Oleg N Kirillov
email: o.kirillov@hzdr.de

Michael L Overton

Pantelis Damianou

Robust stability at the Swallowtail singularity

Keywords: abscissa, optimization, overdamping, asymptotic stability, Whitney umbrella, Swallowtail, caustic, eigenvalue dynamics

Consider the set of monic fourth-order real polynomials transformed so that the constant term is one. In the three-dimensional space of the coefficients describing this set, the domain of asymptotic stability is bounded by a surface with the Whitney umbrella singularity. The maximum of the real parts of the roots of these polynomials is globally minimized at the Swallowtail singular point of the discriminant surface of the set corresponding to a negative real root of multiplicity four. Motivated by this example, we review recent works on robust stability, abscissa optimization, heavily damped systems, dissipation-induced instabilities, and eigenvalue dynamics in order to point out some connections that appear to be not widely known.

INTRODUCTION. ZIEGLER-BOTTEMA DESTABILIZATION PHENOMENON

In 1956, motivated by the paradoxical effect of dissipation on the threshold of oscillatory instability of mechanical structures under non-conservative loadings [START_REF] Ziegler | Die Stabilitätskriterien der elastomechanik[END_REF], Bottema [START_REF] Bottema | The Routh-Hurwitz condition for the biquadratic equation[END_REF] studied the domain of asymptotic stability 1 of a fourth-order real polynomial

q(μ) = μ 4 + b 1 μ 3 + b 2 μ 2 + b 3 μ + b 4 . ( 1
)
He observed that the necessary and sufficient conditions

b 1 > 0, b 2 > 0, b 3 > 0, b 4 > 0, b 2 > b 2 1 b 4 + b 2 3 b 1 b 3 (2)
for the polynomial to be Hurwitz, i.e., to have all its roots in the left half of the complex plane, are not changed by the transformation μ = cλ, where c = 4 √ b 4 > 0. Indeed, after denoting

a i = b i c -i , i = 1, 2, 3, 4, (3)
the inequalities (2) are equivalent to the conditions

a 1 > 0, a 3 > 0, a 2 > 2 + (a 1 -a 3 ) 2 a 1 a 3 > 0 ( 4
)
1 An equilibrium of a dynamical system is said to be Lyapunov stable if all the solutions starting in its vicinity remain in some neighborhood of the equilibrium in the course of time. For asymptotic stability, the solutions are required, additionally, to converge to the equilibrium as time tends to infinity. The first (indirect) method of Lyapunov reduces the study of asymptotic stability of an autonomous (time-independent) system to the problem of location in the complex plane of eigenvalues of the operator of its linearization. In a finite-dimensional case the eigenvalues are roots of a polynomial characteristic equation. Localization of all the roots in the open left half of the complex plane is a necessary and sufficient condition for asymptotic stability of a linearization, which usually implies asymptotic stability of the original non-linear system.

that are necessary and sufficient for the polynomial

p(λ) = λ 4 + a 1 λ 3 + a 2 λ 2 + a 3 λ + 1 ( 5 )
to be Hurwitz. The domain (4) was plotted by Bottema in the (a 1 , a 3 , a 2 )-space; see Figure 1.

Bottema realized that the boundary of the asymptotic stability domain (4) is a ruled surface2 generated by straight lines a 3 = ra 1 , a 2 = r + 1/r, where r ∈ (0, ∞). The boundary has a self-intersection along the ray a 2 ≥ 0 of the a 2 -axis. Two generators pass through each point of the ray; they coincide at a 2 = 2 (r = 1), and for a 2 → ∞ their directions tend to those of the a 1 -and a 3 -axis (r = 0, r = ∞). Remarkably [START_REF] Bottema | The Routh-Hurwitz condition for the biquadratic equation[END_REF][START_REF] Kirillov | Paradoxes of dissipation-induced destabilization or who opened Whitney's umbrella?[END_REF], the point (a 1 , a 3 , a 2 ) = (0, 0, 2) is on , but if one tends to the a 2 -axis along the line a 3 = ra 1 the coordinate a 2 has the limit r + 1/r > 2 except for r = 1, when a 2 = 2. The point (0, 0, 2) marked by the diamond symbol in Figure 1 is the Whitney umbrella singularity [START_REF] Arnold | On matrices depending on parameters[END_REF]. The surface defined by the equation

a 2 = 2 + (a 1 -a 3 ) 2 a 1 a 3 ( 6
)
is just another representation of a standard crosscap surface

f std (u, v) = (u, uv, v 2 ) : R 2 → R 3
, which also bears the name of the Whitney umbrella [START_REF] Bruce | Functions on a crosscap[END_REF].

For given a 1 = a 1,0 and a 2 = a 2,0 > 2, exactly two points correspond at the crosscap [START_REF] Levantovskii | The boundary of a set of stable matrices[END_REF] with the third coordinate determined from the equation Inside the spike bounded by the discriminant surface all the roots of the polynomial (5) are real and negative. The green planar sector is the tangent cone [START_REF] Vishik | The solution of some perturbation problems for matrices and selfadjoint or non-selfadjoint differential equations I[END_REF] to the domain of asymptotic stability at the Whitney umbrella singularity.

a 3,0 = 1 2 a 2,0 ± a 2 2,0 -4 a 1,0 , (7)
where each sign is associated with one of the two sheets of the crosscap that intersect along the interval a 2 ≥ 2; see Figure 1. Writing the equations for the tangent planes at these points

a 1 a 2,0 ± a 2 2,0 -4 -2a 3 = a 1,0 a 2,0 ± a 2 2,0 -4 2 a 2 2,0 ± a 2,0 a 2 2,0 -4 -4 (a 2 -a 2,0 ) (8)
we find that approaching the singular point (0, 0, 2) along the a 2 -axis, i.e., assuming a 1,0 = 0 in [START_REF] Burke | Variational analysis of the abscissa mapping for polynomials via the Gauss-Lucas theorem[END_REF] and then letting a 2,0 tend to 2, results in the collapse of the two planes into one defined by the equality

a 1 = a 3 . ( 9
)
The limit (9) of the tangent planes is the tangent cone to the crosscap at the Whitney umbrella singularity [START_REF] Bruce | Functions on a crosscap[END_REF]. A part of the plane (9) that lies inside the domain of asymptotic stability (4) constitutes a set of all directions leading from the point (0, 0, 2) to the stability region

{(a 1 , a 3 , a 2 ) : a 1 = a 3 , a 1 > 0, a 2 > 2} . ( 10
)
The tangent cone [START_REF] Vishik | The solution of some perturbation problems for matrices and selfadjoint or non-selfadjoint differential equations I[END_REF] to the domain of asymptotic stability at the Whitney umbrella singularity, which is shown in green in Figure 1, is degenerate in the (a 1 , a 3 , a 2 )-space because it selects a set of measure zero on a unit sphere with the center at the singular point [START_REF] Levantovskii | The boundary of a set of stable matrices[END_REF][START_REF] Levantovskii | On singularities of the boundary of a stability region[END_REF][START_REF] Burke | Variational analysis of the abscissa mapping for polynomials via the Gauss-Lucas theorem[END_REF].

In 1971, Arnold [START_REF] Arnold | On matrices depending on parameters[END_REF] considered parametric families of real matrices and studied which singularities on the boundary of the asymptotic stability domain in the space of parameters are not destroyed by small perturbation of the family, i.e., are structurally stable or generic singularities. He composed a list of generic singularities in parameter spaces of low dimension which revealed, in particular, that the Whitney umbrella persists on the stability boundary of a real matrix of arbitrary dimension if the number of parameters is not less than three, i.e., the singularity is of codimension 3.

The singular point (a 1 , a 3 , a 2 ) = (0, 0, 2) corresponds to a double complex conjugate pair of roots λ = ±i of the polynomial [START_REF] Bruce | Functions on a crosscap[END_REF].

At the line of self-intersection of the Whitney umbrella formed by the ray a 2 > 2 of the a 2 -axis there are two distinct complex conjugate pairs of imaginary roots [START_REF] Bottema | The Routh-Hurwitz condition for the biquadratic equation[END_REF] indicating Lyapunov (marginal) stability. However, the threshold of stability a 2 = 2 is extremely sensitive to the coefficients a 1 and a 3 as is seen from [START_REF] Levantovskii | The boundary of a set of stable matrices[END_REF].

