∑ i=2 n (S p ) i0 = +∞ ∑ i=n+1 p i (1 -p) = p n+1 . ( 24
)
Case 3: j is even and j > 0. Then j = ε n-1 . . . ε s 0 . . .

0 s = n-1 ∑ i=s ε i 2 i with s ≥ 1 and ε s = 1. But by Lemma 4.6, (S p ) ij ̸ = 0 if and only if i = 2 m -1 + j where 0 ≤ m ≤ s. Hence i < 2 n = k.
Therefore, in this case

A kj = 0. ( 25
)
Now assume j = k = 2 n . In this case, we have

A kj = |1 -p -λ| + +∞ ∑ i=2 n +1
(S p ) i,2 n . On the other hand, by Lemma 4.6, we deduce that (S p ) i,2 n ̸ = 0 if and only if i = 2 n + 2 m -1 where 0 ≤ m ≤ n and (S p )

2 n +2 m -1,2 n = p m (1 -p). Therefore A kj = +∞ ∑ i=2 n |(S p -λId) i,2 n | = |1 -p -λ| + n ∑ m=0 p m (1 -p). ( 26
)
By ( 23),( 24),( 25) and ( 26), we have for k = 2 n and 0 ≤ j ≤ k,

A kj ̸ = 0 ⇐⇒ j = 0 or j = k = 2 n . Consequently (S p -λId)u (2 n ) α α ≤ C . |w (k) 0 | α A k0 + |w (k) k | α A kk ||w (k) || α α = C . p n+1 + |q 2 n | α ( |1 -p -λ| + n ∑ m=0 p m (1 -p) ) ||w (2 n ) || α α .
We claim that ||w (2 n ) || α goes to infinity as n goes to infinity. Indeed, if not since the sequence ||w (2 n ) || α is a increasing sequence, it must converge. Put w = (q i ) i≥0 with q 0 = 1. It follows that the sequence (w (2 n ) ) n≥0 converges to w in ℓ α which means that there exists a nonzero vector w ∈ l α such that (S p -λId)w = 0. This contradicts Proposition 4.5. Now, since λ belongs to the filled Julia set which is a bounded set and (q n ) n≥0 is a bounded sequence, it follows that ||((S p -λId)u (2 n ) )|| α converge to 0, and the claim is proved. We conclude that λ belongs to the approximate point spectrum of S p and the proof of Proposition 4.10 is complete. □

This ends the proof of Theorem 4.3.

Spectrum of S p acting on the right on ℓ 1 .

Here, we will study the spectrum of S p acting (on the right) in ℓ 1 . We deduce from Proposition 4.4 that the Spectrum of S p on ℓ 1 is contained in the filled Julia set J(f ). On the other hand, using the same proof than Proposition 4.10, we obtain that J(f ) is contained in the approximate point spectrum of S p . This yields that the spectrum of S p acting on ℓ 1 is equal to J(f ). Theorem 4.11. In ℓ 1 , the residual spectrum contains a dense and countable subset of the Julia set ∂(J(f )). The continuous spectrum is not empty and is equal to the relative complement of the residual spectrum with respect to the the filled Julia set J(f ).

Proof. The proof of Proposition 4.8, shows that the residual spectrum of S p in ℓ 1 is equal to the point spectrum of S p (acting on right) in l 1 ′ = l ∞ . By (18) and ( 22), we see that σ r (S p ) = { λ ∈ C, (q n (λ)) and (1/q n (λ)) are bounded and

q 1 = +∞ ∑ i=1 p i-1 (1 -p) q 2 i -1 } = J(f ) ∩ { λ ∈ C, (1/q n (λ)) is bounded and q 1 = +∞ ∑ i=1 p i-1 (1 -p) q 2 i -1 } .
On the other hand we have

q 2 2 n = f (q 2 2 n-1 ) = . . . f n (q 2 1 ) = f n+1 (λ), ∀n ≥ 0. (27) Let n ∈ N and E n = {λ ∈ C, q 2 n (λ) = 1} Claim 1: +∞ ∪ n=0 E n = +∞ ∪ n=0 f -n {1}.
Indeed, let λ ∈ C such that f n (λ) = 1 for some nonnegative integer n ≥ 1. Then, by (27), we have q 2 n-1 = 1 or q 2 n-1 = -1. From (7), we see that

q 2 n-1 = -1 implies q 2 n = 1. Hence f -n {1} ⊂ E n-1 ∪ E n . Since 1 ∈ E n for all integers n ≥ 0, we conclude that, ∪ +∞ n=0 f -n {1} ⊂ ∪ +∞ n=0 E n .
The other inclusion follows from (27).

Claim 2:

+∞ ∪ n=0 E n ⊂ σ r (S p ).
Indeed, assume that n ∈ N and λ ∈ E n . Then by [START_REF] Rudin | Functional analysis[END_REF], we get that

q 2 k = 1, ∀k ≥ n. ( 28
)
But from (28) and ( 8), we have that (q k (λ)) k≥0 and (1/q k (λ)) k≥0 are bounded . Moreover, we have

q 1 = +∞ ∑ i=1 p i-1 (1 -p) q 2 i -1 ⇐⇒ q 2 = +∞ ∑ i=2 p i-2 (1 -p)q 1 q 2 i -1 ⇐⇒ q 2 k = +∞ ∑ i=k+1 p i-k-1 (1 -p) q 2 0 . . . q 2 k-1 q 2 i -1 , ∀k ≥ 0 ⇐⇒ q 2 k = +∞ ∑ i=k+1 p i-k-1 (1 -p) q 2 k q 2 k+1 . . . q 2 i-1
, ∀k ≥ 0.

Thus

q 1 = +∞ ∑ i=1 p i-1 (1 -p) q 2 i -1 ⇐⇒ 1 = +∞ ∑ i=0 p i (1 -p).
From this λ ∈ σ r (S p ) and the claim 2 is proved. But 1 is a repulsor fixed point of f , it follows that ∪ +∞ n=0 f -n {1} is a dense subset of the Julia set ∂J(f ). By this fact combined with claims 1 and 2, we conclude that the residual spectrum contains a dense and countable subset of the Julia set ∂(J(f )).

On the other hand, (p -1) 2 ∈ J(f ) since f ((p -1) 2 )) = (p -1) 2 , but (p-1) 2 ̸ ∈ σ r (S p ) because for any positive integer n, q 2 n ((p-1) 2 ) = (p-1), which implies that lim q n = 0 and hence 1/q n is not bounded. Thus (p -1) 2 ∈ σ c (S p ). This finishes the proof of the theorem. □ Conjecture 4.12. We conjecture that the residual spectrum in ℓ 1 equals the set

∪ +∞ n=0 f -n {1}.

Spectrum of S p acting on the left.

Phillips Theorem combined with Proposition 2.2, Theorems 4.3 and 4.11, leads to the following result. Theorem 4.13. The spectrum of S p (acting on the left) in the spaces c 0 , c, l α where 1 ≤ α ≤ +∞ equals to the filled Julia set J(f ). Precisely:

In c 0 , l α where 1 ≤ α < +∞, the spectrum of S p equals to the continuous spectrum of S p .

In c, the point spectrum of S p equals {1} and the continuous spectrum equals J(f )\{1}.

In ℓ ∞ , the point spectrum equals to the residual spectrum of S p in ℓ 1 .

Fig. 3. Filled Julia set: p = 0.7

5.

