Philippe Rambour
email: philippe.rambour@universite-paris-saclay.fr

Asymptotic of the eigenvalues of Toeplitz matrices with even symbol

Keywords: Mathematical Subject Classification (2020) Primary 47L80 ; Secondary 47A08, 47A10, 47A15, 47G30 Toeplitz matrices, operator eigenvalues

come

Introduction and statement of the main results

For h ∈ L 1 (T) we denote by T N (h) the Toeplitz matrix of order N with symbol h. It is the (N + 1) × (N + 1) matrix such that, for N ≥ k, l ≥ 0, (T N (h)) k+1,l+1 = ĥ(k -l) where ĥ(u) is the Fourier coefficient of order u of h ( [START_REF] Grenander | Toeplitz forms and their applications[END_REF][START_REF] Böttcher | Introduction to large truncated Toeplitz matrices[END_REF]). For h a real valued function the matrix T N (h) is an Hermitian Toeplitz matrix. We here consider symmetric Toeplitz matrix, which is equivalent to assuming that the symbol h is an even function and we denote by λ

(1)

N ≤ λ (2) N ≤ • • • ≤ λ (N +1) N
the eigenvalues of the (N + 1) × (N + 1) matrix T N (h). This paper adresses the asymptotic behavior of the eigenvalues of T N (h) as N goes to the infinity. This is a topic which has attracted mathematicians and physicists for a long time. Toeplitz matrices and their relatives emerge in particular in statistic [START_REF] Dahlhaus | Efficient parameter estimation for self-similar processes[END_REF] and in statistical physics [START_REF] Basor | Toeplitz determinants and Statistical Mechanics[END_REF][START_REF] Basor | The Fisher-Hartwig conjecture and Toeplitz eigenvalues[END_REF] and the questions about the asymptotic behavior of their spectral characteristics, especially their determinants, eigenvalues, and eigenvectors, are always at the heart of the matter. We refer to the papers [START_REF] Deift | Toeplitz matrices and Toeplitz determinants under the impetus of the ising model. some history and some recent results[END_REF] for an extensive list of references. According to the first Szegö limit theorem (see [START_REF] Grenander | Toeplitz forms and their applications[END_REF]) the eigenvalues of T N (h) are asymptotically distributed as the value of h; see [START_REF] Grenander | Toeplitz forms and their applications[END_REF] for L ∞ symbols, [START_REF] Tyrtyshnikov | Toeplitz eigenvalues for Radon measures[END_REF] for L 1 symbols, and [START_REF] Trench | Asymptotic distibution of the spectra of a class of generalized Kac-Murdoch-Szegö matrices[END_REF], [START_REF] Tyrtyshnikov | Distribution of the eigenvalues and singular numbers of toeplitz matrices under weakened requirements on the generating function[END_REF] for more general situations. In the Hermitian case extensive works has been done on the search for eigenvalues (or the extreme eigenvalues) of Toeplitz matrices [START_REF] Widom | On the eigenvalues of certain hermitian operators[END_REF][START_REF] Grenander | Toeplitz forms and their applications[END_REF][START_REF] Serra | On the extreme eigenvalues of Hermitian (block) Toeplitz matrices[END_REF][START_REF] Rinkel | Inverses et propriétés spectrales des matrices de Toeplitz à symbole singulier[END_REF][START_REF] Parter | Extreme eigenvalues of Toeplitz forms and applications to elliptic difference equations[END_REF][START_REF] Parter | On the extreme eigenvalues of truncated Toeplitz matrices[END_REF][START_REF] Parter | On the extreme eigenvalues of Toeplitz matrices[END_REF] and more recently, for instance, [START_REF] Bogoya | Eigenvalues of Hermitian Toeplitz matrices with smooth simple-loop symbols[END_REF][START_REF] Bogoya | From convergence in distribution to uniform convergence[END_REF][START_REF] Bogoya | Eigenvalues of Hermitian Toeplitz matrices with polynomially increasing entries[END_REF][START_REF] Barrera | Eigenvalues of even very nice Toeplitz matrices can be unexpectedly erratic[END_REF][START_REF] Ekström | Eigenvalues and eigenvectors of banded Toeplitz matrices and the related symbols[END_REF][START_REF] Ekström | Exact formulae and matrix-less eigensolvers for block banded symmetric Toeplitz matrices[END_REF][START_REF] Rambour | Une extension d'un résultat de Szegö sur les valeurs propres des matrices de Toeplitz[END_REF]. The results of theorem 1 are consistent with those of theorem 2.3 of [START_REF] Bogoya | Eigenvalues of Hermitian Toeplitz matrices with smooth simple-loop symbols[END_REF]. But the method of proof is different and the first three points of the statement concern functions which are outside the framework of [START_REF] Bogoya | Eigenvalues of Hermitian Toeplitz matrices with smooth simple-loop symbols[END_REF]. The corollaries 1, 2 and 3 (which are immediate consequences of Theorem 1) cannot be obtained from the results of [START_REF] Bogoya | Eigenvalues of Hermitian Toeplitz matrices with smooth simple-loop symbols[END_REF]. These corollaries give results for Toeplitz matrices with trigonometric symbol, which are useful for statistic applications. On the other hand Theorem 1 indicates that the problem of the eigenvalues of Toeplitz matrices is a local problem, related to the variation of the function which is the symbol of the matrix. Here we denote by A + ([0, 2π]) (resp. A -([0, 2π])) the set of even differentiable periodic functions of period 2π, such that f (θ) > 0 (resp. f (θ) < 0) for all θ in ]0, π[. Here we must consider a function f ∈ L 1 (T). For [θ 1 , θ 2 ] ⊂ [0, π] we say that

f ∈ A + ([θ 1 , θ 2 ]) (resp. f ∈ A -([θ 1 , θ 2 ]
) if f is a differentiable, periodic and even function such that f (θ) > 0 (resp. f (θ) < 0) for all θ ∈ [θ 1 , θ 2 ] and also ∀θ ∈

[0, π] f (θ) ∈ [f (θ 1 , f (θ 2 )] ⇐⇒ θ ∈ [θ 1 , θ 2 ] (resp. f (θ) ∈ [f (θ 2 , f (θ 1 )] ⇐⇒ θ ∈ [θ 1 , θ 2 ]).
For ν ≥ 0 we denote by W ν the weighted Wiener algebra of all functions ψ T → C whose Fourier coefficients satisfy

ψ ν = j∈Z | ψ(j)| (|j + 1|) ν < ∞.

Now we define the functions

H(θ 1 , θ 2 ) = f (θ 1 ) -f (θ 2 ) (1 -cos θ 1 ) -(1 -cos θ 2 ) and ρ(θ) = - 1 4π P.V. 2π 0 ln ((H(t, θ)) tan t-θ 2 dt + 1 4π P.V.
Theorem 1 Let f in C 4 [0, 2π] such that f ∈ A + ([θ 1 , θ 2 ]) for an interval [θ 1 , θ 2 ] ⊂ [0, π]. We have the following results. i) If [θ 1 , θ 2 ] ⊂]0, π[ and [a, b] ⊂]θ 1 , θ 2 [ then for N be a sufficiently large integer T N (f ) has k b,N -k a,N + 1 eigenvalues in f ([a, b]). For k an integer in [k a,N , k b,N ] the corresponding eigenvalue is λ(k) N = f kπ N + 2 + c 1 (N + 2) + c 2 (N + 2) 2 + O 1 (N + 2) 3 , uniformly in λ, with c 1 = f kπ N + 2 ρ kπ N + 2 and c 2 = f kπ N + 2 ρ kπ N + 2 ρ kπ N + 1 + 1 2 f kπ N + 2 ρ kπ N + 2 2 .
ii) If θ 1 = 0 and b < θ 2 < π then for a sufficiently large N the matrix

T N (f ) has k b,N eigenvalues in f ([0, b]), with 0 < λ (1)
N < λ

(2)

N < • • • < λ k b,N N ≤ f (b). For k ∈ [1, k b,N
] the corresponding eigenvalue is as in the point i).

iii) If θ 2 = π and 0 < a < π then for a sufficiently large N the matrix

T N (f ) has N +1-k a,N eigenvalues in f ([a, π]) with f (a) ≤ λ (k a,N ) N < λ (k a,N +1) N < • • • < λ N +1 N < f (π). For k ∈ [k a,N , N + 1] the corresponding eigenvalue is as in the point i). iv) If θ 1 = 0 θ 2 = π then for a sufficiently large N the matrix T N (f ) has N + 1 eigenvalues in f ([0, π]) with f (0) < λ (1)
N < λ

(2)

N < • • • < λ (N +1) N < f (π). For k ∈ [1, N + 1
] the corresponding eigenvalue is as in the point i).

Remark 1 Similar results to Theorem 1 holds for the case where

f in A -([θ 1 , θ 2 ]) for an interval [θ 1 , θ 2 ] ⊂ [0, π]. Remark 2 In the point i) of Theorem 1 λ(k) N is not necessarily the k-th largest eigenvalue of f . This is nevertheless true in the cases 0 = θ 1 < θ 2 < π, 0 < θ 1 < θ 2 = π, or 0 = θ 1 < θ 2 = π.
Remark 3 Revisiting the proof of Theorem 1, one can show that under the assumption f ∈ C 3 ([0, 2π]) we obtain the different point of the theorem with the formula

λ(k) N = f kπ N + 2 + c 1 (N + 2) + O log N (N + 2) 2 ,
uniformly in λ and c 1 as in Theorem 1

The following corollaries are tree interesting consequences of Theorem 1. For these corollaries we recall that a function c is said to be a regular function iff c is a 2π-periodic function, with c(θ) > 0 for all θ ∈ [0, 2π] and c ∈ L 1 ([0, 2π]). In the following we denote by λ

(1) N,α ≤ λ (2) N,α ≤ • • • ≤ λ (k) N,α ≤ • • • ≤ λ (N +1) N,α
the eigenvalue of the function h α : θ → |1 -e iθ | 2α c(θ) where c is an even regular function and α a positive real.

Corollary 1 Consider a real α ≥ 2 and a function h α : θ → |1 -e iθ | 2α c(θ) where c is an even regular function such that

• c ∈ C 4 ([0, 2π]) • c (θ) > 0 for all θ ∈]0, π[.
Then for a sufficiently large N the matrix T N (h α ) has N + 1 eigenvalues such that

λ (k) N,α = h α kπ N + 2 + c 1,α (N + 2) + c 2,α (N + 2) 2 + O 1 (N + 2) 3 , for 1 ≤ k ≤ N + 1 and where c 1,α = h α kπ N + 2 ρ α kπ N + 2 c 2,α = h α kπ N + 2 ρ α kπ N + 2 ρ α kπ N + 1 + 1 2 h α kπ N + 2 ρ α kπ N + 2 2 with ρ α (θ) = - 1 4π P.V. 2π 0 ln ((H α (t, θ)) tan t-θ 2 dt + 1 4π P.V. 2π 0 ln ((H α (t, θ)) tan t+θ 2 dt
and

H α (θ 1 , θ 2 ) = h α (θ 1 ) -h α (θ 2 ) (1 -cos(θ 1 )) (1 -cos(θ 2 )) for θ 1 , θ 2 , θ in [0, π].
Using the remark 3 we can write the following corollary.

Corollary 2 Let α be a real such that α ∈ [ 3 2 , 2[ and a function h α : θ → |1 -e iθ | 2α c(θ) where c is an even regular such that

• c ∈ C 3 ([0, 2π]) • c (θ) > 0 for all θ ∈]0, π[. Then T N (h α ) has N + 1 eigenvalues in f ([0, π]) with λ (k) N,α = h α kπ N + 2 + c 1,α (N + 2) + O log N (N + 2) 2 ,
where N + 1 ≥ k ≥ 1 and c 1 as in the corollary 1.