Indeed, fix â1 = â2 and let a 1 = εâ 1 and a 3 = εâ 3 (where 0 < ε 1). Then, using [START_REF] Levantovskii | The boundary of a set of stable matrices[END_REF], we find that the increment to the critical value of the coefficient a 2 is

a 2 (ε) := a 2 -2 = (â 1 -â3 ) 2 â1 â3 ∼ O(1). ( 11
)
In other words, the stability threshold acquires a finite increment a 2 > 0 even if the perturbation of the coefficients a 1 and a 3 is infinitesimally small for almost all combinations of a 1 and a 3 , except in the case a 1 = a 3 . The minimal threshold of marginal stability a 2 = 2 at a 1 = 0 and a 3 = 0 corresponding to the double imaginary root is thus structurally unstable: generic combinations of the parameters a 1 and a 3 at a 2 = 2 lie in the domain of instability [START_REF] Ziegler | Die Stabilitätskriterien der elastomechanik[END_REF][START_REF] Bottema | The Routh-Hurwitz condition for the biquadratic equation[END_REF][START_REF] Kirillov | Paradoxes of dissipation-induced destabilization or who opened Whitney's umbrella?[END_REF]9]. Therefore, looking for extremal values of a parameter (a 2 in our case) at the boundary of asymptotic stability may lead to structurally unstable stability thresholds at the singular points of the stability boundary corresponding to multiple roots.

The more basic fact that multiple roots of a polynomial are sensitive to perturbation of the coefficients is a phenomenon that was studied already by Isaac Newton, who introduced the so-called Newton polygon to determine the leading terms of the perturbed roots as fractional powers of a perturbation parameter. It follows that, in matrix analysis, eigenvalues are in general not locally Lipschitz at points in matrix space with non-semi-simple eigenvalues3 [START_REF] Vishik | The solution of some perturbation problems for matrices and selfadjoint or non-selfadjoint differential equations I[END_REF][START_REF] Lidskiȋ | On the theory of perturbations of nonselfadjoint operators[END_REF][START_REF] Baumgärtel | Analytic Perturbation Theory for Matrices and Operators[END_REF][START_REF] Moro | On the Lidskii-Vishik-Lyusternik perturbation theory for eigenvalues of matrices with arbitrary Jordan structure[END_REF], and, in the context of dissipatively perturbed Hamiltonian systems, [9, [START_REF] Maddocks | Stability theory for dissipatively perturbed Hamiltonian-systems[END_REF][START_REF] Kirillov | Nonconservative Stability Problems of Modern Physics[END_REF]. Thus, it has been well-understood for a long time that perturbation of multiple roots or multiple eigenvalues on or near the stability boundary is likely to lead to instability 4 .

Because of the sensitivity of multiple roots and eigenvalues to perturbation, in engineering and control-theoretical applications a natural desire is to "cut the singularities off " by constructing convex inner approximations to the domain of asymptotic stability [START_REF] Henrion | Ellipsoidal approximation of the stability domain of a polynomial[END_REF][START_REF] Henrion | Inner approximations for polynomial matrix inequalities and robust stability regions[END_REF]. Nevertheless, multiple roots per se are not undesirable. Indeed, multiple roots also occur deep inside the domain of asymptotic stability. Although it might seem paradoxical at first sight, such configurations are actually obtained by minimizing the so-called polynomial abscissa in an effort to make the asymptotic stability of a linear system more robust, as we now explain.

ABSCISSA MINIMIZATION AND MULTIPLE ROOTS

The abscissa5 of a polynomial p is the maximal real part of its roots:

a(p) = max{Re λ : p(λ) = 0}. ( 12
)
We restrict our attention to monic polynomials with real coefficients and fixed degree n: since these have n free coefficients, this space is isomorphic to R n . On this space, the abscissa is a continuous but non-smooth, in fact non-Lipschitz, as well as non-convex, function whose variational properties have been extensively studied using non-smooth variational analysis [START_REF] Burke | Variational analysis of the abscissa mapping for polynomials[END_REF][START_REF] Burke | Optimizing matrix stability[END_REF][START_REF] Lewis | The mathematics of eigenvalue optimization[END_REF][START_REF] Burke | Variational analysis of functions of the roots of polynomials[END_REF]. Now set n = 4, consider the set of polynomials p(λ) defined in [START_REF] Bruce | Functions on a crosscap[END_REF], and consider the restricted set of coefficients

(a 1 , a 3 , a 2 ) : a 1 = a 3 , a 2 = 2 + a 2 1 4 . ( 13
)
For these coefficients, the abscissa a(p) is shown as a function of a 1 by the bold red curve in Figure 2. The roots are

λ 1 = λ 2 = - a 1 4 - 1 4 a 2 1 -16, λ 3 = λ 4 = - a 1 4 + 1 4 a 2 1 -16. (14)
When 0 ≤ a 1 < 4 (a 1 > 4), the roots λ 1,2 and λ 3,4 are complex (real) with each pair being double, that is with multiplicity two. At a 1 = 4 there is a quadruple real eigenvalue -1. So, we refer to the set (13) as a set of exceptional points 6 (abbreviated as the EP-set).

When a 1 > 0, the EP-set (13) (shown by the red curve in Figure 3) lies within the tangent cone [START_REF] Vishik | The solution of some perturbation problems for matrices and selfadjoint or non-selfadjoint differential equations I[END_REF] to the domain of asymptotic stability at the Whitney umbrella singularity (0, 0, 2). The points in the EP-set all define polynomials with two double roots (denoted EP 2 ) except (a 1 , a 3 , a 2 ) = (4, 4, 6), at which p has a quadruple root and is denoted EP 4 ; see Figure 3. Let us consider how the roots move in the complex plane when a 1 and a 3 coincide and are set to specific values and a 2 increases from zero, as shown by black curves in Figure 4. When a 1 = a 3 < 4, the roots that initially have positive real parts and thus correspond to unstable solutions move along the unit circle to the left, cross the imaginary axis at a 2 = 2 and merge with another complex conjugate pair of roots at a 2 = 2 + a 2 1 /4, i.e., at the EP-set. Further increase in a 2 leads to the splitting of the double eigenvalues, with one conjugate pair of roots moving back toward the imaginary axis. By also considering the case a 1 = a 3 > 4, it is clear that when a 1 and a 3 coincide, the choice a 2 = 2 + a 2 1 /4 minimizes the abscissa, with the polynomial p on the EP-set, see Figure 5. Furthermore, when a 1 = a 3 is increased toward 4 from below, the coalescence points (EP 2 ) move around the unit circle to the left. This conjugate pair of coalescence points merges into the quadruple real root λ = -1 (EP 4 ) when a 1 = a 3 = 4 and hence a 2 = 6, as is visible in Figure 6. If a 1 = a 3 is increased above 4 the quadruple point EP 4 splits again into two exceptional points EP 2 , one of them inside the unit circle, Figure 5. Thus, all indications are that the abscissa is minimized by the parameters corresponding to EP 4 , with a quadruple root at -1. In fact, application of the following theorem shows that the abscissa of ( 5) is globally minimized by the EP 4 parameters. Theorem [27, Theorems 7 and 14] 7 . Let F denote either the real field R or the complex field C. Let b 0 , b 1 , . . . , b n ∈ F be given (with b 1 , . . . , b n not all zero) and consider the following family of monic polynomials of degree n subject to a single affine constraint on the coefficients:

P = {λ n + a 1 λ n -1 + . . . + a n -1 λ + a n : b 0 + n j = 1 b j a j = 0, a i ∈ F}.

Define the optimization problem

a * := inf p ∈ P a(p). ( 15
) Let h(λ) = b n λ n + b n -1 n n -1 λ n -1 + . . . + b 1 n 1 λ + b 0 .
First, suppose F = R. Then, the optimization problem [START_REF] Kirillov | Nonconservative Stability Problems of Modern Physics[END_REF] has the infimal value

a * = -max ζ ∈ R : h (i) (ζ) = 0 for some i ∈ {0, . . . , k -1} ,
where h (i) is the i-th derivative of h and k = max{j : b j = 0}. Furthermore, the optimal value a * is attained by a minimizing polynomial p * if and only if -a * is a root of h (as opposed to one of its derivatives), and in this case we can take p * (λ) = (λ -γ) n ∈ P, γ = a * . Second, suppose F = C. Then, the optimization problem [START_REF] Kirillov | Nonconservative Stability Problems of Modern Physics[END_REF] always has an optimal solution of the form

p * (λ) = (λ -γ) n ∈ P, Re γ = a * ,
with -γ given by a root of h (not its derivatives) with largest real part.

In our case, F = R, n = 4 and the affine constraint on the coefficients of p is simply a 4 = 1. So, the polynomial h is given by

h(λ) = λ 4 -1.
Its real root with largest real part is 1, and its derivatives have only the zero root. So, the infimum of the abscissa a over the polynomials ( 5) is -1, and this is attained by

p * (λ) = (λ + 1) 4 = λ 4 + 4λ 3 + 6λ 2 + 4λ + 1, (16)
that is, with the coefficients at the exceptional point EP 4 . There is nothing special about n = 4 here; if we replace 4 by n we find that the infimum is still -1 and is attained by p * (λ) = (λ + 1) n .