Fibonacci Stochastic adding machine (see [5])

Let us consider the Fibonacci sequence (F n ) n≥0 given by the relation

F n = F n-1 + F n-2 ∀n ≥ 2.
Using greedy algorithm, we can write (see [START_REF] Zeckendorff | Représentation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas[END_REF]) every nonnegative integer N in a unique way as

N = k(N ) ∑ i=0 ε i (N )F i where ε i (N ) = 0 or 1 and ε i (N )ε i+1 (N ) ̸ = 0, ∀0 ≤ i ≤ k(N ) -1.
It is known that the addition of 1 in base (F n ) n≥0 (called Fibonacci adding machine) is given by a finite state automaton transductor on A * × A * where A = {0, 1} (see Fig. 4). This transductor is formed by two states ( an initial state I and a terminal state T ). The initial state is connected to itself by 2 arrows. One of them is labeled by [START_REF] Zeckendorff | Représentation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas[END_REF]00) and the other by (101, 000). There are also 2 arrows going from the initial state to the terminal one. One of these arrows is labeled by (00, 01) and the other by (001, 010). The terminal state is connected to itself by 2 arrows. One of them is labeled by (0, 0) and the other by [START_REF] Halmos | A Hilbert space problem book[END_REF][START_REF] Halmos | A Hilbert space problem book[END_REF].

Assume that N = ε n . . . ε 0 . To find the digits of N + 1, we will consider the finite path c = (p k+1 , a k /b k , p k ) . . . In [5], the authors define the stochastic adding machine by the following way: Consider "probabilistic" transductor T p (see Fig. 5) where 0 < p < 1, by the following manner.

The states of T p are I and T . The labels are of the form (0/0, 1), (1/1, 1), (a/b, p) or (a/a, 1 -p) where a/b is a label in T .

The labeled edges in T p are of the form (T, (x/x, 1), T ) where x ∈ {0, 1} or of the form (r, (a/b, p), q) or (T, (a/a, 1 -p), q) where (r, a/b, q) is a labeled edge in T , with q = I.

The stochastic process ψ(N ) is defined by ψ(N ) = ∑ +∞ i=0 r i (N )F i where (r i (N )) i≥0 is an infinite sequence of 0 or 1 without two 1 consecutive and with finitely many non zero terms.

The sequence (r i (N )) i≥0 is defined by the following way: Put r i (0) = 0 for all i, and assume that we have defined (r i (N -1)) i≥0 , N ≥ 1. In the transductor T p , consider a path . . . (T, (0/0, 1), T ) . . . (T, (0/0, 1), T )(p n+1 , (a n /b n , t n ), p n ) . . . (p 1 , (a 0 /b 0 , t 0 ), p 0 ) where p 0 = I and p n+1 = T, such that the words . . . r 1 (N -1)r 0 (N -1) and . . . 00a n . . . a 0 are equal.

We define the sequence (r i (N )) i≥0 as the infinite sequence whose terms are 0 or 1 such that . . . r 1 (N )r 0 (N ) = . . . 00b n . . . b 0 .

We remark that ψ(N -1) transitions to ψ(N ) with probability of p ψ(N -1)ψ(N ) = t n t n-1 . . . t 0 .

Example 1: If N = 10 = 10010, then, in the transductor of Fibonacci adding machine, N corresponds to the path (T, 1/1, T ) (T, 00/01, I) (I, 10/00, I).

In the stochastic Fibonacci adding machine, we have the following paths (see Figure 2):

(1) (T, (1/1, 1), T ) (T, (0/0, 1), T )(T, (0/0, 1), T ) (T, (10/10, 1-p), I). In this case N = 10010 transitions to 10010 with probability 1 -p. (2) (T, (1/1, 1), T ) (T, (00/00, 1 -p), I) (I, (10/00, p), I). In this case N = 10 transitions to 10000 = 8 with probability p(1 -p). By using the transductor T p , we can prove the following result (see [5]).

Proposition 5.1. Let N be a nonnegative integer, then the following results are satisfied.

(1) N transitions to N with probability 1 -p.

(

) If N = ε k . . . ε 2 00, k ≥ 2, then N transitions to N + 1 = ε k . . . 2
tions to N - ∑ m i=0 F 2i = N -F 2m+1 +1 = ε k . . . ε t 00 10 . . . 10 2s-2m 0 . . . 00 2m-1 , 2 ≤ m ≤ s with probability p m-1 (1 -p). (5) If N = ε k . . . ε 3 001, k ≥ 3, then N transitions to N + 1 = ε k . . . ε 3 010
with probability p.

By Proposition 5.1, we construct the transition graph. We also find the transition operator S p associated to the transition graph. Remark 5.2. In [5], the authors prove that the point spectrum of S p in ℓ ∞ is equal to the set

            1 -p p 0 0 0 0 0 0 0 0 0 0 0 . . . 0 1 -p p 0 0 0 0 0 0 0 0 0 0 . . . p(1 -p) 0 1 -p p 2 0 0 0 0 0 0 0 0 0 . . . 0 0 0 1 -p p 0 0 0 0 0 0 0 0 . . . p(1 -p) 0 0 0 1 -p p 2 0 0 0 0 0 0 0 . . . 0 0 0 0 0 1 -p p 0 0 0 0 0 0 . . . 0 0 0 0 0 0 1 -p p 0 0 0 0 0 . . . p 2 (1 -p) 0 0 0 0 p(1 -p) 0 1 -p p 3 0
K p = {λ ∈ C, (q n (λ)) n≥1 is bounded }, where q F 0 (z) = z, q F 1 (z) = z 2 , q F k (z) = 1 p q F k-1 (z)q F k-2 (z) - 1 -p p
, for all k ≥ 2 and for all nonnegative integers n, we have q n (z) = q F k 1 . . . q F km where

F k 1 + • • • + F km is the Fibonacci representation of n. In particular, σ pt(Sp) is contained in the set E p = {λ ∈ C | (q Fn (λ)) n≥1 is bounded } = {λ ∈ C | (λ 1 , λ) ∈ J(g)}
where J(g) is the filled Julia set of the function g : C 2 → C 2 defined by: g(x, y) = ( 1 p 2 (x -1 + p)(y -1 + p), x) and λ 1 = 1 -p + (1-λ-p) 2 p . They also investigated the topological properties of E p .

Proof. The proof is similar to the proof of Proposition 4.10 and will be done in case ℓ α , α > 1. Let λ ∈ E p and let us prove that λ belongs to the approximate point spectrum of S p in ℓ α , α > 1. For every integer k ≥ 2, consider w (k) = (1, q 1 (λ), . . . , q k (λ), 0 . . . 0, . . .) t ∈ l α where (q k (λ)) k≥1 = (q k ) k≥1 is the sequence defined in the proof of Theorem 5. Indeed, if j ∈ {1, . . . F n -1}, then since i ≥ F n , we have (S p -λId) ij = p i,j . If the Fibonacci representation of j is j = ε k . . . ε 2 01 or j = ε k . . . ε t 10 . . . 0, it is easy to see by Proposition 5.1 that p i,j ̸ = 0 implies i < F n .

On the other hand, if j = 0 then B Fn,j = ∑ +∞ l=Fn p l,0 . Since p l,0 ̸ = 0 if and only l = F i -1 and p F i -1,0 = p ⌈i/2⌉ (1-p), we have B Fn,j ≤ 2 Since ||w (Fn) || α goes to infinity as n goes to infinity and (q Fn ) n≥0 is bounded, it follows that ||((S p -λId)u (Fn) )|| α converge to 0. Therefore λ belongs to the approximate point spectrum of S p . Thus the spectrum of S p acting on ℓ α α > 1, contains E p . This finishes the proof for ℓ α α > 1. The case ℓ 1 can be handled in the same way, the details being left to the reader. □ Open questions. We are not yet able to compute the residual and continuous spectrum of S p acting in the Banach spaces ℓ ∞ , c 0 , c or in ℓ α , α ≥ 1. We conjecture that σ(S p ) = E p . Moreover, in the case of ℓ ∞ we conjecture that the residual spectrum is empty and the continuous spectrum is the set E p \ K p . The difficulty here is that the matrix S p is not bi-stochastic. One may also look for a characterization of all real numbers 0 < p < 1 for which E p ̸ = K p .