Now if we remark that

(1 -cos θ 1 )c(θ 1 ) -(1 -cos θ 2 )c(θ 2 ) (1 -cos θ 1 ) -(1 -cos θ 2 ) = c(θ 1 ) + (1 -cos θ 2 ) c(θ 1 ) -c(θ 2 ) (1 -cos θ 1 ) -(1 -cos θ 2 )
we obtain the following corollary.

Corollary 3 Consider a function h 1 : θ → |1 -e iθ | 2 c(θ) where c is an even regular function such that

• c ∈ C 4 ([0, 2π]) • c (θ) > 0 for all θ ∈]0, π[.
Then for all integers k we have

λ (k) N,1 = h 1 kπ N + 2 + c 1 (N + 2) + c 2 (N + 2) 2 + O 1 (N + 2) 3 , where λ (k)
N,1 is the k-th largest eigenvalue of T N (h 1 ) and c 1 , c 2 as in Corollary 2 .

Remark 4 For c = 0 the proof of Theorem 1 provides c 1 = c 2 = 0 and we find again the classical result given in [START_REF] Grenander | Toeplitz forms and their applications[END_REF] Our result can also be compared with that of Trench [START_REF] Trench | Interlacement of the even and odd spectra of real symetric Toeplitz matrices[END_REF] where it is proved that for this class of symbols the eigenvalues are all distinct.

Proof of Theorem 1

In this proof we have used an inversion formula adapted for the Toeplitz matrix with symbol f = g 1 g 2 , g 1 = χ 0 (1 -r χ0 χ) (P N +1 ) -1 , g 2 = (1 -r χ0 χ) (P N +1 ) -1 , where 0 < r < 1, and P N +1 a polynomial with degree N + 1 without zeros on D = {z/|z| ≤ 1}, χ is the function θ → e iθ , χ 0 = e iθ 0 , with θ 0 ∈ R (see [START_REF] Rambour | Une extension d'un résultat de Szegö sur les valeurs propres des matrices de Toeplitz[END_REF] for the statement and the proof of this formula). In the appendix of this article we give this formula and briefly recall how we use it to calculate the inverse of a Toeplitz matrix (N + 1)

× (N + 1) with symbol χ 0 (1 -χ0 χ) (1 -χ0 χ) 1
|P N +1 | 2 . Now we need to recall the definition of the predictor polynomial and its main property.

Definition 1

The predictor polynomial of degree M of a regular function h is the trigonometric polynomial K M defined by

K M = M k=0 (T M (h)) -1 k+1,1 (T M (h)) -1 1,1 χ k .
Property 1 For all integers j, such that -M ≤ j ≤ M we have

1 |K M | 2 (j) = ĥ(j).
For an integer k we denote by β k,N the coefficient of χ k in the predictor polynomial P N of f . The reader can consult [START_REF] Landau | Maximum entropy and the moment problem[END_REF] for the predictor polynomials. We will now give the following definition, which will be used in the demonstration.

Definition 2

We denote by H + is the set of all functions ϕ in L 2 (T) whose Fourier coefficients satisfy φ(j) = 0 for all j < 0.

We can now begin the demonstration of the theorem. Using the assumptions we can write

f (θ) = f 1 (1 -cos θ) where f 1 is a differentiable function strictly increasing on [0, 2]. For all λ in [f (θ 1 ), f (θ 2 )] we put θ λ = f -1 (λ) and λ = f -1 1 (λ), that means θ λ = arccos(1 -λ ), since λ ∈ [0, 2] and θ λ ∈ [0, π]. We have clearly, for λ ∈ [f (θ 1 ), f (θ 2 )], f (θ) -λ = f 1 (1 -cos θ) -λ = ((1 -cos θ) -(1 -cos θ λ )) H λ (θ)
where

H λ : θ → H(θ, θ λ ) is a regular function on [-π, π]. We can write ((1 -cos θ) -(1 -cos θ λ )) = (1 -cos θ) -λ = 1 2 |1 -χ| 2 -2λ . (1)
If χ λ = e iθ λ and χ = e iθ we have 2 and we can write the equation (1) as

χ λ = (1 -λ ) + i 1 -(λ -1)
((1 -cos θ) -(1 -cos θ λ )) = - 1 2 χ λ (1 -χλ χ)(1 -χλ χ). ( 2
)
Remark 5 If we denote by P N +1,λ the predictor polynomial of H λ the main property of these polynomials (see the property 1) allows to write that the (N + 1) × (N + 1) Toeplitz matrices of the respective symbol

((1 -cos θ) -(1 -cos θ λ )) H λ and -1 2 χ λ (1 -χλ χ)(1 -χλ χ) 1 |P N +1,λ | 2
are the same.

For N a fixed integer we denote by T 1,N,λ the quantity (T N (f ) -λI N ) -1 1,1 . We have

T 1,N,λ = det (T N -1 (f ) -λI N -1 ) det (T N (f ) -λI N )
and, since the eigenvalues of (T N -1 (f )) are not in Spec (T N (f )) (see [START_REF] Haugazeau | Application du théorème de sylvester à la localisation des valeurs propres ax = λbx dans le cas symétrique[END_REF][START_REF] Barbaresco | Analyse spectrale par décomposition récursive en sous-espaces propres via les coefficients de réflexion[END_REF]), we have

λ ∈ Spec (T N (f )) ⇐⇒ 1 T 1,N,λ = 0.
Using the remark ( 5) and the inversion formula of Toeplitz matrices obtained in [START_REF] Rambour | Une extension d'un résultat de Szegö sur les valeurs propres des matrices de Toeplitz[END_REF] (the reader can consult the appendix to have an explicit expression of this formula) we obtain the entry

T -1 N (f ) (1,1)
. Hence with the results of the appendix we can write

1 T 1,N,λ = 1 - χ2(N+1) λ τ N (χ λ ) 1 - χ2(N+1) λ τ N (χ λ ) B 2,N,λ -B 1,N,λ . (3)
With

τ N (θ λ ) = PN+1,λ (χ λ )P N +1,λ (χ λ ) PN+1,λ (χ λ )P N +1,λ (χ λ ) ,
and

B 1,N,λ = P N +1,λ (0) P N +1,λ (χ λ ) PN+1,λ (χ λ ) 2 (1 -χ2 λ ) -1 .
For χ λ = -1, 1 the quantity B 1,N,λ is defined and the definition of the predictor polynomial provides that P N +1,λ (0) = det(T N (H λ )) det(T N +1 (H λ )) . Since det (T N (H λ )) = 0 for all N ∈ N * (see [START_REF] Grenander | Toeplitz forms and their applications[END_REF]) and P N +1,λ (e iθ ) = 0 for all θ ∈ R (see [START_REF] Landau | Maximum entropy and the moment problem[END_REF]) we have B 1,N,λ = 0 for χ λ = -1, 1. The constant B 2,N,λ will be is defined for

χ λ = -1, 1. χ λ ∈ {-1, 1} ⇐⇒ λ ∈ {0, 2} ⇐⇒ λ ∈ {min T f, max T f }.
and with the additional remark that neither min f nor max f cannot be eigenvalues of f we can consider that B 2,N,λ is defined. But in any case this constant does not matter in our

demonstration. Indeed if 1 - χ2(N+2) λ τ N (χ λ ) B 2,N,λ -B 1,N,λ = 0 this equality means that det (T N -1 (f ) -λI N -1 ) = 0 and λ is an eigenvalue of T N -1 (f ) so it cannot be an eigenvalue of T N (f ). Hence we can write λ ∈ (Spec (T N (f )) ∩ I) ⇐⇒ χ 2(N +2) λ = τ N (θ λ ), λ ∈ I (4)
Since the function H λ is even, the constant τ N (θ λ ) can be rewritten as

τ N (θ λ ) = P N +1,λ ( χλ ) P N +1,λ (χ λ ) 2 .
As the function θ λ →

P N +1,λ ( χλ )
P N +1,λ (χ λ ) is continuous from I to {z||z| = 1} we have a non negative function ρ N defined and continuous on I such that τ N (θ λ ) = e 2iρ N (θ λ ) Then the equation 4 can be write

λ ∈ (Spec (T N (f )) ∩ I) ⇐⇒ χ 2(N +2) λ = e 2iρ N (θ λ ) , λ ∈ I. (5)
Now according to the construction of χ λ we have necessary, if k 1,N and k 2,N as in the introduction

λ ∈ (Spec (T N (f )) ∩ I) ⇐⇒ λ = 1 -cos ρ N (θ λ ) + kπ (N + 2) for k ∈ [k 1,N , k 2,N ] , λ ∈ I.
(6) Lastly if we consider the equation the equation

θ λ = ρ N (θ λ ) + kπ (N + 2) ( 7
)
it is clear that ( 7) implies ( 6). Now we have to make a more precise study of the function ρ N . If s ≥ 0, then every function f ∈ A(T, s) without zeros on T admits a Wiener-Hopf factorization, that is, there exist function f + and f -such that f (e iθ ) = f + (e iθ )e iγθ f -(e iθ ) with some γ ∈ Z the index of the factorization. The function f + (resp. f -) belongs to set A(T, s) + (resp. A(T, s) -) where

A(T, s) + = f (e iθ ) = +∞ j=0 f (j)e ijθ | f s < +∞ and A(T, s) -= f (e iθ ) = +∞ j=0 f (-j)e -ijθ | f s < +∞ .
Here we have clearly can be written in the form :

1 |P N +1,λ (e iθ )| 2 = 1 P N +1,λ (e iθ ) 1 P N +1,λ (e -iθ
1 |P N +1,λ (e iθ )| 2 + = exp   1 2 log 1 |P N +1,λ (e iθ )| 2 + 1 2πi P.V. T log 1 |P N +1,λ (e iv )| 2 e iv -e iθ dv   that be rewritten as exp   1 2 log 1 |P N +1,λ (e iθ )| 2 - 1 4πi P.V. 2 π 0 log 1 |P N +1,λ (e iu )| 2 tan u-θ 2 du + 1 4π 2 π 0 log 1 |P N +1,λ (e iu )| 2 du   .
that provides

P N +1,λ (e -iθ ) P N +1,λ (e iθ ) = e iρ N,λ (θ) with ρ N,λ (θ) = - 1 4π P.V. 2 π 0 log 1 |P N +1,λ (e iu )| 2 tan u-θ 2 du + 1 4π P.V. 2 π 0 log 1 |P N +1,λ (e iu )| 2 tan u+θ 2 du and finally ρ N (θ λ ) = ρ N,λ (θ λ ). The same methods give, for G λ = (H λ ) + G λ (e iθ ) = (H λ (θ)) + = exp 1 2 log (H λ (v)) + 1 2πi P.V. T log (H λ (v))
e iv -e iθ dv and ρ λ (θ) = -P.V. 1 4π

2 π 0 log (H λ (u)) tan u-θ 2 du + P.V. 1 4π 2 π 0 log (H λ (u)) tan u+θ 2 du, ρ(θ λ ) = ρ λ (θ λ ).
Now we need the following lemmas to relate the two functions ρ N and ρ.

Lemma 1 For all λ ∈ [f (θ 1 , f (θ 2 )] λ = f (0), f (π) the function θ → H λ (θ) is in C 3 ([0, 2π])
and for all j ∈ {0, 1, 2, 3} we have a real K j no depending from λ such that

H (j) λ ∞ ≤ K j Remark 6
The hypothesis λ = f (0), f (π) corresponds to the fact that the maximum and minimum of the function f cannot be eigenvalues of T N (f ). The values f 0) and f (π) are therefore never considered in the proof of Theorem 1 .