SWALLOWTAIL SINGULARITY AS A GLOBAL MINIMIZER OF THE ABSCISSA

Let us look at the dynamics of the roots of the polynomial (5) in the complex plane, when a 1 = a 3 , i.e., when the variation of a 2 happens at a distance from the tangent cone [START_REF] Vishik | The solution of some perturbation problems for matrices and selfadjoint or non-selfadjoint differential equations I[END_REF]. If a 1 ≤ 4 and a 3 ≤ 4 (as in Figures 4,6) the complex roots do not coalesce at EP 2 or EP 4 but instead some of them turn back toward the imaginary axis before reaching the unit circle. Such parameter configurations are therefore not optimal. However, when a 1 > 4 or a 3 > 4 other scenarios of the root movement are possible: for example, formation of triple negative real roots as in Figure 5.

Note, however, that even when the triple negative real root occurs at Re λ < -1, there is another negative real root inside the unit circle. Therefore, in accordance with the above Theorem, configurations with triple roots are not global minimizers of the abscissa of the polynomial [START_REF] Bruce | Functions on a crosscap[END_REF]. Nevertheless, it is instructive to understand the set in the (a 1 , a 3 , a 2 )-space where the roots of the polynomial are real and negative, but not necessarily simple. Consider the discriminant8 of the polynomial [START_REF] Bruce | Functions on a crosscap[END_REF] and equate it to zero:

:= 16a 4 2 -4(a 2 1 + a 2 3 )a 3 2 + (a 2 1 a 2 3 -80a 1 a 3 -128)a 2 2 + 18(a 1 a 3 + 8)(a 2 1 + a 2 3 )a 2 + 256 -27a 4 3 -27a 4 1 -6a 2 1 a 2 3 -192a 1 a 3 -4a 3 1 a 3 3 = 0. ( 17
)
In the (a 1 , a 3 , a 2 )-space, the polynomial (5) has at least one multiple root at the points of the set given by [START_REF] Henrion | Ellipsoidal approximation of the stability domain of a polynomial[END_REF]. For example, in the plane

a 1 = a 3 = (a 2 + 2 + 2a 3 )(a 2 + 2 -2a 3 )(4a 2 -8 -a 2 3 ) 2 ,
and the equation = 0 yields the straight line a 2 = 2(a 3 -1) and parabola a 2 = 2 + a 2 3 /4 that touch each other at a 3 = 4 and a 2 = 6, which corresponds to the EP 4 point in Figure 3. This corresponds to the quadruple negative real root -1 of the polynomial [START_REF] Bruce | Functions on a crosscap[END_REF], which is a global minimum of its abscissa.

Plotting the discriminant surface [START_REF] Henrion | Ellipsoidal approximation of the stability domain of a polynomial[END_REF], we see in Figure 3 that it has a self-intersection along the part of the EP-set (13) selected by the inequality a 2 ≥ 6 and two cuspidal edges defined in the parametric form as

a 1 = a 2 2 ± a 2 a 2 2 -36 -12 a 2 ± a 2 2 -36 3/2 √ 6,
a 3 = a 2 2 ± a 2 a 2 2 -36 + 36 3 6a 2 ± 6 a 2 2 -36 , a 2 ≥ 6.
At the cuspidal edges the polynomial (5) has a triple negative real root and a simple negative real root. For example, at the points

λ 1 = - √ 3 9 , λ 2, 3, 4 = - √ 3,
and

λ 1 = -3 √ 3, λ 2, 3, 4 = - √ 3 3 . ( 18
)
At the point EP 4 with the coordinates (4, 4, 6) in the (a 1 , a 3 , a 2 )space the discriminant surface has the Swallowtail singularity, which is a generic singularity of bifurcation diagrams in threeparameter families of real matrices [START_REF] Arnold | Lectures on bifurcations in versal families[END_REF][START_REF] Galin | Real matrices depending on parameters[END_REF]; see Figure 3. Therefore, the coefficients of the globally minimizing polynomial ( 16) are exactly at the Swallowtail singularity of the discriminant surface of the polynomial [START_REF] Bruce | Functions on a crosscap[END_REF].

We notice, however, that the bifurcation diagram shown in Figure 3 allows us to predict the root configurations that generically exist at the global optimum of the polynomial abscissa when more than one affine constraint restricts the coefficients of the polynomial. Indeed, let the number of affine constraints be κ. Then, according to [START_REF] Galin | Real matrices depending on parameters[END_REF] and [START_REF] Arnold | Lectures on bifurcations in versal families[END_REF], the highest possible codimension of a generic singularity in the restricted space of parameters is n -κ. Therefore, introduction of an additional constraint (κ = 2) generically removes the quadruple real root λ = -1 of codimension 3 from the candidates to be global minimizers: n -κ = 2. Assuming for example a 2 = const., we find only double real or complex conjugate roots when 0 < a 2 < 6 whereas when a 2 > 6 we find only triple real roots. As Figure 7 shows, when a 2 is restricted to 10, the global minimum is at the algebraically smallest of the triple negative roots [START_REF] Henrion | Inner approximations for polynomial matrix inequalities and robust stability regions[END_REF].

In the region inside the "spike" formed by the discriminant surface all the roots are simple real and negative. Owing to this property, this region, belonging to the domain of asymptotic stability (see Figure 1), plays an important role in linear stability theory [START_REF] Veselic | Damped Oscillations of Linear Systems[END_REF], as the next section demonstrates.

HEAVILY DAMPED SYSTEMS IN MECHANICS AND PHYSICS

Consider a linear mechanical system described by the equation [START_REF] Veselic | Damped Oscillations of Linear Systems[END_REF] ẍ

+ Dẋ + Kx = 0, (19)
where x(t) ∈ IR m , t ≥ 0, and with the m × m matrices of viscous damping D and stiffness K both real, symmetric and positive definite. If the magnitude of the damping forces is big enough, one can expect that all the eigenvalues λ of the corresponding matrix polynomial L(λ) = λ 2 I + λD + K will not only have negative real parts but be real and negative. The system [START_REF] Arnold | Lectures on bifurcations in versal families[END_REF] with semisimple real and negative eigenvalues is called heavily damped [START_REF] Bulatovic | On the heavily damped response in viscously damped dynamic systems[END_REF]. The solutions of the heavily damped systems do not oscillate and monotonically decrease, which is favorable for applications in robotics and automatic control [START_REF] Barkwell | Overdamped and gyroscopic vibrating systems[END_REF]. Note that in magnetohydrodynamics (MHD) there exists an inverse problem of excitation of instabilities by changing parameters of a heavily damped system. This is the famous MHD dynamo problem where by controlling the flow of an electrically conducting non-ideal fluid one has to excite and maintain magnetic field. At relatively slow and simple fluid motions the corresponding linearized equations define a heavily damped system with all its solutions decaying and non-oscillating, see [START_REF] Arnold | The growth of a magnetic field in a three-dimensional steady incompressible flow[END_REF][START_REF] Kirillov | Determining role of Krein signature for three-dimensional Arnold tongues of oscillatory dynamos[END_REF][START_REF] Giesecke | Spectral properties of oscillatory and nonoscillatory α 2 -dynamo[END_REF][START_REF] Bouya | Revisiting the ABC flow dynamo[END_REF]. Transition from non-oscillating to oscillating solutions in the dynamo problem is crucial for explanation of the geomagnetic field polarity reversals [START_REF] Stefani | Asymmetric polarity reversals, bimodal field distribution, and coherence resonance in a spherically symmetric mean-field dynamo model[END_REF].

Assume that K is given and that we wish to choose the damping matrix D so as to ensure that all solutions to [START_REF] Arnold | Lectures on bifurcations in versal families[END_REF] decay to zero as rapidly as possible. This problem has been largely solved by Freitas and Lancaster [START_REF] Freitas | On the optimal value of the spectral abscissa for a system of linear oscillators[END_REF]; see also [START_REF] Veselic | Damped Oscillations of Linear Systems[END_REF].

Let α(D, K) denote the spectral abscissa of the matrix polynomial L(λ), that is the abscissa of its characteristic

polynomial: 9 α(D, K) = a(det(L(λ)) = max{Re λ : det(L(λ)) = 0}.
The roots of the characteristic polynomial det(L(λ)) are the eigenvalues of the matrix polynomial L(λ) and also the eigenvalues of the 2m × 2m companion matrix 0 I -K -D .

The optimal damping rate is determined by the solution of the optimization problem

α * = inf D α(D, K).
Remarkably, Freitas and Lancaster [START_REF] Freitas | On the optimal value of the spectral abscissa for a system of linear oscillators[END_REF] established that, for any

K > 0, α * ≥ - 2m √ det K
with equality holding if and only if L(λ) has just one eigenvalue with algebraic multiplicity 2m.

In the case m = 2, this problem reduces to the one studied in the section 2. Since the constant term of the characteristic polynomial is det K, we introduce the quantities [START_REF] Kirillov | Nonconservative Stability Problems of Modern Physics[END_REF]

a 1 = trD 4 √ det K , a 2 = trK + det D √ det K , a 3 = trKtrD -trKD 4 (det K) 3 (20)
and following [37, Section 4], we reduce the characteristic polynomial of L(λ) to the form (5), except that the roots have been scaled by 4 √ det K. Conditions for asymptotic stability of the polynomial (5) are therefore given by (4).