Fig. 1 .

 1 Fig.1. Transition graph of stochastic adding machine in base 2

Fig. 2 . 4 .

 24 Fig.2. Transition operator of stochastic adding machine in base 2

Proposition 4 . 4 .

 44 The spectrum of S p in X, where X ∈ {c 0 , c, l α , 1 ≤ α ≤ +∞}, is contained in the filled Julia set of f .The main idea of the proof of Proposition 4.4 can be found in the Killen-Taylor proof. The key argument is that the S p 2 is similar to the operatorES p ⊕ OS p ,where S p = S p -(1 -p)Id p and E, O denote the even and odd operators acting on X by

Remark 4 . 9 .Proposition 4 . 10 .

 49410 By (19), it follows that λ belongs to σ r,X where X = c 0 or c or ℓ α , α > 1, implies lim |q n (λ)| = +∞. But this contradicts Proposition 4.4, which forces σ r,C 0 (S p ) = σ r,l α (S p ) = ∅. The following equalities are satisfied: σ c,c (S p ) = J(f )\{1}, σ c,c 0 (S p ) = σ c,l α (S p ) = J(f ) for all α > 1.

Fig. 4 .

 4 Fig.4. Transductor of Fibonacci adding machine

( 3 )Fig. 5 .

 35 Fig.5. Transductor of Fibonacci fallible adding machine

Fig. 6 .

 6 Fig.6. Transition graph of stochastic adding machine in Fibonacci base.

5 .

 5 Let u(k) = w (k) ||w(k) || α , then we have the following claim.Claim:lim n→+∞ ||(S p -λId)u (Fn) || α = 0.By using the same proof than Proposition 4.10, we have(S p -λId)u (Fn) j | α B Fn,jwhere D is a positive constant and B Fn,j = ∑ +∞ i=Fn |(S p -λId) ij |. We can prove by the same manner done in Proposition 4.10 that for 0 ≤ j ≤ F n , B Fn,j ̸ = 0 ⇐⇒ j = 0 or j = F n .

  p) = 2p m where m = ⌈(n + 1)/2⌉. Now assume j = F n . In this case, we have B Fn,j = |1 -p -λ| + +∞ ∑ i=Fn+1 p i,Fn . On the other hand, by Proposition 5.1, we deduce that p i,Fn ̸ = 0 if and only ifi = F n + F m -1 where 0 ≤ m ≤ n and p Fn+Fm-1,Fn = p [m/2] (1 -p). Therefore B Fn,Fn = |1 -p -λ| + n ∑ m=0 p ⌈m/2⌉ (1 -p) ≤ |1 -p -λ| + 2 (31) Hence (S p -λId)u (Fn) α α ≤ D 2p m + |q Fn | α (|1 -p -λ| + 2) ||w (Fn) || α α .

Acknowledgment 1 .

 1 The first author would like to express a heartful thanks to Albert Fisher, Eduardo Garibaldi, Paulo Ruffino and Margherita Disertori for stimulating discussions on the subjet. It is a pleasure for him to acknowledge the warm hospitality of the UNESP University in São José of Rio Preto, Campinas University and USP in São Paulo (Brazil) where a most of this work has been done. The second author would like to thanks University of Rouen where a part of this work has been realized.

Fig. 7 .

 7 Fig.7. p = 0.625

  ε 2 01 with probability p. (3) If N = ε k . . . ε t 00 1010 . . . 1010 2s with s ≥ 1 and k ≥ t ≥ 2s + 2, then N transitions to N + 1 = ε k . . . ε t 01 0 . . . 00 If N = ε k . . . ε t 0 0101 . . . 0101 2s , s ≥ 2 and k ≥ t ≥ 2s + 1, then N transitions to N + 1 = ε k . . . ε t 0 1000 . . . 000

		with probability p s+1 , and N
	transitions to N -	2s i=1 F 2i-1 = N -F 2m +1 = ε k . . . ε t 00 10 . . . 10 ∑ m	0 . . . 00	,
		2s-2m	2m
	1 ≤ m ≤ s with probability p m (1 -p).
	(4)

2s with probability p s , and N transi-

1 Research partially supported by French-Brasilian Cooperation (French CNRS and Brasilian CNPq) and Capes-Cofecub Project 661/10 . The second author was Supported by Brasilian CNPq grant 305043/2006-

Proposition 5.3. The operator S p is well defined in the Banach spaces c 0 , c and ℓ α , α ≥ 1. The point spectra of S p acting in the spaces c 0 , and ℓ α associated to the stochastic Fibonacci adding machines are empty sets. In c, the point spectrum equals {1}.

Proof. By Proposition 5.1, we can prove that the sum of coefficients of every column of S p is bounded by a fixed constant M > 0.

Indeed, let n ∈ N and s n = ∑ +∞ i=0 p i,n be the sum of coefficients of the n-th column.

If n = ε k . . . ε 2 01 or n = ε k . . . ε 3 010 (Fibonacci representation), then by 1), 2) and 5) of Proposition 5.1, we have

On the other hand, since S p is a stochastic matrix, then by Proposition 4.1, S p is well defined in the spaces c 0 , c (resp. in ℓ α , α ≥ 1). Now, let λ be an eigenvalue of S p in X where X ∈ {c 0 , c, ℓ α , α ≥ 1} associated to the eigenvector v = (v i ) i≥0 ∈ X. Since the transition probability from any nonnegative integer i to any integer i + k, k ≥ 2 is p i,i+k = 0 (see Proposition 5.1), the operator S p satisfies (S p ) i,i+k = 0 for all i, k ∈ N with k ≥ 2. Thus for every integer k ≥ 1, we have

Then, we can prove by induction on k that for any integer k ≥ 1, there exists a complex number c k = c k (p, λ) such that

Using the fact that the matrix S p is auto-similar, we can prove that c k = q k for all integers k ∈ N (see Theorem 1, page 303, [5]). Since

and (q Fn ) converges to 0 when n goes to infinity, we obtain that the point spectrum of S p acting in c 0 (resp. in ℓ α , α ≥ 1) is empty. Using the same idea than proposition 4.5, we see that σ pt,c = {1}. □ Remark 5.4. By Phillips Theorem and duality, it follows that the spectra of S p acting in X where X ∈ {c 0 , c, l 1 , l ∞ } associated to the stochastic Fibonacci adding machine are equals.

Theorem 5.5. The spectra of S p acting in X where X ∈ {l ∞ , c 0 , c, l α , α ≥ 1} contain the set E p = {λ ∈ C, (q Fn (λ)) n≥0 is bounded }.


Tewfik Lounis
email: tewfik.lounis@gmail.com

ASYMPTOTICALLY OPTIMAL TESTS WHEN PARAMETERS ARE ESTIMATED

Keywords: Local asymptotic normality, Contiguity, Efficiency, Stochastic models, Le Cam's third lemma, Time series models, ARCH models

The main purpose of this paper is to provide an asymptotically optimal test. The proposed statistic is of Neyman-Pearson-type when the parameters are estimated with a particular kind of estimators. It is shown that the proposed estimators enable us to achieve this end. Two particular cases, AR(1) and ARCH models were studied and the asymptotic power function was derived.