Proof of the lemma 1 : If t = 1-cos θ and t λ = 1-cos θ λ we have to prove that the function

H 1,λ : t → f 1 (t)-f 1 (t λ ) t-t λ is in C 3 ([0, 2π]
) and that for all integer j, 0 ≤ j ≤ 3 we have a real K 1,j such that, for all λ in [f (θ 1 ), f (θ 2 )] we have

H (J) 1,λ ∞ ≤ K 1,j . Clearly H 1,λ | ∞ ≤ f (1) ∞ . Now we have for t = t λ H (1) 1,λ (t) = f (1) 1 (t)(t -t λ ) -(f 1 (t) -f 1 (t λ )) (t -t λ ) 2 = f (1)
1 (t λ ) + (t -t λ )f (2) 1 (a 1 ) (t -t λ ) -f (1)
1 (t λ )(t -t λ ) + (t-t λ ) 2 2 f (2) 1 (a 2 ) (t -t λ ) 2
with a 1 and a 2 between t and t λ . That provides

• H (1) 1,λ (t λ ) = f (2) 1 (t λ ) 2 • H (1) 1,λ ∞ ≤ 3 2 f (2) 1 ∞ Now we have, for t = t λ H (2) 1,λ (t) = f (2) 1 (t)(t -t λ ) 2 -2 f (1) 1 (t)(t -t λ ) -(f 1 (t) -f 1 (t λ )) (t -t λ ) 3 = f (2) 1 (t λ ) + f (3) 1 (a 3 )(t -t λ ) (t -t λ ) 2 -2 (d 1,λ (t) -d 2,λ (t)) (t -t λ ) 3 .
with

d 1,λ (t) = f (1) 1 (t λ (t -t λ ) + f (2) 1 (t λ )(t -t λ ) 2 + f (3) 1 (a 4 ) (t -t λ ) 3 2 d 2,λ (t) = f (1) 1 (t λ (t -t λ ) + f (2) 1 (t λ ) (t -t λ ) 2 2 + f (3) 1 (a 5 ) (t -t λ ) 3 6
and a 3 , a 4 , a 5 between t and t λ . That provides

• H (2) 1,λ (t λ ) = f (3) (t λ ) 3 • H (2) 1,λ ∞ ≤ 7 3 f (3) 1 ∞
Finally we can write, always for t = t λ

H (3) 1,λ (t) = f (3) 1 (t)(t -t λ ) 3 -3 f (2) 1 (t)(t -t λ ) 2 -2 f (1) 1 (t)(t -t λ ) -(f 1 (t) -f 1 (t λ )) (t -t λ ) 4 = f (3) 1 (t λ ) + f (4) 1 (a 6 )(t -tλ) (t -t λ ) 3 -3 (d 3,λ (t) -2d 4,λ (t)) (t -t λ ) 4 with d 3,λ (t) = f (2) 1 (t λ )(t -t λ ) 2 + f (3) 1 (t λ )(t -t λ ) 3 + f (4) 1 (a 7 ) (t -t λ ) 4 2 d 4,λ (t) = f (1) 1 (t λ) (t -t λ ) + f (2) 1 (t λ )(t -t λ ) 2 + f (3) 1 (t λ ) (t -t λ ) 3 2 + f (4) 1 (a 8 ) (t -t λ ) 4 6
and a 6 , a 7 , a 8 between t and t λ . This last equalities gives us

• H 1,λ (t λ ) = 1 4 f (4) 1 (t λ ) • H 1,λ ∞ ≤ 15 4 f 1 ∞ .
that ends the proof. 2

Lemma 2 We have a real H no depending from k and λ such that

1 G λ (k) ≤ H k 3 , for k > 0, and
G λ Ḡλ (k) ≤ H k 3 . for k = 0 Proof : We can observe that, for 0 ≤ j ≤ 3 (π + (log H λ )) (j) = π + (log H λ ) (j)
. Hence, with the lemma 1, we have, for 0 ≤ j ≤ 3,

(π + (log H λ )) (j) 2 ≤ (log H λ ) (j) 2 ≤ T j (8) If m 0 is the minimum of H on [0, 2π] × [θ 1 , θ 2 ]
it is clear that for all j ∈ {0, 1, 2, 3} T j is only depending from the constants m 0 , K 0 , K 1 , K 2 , K 3 , of the lemma 1. Hence T j is no depending from λ.

On the other hand since log

H λ ∈ C 3 ([0, 2π]) we have, for all n ≥ 0 | log H λ (n)| ≤ (log H λ ) (3) 2 n 3 ≤ T 3 n 3 and | (log H λ ) (1) (n)| ≤ (log H λ ) (3) 2 n 2 ≤ T 3 n 2 . Hence exp (π + (log H λ )) ∞ ≤ exp   T 3 n≥0 1 n 3   = M 1 and (π + (log H λ )) (1) ∞ ≤ T 3 n≥0 1 n 2 = M 2 . Now if we put π + (log H λ ) = F λ we can write (exp -F λ ) (3) = -F (3) λ + 3F (1)
λ F

(2)

λ -F (1) λ 3 exp F λ .
Clearly we have the inequalities

F (3) λ exp F λ 2 ≤ F (3) λ 2 exp F λ ∞ ≤ T 2 M 1 F (1)
λ F

(2)

λ exp F λ 2 ≤ F (1) λ ∞ exp F λ ∞ F (2) λ 2 ≤ M 1 M 2 T 2 (F (1)
λ ) 3 exp F λ 2 ≤ (F (1) λ ) 3 ∞ exp F λ ∞ ≤ M 3 2 M 1
according to [START_REF] Böttcher | Introduction to large truncated Toeplitz matrices[END_REF] this means that (exp F λ ) (3) 2 , is bounded by a constant H 1 no depending from λ. This result implies

| exp -F λ (n)| ≤ (exp -F λ ) (3) 2 n 3 ≤ H 1 n 3
for all n ≥ 0, and H no depending from λ and n. On the other hand for n > 0 we have

G λ Ḡλ (n) = h≥0 G λ (h + n) 1 Ḡλ (-h) ≤ H 2 1 h>0 1 h 3 n 3 + 1 n 3 | 1 Ḡλ (0)|,
and

G λ Ḡλ (-n) = k≥0 G λ (k) 1 Ḡλ (-(k + n)) ≤ H 2 1 k>0 1 k 3 n 3 + 1 n 3 | 1 Ḡλ (0)|,
with the additional remark that | 1 Ḡλ (0)| ≤ 1 m 0 with m 0 as previous we have the lemma with

H = max H 1 , H 2 1 k>0 1 k 3 + 1 m 0 , where m 0 = min{H(θ 1 , θ 2 )|(θ 1 , θ 2 ) ∈ [0, 2π]×[f (θ 1 ), f (θ 2 )]}. 2 Lemma 3 If β k,λ = 1 G λ (k) we have, for a sufficient large N ((T N ((H λ )) -1 k,1 = β 0,λ β k,λ + R k,N,λ with |R k,N,λ | ≤ M N 2 (N +1-k) 2
where M is no depending from λ and k.

Proof :

Using the inversion formula given in the appendix of this paper we obtain, for

H λ = G λ Ḡλ , G λ ∈ H + (T). (T N (H λ )) -1 l+1,k+1 = π + χ l Ḡλ | χ k Ḡλ - +∞ s=0 H * Φ N,λ H Φ N,λ s π + ΦN,λ π + χ l Ḡλ |π + ΦN,λ π + χ k Ḡλ . with Φ N,λ = G λ Ḡλ χ N +1 , and ΦN,λ = λ G λ χ -(N +1)
and (see the appendix for H + , π + , π -)

H Φ N,λ (Ψ) = π -(Φ N,λ Ψ) for Ψ ∈ H + H * Φ N,λ (Ψ) = π + ( ΦN,λ Ψ) for Ψ ∈ (H + ) ⊥ . For l = 0 this formula becomes (T N (H λ )) -1 k+1,1 = π + 1 Ḡλ | χ k Ḡλ - +∞ s=0 H * Φ N,λ H Φ N,λ s π + ΦN,λ π + 1 Ḡλ |π + ΦN,λ π + χ k Ḡλ .
In the next of the proof we use the following notations :

1 G λ = u≥0 β u,λ χ u G λ Ḡλ = u∈Z γ u,λ χ u .
With lemma 2 we have a positive constant H such that

|β u,λ | ≤ H u 3 ∀u ∈ N and |γ u,λ | ≤ H u 3 ∀u ∈ Z .
With these notations we obtain

π + 1 Ḡλ | χ k Ḡλ = β0,λ β k,λ π + ΦN,λ π + 1 Ḡλ = π + ΦN,λ β0,λ = β0,λ v≥N +1 γ-v,λ χ v-N -1 , π + ΦN,λ π + χ k Ḡλ = k w=0 βw,λ   v≥N +1-k+w γ-v,λ χ v-N -1+k-w   .
Hence, with lemma 2 we obtain

π + ΦN,λ π + 1 Ḡλ 2 ≤ H (N + 1) -2 and π + ΦN,λ π + χ k Ḡλ 2 ≤ H((N + 1 -k) -2 .
On the other hand for ψ = w≥0 α w χ w a function in H + we have, with the continuity of the projection π -,

H Φ N,λ (ψ) = w≥0 α w v>N +1+w γ -v,λ χ v+w+N +1
that provides

H Φ N,λ (ψ) 2 ≤ w≥0 |α w | v>N +1+w |γ -v,λ | ≤ ψ 2   w≥0 v>N +1+w |γ -v,λ | 2   1/2 ≤ H ψ 2 (N + 1) -3/2
that means H Φ N,λ ≤ H(N + 1) -3/2 . Clearly we have also H Φ N,λ ≤ H(N + 1) -3/2 and we can write

+∞ s=0 H * Φ N,λ H Φ N,λ s π + ΦN,λ π + 1 Ḡλ 2 ≤ H (1 -H 2 (N + 1) -3 ) 2 (N + 1) -2 .
And finally we can write

(T N (H λ )) -1 1,k+1 = β0,λ β k,λ + O (N + 1) -2 (N + 1 -k) -2 with O (N + 1) -2 (N + 1 -k) -2 = 2H 2 (N + 1) -2 (N + 1 -k) -2
uniformly in λ that is the expected result with M = 2H 2 . 2

Remark 7 As the coefficient β 0,λ is real the form of τ N (χ λ ) allows to assume that β 0,λ = 1 is the rest of our demonstration.

Lemma 4 For a sufficiently large N we have two constants, no depending from λ such that

1 P N +1,λ 0 ≤ A 1 and G λ 0 ≤ A 2
Proof : As previously we have

P -1 N +1,λ 0 ≤ +∞ u=1 1 u 2 (P -1 N +1,λ ) (2) 2 .
We write

(P -1 N +1,λ ) (2) = D N +1,λ P 3 N +1,λ
.

The lemmas 2 and 3 provide that D N +1,λ is bounded by a constant no depending from λ and θ. Since G λ ((θ) = 0 we have a real m 1 > 0 such that for all θ ∈ [0, 2π] and λ ∈ [f (θ 1 ), f (θ 2 )] we can write |G λ (θ)| > m 1 . That implies, with the lemma 3, that we have a real m 1 > 0, no depending from θ and λ and such that

1 |P N +1,λ (e iθ )| 3 < m 1
, that provides the constant A 1 . Lastly we can obtain, as in the lemma 2, | G λ (k)| ≤ M 0 k 3 with M 0 a constant no depending from λ, that provides the existence of the real A 2 .