Solving the system of equations a 1 = 4, a 2 = 6, a 3 = 4, where the coefficients a 1 , a 2 , and a 3 are defined by [START_REF] Troger | Zur korrekten modellbildung in der dynamik diskreter systeme[END_REF], yields an explicit expression for the damping matrix D that minimizes the spectral abscissa α(D, K) when m = 2 in terms of the coefficients k 11 , k 22 , k 12 and the eigenvalues ω 2 1 and ω 2 2 (ω 1,2 > 0) of the given matrix K. As shown by Freitas and Lancaster [START_REF] Freitas | On the optimal value of the spectral abscissa for a system of linear oscillators[END_REF], but expressed here in different notation, given K > 0, we find that α(D, K) attains its global minimum with L(λ) having just one eigenvalue λ = -

D ± = 4k 12 √ ω 1 ω 2 ± (k 11 -k 22 )(ω 1 -ω 2 ) (ω 1 + ω 2 ) 2 -2k 12 k 11 -k 22 1 1 2k 12 k 11 -k 22 + - 4 √ ω 1 ω 2 (ω 1 ω 2 -k 11 ) k 11 -k 22 0 0 4 √ ω 1 ω 2 (ω 1 ω 2 -k 22 ) k 11 -k 22 . ( 21
)
Notice that D ± may be either positive definite or indefinite and that D + = D -if and only if K = κI, where κ is a scalar [START_REF] Freitas | On the optimal value of the spectral abscissa for a system of linear oscillators[END_REF].

Now we can give the following interpretation of the stability diagram of Figure 1. The dissipative system ( 19) is asymptotically stable inside the domain (4) with the coefficients a 1 , a 2 , and a 3 defined by [START_REF] Troger | Zur korrekten modellbildung in der dynamik diskreter systeme[END_REF]. The boundary of the domain (4) has the Whitney umbrella singular point at a 1 = 0, a 3 = 0, and a 2 = 2. The domain corresponding to heavily damped systems of the form [START_REF] Arnold | Lectures on bifurcations in versal families[END_REF] is confined between three hypersurfaces of the discriminant surface [START_REF] Henrion | Ellipsoidal approximation of the stability domain of a polynomial[END_REF] and has a form of a trihedral spike with the Swallowtail singularity at its cusp at a 1 = 4, a 2 = 6, and a 3 = 4; see Figure 3. The Whitney umbrella and the Swallowtail singular points are connected by the EP-set given by [START_REF] Moro | On the Lidskii-Vishik-Lyusternik perturbation theory for eigenvalues of matrices with arbitrary Jordan structure[END_REF]. At the Swallowtail singularity of the boundary of the domain of heavily damped systems, the abscissa of the characteristic polynomial of the damped potential system [START_REF] Arnold | Lectures on bifurcations in versal families[END_REF] attains its global minimum.

Therefore, by minimizing the spectral abscissa one finds points at the boundary of the domain of heavily damped systems. Furthermore, the sharpest singularity at this boundary corresponding to a quadruple real eigenvalue λ = -4 √ det K < 0 with the Jordan block of order four is the very point where all the modes of the system [START_REF] Arnold | Lectures on bifurcations in versal families[END_REF] with m = 2 are decaying to zero as rapidly as possible when t → +∞.

DISCUSSION

Typical singularities of the contour plots of the abscissa (or decrement diagrams) in the plane of two parameters were listed already by Arnold [START_REF] Arnold | Lectures on bifurcations in versal families[END_REF]. Let us have a closer look at the decrement diagram of the polynomial [START_REF] Bruce | Functions on a crosscap[END_REF] shown in Figure 8. The additional constraint (a 2 = 10) removes the quadruple real root from the generic singularities; hence, the minimizer of the abscissa is the EP 3 point corresponding to the triple root λ = -√ 3 3 (see Figure 7, which shows the graph of the abscissa in three dimensions). The lower (curved) contours in Figure 8 correspond to the real parts of a complex-conjugate pair. The upper contours are straight lines and correspond to a simple negative real root. The contours cross either at the points of the discriminant set [START_REF] Henrion | Ellipsoidal approximation of the stability domain of a polynomial[END_REF] corresponding to the double negative real roots (the lower part of the red cuspidal curve in Figure 8) or at the points where both complex conjugate roots and the real root have the same real part; see [START_REF] Arnold | Lectures on bifurcations in versal families[END_REF].

What are the relationships among the decrement diagram, stability boundary, discriminant set, and abscissa minimization? For a detailed answer it is simpler to consider a two-parameter cubic analog of the polynomial family [START_REF] Bruce | Functions on a crosscap[END_REF] p(λ) := λ 3 + a 1 λ 2 + a 2 λ + 1 [START_REF] Burke | Optimizing matrix stability[END_REF] and to look closely not only at the active roots, that is those whose real parts coincide with the abscissa a(p), but also at the remaining inactive roots, whose real part is algebraically smaller than a(p). The polynomial [START_REF] Burke | Optimizing matrix stability[END_REF] is Hurwitz if and only if

a 1 a 2 > 1, a 1 > 0, a 2 > 0. ( 23
)
The discriminant curve of the polynomial ( 22) The open circle marks the location of the EP 3 singularity of the bifurcation diagram, where the abscissa reaches its minimum. The contours corresponding to simple real roots are straight lines that sweep out the cuspidal area confined between the boundaries of the bifurcation diagram. In the limit, these contours are tangent to the black cuspidal curve and may be viewed as rays forming a caustic [START_REF] Arnold | Lectures on bifurcations in versal families[END_REF][START_REF] Arnold | Singularities of Caustics and Wave Fronts[END_REF][START_REF] Ehlers | The theory of caustics and wave front singularities with physical applications[END_REF]. The curved green contours correspond to real parts of a complex conjugate pair of eigenvalues. The black hyperbola marks the stability boundary with the stable region above it and to the right. Along the stability boundary, a few (negative) gradient vectors of the real part of the complex conjugate pair of active eigenvalues are marked at a 1 = 0.25, a 1 = 1, and a 1 = 4.

(a 1 a 2 + 9) 2 -4(a 3 1 + a 3 2 + 27) = 0 ( 24
Theorem of section 2 the polynomial abscissa a(p) has its global minimum at the singular EP 3 point of the discriminant curve corresponding to a triple root λ = -1.

Denote

x = (a 1 , a 2 ), y = Re(∇ x λ), (25)
where the gradient of a root is given by the expression

∇ x λ = - ∇ x p ∂ λ p . ( 26
)
For example, taking into account that at the points of the stability boundary (a 1 a 2 = 1) the roots are

λ 1, 2 = ±i √ a 1 a 1 , λ 3 = -a 1 ,
we find that the gradient of the real part of the complex-conjugate pair is

y(λ 1, 2 ) = - 1 2 1 1 + a 3 1 , a 2 1 1 + a 3 1 . ( 27
)
Figure 9 is plotted in the (a 1 , a 2 ) parameter space. The black hyperbola shows the stability boundary, with the stable region [START_REF] Lewis | The mathematics of eigenvalue optimization[END_REF] above and to the right. The small black strokes drawn at right angles to the hyperbola indicate some of the (negative) gradients [START_REF] Blondel | Explicit solutions for root optimization of a polynomial family with one affine constraint[END_REF].

The green curves and line segments in Figure 9 show the contours of the maximum of the real parts of the three roots of ( 22), i.e., they comprise a contour plot of the polynomial abscissa (the decrement diagram). More specifically, the green curves in the stable region correspond to a complex conjugate pair of active roots; in this region, as well as in the unstable region, the contours of the remaining inactive simple real root are shown by gray line segments. The green line segments appearing in the top part of the figure correspond to a simple real active root; in this region the contours of the real part of the complex conjugate pair of inactive roots are marked by the red curves.

Inside the cuspidal area whose boundary is exactly the discriminant curve given by [START_REF] Burke | Variational analysis of functions of the roots of polynomials[END_REF], all three roots are real: the contours of the active root are marked by green segments, while the contours of the two inactive real roots are shown by red and gray line segments. In fact, the rays defined by the green, gray, and red line segments shown in the figure sweep out the cuspidal area. The boundary of the region of real roots can thus be viewed as a caustic or an evolute of the rays defining the contours of the negative real roots [START_REF] Arnold | Lectures on bifurcations in versal families[END_REF]. The open circle at the cusp is the EP 3 singularity, where the abscissa reaches its global minimum with all three roots equal to -1. 10 For example, movement of eigenvalues in parametric families of real symmetric and Hermitian matrices possesses physical interpretation as dynamics of Pechukas-Yukawa gas [START_REF] Yukawa | Lax form of the quantum mechanical eigenvalue problem[END_REF] which is an integrable multidimensional system; the latter fact in its turn is used for analysis and design of numerical algorithms for eigenvalue computation [START_REF] Bloch | Steepest descent, linear programming, and Hamiltonian flows[END_REF][START_REF] Brockett | Dynamical systems that sort lists, diagonalize matrices, and solve linear programming problems[END_REF][START_REF] Iserles | Lie-group methods[END_REF][START_REF] Chu | Linear algebra algorithms as dynamical systems[END_REF]. Notice also that computation of multiple roots and eigenvalues is nowadays an important applied problem related to the so-called "physics of exceptional points" [START_REF] Berry | Physics of nonhermitian degeneracies[END_REF][START_REF] Heiss | The physics of exceptional points[END_REF] in systems described by non-Hermitian Hamiltonians [START_REF] Moiseyev | Non-Hermitian Quantum Mechanics[END_REF][51][START_REF] Kokoouline | Laser-controlled rotational cooling of Na 2 based on exceptional points[END_REF].