1. Introduction. Local asymptotic normality (LAN) for the log likelihood ratio was studied for a several classes of nonlinear time series model, from a LAN the contiguity property follows, for more details the interested reader may refer to [START_REF] Bickel | On adaptive estimation[END_REF], [START_REF] Swensen | The asymptotic distribution of the likelihood ratio for autoregressive time series with a regression trend[END_REF], and [START_REF] Cassart | Optimal detection of Fechner-asymmetry[END_REF]. Applying the contiguity property, we construct a statistic for testing a null hypothesis H 0 against the alternative hypothesis H (n) 1 , often a various classical test statistics depends on the central sequence which appears in the expression of the log likelihood ratio, in the case when the parameter of the time series model is known we obtain good properties of the test, precisely, the optimality, see for instance [9, Theorem 3]. However, in a general case, particularly in practice, the parameter is unspecified, in the expression of the estimate central sequence appears an additional term which is non degenerate asymptotically. The latter, alters the power function of the constructed test. In order to solve this very problem, and on a basis of an estimator of the unknown parameter, we introduce and define another estimator which does not effects asymptotically the power function of the test, more precisely the additional term is absorbed. The principle of this construction is to modify one of the component of the first estimator in order to avoid the additional term, the details of this method are expanded further in the section 2. The main purpose of this paper is to investigate the problem of testing two hypothesis corresponding to a stochastic model which is described in the following way. Let {(Y i , X i )} be a sequence of stationary and ergodic random vectors with finite second moment such that for all i ∈ Z, where Y i is a univariate random variable and X i is a d-variate random vector. We consider the class of stochastic models

Y i = T (Z i ) + V (Z i ) ǫ i , i ∈ Z, (1.1)
where, for given non negative integers q and s, the random vectors Z i is equal to (Y i-1 , Y i-2 , . . . , Y i-s , X i , X i-1 , . . . , X i-q ), the ǫ i 's are centered i.i.d. random variables with unit variance and density function f (•), such that for each i ∈ Z, ǫ i is independent of the filtration F i = σ(Z j , j ≤ i), the realvalued functions T (•) and V (•) are assumed to be unknown. We consider the problem of testing whether the bivariate vector of functions (T (•), V (•)) belongs to a given class of parametric functions or not. More precisely, let

M = (m(ρ, •), σ(θ, •)) , (ρ ′ , θ ′ ) ′ ∈ Θ 1 × Θ 2 , Θ 1 × Θ 2 ⊂ R ℓ × R p , Θ1 = ∅, Θ2 = ∅,
where for all set A, Å denotes the interior of the set A and the script " ′ " denotes the transpose, ℓ and p are two positive integers, and each one of the two functions m(ρ, •) and σ(θ, •) has a known form such that σ(θ, •) > 0. For a sample of size n, we derive a test of

H 0 : [(T (•), V (•)) ∈ M] against H 1 : [(T (•), V (•)) / ∈ M] . (1.2)
It is easy to see that the null hypothesis H 0 is equivalent to

H 0 : [(T (•), V (•)] = m(ρ 0 , •), σ(θ 0 , •) , (1.3)
while the alternative hypothesis H 1 is equivalent to

H 1 : [(T (•), V (•)] = m(ρ 0 , •), σ(θ 0 , •) , for some (ρ ′ 0 , θ ′ 0 ) ′ ∈ Θ 1 × Θ 2 .
In the sequel, our study will be focused on the following alternative hypotheses. For all integers n ≥ 1, the alternative hypothesis

H (n) 1
is defined by the following equation

H (n) 1 : [(T (•), V (•)] = m(ρ 0 , •) + n -1 2 G(•), σ(θ 0 , •) + n -1 2 S(•) , (1.4)
where G(•) and S(•) are two specified real functions. The situation is different in the case when the used statistic is the Neyman-Pearson test which is based on the log-likelihood ratio Λ n defined as follows

(1.5) Λ n = log f n f n,0 = n i=1 log(g n,i ),
where f n,0 (•) and f n (•) denote the probability densities of the random vector (Y 1 , . . . , Y n ) corresponding to the null hypothesis and the alternative hypothesis, respectively. The use of the Neyman-Pearson statistics needs to resort to the following conditions:

Under the hypothesis H 0 , there exists a random variable V n such that

V n D -→ N (0, τ 2 ),
where D -→ denotes the convergence in distribution and some constant τ > 0 depending on the parameter

φ 0 = (ρ ′ 0 , θ ′ 0 ) ′ , such that Λ n = V n (φ 0 ) - τ 2 (φ 0 ) 2 + o P (1). (1.6)
The equality (1.6) is a modified version of the LAN given by [9, Theorem 1]. We mention that there exist other versions of the LAN, we may refer to [START_REF] Cam | Locally asymptotically normal families of distributions. Certain approximations to families of distributions and their use in the theory of estimation and testing hypotheses[END_REF], [START_REF] Hall | On large-sample estimation and testing in parametric models[END_REF], and the references therein. On the basis of the LAN, an efficient test of linearity based on Neyman-Pearson-type statistics was obtained in a class of nonlinear time series models contiguous to a first-order autoregressive process AR(1) and its asymptotic power function is derived (see, [9, Theorem 1 and Theorem 3]).The expression of the obtained test depends on the central sequence V n (φ 0 ) which itself depends on the parameter φ 0 . In a general case the parameter φ 0 is unspecified, so, in order to estimate it, we introduce, under some assumptions, an estimate preserving, asymptotically, the power on Neyman-Pearson test when we replace, in the expression of the statistics, the parameter φ 0 by an appropriate estimator, φn . Say, this estimator will be constructed on the tangent space with the direction of the partial derivatives of the central sequences in φn , where φn is a √ n-consistent estimator of φ 0 .

In the sequel, φn will be called a modified estimate (M.E.). This paper is organized as follows: Section 2 describes the methodology used to construct the M.E. In Section 3, we give the asymptotic properties of the proposed estimate. In Section 4, we conduct a simulation in order to evaluate the power of the proposed test. All mathematical developments are relegated to the Section 5.

2. Estimation with modifying one component . Consider the problem of testing the two hypothesis H 0 against H (n) 1 which are given in (1.3) and (1.4) respectively and corresponding to the stochastic model (1.1). We assume that the LAN (1.6) of the model (1.1) is established, for example refer to [START_REF] Hwang | Nonlinear time series contiguous to AR(1) processes and a related efficient test for linearity[END_REF].

Let φn = (ρ ′ n , θ′ n ) ′ a √ n-consistent estimate of the parameter φ 0 = (ρ ′ 0 , θ ′ 0 ) ′ , where ρ′ n =
ρn,1 , . . . , ρn,ℓ , θ′ n = θn,1 , . . . , θn,p , ρ 0 ′ = ρ 1 , . . . , ρ ℓ and θ 0 ′ = θ 1 , . . . , θ p .