2 Lemma 5 For s 0 = 0 we have ln

1 |P N +1,λ | 2 -ln (H λ ) s 0 = O 1 N 2 uniformly in λ.
Proof : Using the lemma 3, we obtain

P N +1,λ - 1 G λ s 0 ≤ M (N + 1) -2 N k=0 1 (N + 1 -k) 2 + +∞ k=N +1 |β k,λ | Hence P N +1,λ - 1 G λ s 0 ≤ M + H (N + 1) 2 (9)
where M and β k,λ as in the lemma 3 and H is the real no depending from N and from λ which has been introduced in the lemma 2. always with M and H no depending from λ and the norm

P N +1,λ -1 G λ s 0 is bounded by O 1 N 2 . Now since ΨΦ s 0 ≤ Ψ s 0 Φ s 0 we have 1 P N +1,λ -G λ s 0 ≤ P N +1,λ - 1 G λ s 0 G λ P N +1,λ s 0 ≤ A M + H (N + 1) 2 (10)
with

A = A 1 A 2 , A 1 , A 2 as in the lemma 4.
Then, according to (10)

1 |P N +1,λ | 2 -H λ s 0 ≤ 1 |P N +1,λ | 2 - 1 P N +1,λ Ḡλ s 0 + 1 P N +1,λ Ḡλ -H λ s 0 ≤ (A 1 +A 2 ) M + H (N + 1) 2 Since 1 |P N +1,λ | 2 -H λ s 0 = O 1 N 2 , uniformly in λ. Now observe that ln 1 |P N +1,λ | 2 -ln (H λ ) s 0 = ln 1 + 1 |P N +1,λ | 2 -H λ H λ s 0 that is also ln 1 + 1 |P N +1,λ | 2 -H λ H λ s 0 ≤ n≥1 1 n 1 |P N +1,λ | 2 -H λ s 0 n 1 H λ s 0 n . Now we have 1 H λ s 0 ≤ 1 H λ s 0 ≤ 1 | H λ (0)| ≤ 1 m 0
with m 0 as in the lemma 2. That gives us ln 1 +

1 |P N +1,λ | 2 -H λ H λ s 0 ≤ n≥1 1 n 1 |P N +1,λ | 2 -H λ s 0 n 1 m n 0 .
Since m 0 no depending from λ we can conclude

ln 1 |P N +1,λ | 2 -ln (H λ ) s 0 = ln 1 + 1 |P N +1,λ | 2 -H λ H λ s 0 = O 1 N 2
uniformly in λ.

2

Then the fact that the Cauchy singular operator is bounded on the Wiener classes A(T, s), s ≥ 0, with ρ N -

ρ s 0 = O( 1 N 2 ) that implies that |ρ N (θ λ ) -ρ(θ λ )| = O( 1 N 2 ) uniformly in λ. Hence we have two reals m, M with M ≥ m such that m ≤ ρ N (θ) ≤ M for all θ ∈ [θ 1 , θ 2 ]
and all N ∈ N. Then for an integer k ∈ {k a,N , • • • , k b,N } we have, for a sufficiently large N , and for all θ ∈ [θ 1 , θ 2 ],

θ 1 < kπ + m N + 2 < kπ + ρ N (θ) N + 2 < kπ + M N + 2 < θ 2
and we have two reals θ 0 and θ 0 such that

θ 0 < kπ+m N +2 < kπ+ρ N (θ) N +2
and kπ+ρ N (θ)

N +2 < kπ+M N +2 < θ 0 . Hence the function Ψ k : θ λ → ρ N (θ λ )+kπ N +2
-θ λ changes sign on [θ 1 , θ 2 ] then, by continuity of the function ρ N on this interval, the equation 7 has at less one solution on this interval. If θ 1 = 0 we can remark that we have necessary, for all k ∈ {1, • • • , k 2,N } and for all θ ∈ [0, θ 2 ] the inequality

0 ≤ 1 -cos ρ N (θ) + kπ N + 2
and always :

1 -cos ρ N (θ) + kπ N + 2 ≤ 1 -cos kπ + M N + 2 ≤ 1 -cos θ 0 .
Hence we can conclude that the equation ( 6 and always :

1 -cos θ 0 < 1 -cos kπ + m N + 2 < 1 -cos kπ + ρ N (θ) N + 2 .
Hence we can conclude that the equation ( 6) has at less one solution on [θ 0 , π]. Finally we have obtain the eigenvalues in the interval [f (a), f (b)]. Now we have to obtain the asymptotic behavior of these eigenvalues and to prove that we have k b,N -k a,N + 1 eigenvalues in this interval Now we need the two following lemmas :

Lemma 6 The function ρ is in C 2 ([θ 1 , θ 2 ]).
Proof : We prove the result for the function A simple calculus provides us I 1,θ = 0. On the other hand we can observe that the function

I : θ → P.V.
Ψ : (t, θ) → log(H(t,θ))-log(H(θ,θ)) tan( t-θ 2 )
can be write log(H(t,θ))-log(H(θ,θ)) are continuous on [0, 2π] × [θ 1 , θ 2 ] we can conclude that the function (t, θ) → ∂Ψ ∂θ is defined and continuous on [0, 2π]×[θ 1 , θ 2 ], that conclude this demonstration for the existence of ρ (1) . For ρ (2) the function ∂Ψ 1 ∂θ (t, θ) which is defined for all t = θ and is (t -θ) 2 Ψ 2 (t, θ) + 2(-θ)Ψ 3 (t, θ) + 2Ψ 3 (t, θ) (t -θ) 3 where, with same notation as for ρ ,

Ψ 1 (t, θ) = (log H) θ 2 (t, θ) -(log H) t 2 (θ, θ) -(log H) θ 2 (θ, θ) -2(log H) t,θ (θ, θ) Ψ 2 (t, θ) = (log H) θ (t, θ) -(log H) t (θ, θ) -(log H) θ (θ, θ) Ψ 3 (t, θ) = (log H)(t, θ) -(log H)(θ, θ)
and for t = θ we see that

∂ 2 Ψ 1 ∂θ 2 = (log H) (3)
θ 2 ,t (θ, θ) + (log H) (3)
t 3 (θ, θ) + (log H) (3)
t 2 θ (θ, θ)(θ, θ).
Then the same arguments as previously allow us to conclude the proof. 2

Now we can easily obtain the formula announced in the statement of the Theorem 1. For λ an eigenvalue in ]f (θ 1 ), f (θ 2 )[ we have, following the equation (7

) λ = f kπ+ρ N (θ λ ) N +2
that is also

λ = f kπ+ρ(θ λ ) N +2 + R N,λ (N +2) with R N,λ (N +2) = O 1 (N +2) 3
uniformly in λ according to the lemma 5. Putting d = πk N +2 we have by Taylor's theorem,

λ = f (d) + f (d) ρ(θ λ ) + R N,λ N + 2 + 1 2 f (d) ρ(θ λ ) + R N,λ N + 2 2 + 1 6 f (3) d + h 1 ρ(θ λ ) + R N,λ N + 2 ρ(θ λ ) + R N,λ N + 2 3 , with 0 < h 1 < 1. That provides λ = f (d)+f (d) ρ(θ λ ) N + 2 + 1 2 f (d) ρ(θ λ ) N + 2 2 + 1 6 f (3) d + h 1 ρ(θ λ ) N + 2 ρ(θ λ ) N + 2 3 +O 1 (N + 2) 3 , (11)
where the quantityO 1 (N +2) 3 is bounded uniformly in λ. On the other hand, with the equation [START_REF] Bogoya | From convergence in distribution to uniform convergence[END_REF]

θ λ = kπ+ρ(θ λ ) N +2 + R N,λ
N 2 ) and we can write, always by Taylor's theorem,

ρ(θ λ ) = ρ(d) + ρ (d) ρ(θ λ ) + R N,λ N + 2 + 1 2 ρ (d + h 2 ρ(θ λ ) + R N,λ N + 2 ρ(θ λ ) + R N,λ N + 2 2 , (12)
with 0 < h 2 < 1, that implies

ρ(θ λ ) = ρ(d) + ρ (d) ρ(d) N + 2 + O( 1 (N + 2) 2 ). ( 13
)
with

O( 1 (N + 2) 2 ) = (ρ (d)) 2 ρ(θ λ ) (N + 2) 2 + 1 2(N + 2) ρ (d + h 2 ρ(θ λ ) N + 2 ρ(θ λ ) N + 2 2 + 1 2 ρ (d + h 2 ρ(θ λ ) N + 2 ρ(θ λ ) N + 2 2 + RNλ ,
where the quantity RN,λ can be bounded with R N,λ , and the maximum of the functions ρ, ρ and ρ Combining the equation ( 11) and ( 13) we obtain

λ = f (d) + f (d)ρ(d) N + 2 + f (d)ρ (d)ρ(d) (N + 2) 2 + 1 2 f (d)ρ 2 (d) (N + 2) 2 + R N,d , (14)
with

R N,d = f (d) ρ(θ λ) -ρ(d) -ρ (d) ρ(d) N +2 (N + 2) 3 + f (d) (N + 2) 2 (ρ(θ λ ) -ρ(d)) 2 + 1 6 f (3) d + h 1 ρ(θ λ ) N + 2 ρ(θ λ ) N + 2 3 + O( 1 (N + 2) 3 ) = O 1 (N + 2) 3
uniformly in λ. To achieve the proof of the theorem we have to be sure that the eigenvalues found are distinct. To do this we need the following two lemmas. Proof : Assume λ N and λ N two solutions of (7) for a same integer k. By [START_REF] Haugazeau | Application du théorème de sylvester à la localisation des valeurs propres ax = λbx dans le cas symétrique[END_REF] we have |λ -λ | = o( 1 (N +2) 2 ). By [START_REF] Barbaresco | Analyse spectrale par décomposition récursive en sous-espaces propres via les coefficients de réflexion[END_REF] we know that we have an eigenvalue, λ N +1 with the bound λ N < λ N +1 < λ N that implies |λ N -λ N +1 | = o( 1 (N +2) 2 ). By [START_REF] Haugazeau | Application du théorème de sylvester à la localisation des valeurs propres ax = λbx dans le cas symétrique[END_REF] we have |λ N -λ N +1 | ≥ O( 1 N ), that is a contradiction with the previous estimation. In [START_REF] Rambour | Une extension d'un résultat de Szegö sur les valeurs propres des matrices de Toeplitz[END_REF] we have state the following inversion formula Theorem 2 Let P N +1 a trigonometric polynomial with degree N + 1 and without zeros on the united disc D. Let ω = r χ0 , 0 < r < 1, |χ 0 | = 1 and also f r = g 1, g 2, with g 1 = χ 0 (1 -ωχ)(P N +1 ) -1 and g 2 = (1 -ω χ) P N +1 -1 . Then for all polynomial P in P N we have

Lemma 7 For k, k + 1 in ]f (θ 1 ), f (θ 2 )[ we have λ(k+1) N - λ(k) N > 0 and | λ(k) N - λ(k+1) N | = O( 1
T N (f r ) -1 (P ) = 1 g 1 π + P g 2 - 1 g 1 π + Φ N +∞ s=0 H * Φ N H Φ N s π + ΦN π + P g 2 . with          Φ N = g 1 g 2 χ N +1 , ΦN = g 2 g 1 χ -(N +1) , H Φ N (Ψ) = π -(Φ N Ψ) H * Φ N (Ψ) = π + ( ΦN Ψ) for Ψ ∈ H + .
For the proof Ttheorem 2 we have to know T N (f r ) - 1 1,1 that is also T N (f r ) -1 (1)|1 . Write T N (f ) -1 (1)|1 = x 0 -y 0 . The previous theorem provides

x 0 = π + 1 g 2 | 1 ḡ1 = χ 0 1 P N +1 (0) 2 .
To obtain y 0 we need the terms π + ΦN π + 1 g 2 and π + ΦN π + 1 ḡ1

. We have

π + ΦN π + 1 g 2 = P N +1 (0)π + g 2 g 1 χ -N -1 = C 1 1 1 -ωχ with C 1 = P N +1 (0) χ0 P N +1 ( 1 ω ) PN+1 (ω) ω N +1 (1 -ω 2 ).
Likewise we can write

π + ΦN π + 1 ḡ1 = C 1 1 1 -ωχ , with C 1 = P N +1 (0) χ0 P N +1 ( 1 ω ) PN+1 (ω)
ωN+1 (1 -ω2 ).