Despite the lack of smoothness and convexity of the abscissa and spectral abscissa functionals, numerical methods have been developed that are quite practical for finding local minimizers for arbitrary polynomial and matrix parameterizations-even when these minima correspond to multiple roots or multiple eigenvalues-provided the number of variables and the optimal multiplicities are not too large [START_REF] Burke | Stabilization via nonsmooth, nonconvex optimization[END_REF][START_REF] Burke | A robust gradient sampling algorithm for nonsmooth, nonconvex optimization[END_REF][START_REF] Lewis | Nonsmooth optimization via quasi-Newton methods[END_REF][START_REF] Overton | Stability optimization for polynomials and matrices[END_REF]. It may be interesting to interpret these or related methods in terms of rays and wave fronts propagating in the parameter space. Then, the powerful formalism of the theory of caustics and wave front singularities [START_REF] Arnold | Singularities of Caustics and Wave Fronts[END_REF][START_REF] Nye | Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations[END_REF][START_REF] Ehlers | The theory of caustics and wave front singularities with physical applications[END_REF] could give a geometrical and even physical 10 insight to the remarkable fact that globally minimizing the abscissa or spectral abscissa typically leads to multiple roots or multiple eigenvalues. Such an interpretation of polynomial root and matrix eigenvalue optimization algorithms as dynamical systems could be a line of research that is mutually fruitful for numerical analysis, stability theory, and mathematical physics.

FIGURE 1 |

 1 FIGURE 1 | A singular boundary of the domain of asymptotic stability (4) of the polynomial (5) with the Whitney umbrella singularity at the point (a 1 , a 3 , a 2 ) = (0, 0, 2), marked by the diamond symbol. Inside the domain of asymptotic stability a part of the discriminant surface of the polynomial is situated with the Swallowtail singularity at the point (4, 4, 6) marked as an open circle.Inside the spike bounded by the discriminant surface all the roots of the polynomial (5) are real and negative. The green planar sector is the tangent cone[START_REF] Vishik | The solution of some perturbation problems for matrices and selfadjoint or non-selfadjoint differential equations I[END_REF] to the domain of asymptotic stability at the Whitney umbrella singularity.

FIGURE 2 |FIGURE 3 |

 23 FIGURE 2 | The real parts of the roots of the polynomial (5) with the coefficients in the set (13) and (shown by the bold red curve) the abscissa (12).

FIGURE 4 |FIGURE 5 |

 45 FIGURE 4 | Trajectories of roots of the polynomial (5) when a 2 increases from 0 to 15 and: a 1 = a 3 = 3.7 (black); a 1 = 3.7, a 3 = 3.6 (red); a 1 = 3.6, a 3 = 3.7 (green).

FIGURE 6 | 5 ) when a 2 increases from 0 to 15 and: a 1 = a 3 = 4 (black); a 1 = 4 , a 3 =

 65134143 FIGURE 6 | Trajectories of roots of the polynomial (5) when a 2 increases from 0 to 15 and: a 1 = a 3 = 4 (black); a 1 = 4, a 3 = 3.9 (red); a 1 = 3.9, a 3 = 4 (green). The global minimum of the abscissa is attained when all the roots coalesce into the quadruple root λ = -1 (EP 4 ).

√ 3 ,

 3 10 we have, respectively,

FIGURE 7 |√ 3 .

 73 FIGURE 7 | When a 2 is restricted to 10, the abscissa of (5) reaches its minimum at the triple negative real root -√ 3 3 when a 1 = 4 √ 3 and a 3 = 28 9 √ 3. The red curve is a cross-section of the discriminant surface (17) by the plane a 2 = 10.

4 √

 4 det K = -√ ω 1 ω 2 with algebraic multiplicity 4 and geometric multiplicity 1, if D = D + or D = D -, where[START_REF] Kirillov | Nonconservative Stability Problems of Modern Physics[END_REF]

FIGURE 8 |

 8 FIGURE 8 |The gray curves show the contour plot (decrement diagram) of the abscissa a of the polynomial (5) when a 2 is restricted to 10. The green circle at the cusp of the red discriminant curve[START_REF] Henrion | Ellipsoidal approximation of the stability domain of a polynomial[END_REF] marks the location of the EP 3 singularity, where the abscissa reaches its minimum in this case.

FIGURE 9 |

 9 FIGURE 9 | The green curves show the contour plot (decrement diagram) of the abscissa a of the cubic polynomial (22) with contour plots of the real parts of the inactive roots superimposed (red and gray curves).The open circle marks the location of the EP 3 singularity of the bifurcation diagram, where the abscissa reaches its minimum. The contours corresponding to simple real roots are straight lines that sweep out the cuspidal area confined between the boundaries of the bifurcation diagram. In the limit, these contours are tangent to the black cuspidal curve and may be viewed as rays forming a caustic[START_REF] Arnold | Lectures on bifurcations in versal families[END_REF][START_REF] Arnold | Singularities of Caustics and Wave Fronts[END_REF][START_REF] Ehlers | The theory of caustics and wave front singularities with physical applications[END_REF]. The curved green contours correspond to real parts of a complex conjugate pair of eigenvalues. The black hyperbola marks the stability boundary with the stable region above it and to the right. Along the stability boundary, a few (negative) gradient vectors of the real part of the complex conjugate pair of active eigenvalues are marked at a 1 = 0.25, a 1 = 1, and a 1 = 4.

A surface is ruled if every one of its points lies on a line that belongs to the surface; the ruled surface is literally swept out by these lines. Familiar examples include a conical surface or a hyperboloid of one sheet. www.frontiersin.org December 2013 | Volume 1 | Article 24 | 1

Those whose algebraic multiplicity exceeds their geometric multiplicity (the number of linearly independent associated eigenvectors) and that are therefore associated with Jordan blocks of size 2 or higher in the Jordan canonical form of the matrix. Frontiers in Physics | Mathematical Physics December 2013 | Volume 1 | Article 24 | 2

In fact, even simple eigenvalues of non-Hermitian (more specifically, nonnormal) matrices are often highly sensitive to perturbation, and when this is the case, eigenvalues may not be a useful tool for studying system behavior, as they model only asymptotic behavior of linear systems, not transient or non-linear behavior. See[START_REF] Trefethen | Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators[END_REF] for further discussion of non-normality and how alternatives to eigenvalues, notably pseudospectra, may be used to model physical systems; a particularly well-studied example is the transition to turbulence in high Reynolds number fluid

flows.[START_REF] Bruce | Functions on a crosscap[END_REF] The contour plot of the abscissa in the plane of two parameters is known also as the decrement diagram [

[START_REF] Arnold | Lectures on bifurcations in versal families[END_REF][START_REF] Troger | Zur korrekten modellbildung in der dynamik diskreter systeme[END_REF].[START_REF] Levantovskii | The boundary of a set of stable matrices[END_REF] In the sense of T. Kato[START_REF] Kirillov | Nonconservative Stability Problems of Modern Physics[END_REF][START_REF] Berry | Physics of nonhermitian degeneracies[END_REF][START_REF] Heiss | The physics of exceptional points[END_REF].

www.frontiersin.org December 2013 | Volume 1 | Article 24 | 3

As explained in[START_REF] Blondel | Explicit solutions for root optimization of a polynomial family with one affine constraint[END_REF], most of this result when F = R was established by R. Chen in a 1979 Ph.D. thesis. Frontiers in Physics | Mathematical Physics December 2013 | Volume 1 | Article 24 | 4

The discriminant is a function of the coefficients of a polynomial which vanishes if and only if the polynomial has a repeated (real or complex) root. When the discriminant of a real polynomial changes its sign, a complex conjugate pair of roots either originates or disappears.

www.frontiersin.org December 2013 | Volume 1 | Article 24 | 5

In[START_REF] Freitas | On the optimal value of the spectral abscissa for a system of linear oscillators[END_REF], a slightly different notation is used, with their spectral abscissa and ours having opposite sign. Frontiers in Physics | Mathematical Physics December 2013 | Volume 1 | Article 24 | 6

www.frontiersin.org December 2013 | Volume 1 | Article 24 | 7

Frontiers inPhysics | Mathematical Physics December 2013 | Volume 1 | Article 24 | 8

www.frontiersin.org December 2013 | Volume 1 | Article 24 | 9

ACKNOWLEDGMENTS FUNDING

The second author was supported in part by the U.S. National Science Foundation under grant DMS-1317205.