Our purpose is to construct another estimate φ′ n of the parameter (ρ ′ 0 ; θ ′ 0 ) ′ , such that the following fundamental equality is fulfilled

V n ( φn ) -V n ( φn ) = D n , (2.1)
where D n is a specified bounded random function. In the sequel, the functions (ρ, •) → m(ρ, •) and (θ, •) → σ(θ, •) are assumed to be twice differentiable. Our goal, is to find an estimate φn satisfying (2.1) pertaining to the tangent space Γ n , such that, for (X ′ , Y ′ ) ′ ∈ R ℓ × R p , the following equation holds With the connection with the equality (2.1), the new estimate is then given by imposing that the value (X ′ , Y ′ ) ′ satisfied the following identity

Γ n : V n ((X, Y )) -V n ( φn ) = ∂V ′ n ( φn ). (X -ρn ) ′ , (Y -θn )
D n = ∂V n ( φn ) ′ . (X -ρn ) ′ , (Y -θn ) ′ ′ . (2.2)
Clearly, the equation (2.2) has ℓ + p unknown values, so it has an infinity of solutions, after modification of the j n -th component of the first estimate ρn , we shall propose an element in tangent space Γ n which satisfies the equality (2.2). We obtain then a new estimate φ′ n = φ explains that we obtain the new estimate φn of the parameter φ 0 when we change in the expression of the estimate φn the j n component with respect to the first estimate ρn corresponding to the step n of the estimation. It follows from the equality (2.1) combined with the constraint (2.2) that

(1,jn) n ′ = (ρ ′ n , θ′ n ) ′ of
V n (φ (1,jn) n ) -V n ( φn ) = ℓ s=1 ∂V n (
n ′ = (ρ ′ n , θ′ n ) ′
, such that for t ∈ {1, . . . , p} θn,t = θn,t if t = k n and θn,kn = θn,kn .

we obtain ) is called a modified estimate in j n -th component with respect to the first estimate (respectively, in k n -th component with respect to second estimate), we denote this estimate by (M.E.).

V n (φ (2,kn) n ) -V n ( φn ) = ∂V n (
Remark 2.1. For each step n of the estimation corresponding a value of the position j n or k n of the component where the estimate was modified.

Properties of the (M.E.).

Consistency. Throughout, φn is a

√ n-consistent estimate of the unknown parameter φ 0 . The conditions (2.4) and (2.7) are not sufficient to get the consistency of the modified estimate (M.E.). In order to get its consistency, we need to resort to one of the following additional conditions .

(C.1) 1 √ n ∂V n ( φn ) ∂ρ jn P -→ c 1 as n → ∞, (C.2) 1 √ n ∂V n ( φn ) ∂θ kn P -→ c 2 as n → ∞,
where c 1 and c 2 are two constantes, such that c 1 = 0 and c 2 = 0.

Our first result concerning the consistency of the proposed estimate is summarized in the following proposition.

Proposition 3.1. Under (2.4) and (C.1) ((2.7) and (C.2), respectively), the estimate φ

(1,jn) n (φ (2,kn) n , respectively) is a √ n-consistent estimator of the unknown parameter φ 0 .
In practice, it is not easy to verify the condition (C.1) (respectively, (C.2)), in the case when the unknown parameter φ 0 is univariate, a sufficient condition will be stated in Lemma (3.1), in this case, we need the following assumption:

(C.3) : For all real sequence (η n ) n≥1 with values in the interval [0, 1], we have:

1 √ n Vn (η n φ 0 + (1 -η n ) φn )) = O P (1),
where Vn is a second derivative of V n .

Remark 3.1. In a problem of testing the two hypothesis H 0 against H (n) 1 , and when the error ǫ i 's are centered i.i.d. and ǫ 0 D -→ N (0, 1), a large classe of time series model satisfied the condition (C.3), for instance, we cite the nonlinear time series contiguous to AR(1) processes , the details are expanded further later in the proofs of the Propositions (3.3) and (3.4) . Now, we may state the sufficient condition which implies assumptions (C.1) corresponding to the case when the parameter of the time series model is univariate.

Lemma 3.1. Let φn be a √ n-consistent estimate of the parameter φ 0 . Let c 1 be a constant, such that c 1 = 0, then we have:

(i) Under (C.3), if 1 √ n Vn (φ 0 ) P -→ c 1 , as n → ∞, then ∀A > 0, P 1 √ n Vn ( φn ) -c 1 > A → 0, as n → ∞.
3.2. Absorbtion of the error. Consequently, with the modified estimate and in the case when the error between two central sequences is bounded, it is possible to absorb this error, this result is stated and proved in the following proposition. Proposition 3.2. Let φ′ n be an estimate ( √ n consitency) of the parameter (ρ ′ , θ ′ ) ′ . We assume that there exists a known bounded function D n , such that

V n ( φn ) = V n (φ 0 ) -D n + o P (1). (3.1)
Then, there exists an estimate φn ′ of (ρ ′ , θ ′ ) ′ such that

V n ( φn ) = V n (φ 0 ) + o P (1).
Remark 3.2. The equality (3.1) gives the link between the estimated central sequences V n ( φn ) and the central sequence V n (φ 0 ). Sometimes it is not easy to establish the form of the function D n , in the next section, we propose, under some assumptions, how to specify this function in the two cases, i.e., the case when the problem of testing the linearity and nonlinearity of the s-th order and the case of time series model with conditional heteroscedasticity respectively corresponding to the equalities

Y i = ρ 0 Y i-1 + αG(Y (i -1)) + ǫ i and Y i = ρ 0 Y i-1 + αG(Y (i -1)) + 1 + βB(Y (i -1)) ǫ i , respectively, where Y (i -1) = Y i-1 , Y i-2 , .
. . , Y i-s , α and β are real parameters and the ǫ i 's are centered i.i.d. random variables with unit variance and density function f (•).

Throughout, we assume that the function f (•) is positive with a third derivative, we denote by ḟ (•), f (•) and f (3) (•) the first, the second and the third derivative respectively. For all x ∈ R, let

M f (x) = ḟ (x) f (x) .
According to the notation (1.5), we suppose that the three following conditions are satisfied :

• (L.1): max 1≤i≤n |g n,i -1| = o P (1), • (L.2): there exists a positive constante τ 2 such that n i=1 (g n,i -1) 2 = τ 2 + o P (1), • (L.3): there exists a-F n mesurable V n satisfying n i=1 (g n,i -1) = V n + o P (1), where V n D -→ N (0, τ 2 ).
Conditions (L.1), (L.2) and (L.3) imply under H 0 the local asymptotic normality LAN corresponding to the equality (1.6), for more details see ([9, Theorem 1]). This last theorem is the fundamental tool used later to aim to establish the LAN for the considering models.

3.3.

Link between central sequences in nonlinear time series contiguous to AR(1) processes. Consider the s-th order (nonlinear) time series

Y i = ρ 0 Y i-1 + α G(Y (i -1)) + ǫ i , |ρ 0 | < 1. (3.2)
In this case and with the comparison to the equality (1.1), we have

Z i = Y i , T (Z i ) = ρ 0 Y i-1 + α G(Y (i -1)) and V (Z i ) = 1.
In the sequel, it will be assumed that the model is a stationary and ergodic time series with finite second moment. We consider the problem of testing the null hypothesis H 0 : α = 0 against the alternative hypothesis H

(n) 1 : α = n -1 2
, with the comparison to (1.3) and (1.4), we have

m(ρ 0 , Y i-1 ), σ(θ 0 , Y i-1 ) ′ = ρ 0 Y i-1 , 1 ′ , M = {m(ρ, •), ρ ∈ Θ 1 } , Z i ′ = Y i-1 , . . . , Y i-s and S(•) = 0.
Note that this problem of testing is equivalent to test the linearity of the s-th AR(1) time series model when (α = 0) against the nonlinearity of the s-th AR(1) time series model when (α = n -1 2 ). Throughout, the scripts " • ℓ+p " , " • ℓ " and " • p " denote the euclidian norms in R ℓ+p , R ℓ and R p respectively. It will be assumed that the conditions (A.1) and (A.2) are satisfied, where • (A.1): There exists positive constants η and c such that for all u with u ℓ+p > η, G(u) ≤ c u ℓ+p . • (A.2): for a location family {f (ǫ ic), -∞ < c < -∞}, there exist a square integrable functions Ψ 1 , Ψ 2 and a constant δ such that for all ǫ i and |c| < δ, such that :

d k f (ǫ i -c) f (ǫ i ) dc k ≤ Ψ k (ǫ i ), for k = 1, 2.
Under the conditions (A.1) and (A.2) the LAN of the time series model (3.2) was established in ([9, Theorem 2]), the proposed test T n is the Neyman-Pearson statistic which is given by the following equality