Hence

y 0 = C 1 C 1 (I -H Φ * N H Φ N ) 1 1 -ωχ 1 1 -ωχ .
We have now to use the lemma Lemma 9 1 1-ωχ is an eigenvector of H Φ N H Φ N for the eigenvalue τ N,r (ω)ω 2(N +2) with τ N,r (ω) = P N +1 ( 1 ω )P N +1 ( 1 ω )

P N +1 (ω)P N +1 (ω) , with |τ N,r (ω)| = 1 for 0 < r < 1.

It is the lemma 1 of [START_REF] Rambour | Une extension d'un résultat de Szegö sur les valeurs propres des matrices de Toeplitz[END_REF]. We obtain

y 0 = C 1 C 1 1 1 -ω 2N +2 τ N,r (ω) 1 1 -ω 2 .
If now we consider the function f 1 defined by the product f 1 = g1 g2 with g1 = χ 0 (1-χ0 χ) 1 P N +1 and g2 = (1 -χ0 χ) 1

PN+1

, then for a fixed N lim r→1 (T N f r ) -1

1,1 = (T N f ) -1 1,1 . Indeed

(T N f r ) -1 (T N f ) = (T N f r ) -1 (T N f r ) + (T N f r ) -1 (T N (f 1 -f r )) .
And lim r→1 (T N (f -f r )) = 0 that implies lim r→1 (T N f r ) -1 (T N f ) = I N . Hence we can conclude that

(T N (f )) -1 1,1 = 1 -χ 2(N +2) 0 τ N (χ 0 ) B 2,N -B 1,N 1 -χ 2(N +2) 0 τ N , with B 1,N = C 1 C 1 (1 -χ2 0 ), B 2,N = χ 0 1 P N +1 (0) 2
, and τ N (χ 0 ) = P N +1 ( χ0 )P N +1 ( χ0 ) P N +1 (χ 0 )P N +1 (χ 0 )

tan t+θ 2 dt for θ 1

 21 , θ 2 , θ in [0, π]. Lastly for an interval [a, b] ⊂]θ 1 , θ 2 [ and N an integer we denote by k a,N and k b,N the integers such that : k a,N = min{k| kπ N +2 ∈ [a, b]}, k b,N = max{k| kπ N +2 ∈ [a, b]}. Now we can give the following theorem.

) and 1

 1 |P N +1,λ (e iθ )| 2 + = 1 P N +1,λ (e iθ ) and 1 |P N +1,λ (e iθ )| 2 -= 1 P N +1,λ (e -iθ ) , with index zero. Now it is well known that in the Wiener-Hopf factorization 1 |P N +1,λ (e iθ )| 2 +

  )has at less one solution on [0, θ 2 ]. If θ 2 = π we can remark that we have necessary, for all k ∈ {k 1,N , • • • , π} and for all θ ∈ [θ 1 , π] the inequality 1 -cos ρ N (θ) + kπ N + 2 ≤ 2

  quite the same for the function θ → P.V.

  (t, θ)) tan t+θ 2 dt.

First

  we write I(θ) = I 1,θ + I 2,θ with I 1,θ = P.V.

  t, θ)) -log (H(θ, θ)) tan t-θ 2 dt.

2 Lemma 8

 28 N ) Proof : This lemma follows directly from[START_REF] Haugazeau | Application du théorème de sylvester à la localisation des valeurs propres ax = λbx dans le cas symétrique[END_REF]. For a fixed k the equation[START_REF] Bogoya | From convergence in distribution to uniform convergence[END_REF] has one and only one solution in [θ 1 , θ 2 ].

2

 2

3 Appendix 3 . 1

 331 Inversion formula for Toeplitz matrices. We denote by π + and π -the operator defined by π + n∈Z a n χ n = n∈N a n χ n and π - n∈Z a n χ n = n∈N * a -n χ -n .

  function H we can say that the function θ → H(t, θ) is in C 3 ([θ 1 , θ 2 ]) for all t in [0, 2π]. Hence if Ψ 1 is the function defined by Ψ 1 : θ → log(H(t,θ))-log(H(θ,θ))We see that for t = θ the function∂Ψ 1 ∂θ is equal to ∂ 2 log H ∂θ 2 (θ, θ) . Since the functions log H,

				t-θ
				t-θ 2	2 tan( t-θ 2 )	. Thanks to the
	symmetry of the t-θ	the function
	∂Ψ 1 ∂θ (t, θ) is defined for all θ = t and is equal to
	(t -θ) ((log H) θ (t, θ) -(log H) t (θ, θ) -(log H) θ (θ, θ)) + ((log H)(t, θ) -(log H)(θ, θ)))
			(t -θ) 2
	where we have denoted by (log H) t the quantity ∂(log H) ∂t	and by (log H) θ the quantity ∂(log H) ∂θ
	∂(log H) ∂t	, ∂(log H) ∂θ	, and ∂ 2 log H ∂θ 2


Camille Cornand
email: cornand@gate.cnrs.fr

Maria Alejandra

Erazo Diaz
email: erazodiaz@gate.cnrs.fr

Adam Zylbersztejn
email: zylbersztejn@gate.cnrs.fr

Aurélien Baillon

Muhammed Bulutay

Brice Corgnet

Sibilla Di Guida

Jan Engelmann

Nobuyuki Hanaki

Frank Heinemann

Roberto Hernán-González

Ian Kra- Jbich

Charles Noussair

Luca Polonio

Eli Spigelmann

Marie Claire Villeval

Trading and Cognition in Asset Markets: An Eye-tracking Experiment *

Keywords: Experiment, Asset market, Attention, Information acquisition, Eye-tracking. JEL codes: C92, G41

We use an experimental asset market with eye-tracker measurements for a novel exploration of the cognitive validity of a classic heterogeneous trader taxonomy. Following a top-down approach, we assume that the patterns of attention and information acquisition are governed by one of the three trading strategies, either feedback, passive, or speculative. In line with our first hypothesis, speculators seek information about market expectations. Notwithstanding the two other hypotheses, feedback traders reveal patterns of attention and information acquisition that could ex ante be expected from passive traders, and vice versa.

Introduction

Expectations are critical for asset price dynamics. To formalize the expectation-formation process in an internally consistent manner, standard economic theory posits that agents form Rational Expectations (RE). Although appealing from the theoretical perspective, this approach has found little support in experimental evidence (see [START_REF] Arifovic | Heterogeneous Agent Modeling: Experimental Evidence, chapter 4[END_REF], for a survey). The failure of RE has led to the development of a variety of expectation-formation heuristics as a descriptive attempt to rationalize behavioral data. Relying on [START_REF] Delong | Positive feedback investment strategies and destabilizing rational speculation[END_REF], [START_REF] Haruvy | The effect of short selling on bubbles and crashes in experimental spot asset markets[END_REF] classify traders according to three types -feedback, passive and speculators -in an asset market experiment studying the effect of short sales on both the incidence and magnitude of market bubbles. These three trader types apply strategies requiring different sources of information: either past, present or future market outcomes. Feedback trader's decisions are based on the momentum: their demand for assets increases when prices rise.1 Passive traders account for fundamental values: they buy (sell) when prices are below (above) fundamentals. Finally, speculator traders base their decisions on the expected price fluctuations in near future: they purchase more when there is an expected increase in prices. 2 In this study, we use an experimental asset market with eye-tracker measurements to explore the cognitive underpinnings of this classic behavioral taxonomy. We are interested in assessing the degree of consistency between the observed behavior and the patterns of attention and information acquisition.

Previous research has shown that ways in which individuals sample and process information predict subsequent decisions. 3 The broad use of the eye-tracking technique across domains has proven its adequacy to investigate different cognitive aspects in decision-making processes (see [START_REF] Rahal | Understanding cognitive and affective mechanisms in social psychology through eye-tracking[END_REF], for a review). In economics, this technique has been employed in studies of strategic interactions in games (e.g. [START_REF] Knoepfle | Studying learning in games using eye-tracking[END_REF][START_REF] Devetag | An eye-tracking study of feature-based choice in one-shot games[END_REF][START_REF] Polonio | Testing the level of consistency between choices and beliefs in games using eye-tracking[END_REF][START_REF] Marchiori | Plasticity of strategic sophistication in interactive decision-making[END_REF] as well as financial decision-making (e.g. [START_REF] Gödker | Attention to extreme returns[END_REF][START_REF] Bose | Decision weights for experimental asset prices based on visual salience[END_REF]. However, the eye-tracking technique is barely applied in market settings. To our knowledge, [START_REF] Powell | Information and subject focus[END_REF] offers the only asset market experiment with eyetracking. While he provides evidence that the information sought by subjects predicts trading behavior, his design falls short of linking the observed trading behavior to the type-specific patterns of information seeking. 4 We conduct an asset market experiment in which we analyze trading and forecasts decisions to classify subjects as one of the three trade types (feedback, passive, and speculators) and evaluate the patterns of attention and information acquisition by monitoring cognition through eye-tracking.

We find limited support for the existence of a relationship between trading strategies and the patterns of attention and information acquisition. Our experimental data suggests that, in line with our initial hypothesis, speculators base their decisions on the incoming information about market expectations. Notwithstanding the two other hypotheses we formulated, feedback traders reveal patterns of attention and information acquisition that could ex ante be expected from passive traders, and vice versa.

The remainder of the paper is organized as follows. Section 2 presents our empirical strategy and research hypotheses. Section 3 describes the design of our experiment. In Section 4, we outline the main results. Section 5 concludes.

Empirical strategy

Our empirical strategy is based on two implicit assumptions. First, we assume that each participant to our experiment represents one of the three trader types which can be identified from the observed decisions. Second, the strategy underlying each trader type governs the patterns of attention resulting in a top-down information acquisition process.

Behavioral measurements

We classify traders using the standard taxonomy proposed by [START_REF] Haruvy | The effect of short selling on bubbles and crashes in experimental spot asset markets[END_REF]. There are three types of traders -feedback, passive, and speculator -characterized by a demand function in period t:

• feedback trader : constant+β(p t-1 -p t-2 ), where p t-1 and p t-2 are the average transaction prices in periods t -1 and t -2;

• passive trader : constant -α(p t-1 -f v t-1 ), where f v is the fundamental value;

• speculator trader : constant + γE(p t+1 -p t );

where α ⩾ 0, β ⩾ 0, and γ ⩾ 0. Although different from [START_REF] Haruvy | The effect of short selling on bubbles and crashes in experimental spot asset markets[END_REF], our classification method remains descriptive.5 Each demand function yields a (constrained) linear regression model which can be estimated from the experimental data. For each subject, we estimate three regression models, compare their goodness of fit (R 2 ) and assign the type associated with the best-fitting model.