Conflict of Interest Statement:

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


Ivan Nourdin

David Nualart

Fisher Information and the Fourth Moment Theorem

Keywords: Fisher information, total variation distance, relative entropy, Fourth Moment Theorem, Fractional Brownian motion, Malliavin calculus

. To illustrate the power of our approach we prove a local limit theorem together with some rates of convergence for the normal convergence of a standardized version of the quadratic variation of the fractional Brownian motion.

Introduction

Measuring the discrepancy between the law of a given real-valued random variable F and that of its Gaussian counterpart N is arguably an important and recurrent problem both in probability and statistics. For instance, one faces this situation when trying to prove a central limit type theorem, or when wanting to check the asymptotic normality of an estimator. And quite often, the choice of a suitable probability metric reveals to be a crucial step.

In the present paper, we are concerned with this question within the framework of the Malliavin calculus. More precisely, we will focus on the Wiener chaos of a given order and, as a way to measure the proximity betweens laws, we will work either with the L r -distance between densities (especially for r = 1 and r = ∞), or with the relative entropy D(F N),

or with the relative Fisher information J(F ) -1. These three notions, that we will recall now, are strongly related to each other.

Let F be a centered real-valued random variable with unit variance and density p F . We suppose throughout that all needed assumptions on p F (such as its strictly positivity, differentiability, etc.) are always satisfied when required. Let also N ∼ N(0, 1) be standard Gaussian, with density p N (x) = e -x 2 /2 / √ 2π, x ∈ R. The L r -distance between densities of F and N is given by

p F -p N r = R |p F (x) -p N (x)| r dx 1 r , r ∈ [1, ∞);
(1.1)

p F -p N ∞ = sup x∈R |p F (x) -p N (x)| (assuming, say, that p F is continous).
Actually, in what follows we will only consider the particular cases r = 1 and r = ∞. This is because the bounds we will produce are going to be of the same order. So, a bound for the L r -distance will simply follow from the crude estimate:

p F -p N r p F -p N 1/r 1 p F -p N 1-1/r ∞ .
When r = 1 in (1.1), it is an easy exercise (sometimes referred to as the Scheffé's theorem) to show that p F -p N 1 = 2d T V (F, N), where d T V (F, N) is the total variation distance defined as

d T V (F, N) = sup A∈B(R) |P (F ∈ A) -P (N ∈ A)|. (1.2)
It is clear from its very definition (1.2) that d T V (F, N) represents a strong measure on how close the laws of F and N are.

The relative entropy D(F N) of F with respect to N is given by

D(F N) = R p F (x) log(p F (x)/p N (x))dx. (1.3)
Our interest in this quantity comes from its link with the total variation distance, as provided by the celebrated Csiszár-Kullback-Pinsker inequality, according to which:

2 d T V (F, N) 2 D(F N). (1.4)
(In particular, note that D(F N) 0.) See, e.g., [START_REF] Bolley | Weighted Csiszár-Kullback-Pinsker inequalities and applications to transportation inequalities[END_REF] for a proof of (1.4) and original references.

Inequality (1.4) shows that bounds on the relative entropy translate directly into bounds on the total variation distance. Hence, it makes perfectly sense to quantify the discrepancy between the law of F and that of the standard Gaussian N in terms of its relative entropy. Actually, one can go even further by considering the Fisher information J(F ) of F . Let us recall its definition. Let s F (F ) denote the score associated to F . This is the F -measurable random variable uniquely determined by the following integration by parts:

E[φ ′ (F )] = -E[s F (F )φ(F )]
for all test function φ : R → R.

(1.5)

When it makes sense, it is easy to compute that s

F = p ′ F /p F . Set J(F ) = E[s F (F ) 2 ]
if the random variable s F (F ) is square-integrable and J(F ) = +∞ otherwise. In the former case, it is a straightforward exercise to check that

J(F ) -1 = E[(s F (F ) + F ) 2 ].
In particular, J(F ) 1 = J(N) with equality if and only if F is standard Gaussian. Our interest in the relative Fisher information J(F ) -1 comes from its link with the relative entropy through the following de Bruijn's formula (stated in an integral and rescaled version due to Barron [START_REF] Barron | Entropy and the central limit theorem[END_REF]; see also [START_REF] Johnson | Information theory and the central limit theorem[END_REF]Theorem C.1]). Assume, without loss of generality, that F and N are independent; then

D(F N) = 1 0 J( √ tF + √ 1 -tN) -1 2t dt. (1.6)
Since from, e.g., [START_REF] Johnson | Information theory and the central limit theorem[END_REF]Lemma 1.21] one has J(

√ tF + √ 1 -tN) tJ(F ) + (1 -t)J(N) = 1 + t(J(F ) -1), we deduce that D(F N) 1 2 (J(F ) -1). (1.7)
By comparing (1.7) with (1.4), we observe that the gap between J(F ) and 1 = J(N) is an even stronger measure of how the law of F is close to the standard Gaussian N. This claim is even more supported by the Shimizu's inequality [START_REF] Shimizu | On Fisher's amount of information for location family[END_REF], which gives a L ∞ -bound between p F and p N provided p F is continuous and satisfies x 2 p F (x) → 0 as x → ±∞:

p F -p N ∞ J(F ) -1.
(1.8) (In the original statement of Shimizu [START_REF] Shimizu | On Fisher's amount of information for location family[END_REF], there is actually an extra factor (1 + 6/π) in the right-hand side of (1.8); but this latter was removed by Ley and Swan in [START_REF] Ley | Stein's density approach and information inequalities[END_REF]).

Let us now come to the description of the main results contained in the present paper. From now on, we will systematically assume that F belongs to a Wiener chaos H q of order q 2, that is, has the form of a qth multiple Wiener-Itô integral (see Section 2 below for precise definitions). Our first result is the following, with DF the norm of the Malliavin derivative of F (again, see Section 2 for details).

Theorem 1.1 Let q 2 be an integer and let F ∈ H q have variance one. Assume in addition that ε > 0 and η 1 satisfy

E[ DF -4-ε ] η.
(1.9)

Then, there exists a constant c > 0, depending on q, ε and η but not on F , such that

J(F ) -1 c(E[F 4 ] -3). (1.10)
In the next result, we take advantage of the conclusion (1.10) of Theorem 1.1 to complete the current state of the art related to the Fourth Moment Theorem of Nualart and Peccati [START_REF] Nualart | Central limit theorems for sequences of multiple stochastic integrals[END_REF]. See also the discussion located just after the statement of Corollary 1.2.

Corollary 1.2 Fix an integer q 2, and let (F n ) ⊂ H q be a sequence of random variables satisfying E[F 2 n ] = 1 for all n. Then, the following four assertions are equivalent as n → ∞:

(a) E[F 4 n ] → 3; (b) F n law → N ∼ N(0, 1); (c) d T V (F n , N) = 1 2 p Fn -p N 1 → 0; (d) D(F n N) → 0.
Moreover, there exists c 1 , c 2 , c 3 , c 4 > 0 (independent of n) such that, for all n large enough,

c 1 max{|E[F 3 n ]|, E[F 4 n ] -3} d T V (F n , N) c 2 max{|E[F 3 n ]|, E[F 4 n ] -3} (1.11) c 3 E[F 4 n ] -3;
(1.12)

D(F n N) c 4 (E[F 4 n ] -3)| log(E[F 4 n ] -3)|. (1.13)
Suppose in addition that, for some ε > 0,

lim sup n→∞ E[ DF n -4-ε ] < ∞. (1.14)
Then, the four previous assertions (a) -(d) are equivalent to the following two further assertions:

(e) p Fn -p N ∞ → 0;