T n = I V n (ρ 0 ) τ (ρ 0 ) ≥ Z(α) , where τ 2 = E(M 2 f (ǫ 0 ))E(G 2 (Y (0))),
and Z(α) is the (1α)-quantile of a standard normal distribution Φ(•). In this case, the central sequence is given by the following equality

V n (ρ 0 ) = - 1 √ n n i=1 M f (ǫ i )G(Y (i -1)), where τ 2 = E(M 2 f (ǫ 0 ))E(G 2 (Y (0))),
and such that under

H 0 , V n (ρ 0 ) D -→ N (0, τ 2 )
. The asymptotic power of the test is derived and equal to 1 -Φ(Z(α)τ 2 ), recall that when ρ 0 is known, this test is asymptotically optimal, for more details see [START_REF] Hwang | Nonlinear time series contiguous to AR(1) processes and a related efficient test for linearity[END_REF]Theorem 3]. Our aim is to specify the form of the function D n which is defined in (3.1), the parameter ρ 0 is estimated by the √ n-consistent estimator ρn and the residual ǫ i is estimated by ǫi,n = Y i -Y i-1 ρn . We have the following statement:

Proposition 3.3. Assume that the conditions (A.1) and (A.2) hold and ǫ i 's are centered i.i.d. and ǫ 0 D -→ N (0, 1). We have

V(ρ n ) = V n (ρ 0 ) -D n + o P (1), (3.3)
where

D n = -c 1 √ n(ρ n -ρ 0 ), (3.4) ρn = D n Vn (φ n ) + ρn and c 1 = -E Y 0 G(Y (0)) . (3.5) Remark 3.3.
• The use of the ergodicity of the model imposes to require the condition E Y -1 G(Y 0 ) < ∞, therefore we choose the function G(•) in order to get this condition. For instance, we shall choose

G(Y (i -1)) = 2a 1+Y 2 i-1
, where a = 0.

• With this choice of the function G, the condition (A.1) remains satisfied, in fact, we can remark that |G(u)| ≤ 2|a|, then for all u with

u ℓ+p ≥ η we have G(u) ≤ 2a × u ℓ+p × 1 u ℓ+p ≤ 2a η × u ℓ+p , therefore, we shall choose c = 2a η .

An extension to ARCH processes. Consider the following time series model with conditional heteroscedasticity

Y i = ρ 0 Y i-1 + α G(Y (i -1)) + 1 + βB(Y (i -1)) ǫ i , i ∈ Z. (3.6)
It is assumed that the model (3.6) is ergodic and stationary. It will be assumed that the conditions (B.1), (B.2) and (B.3) are satisfied, where • (B.1): The fourth order moment of the stationary distributions of (3.6) exists.

• (B.2):

There exists a positive constants η and c such that for all u with

u ℓ+p > η, B(u) ≤ c u 2 ℓ+p . • (B.3): for a location family {b -1 f ( ǫ i -a b ), -∞ < a < -∞, b > 0}
, there exists a square integrable function ϕ(•), and a strictly positive real ς, where ς > max(|a|, |b -1|), such that,

∂ 2 b -1 f ǫ i -a b f (ǫ i ) ∂a j ∂b k ≤ ϕ(ǫ i ),
where j and k are two positive integers such that j + k = 2.

We consider the problem of testing the null hypothesis H 0 against the alternative hypothesis H

(n) 1 such that H 0 : m(ρ, Z i ) = ρ 0 Y i-1 and σ(θ 0 , •) = 1, H (n) 1 : m(ρ, Z i ) = ρ 0 Y i-1 + n -1 2 G(Y (i -1)) and σ(θ 0 , Z i ) = 1 + n -1 2 B(Y (i -1)).
Remark that H 0 , H

correspond to α = β = 0 (linearity of (3.6)) and α = β = n -1 2 (non linearity of (3.6)) with the comparison to the equality (1.1), we have

Z i = Y i , T (Z i ) = ρ 0 Y i-1 + α G(Y (i -1)) and V (Z i ) = 1 + βB(Y (i -1)).
Note that when n is large, we have

σ(θ 0 , Z i ) = 1 + n -1 2 B(Y (i -1)) ∼ 1 + n -1 2 2 B(Y (i -1)) = 1 + n -1 2 S(Y (i -1)).
Under the conditions (A.1), (B.1), (B.2), and (B.3), the LAN was established in [9, Theorem 4], an efficient test is obtained and its power function is derived. In this case, the central sequence is given by the following equality

V n (ρ 0 ) = - 1 √ n n i=1 M f (ǫ i )G(Y (i -1)) + n i=1 (1 + ǫ i M f (ǫ i ))B(Y (i -1)) , such that under H 0 , V n (ρ 0 ) D -→ N (0, τ 2 ),
where

τ 2 = I 0 E (G(Y (0)) 2 + (I 2 -1) 4 E (B(Y (0)) 2 + I 1 E (G(Y (0))B(Y (0))
where

I j = E ǫ j 0 M 2 f (ǫ 0 ) and j = 0, 1, 2.
The proposed test is then given by

T n = I V n (ρ 0 ) τ (ρ 0 ) ≥ Z(α) . (3.7)
By the subsisting ρ 0 by its √ n-consistent estimator ρn in the expression of the central sequence, we shall state the following proposition:

Proposition 3.4. Suppose that the conditions (A.1), (B.1), (B.2) and (B.3) hold and ǫ i 's are centered i.i.d. and ǫ 0 D -→ N (0, 1). We have

V(ρ n ) = V n (ρ 0 ) -D n + o P (1), (3.8)
where

D n = -c 1 √ n(ρ n -ρ 0 ), (3.9) ρn = D n Vn (φ n ) + ρn and c 1 = -E Y 0 G(Y (0)) . (3.10)
3.5. Optimality of the proposed test. Throughout, Tn and τ are the statistics test and the constant respectively obtained with the subsisting of the unspecified parameter φ 0 by its modified estimate φn in the expression of the test (3.7) and the constant τ appearing in the expression of the log likelihood ratio (1.6) respectively. We assume in the problem of testing the two hypothesis H 0 against H (n) 1 that the LAN of the the model (1.1) is established, in order to prove the optimality of the proposed test. To this end, we need the following assumption :

(E.1) There exists a √ n-estimate φn of the unknown parameter φ 0 and a random bounded function D n , such that

V n ( φn ) = V n (φ 0 ) -D n + o P (1).
It is now obvious from the previous definitions that we can state the following theorem: Theorem 3.1. Under LAN and the conditions (2.4) (respectively, (2.7)), (C.1) ((C.2), respectively) and (E.1) the asymptotic power of Tn under

H n 1 is equal to to 1 -Φ(Z(α) -τ 2 ).
Furthermore, Tn is asymptotically optimal.

We shall now apply this last theorem in order to conduct simulations corresponding to the representation of the derived asymptotic power function. The concerned model is the Nonlinear time series contiguous to AR [START_REF] Bertail | Un test bootstrap dans un modèle AR(1)[END_REF] processes with an extension to ARCH processes.

Simulations.