Eye-tracker measurement of attention

The literature on attention distinguishes between two coexisting processes underlying visual information acquisition: top-down and bottom-up. [START_REF] Coricelli | The Process of Choice in Games[END_REF] note that the fact that attention can be mediated by bottom-up or top-down mechanisms has important implications for the interpretation of the process data because an observed information search pattern may be the result of a predetermined information search strategy (top-down analysis) or mainly determined by some features of the visual scene (bottom-up analysis). (p. 77). As mentioned in the opening paragraph of this section, herein we focus on the top-down processes. Although we cannot fully rule out the parallel presence of bottom-up factors, our design is aimed at minimizing their potential role through a standardized and balanced screen free of attractors or focal points (see Figure 1). 6We use the SR-Research EyeLink 1000 Plus eye-tracking system, a high-accuracy videobased eye-tracker with binocular sampling rate of up to 2000 Hz. Following standard protocols, the eye-to-monitor distance is 1.8 times the display width. 7 For the sake of the quality of eye-movement data, all eye-tracked participants use a chinrest and a forehead rest throughout the experimental session. In addition, we perform two calibrations during the experimental session: one before the beginning of the training task, and another one in the middle of the experiment. Our analysis of the eye-tracking data uses a standard metric: fixations location and their duration. During fixations the eyes extract information from the visual scene for further processing. We consider fixations lasting for at least 50 milliseconds to calculate dwell times, i.e. the overall duration of all the fixations falling into a given Area of Interest (AOI). 8 The Areas of Interest (AOIs) on the screen, in turn, are determined using the aforementioned trader typology. AIOs relate to backward-looking information consisting in realized prices (echoing feedback trader's demand function), current information consisting in fundamental values (echoing passive trader's demand function), and forward-looking information consisting in market expectations of price evolution (echoing speculative trader's demand function). Each AOI has a rectangular shape with an area of 20800 pixels. AOIs never overlap and all the fixations that are not located inside of the AOIs are discarded for the analysis. Note that the set of AOIs expands as the experimental asset market unfolds over periods (see the next section for details).

Hypotheses

While entering the demand function of all the three types of traders, the past prices constitute the unique determinant of the feedback type's decisions. This leads us to formulating our first hypothesis:

Hypothesis 1: Compared to other types, feedback traders devote more attention to the information about past prices.

Second, fundamental values only enter the demand function of the passive traders, making this information particularly relevant for this type:

Hypothesis 2: Compared to other types, passive traders devote more attention to the information about fundamental values.

Finally, an analogous intuition applies to a speculator's use of the information about market expectations:

Hypothesis 3: Compared to other types, speculators devote more attention to the information about the market expectations on price evolution.

Experimental design

We recruited 186 students to participate in 31 computerized (zTree, [START_REF] Fischbacher | z-tree: Zurich toolbox for ready-made economic experiments[END_REF] experimental sessions conducted at the GATE-Lab in Lyon. 9 Each session involves 6 participants: 5 that are seated in a regular cubicle, and one that is seated in a separate eye-tracker room. The usual duration is around 2h (which includes the experimental instructions and the postexperimental questionnaires). The average age is 21 years, 32.25% of our participants are female. The average payoff is 29 EUR (including the 5 EUR show-up fee).

An experimental session unfolds as follows. First, one subject (among those not wearing glasses or contact lenses) is randomly assigned for eye-tracking. Then, everyone is seated in front of their computers and instructions are given to the subjects and read aloud which is followed by a series of comprehension questions. 10 The experiment does not continue unless 8 Dwell times have been previously used to study eye-movement in both strategic interactions (e.g. [START_REF] Halevy | How decisions happen: Focal points and blind spots in interdependent decision making[END_REF][START_REF] Polonio | Strategic sophistication and attention in games: An eye-tracking study[END_REF][START_REF] Peshkovskaya | The socialization effect on decision making in the prisoner's dilemma game: An eye-tracking study[END_REF] and individual decisions (e.g. [START_REF] Engelmann | Top down or bottom up? disentangling the channels of attention in risky choice[END_REF].

9 This project has been approved by the GATE-Lab Review Board for ethical standards in research, under the reference number 2020-09. The design and behavioral conjectures have been preregistered at AsPredicted (#106714). 10 The complete instructions can be found in Appendix 6.1. every participant managed to correctly answer every comprehension question. We then launch a practice market for four periods to familiarize the participants with trading decisions and the market environment. We provide feedback after each period, yet this part is not considered for the final earnings (which is common knowledge).

The main part of the experiment consists of two independent markets (that only differ in terms of calibration, as summarized in Table 1), each lasting for 8 periods. 11 As illustrated in Figure 1, the number of periods we use is also compatible with the requirements of the eye-tracking technology: it allows us to construct distinguishable and non-overlapping AOIs.

At the beginning of each market, all subjects receive the same initial endowment in terms of i) assets holdings and ii) cash to spend in Experimental Currency Units (ECU). 12 We implement a call market environment, as in Van Boening et al. (1993), [START_REF] Haruvy | Traders' expectations in asset markets: experimental evidence[END_REF], [START_REF] Akiyama | It is Not Just Confusion! Strategic Uncertainty in An Experimental Asset Market[END_REF][START_REF] Hanaki | Effects of different ways of incentivizing price forecasts on market dynamics and individual decisions in asset market experiments[END_REF]. In a call market, all buying and selling orders are submitted simultaneously, aggregated into the market demand and supply curves that determine the market clearing price (provided that they intersect). The market is cleared at a uniform price for all transactions of each period.

The calibration of Market 1 has been directly adopted from "design 4" in the seminal study of [START_REF] Smith | Bubbles, crashes, and endogenous expectations in experimental spot asset markets[END_REF]. At the end of each period and independently for each asset, the computer randomly picks one of the four equiprobable dividend values: 0, 8, 28, 60 ECU. Thus, the expected dividend in any period is equal to 24 ECU. Dividends are the only source of value for the asset. Therefore, the fundamental value of an asset during period t is equal to the expected future dividend stream (24 × (8 -t)) ECU. The initial cash endowment in this market is 1040 ECU.

Then, Market 2 is reparameterized in terms of the dividends: the distribution now contains five possible values (0, 1, 8, 28, and 98 ECU), so that in a given period t the expected dividend equals 27 ECU and the fundamental value of is given by 27 × (8 -t) ECU. Subjects receive an initial cash endowment of 1190 ECU.

Each period consists of two tasks (and distinct sources of profit): i) forecasting market prices and ii) asset trading. For each task, profits are aggregated over all market periods; for trading, this includes the earnings from both asset trading and asset dividends. These profits are not transferred between Market 1 and Market 2. Following [START_REF] Hanaki | Effects of different ways of incentivizing price forecasts on market dynamics and individual decisions in asset market experiments[END_REF], subjects are paid based on either their trading or forecasting performance: the final payoff at the end of the experiment is randomly determined by the computer and corresponds to a single source of earnings (either price forecasting or asset trading) in one of the two markets. 13 11 As previously observed by [START_REF] Hanaki | Effects of different ways of incentivizing price forecasts on market dynamics and individual decisions in asset market experiments[END_REF], prices in market experiments tend to converge to the asset's fundamental value as traders gain experience over periods. To avoid such convergence (which would make the behaviors from the later market periods highly homogeneous and thus hardly exploitable for our purposes) and increase the number of observation per subject, we opted for having two short and distinct (yet related) markets. Market 2 was calibrated to ensure enough similarities with Market 1 for making reasonable comparisons of decisions across markets.

12 The exchange rate is such that 1 ECU = 0.015 euros. 13 Hanaki et al. (2018) show that this payment method does not induce mispricing, as opposed to the alternative of paying subjects for both trading and forecasting performance. In the forecast task happening in the first stage of every period (except for the final period 8), subjects are asked to provide their forecast regarding two market prices: the price realized in the second stage of the present period and the price realized in the following period. For instance, in the beginning of period 1 subjects are first asked to indicate their forecast for the market price in period 1 and the market price in period 2. Since the market ends after eight periods, in period 8 subjects only indicate their forecast for the final period. If the forecast lies within a fixed interval of ±25 around the actual realized price, the subject earns 250 ECU, so that the maximum earning from a total of 15 forecasts (two per period in periods 1-7 and one in period 8) amounts to 250 × 15 = 3750 ECU.

The asset trading task constitutes the second stage of every period during which each subject has the opportunity to submit one buy order and one sell order on the market. A buy order consists in providing the maximum price at which the subject is willing to buy and the maximum number of assets she wishes to buy at that price. A sell order consists in indicating the minimum price at which she is willing to sell and the number of assets she wishes to sell at such price.

Subjects do not observe other traders' orders before the end of every period. Once all subjects submit their orders, the computer calculates the market price. The market price is defined as the lowest price at which there is an equal number of assets offered for buying and selling, or the lowest price at which there is a greater number of assets offered for selling than for buying. Subjects who submit a buy (sell) order at a price equal to or above (equal to or below) the market price purchase (sell) assets. If there are any ties between accepted buy or sell orders, the computer randomly selects those that trade.

While deciding about their buying and selling orders, subjects have access to different bits of information displayed on the screen (see Figure 1):

• feedback information about the previous periods: i) realized prices, ii) aggregate earning from the two sources of profit, and iii) last realized dividend;

• fundamental values of assets;

• median price forecasts made by the six market participants.

Such design of the information structure enables us to operationalize the empirical strategy laid out in Section 2: previously realized prices are the sole determinant of feedback trader's behavior, passive trader also takes into account the fundamental value, while a speculator only cares about market expectations on price evolution which is proxied by the median price forecast. 14 To avoid data contamination due to a possible center bias (Tseng et al., 2009) or bottomleft/top-right biases [START_REF] Hagenbeek | Left-right and upper-lower visual field asymmetries for face matching, letter naming, and lexical decision[END_REF], the position of the columns containing the "realized prices", "fundamental values", and "median forecasts" was randomized betweensubjects. Subjects introduce their trading choices at the bottom of the screen.

Finally, before leaving the lab, subjects perform the standard 3-item Cognitive Reflection Test (CRT, Frederick, 2005) and fill in a short socio-demographic questionnaire. 15 14 We are grateful to Brice Corgnet and Charles Noussair for suggesting the use of median forecast as a proxy of market expectations. This approach also follows the recent learning-to-forecast literature (e.g., [START_REF] Petersen | Central bank communication that works: Lessons from lab experiments[END_REF][START_REF] Arifovic | Stabilizing expectations at the zero lower bound: Experimental evidence[END_REF] in the sense that making use of median rather than average forecasts minimizes the ability of a single participant to manipulate aggregate expectations. We also note that this approach does not lack external validity. There are different publicly available surveys that provide median expectations to market participants (such as the Survey of Professional Forecasters for inflation expectations that provides median forecasts).

15 In Appendix 6.2 we show that there is no systematic association between trader types and cognitive abilities (measured by the CRT score) or socio-demographic characteristics. Furthermore, in Appendix 6.3, we report the systematic association between patterns of attention and these two sets of individual characteristics.

Results

In what follows, we first conduct the behavioral classification of traders based on the decisions observed in the experimental asset market. Second, following our Hypotheses 1-3, we measure the relationship between these trading strategies and the patterns of attention revealed through the eye-tracking data. Overall, we find limited empirical support for our research hypotheses: the data corroborate only Hypothesis 3 according to which speculators base their decisions on the incoming information about market expectations.

Behavioral classification of traders

The left-hand side of Table 2 presents the results of our classification exercise for the final sample of 182 subjects. 16 The decisions of 46.45%, 28.96%, 16.94%, and 7.65% of traders are best described as feedback, passive, speculators, and "other", respectively. 17 Altogether, the outcomes of our classification exercise are close to those reported in [START_REF] Haruvy | The effect of short selling on bubbles and crashes in experimental spot asset markets[END_REF].