(f ) J(F n ) → 1.
More precisely, one has the existence of c 5 > 0 (independent of n) such that, for all n large enough, .16) Equivalence between (a) and (b) in Corollary 1.2 is known as the Fourth Moment Theorem. This striking result, discovered by Nualart and Peccati in [START_REF] Nualart | Central limit theorems for sequences of multiple stochastic integrals[END_REF], has been the starting point of a new and fruitful line of research, consisting in using the Malliavin calculus to prove limit theorems. It has led to a burst of new research in many different fields, such as information theory [START_REF] Nourdin | Entropy and the fourth moment phenomenon[END_REF], stochastic geometry [START_REF] Lachièze-Rey | Fine Gaussian fluctuations on the Poisson space I: contractions, cumulants and random geometric graphs[END_REF][START_REF] Reitzner | Central Limit Theorems for U-statistics of Poisson Point Processes[END_REF], Markov operator [START_REF] Azmoodeh | Fourth Moment Theorems for Markov Diffusion Generators[END_REF][START_REF] Ledoux | Chaos of a Markov operator and the fourth moment condition[END_REF], random matrices of large size [START_REF] Nourdin | Universal Gaussian fluctuations of non-Hermitian matrix ensembles: from weak convergence to almost sure CLTs[END_REF], free probability [START_REF] Kemp | Wigner chaos and the fourth moment[END_REF][START_REF] Nourdin | Poisson approximations on the free Wigner chaos[END_REF], q-calculus [START_REF] Arizmendi | Convergence of the fourth moment and infinite divisibility[END_REF][START_REF] Deya | Nourdin: Fourth Moment Theorem and q-Brownian Chaos[END_REF], computer science [START_REF] De | Deterministic Approximate Counting for Juntas of Degree-2 Polynomial Threshold Functions[END_REF][START_REF] De | Efficient deterministic approximate counting for low-degree polynomial threshold function[END_REF], cosmology [START_REF] Marinucci | Ergodicity and Gaussianity for Spherical Random Fields[END_REF][START_REF] Marinucci | On Nonlinear Functionals of Random Spherical Eigenfunctions[END_REF], statistics [START_REF] Bardet | Moment bounds and central limit theorems for Gaussian subordinated arrays[END_REF][START_REF] Hu | Parameter estimation for fractional Ornstein-Uhlenbeck processes[END_REF], or spin glasses [START_REF] Nourdin | Comparison inequalities on Wiener space[END_REF][START_REF] Viens | Stein's lemma, Malliavin calculus, and tail bounds, with application to polymer fluctuation exponent[END_REF], to name a few. One can also consult the constantly updated webpage http://www.iecn.u-nancy.fr/~nourdin/steinmalliavin.htm for literally hundreds of results related to the Fourth Moment Theorem and its ramifications.

p Fn -p N 2 ∞ c 5 (E[F 4 n ] -3) (1.15) J(F n ) -1 c 5 (E[F 4 n ] -3). ( 1
Equivalence between (a) and (c) in Corollary 1.2, coming from the bound (1.12), is due to Nourdin and Peccati [START_REF] Nourdin | Stein's method on Wiener chaos[END_REF]. By combining Malliavin calculus with the Stein's method, they were indeed able to show that, for any random variable

F ∈ H q such that E[F 2 ] = 1, d T V (F, N) 2E|1 - 1 q DF 2 | 4q -4 3q E[F 4 ] -3, (1.17)
see [START_REF] Nourdin | Normal Approximations using Malliavin Calculus: from Stein's Method to the Universality[END_REF]Theorem 5.2.6]. (Note that E[F 4 ] > 3 in (1.17): see, e.g., [START_REF] Nourdin | Normal Approximations using Malliavin Calculus: from Stein's Method to the Universality[END_REF]Lemma 5.2.4].) The refinement (1.11) of (1.12), leading to optimal rates, is taken from [START_REF] Nourdin | The optimal fourth moment theorem[END_REF].

Equivalence between (a) and (d) in Corollary 1.2, as well as the bound (1.13), was shown by Nourdin, Peccati and Swan in [START_REF] Nourdin | Entropy and the fourth moment phenomenon[END_REF]. Their strategy of proof relied on the discovery a novel representation formula for the relative entropy, namely,

D(F N) = 1 2 1 0 t 1 -t E E[N(1 - 1 q DF 2 )| √ tF + √ 1 -tN] 2 dt
for any F ∈ H q with unit variance and where N ∼ N(0, 1) is supposed to be independent of F . When (1.14) is satisfied for ε = 2, the inequality (1.15) (leading to the equivalence between (a) and (e) in Corollary 1.2) was proved by Hu, Lu and Nualart in [START_REF] Hu | Convergence of densities of some functionals of Gaussian processes[END_REF], after adapting Stein's method to handle the supremum distance. Note that combining our Theorem 1.1 with Shimizu inequality (1.8) allows to recover (1.15) (which corresponds to Theorem 4.1 in [START_REF] Hu | Convergence of densities of some functionals of Gaussian processes[END_REF]).

Finally, inequality (1.16) (leading to the equivalence between (a) and (f ) in Corollary 1.2) is new and will be a direct consequence of Theorem 1.1. It is worth noting at this stage that validity of (1.9) is, unfortunately, far to be a small assumption. Let us discuss this point a little bit more. As it is well-known, the Bouleau-Hirsch criterion (see, e.g., [START_REF] Nualart | The Malliavin calculus and related topics[END_REF]Theorem 2.1.3]) asserts that any (smooth and bounded enough) random variable F in the Wiener space admits a density as soon as P ( DF > 0) = 1. This latter condition is always satisfied for F ∈ H q with unit variance, see [START_REF] Shigekawa | Derivatives of Wiener functionals and absolute continuity of induced measures[END_REF]. In fact, one can prove a far better statement, see [29, (3.19)]: there exists a constant c q > 0 such that, for all x > 0 and all F ∈ H q with unit variance,

P ( DF 2 x) ≥ 1 -c q x 1 2q-2 .
(1.18)

As a consequence, using that

E[ DF -r ] = ∞ 0 P ( DF 2 u -2 r )du 1 + ∞ 1 P ( DF 2 u -2 r )du, (1.19)
one deduces from (1.18) that, for all F ∈ H q with unit variance,

E[ DF -r ] 1 + c q 1 r(q-1) -1 provided r < 1 q-1 . (1.20)
Unfortunately, one cannot deduce (1.14) from (1.20). It means that verifying (1.14) has to be made on a case-by-case basis, and heavily depends on the particular sequence (F n ) we are dealing with. In [START_REF] Hu | Convergence of densities of some functionals of Gaussian processes[END_REF], one can find an application for the least squares estimator of the parameter θ in the Ornstein-Uhlenbeck process dX t = θX t dt + dW t , where W is a standard Brownian motion. In the present paper, we consider a more involved application to the quadratic variation of a fractional Brownian motion B H of index H. We obtain optimal rates for the relative Fisher information when H < 5 8 , and (possibly suboptimal) rates when H < 3 4 . More precisely, let us introduce the so-called fractional Gaussian noise associated with B H , which is the Gaussian sequence given by

ξ k = B H (k + 1) -B H (k), k ∈ N ∪ {0}.
(1.21)

Set

F n := 1 √ n v n n-1 k=0 (ξ 2 k -1), (1.22)
with

v n > 0 chosen so that E[F 2 n ] = 1.
It is well-known (it is indeed a very particular case of the Breuer-Major Theorem [START_REF] Breuer | Central limit theorems for non-linear functionals of Gaussian fields[END_REF], see also [START_REF] Nourdin | Selected Aspects of Fractional Brownian Motion[END_REF]Theorem 7.2]) that, as n → ∞,

F n law → N(0, 1) ⇐⇒ H ∈ (0, 3/4].
(1.23)

In Section 4, we will show that (1.14) is satisfied for F n defined by (1.22). Then, as a consequence of (1.15) and (1.16) on one hand and of the estimates for E[F 4 n ] -3 computed in [START_REF] Biermé | Optimal Berry-Esseen rates on the Wiener space: the barrier of third and fourth cumulants[END_REF] on the other hand, we will be able to deduce the following local limit theorem for F n .

Theorem 1.3 Let F n be as in (1.22). Then, their exists c, C > 0 independent of n such that, for all n large enough,

p Fn -p N ∞ J(F n ) -1 C ×    n -1 2 if 0 < H < 5 8 n -1 2 log 3 2 n if H = 5 8 n 4H-3 if 5 8 < H < 3 4 and J(F n ) -1 c n -1 2 if H < 5 8 .
A brief outline of the paper is as follows. In Section 2, we introduce the language of the Malliavin calculus, which is the framework in which our study takes place. We also recall the Carbery-Wright inequality, which will play a key role in the proof of Theorem 1.3. Proofs of Theorem 1.1 and Corollary 1.2 are presented in Section 3. Finally, Section 4 contains the proof of Theorem 1.3.

Notation and preliminaries

The language of Gaussian analysis and Malliavin calculus

We start by briefly recalling some basic notation and results connected to Gaussian analysis and Malliavin calculus. The reader is referred to [START_REF] Nourdin | Normal Approximations using Malliavin Calculus: from Stein's Method to the Universality[END_REF][START_REF] Nualart | The Malliavin calculus and related topics[END_REF] for details or missing proofs.

Let H be a real separable Hilbert space with inner product •, • H . The norm of H will be denoted by • = • H . Recall that we call isonormal Gaussian process over H any centered Gaussian family X = {X(h) : h ∈ H}, defined on a probability space (Ω, F , P ) and such that E[X(h)X(g)] = h, g H for every h, g ∈ H. Assume from now on that F is the σ-field generated by X.