In this section, we assume that ǫ i 's are centered i.i.d. and ǫ 0 D -→ N (0, 1), in this case, we have E(ǫ i ) = 0, E(ǫ 2 i ) = 1, and E(ǫ 4 i ) = 3. We treat the case when the unknown parameter φ 0 = ρ 0 ∈ Θ 1 ⊂ R, under H 0 , the considering time series model can also rewritten

Y i = ρ 0 Y i-1 + ǫ i where |ρ 0 | < 1. (4.1)
4.1. Nonlinear time series contiguous to AR(1) processes. To evaluate the performance of our estimator, we provide simulations with comment in this section. In the case when the parameter ρ 0 is known, the test T n is optimal and its power is asymptotically equal to 1-Φ(Z(α)-τ 2 ), for more details see [START_REF] Hwang | Nonlinear time series contiguous to AR(1) processes and a related efficient test for linearity[END_REF]Theorem 3]. In a general case, when the parameter ρ 0 is unspecified, firstly, we estimate it with the least square estimates ρn =

n i=1 Y i Y i-1 n i=1 Y 2 i-1
, secondly, with the use of the (M.E.) under the conditions (2.4) and (C.1), the modified estimate ρn exists and remains √ n-consistent, making use of (2.5) in connection with the Proposition (3.3) it follows:

ρn = D n Vn (ρ n ) + ρn = -c 1 (ρ n -ρ 0 ) Vn(ρn) √ n + ρn , (4.2)
with the substitution of the parameter ρ 0 by its estimator ρn in (3.7), we obtain the following statistics test

Tn = V n (ρ n ) τ (ρ n ) ≥ Z(α) where τ 2 = E(M 2 f (ǭ 0,n ))E(G 2 (Y 0 )),
and ǭ0,n = Y 0 -Y -1 ρn .

It follows from Theorem (3.1) that Tn is optimal with an asymptotic power function equal to 1 -Φ(Z(α)τ 2 (ρ n )).

We choose the function G like this G :

x 1 , x 2 , •••, x s , x s+1 , x s+2 , •••, x s+q -→ 5a 1+x 2 1
where a = 0. In our simulations, the true value of the parameter ρ 0 is fixed at 0.1 and the sample sizes are fixed at n = 30, 40, 80 and 400, for a level α = 0.05, the power relative for each test estimated upon m = 1000 replicates, we represent simultaneously the power test with a true parameter ρ 0 , the empirical power test which is obtained with the replacing the true value ρ 0 by its estimate (M.E.) ρn corresponding to the equality (4.2), and the empirical power test which is obtained with the subsisting the true value ρ 0 by its least square estimator LSE ρn (an estimator with no correction), we remark that, the two representations with the true value and the modified estimate M.E. are close for large n.

ARCH processes.

With the substitution of the parameter ρ 0 by its modified estimate ρn , in (3.7), we obtain the following test

Tn = I V n (ρ n ) τ (ρ n ) ≥ Z(α) , such that τ 2 = Ī0,n E (G(Y (0)) 2 + ( Ī2,n -1) 4 E (B(Y (0)) 2 + Ī1,n E (G(Y (0))B(Y (0)) , Īj,n = E ǭj 0,n M 2 f (ǭ 0,n ) , j = 0, 1, 2, and ǭ0,n = Y 0 -Y -1 ρn .
In our simulations, the true value of the parameter ρ 0 is fixed at 0.1 and the sample sizes are fixed at n = 30, 40, 80 and 200, for a level α = 0.05, the power relative for each test estimated upon m = 1000 replicates. We choose the functions G and B like this

G = B : x 1 , x 2 , • • •, x s , x s+1 , x s+2 , • • •, x s+q -→ 3.5a 1+x 2 1
where a = 0. We represent simultaneously the power test with a true parameter ρ 0 and the empirical power test which is obtained with the subsisting the true value ρ 0 by its estimate (M.E.) ρn corresponding to the equality (4.2),we represent simultaneously the power test with a true parameter ρ 0 , the empirical power test which is obtained with the subsisting the true value ρ 0 by its estimate (M.E.) ρn corresponding to the equality (4.2), and the empirical power test which is obtained with the subsisting the true value ρ 0 by its least square estimator LSE ρn (estimator with no correction), we remark that, when n is large, we have a similar conclusion as the previous case . Remark 4.1. We mention that the limiting distributions appearing in Proposition (3.3) and Proposition (3.4) depend on the unknown quantity b n = (ρ nρ 0 ), i.e., in practice ρ 0 is not specified, in general. To circumvent this difficulty, we use the Efron's Bootstrap in order to evaluate b n , more precisely, the interested reader may refer to the following references : [START_REF] Efron | Bootstrap methods: another look at the jackknife[END_REF] for the description of the Bootstrap methods, [START_REF] Bertail | Un test bootstrap dans un modèle AR(1)[END_REF], [START_REF] Kvam | Nonparametric statistics with applications to science and engineering[END_REF] for the Bootstrap methods in AR(1) time series models and [START_REF] Fryzlewicz | Normalized leastsquares estimation in time-varying ARCH models[END_REF] for the ARCH models. x is continuous on R -{0}, it follows that the random variable Consider again the equality (5.1), since the function (x, y) → x + y is continuous on R ℓ+p ×R ℓ+p , it results from (5.3) that φ (1,jn) n converges in probability to φ 0 as n → ∞. Notice that the last previous convergences in probability follow immediately with the use of the continuous mapping theorem, for more details, see [START_REF] Billingsley | Convergence of probability measures[END_REF] or [START_REF] Van Der Vaart | Asymptotic statistics, volume 3 of Cambridge Series in Statistical and Probabilistic Mathematics[END_REF]. By following the same previous reasoning, we shall prove the consistency of the estimate φ We deduce that √ n(φ (1,jn) n φ 0 ) = O P (1). (5.5) Notice that with a similar argument and with changing φ In order to prove Lemma 3.1, we need to stated the following classical lemmas: Lemma 5.1. Let (X i ) i∈{1,...,l} be a sequence of a positive random variables on the probability space (Ω, F, P ), (α i ) i∈{1,...,l} a sequence of a positive (strictly) reals such that l i=1 1 α i = 1, then we have, for each ǫ > 0,

1 1 √ n ∂Vn( φn) ∂ρ jn P -→ 1 c 1 , then the couple 1 √ n D n ;
P l i=1 X i > ǫ ≤ l i=1 P X i > ǫ α i .
Lemma 5.2. Let (X n ) n≥0 be a sequence of a random variables on the probability space (Ω, F, P ), such that X n = O P (1), then X 2 n = O P (1).

Proof of the Lemma 5.1. Firstly, we remark that, ∀ǫ > 0, we have

l i=1 X i > ǫ ⊂ n i=1 X i > ǫ α i .
In fact, we suppose there exists

ω ∈ l i=1 X i > ǫ and ω / ∈ n i=1 X i > ǫ α i , then for each i ∈ {1, . . . , l}, we have X i (ω) ≤ ǫ α i , which implies that l i=1 X i (ω) ≤ ǫ,
hence a contradiction. With the use of the σ-additivity, we obtain

P l i=1 X i > ǫ ≤ P n i=1 X i > ǫ α i ≤ l i=1 P X i > ǫ α i .
Proof of Lemma 5.3. For all A > 0, we have:

P (|Y n | > A) = P (|X n -c| > A) → 0, as n → ∞.
Proof of Lemma 5.2. For all ǫ > 0, ∃M 1 > 0 such that: sup α (P (|X α | > M 1 ) < ǫ, this implies that sup α (P |X α | 2 > M 2 1 < ǫ, therefore with the choice of M = M 1 , we obtain the result.