The right-hand side of Table 2 breaks down these data into two categories: the non-eyetracked and the eye-tracked individuals. Fisher's exact test suggests that the two samples are not different (p -value = 0.795) meaning that the randomization procedure of the assignment to the eye-tracking condition has been successful and the behavior of eye-tracked subjects is representative of the larger population of traders in the experiment. In the remainder of this section we focus on the sub-sample of eye-tracked participants who where using the eye-tracker device while performing the experiment.

Trader types and patterns of attention

We use dwell times to measure the patterns of attention. Following [START_REF] Polonio | Testing the level of consistency between choices and beliefs in games using eye-tracking[END_REF] we ignore fixations located outside the AOIs and transform absolute dwell times into relative ones, i.e the share of time spent looking at a given AOI out of the total time spent looking at all the AOIs on the screen. In addition, we aggregate these relative dwell times corresponding to each of the three information sets presented on the screen: realized prices, fundamental values 16 Out of the initial sample of 186 subjects, three were removed as they abstained from over the course of the experiment. Another subject was removed from the analyses due to the incompleteness of the eye-tracking data.

17 "Other" corresponds to a null R 2 coefficient in each of three models discussed in Section 2. For other categories, let us note that the demand function of feedback traders in period t depends on the average realized price in periods t-1 and t-2. Since realized prices of periods t-1 and t-2 are only available after period 3, we discard periods 1 and 2 from this analysis. Furthermore, a speculator's demand relies on the market expectation of price evolution. Since period 8 is the last period of the market, we do not use it in the classification exercise. As a result, for each participant we have a total of 10 periods (5 per market). Regression models control for the (average) period and market fixed effects. and median forecasts. We then perform the analysis of patterns of attention based on the three main sets of information along with their relative dwell times.

Figure 2 gives an overview of the mean relative dwell times for each set of information across trader types. We interpret this descriptive evidence as pointing to differences in the patterns of attention which, however, do not mesh well with our initial hypotheses. On average, passive traders spend more time than others looking at the realized prices, while feedback traders devote more attention to the fundamental values (notwithstanding Hypotheses 1 and 2). Speculators, in turn, show the highest interest in inspecting median forecasts (in line with Hypothesis 3). 3 shows the results from between-subject comparisons based on the two-sided Wilcoxon ranksum test. 18Notwithstanding Hypothesis 1, the median relative dwelling time for realized prices is significantly shorter in feedback than in passive traders (p -value = 0.036) and not significantly different as compared to speculators (p -value = 0.727).

The results of nonparametric tests also contradict Hypothesis 2. Passive traders pay less attention to the fundamentals than feedback traders (p -value = 0.046), and do not differ significantly in this regard from speculator traders (p -value = 0.532).

Solely Hypothesis 3 finds full empirical support both in descriptive and statistical terms: speculator traders spend substantially more time looking at the median forecasts than both feedback (p -value = 0.037) and passive traders (p -value = 0.074).19

Conclusion

Our study provides a first piece of empirical evidence on (and limited support for) the cognitive validity of the classic heterogeneous trader classification due to [START_REF] Delong | Positive feedback investment strategies and destabilizing rational speculation[END_REF]; [START_REF] Haruvy | The effect of short selling on bubbles and crashes in experimental spot asset markets[END_REF]. We see these results as a promising starting point for future research.

Our experimental design is embedded in a predefined behavioral taxonomy of traders and focuses on the top-down processes in which trading strategy governs attention and information acquisition. Future designs could go beyond these two paradigms. First, they could allow for richer structures of heterogeneity in strategies, including multiple types and switching heuristics. Such enriched taxonomy has been shown to perform well in describing the patterns of belief formation in markets (see, e.g., [START_REF] Hommes | Behavioral and experimental macroeconomics and policy analysis: A complex systems approach[END_REF][START_REF] Bulutay | Learning to deal with repeated shocks under strategic complementarity: An experiment[END_REF]. Relatedly, the reason for the discrepancy between the hypothesized and the observed decisions may not be purely behavioral (e.g., due to the substantial heterogeneity in trading strategies, the role of trembles, or the presence of heuristic switching), but also cognitive: strategies may be also governed by the bottom-up processes in attention and information acquisition. Addressing these points is a challenge for future experimental designs.

Tseng, P.-H., Carmi, R., Cameron, I. G., Munoz, D. P., and Itti, L. (2009). Quantifying center bias of observers in free viewing of dynamic natural scenes. Journal of Vision, 9(7):4-4.

Van Boening, M. V., Williams, A. W., and LaMaster, S. (1993). Price bubbles and crashes in experimental call markets. Economics Letters, 41(2):179-185.

6 Appendix

English translation of the instructions

General instructions

In this experiment, you will participate in a market where you trade units of a fictitious asset with 5 other participants.

The experiment consists of two rounds, each round represents a different market. Each market consists of 8 periods. In each period you have the opportunity to forecast the market price for such period and to trade in the market (i.e., to buy and sell). Specific instructions for your forecasts and trading tasks will be provided later in the instructions.

You will receive 5 euros, provided that you make all the choices and complete a short questionnaire at the end of the experiment. You can earn extra money depending on the accuracy of your forecasts and your trading decisions. Your objective in this experiment is to make as much profit as you can.

You will see two different screens in each period. In the first screen, you will enter your forecasts of the market prices. In the second screen you will be able to trade your assets in the market, you can also obtain dividends for the assets you hold in each period (earnings from dividends are included in the trading profit). You will have two separate sources of profit: forecasts and trading. The total profit from each source is the sum of all your earnings along the 8 periods of each market. However, your profit does not transfer across markets, meaning that at the end of the experiment, you will have 4 different possible payoffs (i.e., 2 profits from forecasts and 2 profits from trading), each of the forecast and trading profits correspond to your decisions in the 2 markets.

Your final payoff is determined as follows. At the end of the experiment, the computer will first select randomly one of the markets, with equal probability for each market being selected. Then, the computer will select either the sum of your forecast profit along the 8 periods within the selected market or the sum of your trading profit (which includes profit from dividends) along the 8 periods within the selected market, once again with equal chance of being selected. As such, it is in your best interest to make each decision as if it was the one that will be chosen.

Payoffs for your decisions will be expressed in experimental currency units (ECU). Please note that each ECU is equal to 0.015 euros.

Earning profit from forecasting

You can earn money by forecasting the market price of the current and the next period. Later in the instructions, we present a detailed explanation of how the market price is determined. For now, we just explain to you how you can make profit from forecasting.

Before starting to trade your assets, you will be asked to forecast the market price for the current period and for the next period to come. If your forecast lies within a fixed interval of ±25 around the actual realized market price, you earn 250 ECU. Note that in the last period (period 8), you will be asked to forecast only the price for such period.

Example:

For example, suppose you are in period 2. You forecast that the market price is going to be 90 ECU in the current period (period 2) and 120 ECU in the next period (period 3). Assume that after all the transactions the realized market price in period 2 is 100 ECU and the realized market price in period 3 is 180 ECU.

You will earn 250 ECU for your forecast regarding period 2 and 0 ECU for your forecast regarding period 3. For period 2, any forecast between 75 and 125 leads to earnings. For period 3, any forecast between 155 and 205 leads to earnings. The maximum earnings from forecasting you can accumulate during a market is thus 250 × 15 = 3750 ECU.

Experimental interface

Figure 3: First screen in each period Figure 3 illustrates the interface of the first screen of each period. In this stage, your task is to forecast the market price of the current period and the next period, according to the rules stated in the instructions previously presented. Remember that in the last period (period 8), you will be asked to forecast only the price for such period. Please click on the OK button to confirm your forecast. You can only submit once.

Earning profit from trading

Trading assets generates two sources of profits: one from buying and selling in the market and another one from dividends.

How to buy and sell assets

At the beginning of every market, you receive an endowment of a number of assets and ECU. You can use this endowment to trade in the market. Every trader begins this market with the identical endowment. To earn profit from trading, you need to buy assets at a lower price and sell these at a higher price.

For example, suppose you buy an asset for 100 ECU, and then the price of the asset increases to 120 ECU. If you sell the asset, you will earn 120 (selling price) -100 (purchase price) = 20 ECU profit. In contrast, suppose you buy an asset for 100 ECU, and then the price of the asset decreases to 80 ECU. If you sell the asset, you will make 80 (selling price) -100 (purchase price) = 20 ECU loss.

If you want to buy assets, you need to submit the highest price at which you are willing to buy one asset and the maximum number of assets you wish to buy. This is called a buy order.

If you want to sell assets, you need to submit the lowest price at which you are willing to sell one asset and the maximum number of assets you wish to sell. This is called a sell order.

In practice, the price you actually pay for an asset may be lower than the maximum price you are willing to pay. This is because the market price is set based on all the orders placed by market participants. If the market price is greater than the maximum you are willing to pay, your order will not be processed.

The orders of all traders will be aggregated in the end of every period to determine the single price for all assets in each period. There are two ways to determine the market price. The implemented way to determine the market price depends on the buy/sell orders done by all the participants. We will explain each of these ways separately.

First way to determine the market price

The market price is the lowest price at which there is an equal number of assets offered for buying and selling.

We illustrate how the market price is set through this first way by using the following example.

Consider the following buy/sell orders placed by four traders:

-Trader 1: One sell order, which can be executed at 10 ECU or higher -Trader 2: Two sell orders, which can be executed at 40 ECU or higher -Trader 3: One buy order, which can be executed at 60 ECU or lower -Trader 4: One buy order, which can be executed at 20 ECU or lower A seller is willing to sell at the price requested or higher. A buyer is willing to buy at the price specified or lower. As shown in Figure 4, there is only one asset supplied at 10 ECU or higher. If the price rises to 40 ECU, the number of assets supplied increases to three. On the other hand, only one asset is demanded at 60 ECU. If the price falls to 20 ECU, the quantity demanded increases to two. Therefore, the quantity demanded is equal to the quantity supplied at prices between 21 ECU and 39 ECU. The market price is set to the minimum price of this interval, i.e., 21 ECU.

Second way to determine the market price

If there is no price at which the number of assets offered for buying is precisely the same as the number of assets offered for selling, and some of the assets offered for buying are at a lower price than the price at which assets are offered for selling, the market price is defined as follows. The market price is the lowest price at which there is a greater number of assets offered for selling than for buying.

We illustrate how the market price is set through this second way by using the following example.

Consider the following buy/sell orders placed by five traders:

-Trader 1: One sell order, which can be executed at 10 ECU or higher -Trader 2: One sell order, which can be executed at 30 ECU or higher -Trader 3: One sell order, which can be executed at 30 ECU or higher -Trader 4: One buy order, which can be executed at 60 ECU or lower -Trader 5: One buy order, which can be executed at 30 ECU or lower As shown in Figure 5, only one asset is supplied at 10 ECU or higher as in the previous example. If the price rises to 30 ECU, the number of assets that are supplied increases to three. However, there is only one asset demanded at 60 ECU or lower. If the price falls to 30 ECU, the quantity demanded increases to two. As a result, two transactions can be completed at 30 ECU. In this case, the market price is set to 30 ECU. The orders that will be fulfilled are determined as follows.

Priority is given to Trader 1, because he/she requested a price lower than the market price. In addition to the order of Trader 1, the order of either Trader 2 or Trader 3 will be fulfilled, since the traded quantity is two. The chosen order to trade between Trader 2 or Trader 3 is determined randomly by the computer.

Dividends

Other than forecasting and trading, you can also earn money from dividends. The assets that you have purchased in one period are at your disposal at the next period. For example, if you happen to own 5 assets at the end of period 2, you own the same 5 assets at the beginning of period 3. For every asset you own, you receive a dividend at the end of each of the 8 periods. The dividend is added automatically to your ECU account at the end of each period. After the dividend of period 8 has been paid, the market closes, and you will not receive any further dividends for the assets you own.

The computer randomly selects, with equal probability, the amount of the dividend each asset pays from a set of different possible values. This random selection is done at the end of each period. The different possible values of the dividends do not change within each market.

Example:

Suppose that each asset pays a dividend of either 0, 10, 18, or 56 ECU, with equal probability. This means that the average dividend is 21 ECU.

For example, if you own 5 assets at the end of period 2, and the computer randomly chose a dividend of 10 ECU for this period. Then, in period 3 you receive 50 ECU = 5 (assets) × 10 ECU as profit from dividends.

The total profit from trading assets consists of buying and selling them in the market plus the accumulated earnings from dividends.

Experimental interface

Figure 6: Second screen in each period Figure 6 illustrates the interface of the second screen of each period. This image corresponds to what you will see in period 1. In such stage of the experiment, your can trade in the market, according to the rules stated in the instructions previously presented. The red numbers in Figure 1, are not part of the experimental interface. However, we include these numbers for the sake of illustration. Below, you will find the explanation of the information corresponding to each number. Please read them carefully. Be aware that the position of columns 2, 3, and 4 might vary in the experiment.

1. This column shows the trading period corresponding to the information you will see in the other columns.

2. This column shows the realized market prices until the period you are in. If the selling and buying orders do not reach a realized market price, three dots (. . .) will appear. Note that since Figure 6 corresponds to period 1, no information is available given the lack of previous transactions in the market. Therefore, there is no realized market price to display.

3. This column shows the average holding value of the asset. This information is shown to facilitate your choices. It shows how one unit of the asset pays on average, if you hold it from the current period until the last period, i.e. period 8 of this market. These values are calculated as follows: average dividend × number of remaining periods. As Figure 6 shows, you will observe the average holding value of all the periods in the market from the first period until the end along the entire market.

4. This column shows the median forecasts provided by all the participants in the experiment.

5. This shows your earnings separately for each task. First, it shows how much you earned so far in the current market from forecasting. Recall that for each forecast that lies within a fix interval of ±25 around the actual realized market price, you earn 250 ECU. Second, it shows how much you earned in the current period from your trading. Third, how much you earned in the current period from dividends.

6. This shows how much money (ECU) you have at your disposal, which is the sum of your earnings from trading plus your earnings from dividends. This is the maximum amount you can spend on buying assets. Also, this shows the number of assets you currently have. This is the maximum number of assets you can sell.

7. This shows the potential dividend values that can be realized in the current market. Also, you can look at the realized value of the last period's dividend. Note that you will only observe the value of the last realized dividend after period 1.

8. This is where you enter the highest price you are willing to pay to buy an asset in the current period. Recall that if the market price turns out to be greater than the highest you are willing to pay, your order will not be processed. Also, here is where you enter the highest number of assets you want to buy in this period. If you do not want to purchase any asset, enter 0. 9. Here is where you enter the lowest price at which you would be prepared to sell an asset in the current period. Recall that if the market price turns out to be lower than your lowest selling price, your order will not be processed. Also, here is where you enter the number of assets you want to sell in this period. If you do not want to sell any of your assets, enter 0.

10. By clicking this button, you confirm your selling and buying orders and move to the next period.

Trader types, cognitive abilities and socio-demographic variables

Table 4 presents the number of traders classified as feedback, passive, or speculators according to the CRT score. We do not reject the null hypothesis that distributions are the same across the four scores (χ 2 test, p -value = 0.591). Note. For each trader type, each cell provides the number (fraction) of subjects with a given CRT score.

Table 5 shows the number of traders classified as either feedback, passive or speculators by gender. We find no gender difference in the distribution of types (χ 2 test, p -value = 0.244). Looking at Table 6, we do not find a statistically significant relationship (χ 2 test, p -value = 0.581) between the discipline in which subject majored (economics/ finance vs. other disciplines) and trader types.

Finally, the analysis of variance (one-way ANOVA) shows that the mean age does not significantly differ across trader types (p -value = 0.935). Note. For each trader type, each cell provides the number (fraction) of subjects with a given major.

Patterns of attention, cognitive abilities and socio-demographic variables

To analyze the relationship between patterns of attention, cognitive abilities and demographic variables, we run linear regressions taking the relative dwell times for each of the three sets of information presented in the experiment (realized prices, fundamentals and median forecasts) as dependent variable. Results are summarized in Table 7. Model 1 shows that there is a significant positive correlation between relative dwell times in prices and CRT scores. Holding everything else constant, one unit increase in the CRT score increases the relative share of dwell time in realized prices by 6.4 percentage points. All remaining coefficients across the three models lack statistical significance at the 5% level.

Figure 1 :

 1 Figure 1: Experimental screen for stage 2 and AOIs

Figure 2 :

 2 Figure 2: Mean relative dwell times in AOIs over trader types

Figure 4

 4 Figure 4 summarizes these orders.

Figure 4 :

 4 Figure 4: Graphical example: first way to determine the market price

Figure 5

 5 Figure 5 summarizes these orders.

Figure 5 :

 5 Figure 5: Graphical example: second way to determine the market price

Table 1 :

 1 Summary of parameters in Markets 1 and 2.

	Market 1	Market 2

Table 2 :

 2 Trader type classification

	Trader type Entire sample Not eye-tracked Eye-tracked
	Feedback	84 (46.15%)	69 (45.39%)	15 (50%)
	Passive	53 (29.12%)	43 (28.29%)	10 (33.33%)
	Speculator	31 (17.03%)	27 (17.76%)	4 (13.33%)
	Other	14 (7.69%)	13 (8.55%)	1 (3.33%)
	Total	182 (100%)	153 (100%)	30 (100%)

Table 3 :

 3 Trader types and patterns of attention

		Prices Fundamentals Median forecasts
			Median dwell time
	Feedback	0.204	0.343	0.364
	Passive	0.311	0.303	0.434
	Speculators	0.200	0.216	0.610
			p -values
	Feedback vs. passive	0.036	0.046
	Feedback vs. speculators 0.727		0.037
	Passive vs. speculators		0.532	0.074
	Note. Median dwell times in bold correspond to the hypothesized dominant source
	of information of each trader type.

Table 4 :

 4 Trader types and cognitive skills

	CRT score	Feedback	Passive	Speculator
	0	16 (19.05%) 8 ( 15.09%) 5 (16.13%)
	1	24 (28.57%) 9 (16.98%) 10 (32.26%)
	2	18 (21.43%) 16 (30.19%) 8 (25.81%)
	3	26 (30.95%) 20 (37.74%) 8 (25.81%)

Table 5 :

 5 Trader types and genderNote. For each trader type, each cell provides the number (fraction) of subjects of a given gender.

	Gender	Feedback	Passive	Speculator
	Male	60 (71.43%) 35 (66.04%) 17 (54.84%)
	Female 24 (28.57%) 18 (33.96%) 14 (45.16%)

Table 6 :

 6 Trader types and field of study

	Major	Feedback	Passive	Speculator
	Economics or finance 27 (32.14%) 17 (32.08%)	7 (22.58%)
	Other disciplines	57 (67.86%) 36 (67.92% ) 24 ( 77.42%)

Table 7 :

 7 Relative dwell times, with CRT and demographics; regression analysis Note. Binary variable female is set to 1 for females; binary variable econ/finance is set to 1 if subject majored in economics or finance. Standard errors are clustered at the individual level. Asterisk indicates statistical significance at the 5% level. N = 30.

		Model 1: Dwells in prices	Model 2: Dwells in fundamentals	Model 3: Dwells in median forecasts
		Coefficient Std. Error	Coefficient	Std. Error	Coefficient	Std. Error
	CRT	0.064 *	(0.027)	-0.000	(0.040)	-0.063	(0.039)
	Age	0.006	(0.010)	-0.014	(0.016)	0.007	(0.015)
	Female	-0.015	(0.065)	0.030	(0.086)	-0.014	(0.083)
	Econ/finance	-0.002	(0.012)	-0.022	(0.015)	0.025	(0.423)

Decision-making errors made by feedback traders foster market bubbles(Lei et al.,

2001).2 The presence of speculator traders is in line with the hypothesis of the lack of common knowledge of expectations[START_REF] Smith | Economics in the laboratory[END_REF].3 For example, gaze direction during the search process can predict choices in moral dilemmas[START_REF] Pärnamets | Biasing moral decisions by exploiting the dynamics of eye gaze[END_REF]. Patterns of attention have also been used to study food choices (e.g.[START_REF] Krajbich | Visual fixations and the computation and comparison of value in simple choice[END_REF][START_REF] Towal | Simultaneous modeling of visual saliency and value computation improves predictions of economic choice[END_REF].[START_REF] Krajbich | The attentional drift-diffusion model extends to simple purchasing decisions[END_REF] show that attention dedicated to looking at a product rather than its price is predictive of purchasing decisions.4 Our study differs from his in several ways. First, his market is a continuous double auction while ours is a call market. Thus, we have a single market price per period (whenever a clearing prince exists). Second, unlike us, he does not elicit market expectations and does not provide visual information allowing to discriminate speculators from the two other types.

Their method can be summarized as follows (p. 1143): [e]ach subject receives a score with respect to each of the three types, with the score for each type equal to the number of periods in which subject's behavior is consistent with that type. A subject is classified as the type that receives the highest score, provided that the score is greater than 8 [out of 15], and as "other" otherwise. Ties are broken by assigning the corresponding fraction to each type. In addition, [t]he proportion classified as "other" is randomly assigned to the three behavioral types with probabilities equal to their proportions in the population. Our comparison, in turn, is based on a goodness-of-fit criterion thus ruling out the problem of ties and providing an unambiguous outcome in principle (unless all models have null capacity to explain the data).

On the same page,[START_REF] Coricelli | The Process of Choice in Games[END_REF] note that [i]n eye-tracking experiments, the characteristics of the task and of the decision maker may significantly affect how attention is allocated in a visual scene. For example, a bottom-up analysis may be promoted by the presence of attractors or focal points [. . .].

For eye-tracking setup standard protocol, see EyeLink (2013).

We note that within-subject comparisons are also possible, yet less informative for our analyzes. The reason for which we avoid these comparisons is that the number of AOIs is not constant across our variables of interest (i.e., realized prices, fundamentals and median forecasts) and evolves over periods, as can be seen in Figure1. Thus, relative dwell times may vary within-subject in a purely mechanical manner: independently of their subjective strategic relevance and the level of attention they attract, exploring some sources of information simply require inspecting more AOIs (and thus is more time-consuming) than others.

Although the magnitude of these differences is meaningful on its own, we note that these comparisons may suffer from low statistical power given that only 4 in 30 eye-tracked subjects are classified as speculators. As shown in Table2, speculators constitute the rarest type, accounting for 17% of all traders. Taking these figures at face value, increasing the number of eye-tracked speculators in the dataset is extremely resource-intensive. In expectancy, increasing the sub-sample of eye-tracked speculators by one observation would require running more than 5 additional experimental sessions of 6 subjects.

We also thank Mateus Joffily, Marie-Sophie Roul and Quentin Thévenet for valuable technical support. This research was performed within the framework of the LABEX CORTEX (ANR-11-IDEX-007) operated by the French National Research Agency (ANR). The authors are thankful to the ANR (ANR-18-FRAL-0 0 08-01 ASUR), the ASFEE (2022 Young Researcher Prize), the GATE laboratory and Université Lyon 2 for financial support. The authors declare no competing interests. Adam Zylbersztejn acknowledges VISTULA Fellowship.