Proof of Proposition 3.3. ǫ i 's are centered i.i.d. and ǫ 0 D -→ N (0, 1), making use of the results of [9, Theorem 2], we have

V n (ρ 0 ) = - 1 √ n n i=1 M f (ǫ i )G(Y (i -1)).
The estimated central sequence is

V n (ρ n ) = - 1 √ n n i=1 M f (ǫ i,n )G(Y (i -1)).
By Taylor expansion with order 2, we have : Note that

V n (ρ n ) -V n (ρ 0 ) = Vn (ρ n )(ρ n -ρ 0 ) + 1
R n = 1 2 Vn ( ρn )(ρ n -ρ 0 ) 2 = 1 2 √ n 1 √ n Vn ( ρn ) √ n(ρ n -ρ 0 ) 2 .
Since the estimator ρn is √ n-consistent and with the use of Lemma (5.2), it results that √ n(ρ nρ 0 ) By a simple calculus and since the the function f is the density of the standard normal distribution, it is easy to prove that the quantity 2 Ṁf (ρ) is bounded, therefore, there exists a positive constant w such that 2 Ṁf (ρ) ≤ w, then

| 1 √ n Vn (ρ)| ≤ w 1 n n i=1 Y 2 i-1 |B(Y (i -1))|.
With the choice B(Y (i -1)) = 2a

1+Y 2 i-1
with a = 0, it results that

| 1 √ n Vn (ρ)| ≤ 2w|a| 1 n n i=1 Y 2 i-1 .
By the use of the ergodicity of the model and since the model is with finite second moments, it follows that the random variable It follows from the convergence in probability of the estimate φn to φ 0 , the continuity of the function τ : • -→ τ (•) and the application of the continuous mapping theorem see, for instance ( [START_REF] Van Der Vaart | Asymptotic statistics, volume 3 of Cambridge Series in Statistical and Probabilistic Mathematics[END_REF]) or [START_REF] Billingsley | Convergence of probability measures[END_REF], that asymptotically, the power of the test is not effected when we replace the unspecified parameter φ 0 by it's estimate, φn , hence the optimality of the test. The power function of the test is asymptotically equal to 1 -Φ(Z(α)τ 2 ( φn )), the proof is similar as [9, Theorem 3].

(2. 5 ).

 5 In summary, we define the modified estimate by φ′ n = φ(1,jn) n ′ = ρn,1 , . . . , ρn,jn-1 , ρn,jn , ρn,jn+1 , . . . , ρn,ℓ , θn,1 , . . . , θn,p ′ With a same reasoning as the previous case and after modifying the k nth component with respect to the second estimate, we shall define a new estimate φn ′ = φ(2,kn)

5 .

 5 Proof of the results. Proof of the Proposition 3.1. Consider the following fundamental decomposition: (φ (1,jn) n ) ′ = ( φn ) ′ + (O jn ) ′ , (5.1) where O ′ jn = (O jn,i ) ′ i∈{1,...,ℓ+p} , such that O jn,i = 0 when i = j n , and O jn,jn = ρn,jnρn,jn . Firstly, we have φn P -→ φ 0 , secondly we can deduce from (2.5) that: O jn,jn = D n ∂Vn( φn) ∂ρ jn D n is bounded, we can remark that 1 √ n D n P -→ 0, from (C.1), there exists some constante c 1 = 0, such that 1 √ n ∂Vn( φn) ∂ρ jn P -→ c 1 , from (2.4) and since the function x → 1

converges in probability to the couple 0 ; 1 c 1 ,

 1 since the function (x, y) → xy is continuous on R × R, it result from (5.2), that the random variable O jn,jn P -→ 0 c 1 = 0, therefore O jn ′ = (0, . . . 0, O jn,jn , 0 . . . 0) ′ P -→ (0, . . . 0, 0, 0 . . . 0) ′ . (5.3)

.

  Note that φ (1,jn) n is √ n-consistent estimate of the parameter φ 0 and √ n(φ (1,jn) n φ 0 ) = O P (1),where O P (1) is bounded in probability in R ℓ+p . In fact, it follows from (5.1) that √ n(φ(1,jn) nφ 0 ) = √ n( φnφ 0 ) + √ nO jn = O P (1condition (C.1), it results that √ nO jn = O P 1 (1), where O P 1 (1) is bounded in probability in R.

2 Y

 2 Vn ( ρn )(ρ nρ 0 ) 2 , (5.11)where ρn is a point between ρ 0 and ρn and Vn i-1 G(Y (i -1)).

2 =

 2 O P (1), from the assumption (C.3), it follows that R n = o P (1), finally we deduce that,V n (ρ n ) -V n (ρ 0 ) = Vn (ρ n )(ρ nρ 0 ) + o P (1). (5.12)Proof of Proposition 3.4. The assumption (C.1) remains satisfied and the proof is similar as the proof of Proposition(3.3), in this case, for all ρ ∈ Θ 1 , B(Y (i -1))2 Ṁf (ρ).

1

 1

  where k is some constant, this implies that the condition (C.3) is straightforward .Proof of theTheorem 3.1. From the conditions (2.4) ((2.7), respectively), (C.1) ((C.2), respectively), it results the existence and the √ n-consistency of the modified estimate estimate φn corresponding to the equation (2.5) ((2.8), respectively). The combinaison of the condition (E 1 ) and the Proposition (3.2) enable us to get under H 0 the following equality V n ( φn ) = V n (φ 0 ) + o P (1). This last equation implies that with o P (1), the estimate central and central sequences are equivalent, in the expression of the test (3.3), the replacing of the central sequence by the estimate central sequence has no effect. LAN implies the contiguity of the two hypothesis (see, [5, Corrolary 4.3]), by Le Cam third lemma's (see for instance, [8, Theorem 2]), under H (τ 2 , τ 2 ).

  , . . . , ρn,ℓ , θn,1 , . . . , θn,kn-1 , θn,kn , θn,kn+1 , . . . , θn,p

	it follows from the equality (2.2) combined with (2.7), that
	(2.8)	θn,kn =	D n ∂Vn( φn)	+ θn,kn .
			∂θ kn
	In summary, we obtain the modified estimate
	φ′ n = φ (2,kn) n ′ = ρn,1 The estimate φ (1,jn) n (respectively, φ (2,kn) n
	(2.6)			φn ) ∂θ kn	( θn,kn -θn,kn ).
	Under the following condition
	(2.7)	∂V n ( φn ) ∂θ kn	= 0,

′

.

Proof of the Lemma 3.1. In this case φ 0 = ρ 0 ∈ Θ 1 ⊂ R, we denote by ρn the √ n-consistent estimator of ρ 0 .

Let A > 0, from the triangle inequality combined with the Lemma (5.1),we obtain:

Firstly, we have

where ρn is a point between ρ 0 and ρn , then there exists a sequence η n with values in the interval [0, 1], such that ρn =

this last inequality enable us to concluded that ρn is √ n-consistency estimator of ρ 0 , it follows from (C.3) applied on the equality (5.9) that

Thus we obtain (i).

Proof of Proposition 3. In order to prove the Proposition (3.3), we need a following classical result. Lemma 5.3. Let (Ω, F, P ) be a probability space, (X n ) n≥1 is a sequence of real random variables on Ω. If X n converges in probability to a constant c, then, there exists a sequence of random variable (Y n ) n , with X n = c + Y n , such that, Y n converges in probability to 0.

This implies that

, (5.13) where ρn is between ρn and ρ 0 , and Vn is the second derivative of V n . From the assumption (C.3), we have

With the use of (5.14), the equality (5.12) can also rewritten

therefore from the Lemma (5.3), there exists a random variable X n , X n

We deduce from the equality (5.15) and the √ n-consistence of the estimator ρn , that

V n (ρ n ) -V n (ρ 0 ) = c 1 √ n(ρ nρ 0 ) + o P (1) = -D n + o P (1), (5.16) where D n = -c 1 √ n(ρ nρ 0 ). Recall that the second derivative Vn is equal to 0, this implies that the assumption (C.3) is satisfied.