For any integer q ∈ N ∪ {0}, we denote by H q the qth Wiener chaos of X. We recall that H 0 is simply R whereas, for any q 1, H q is the closed linear subspace of L 2 (Ω) generated by the family of random variables {H q (X(h)), h ∈ H, h H = 1}, with H q the qth Hermite polynomial given by

H q (x) = (-1) q e x 2 2 d q dx q e -x 2 2 .
For any q 1, we denote by H ⊗q (resp. H ⊙q ) the qth tensor product (resp. the qth symmetric tensor product) of H. Then, the mapping I q (h ⊗q ) = H q (X(h)) can be extended to a linear isometry between H ⊙q (equipped with the modified norm √ q! • H ⊗q ) and H q . For q = 0 and x ∈ R, we write I 0 (x) = x. In the particular case where H = L 2 (A, A , µ), where (A, A ) is a measurable space and µ is a σ-finite and non-atomic measure, one has that H ⊙q = L 2 s (A q , A ⊗q , µ ⊗q ) is the space of symmetric and square integrable functions on A q . Moreover, for every f ∈ H ⊙q , the random variable I q (f ) coincides with the multiple Wiener-Itô integral (of order q) of f with respect to X.

Recall that L 2 (Ω) = ∞ q=0 H q , meaning that every square-integrable random variable F measurable with respect to F admits a unique decomposition of the type

F = E[F ] + ∞ q=1 I q (f q ), (2.24)
where the series converges in L 2 (Ω), and f q ∈ H ⊙q , for q 1. Identity (2.24) is the socalled Wiener-Itô chaotic decomposition of F . According to a classical result of Shigekawa [START_REF] Shigekawa | Derivatives of Wiener functionals and absolute continuity of induced measures[END_REF], when F is not zero and when the kernels f q in (2.24) all equal zero except for a finite number, then the distribution of F necessarily admits a density with respect to the Lebesgue measure.

Let {e i , i 1} be a complete orthonormal system in H. Given f ∈ H ⊙p and g ∈ H ⊙q , for every r = 0, . . . , p ∧ q, the contraction of f and g of order r is the element of H ⊗(p+q-2r) defined by

f ⊗ r g = ∞ i 1 ,...,ir=1 f, e i 1 ⊗ • • • ⊗ e ir H ⊗r ⊗ g, e i 1 ⊗ • • • ⊗ e ir H ⊗r .
Note that, in the particular case where H = L 2 (A, A , µ) (with µ non-atomic), one has that (f ⊗ r g)(t 1 , . . . , t p+q-2r ) = A r f (t 1 , . . . , t p-r , s 1 , . . . , s r ) g(t p-r+1 , . . . , t p+q-2r , s 1 , . . . , s r )dµ(s 1 ) . . . dµ(s r ).

Moreover, f ⊗ 0 g = f ⊗ g equals the tensor product of f and g while, for p = q, f ⊗ p g = f, g H ⊗p . The contraction f ⊗ r g is not necessarily symmetric, and we denote by f ⊗ r g its symmetrization. We have the following product formula: if f ∈ H ⊙p and g ∈ H ⊙q then

I p (f )I q (g) = p∧q r=0 r! p r q r I p+q-2r (f ⊗ r g).
(2.25)

We will now introduce some standard operators from Malliavin calculus. Let S be the set of all cylindrical random variables of the form

F = g(X(h 1 ), . . . , X(h n )),
where n 1, h i ∈ H, and g is infinitely differentiable such that all its partial derivatives have polynomial growth. The Malliavin derivative of F is the element of L 2 (Ω; H) defined by

DF = n i=1 ∂g ∂x i (X(h 1 ), . . . , X(h n ))h i .
By iteration, for every m 2, we define the mth derivative D m F which is an element of L 2 (Ω; H ⊙m ). For m 1 and p 1, D m,p denote the closure of S with respect to the norm • m,p defined by

F p m,p = E[|F | p ] + m j=1 E D j F p H ⊗j .
One can then extend the definition of D m to D m,p . When m = 1, one simply write D instead of D 1 . As a consequence of the hypercontractivity property of the Ornstein-Uhlenbeck semigroup (see, e.g., [24, Theorem 2.7.2]), all the • m,p -norms are equivalent in any finite sum of Wiener chaoses. This is a crucial result that will be used all along the paper.

The Malliavin derivative D satisfies the following chain rule: if ϕ : R n → R is in C 1 b (that is, belongs to the set of continuously differentiable functions with a bounded derivative) and if {F i } i=1,...,n is a vector of elements of D 1,2 , then ϕ(F 1 , . . . , F n ) ∈ D 1,2 and

Dϕ(F 1 , . . . , F n ) = n i=1 ∂ϕ ∂x i (F 1 , . . . , F n )DF i .
Also, when H = L 2 (A, A , µ) (with µ non-atomic), one has, for any f ∈ L 2 s (A q , A ⊗q , µ ⊗q ),

D x (I q (f )) = qI q-1 (f (•, x)), x ∈ A.
The divergence operator δ, which will play a crucial role in our approach, is defined as the adjoint of D. Denoting by dom δ its domain, one has the so-called integration by parts formula: for every D ∈ D 1,2 and every u ∈ domδ,

E[F δ(u)] = E[ DF, u H ].
(2.26)

We will moreover need the following two properties. For every F ∈ D 1,2 and every u ∈ domδ such that F u and F δ(u) + DF, u H are square integrable, one has that F u ∈ domδ and

δ(F u) = F δ(u) -DF, u H . (2.27)
Also, one has a commutation relationship between the Malliavin derivative and the Skorohod integral:

Dδ(u) = u + δ(Du), (2.28)
for any u ∈ D 2,2 (H). In particular, for such an u,

E[δ(u) 2 ] = E[ u 2 H ] + E[ Du 2 H ⊗2 ]. (2.29)

Carbery-Wright inequality

In the proof of Theorem 1.3, we will make use of the following inequality due to Carbery and Wright [8, Theorem 8]: there is an absolute constant c > 0 such that, for any d, n 1, any polynomial Q : R n → R of degree at most d and any Gaussian random vector (X 1 , . . . , X n ),

E[|Q(X 1 , . . . , X n )|] 1 d P (|Q(X 1 , . . . , X n )| x) c d x 1 d , x > 0.
(2.30)

Proofs of Theorem 1.1 and Corollary 1.2

In what follows, c denote positive constants which may depend of q, ε and η but not of F , and whose values may change from one appearance to the next. Also, •, • ( • , respectively) always stands for inner product (the norm, respectively) in an appropriate tensor product H ⊗s .

Proof of Theorem 1.1

Observe first that, without loss of generality, we may and will assume that X is an isonormal process over some Hilbert space of the type H = L 2 (A, A , µ) (with µ non atomic).

Due to (1.9) and the fact that F has moments of all order by hypercontractivity, it is straightforward to check that DF DF -2 ∈ domδ with E[δ(DF DF -2 ) 2 ] < ∞. Let φ : R → R be a test function. We have, on one hand,

E[δ(DF DF -2 )φ(F )] = E[ Dφ(F ), DF DF -2 ] = E[φ ′ (F )].
After setting Σ = 1 -1 q DF 2 and because δDF = qF for any F ∈ H q , we deduce from (1.5) that

s F (F ) + F = -E δ DF DF -2 - 1 q F = -E δ(DF DF -2 Σ) F .
Using the formula

δ(GDF ) = Gδ(DF ) -DF, DG = qF G -DF, DG , one can write for G = DF -2 Σ, -δ(DF DF -2 Σ) = -qF DF -2 Σ + DF, DG .
Notice that

DG = -2 DF -4 (D 2 F ⊗ 1 DF )Σ - 2 q DF -2 (D 2 F ⊗ 1 DF ).
Therefore,

DF, DG = -2 DF -4 D 2 F, DF ⊗ DF )Σ - 2 q DF -2 DF, D 2 F ⊗ 1 DF .
This leads to the estimate

| DF, DG | ≤ 2 DF -2 D 2 F |Σ| + 2 q DF -1 D 2 F ⊗ 1 DF .
As a consequence,

δ(DF DF -2 Σ) 2 ≤ 2q 2 F 2 DF -4 Σ 2 +16 DF -4 D 2 F 2 Σ 2 + 16 q 2 DF -2 D 2 F ⊗ 1 DF 2 .
Thus, using among other properties the hypercontractivity for F , D 2 F 2 and Σ,

J(F ) -1 = E[(s F (F ) + F ) 2 ] E δ(DF DF -2 Σ) 2 c E[ DF -4-ε ] 4 4+ε E[Σ 2 ] + E[Σ 2 ]E[ D 2 F 4 ] 1 2 + E[ D 2 F ⊗ 1 DF 4 ] 1 2 . (3.31)
Now, use the product formula to get that, for any x ∈ A,

(D 2 F ⊗ 1 DF )(x) = q 2 (q -1) A I q-2 (f (x, y, •))I q-1 (f (y, •))dµ(y) = q 2 (q -1) q-2 r=0 r! q -1 r q -2 r I 2q-3-2r A f (x, y, •) ⊗ r f (y, •)dµ(y) = q 2 (q -1) q-1 r=1 (r -1)! q -1 r -1 q -2 r -1 I 2q-1-2r ((f ⊗ r f )(x, •)).
As a result, using again the product formula and with c q,r,s,a some constant whose exact value is useless here,

D 2 F ⊗ 1 DF 2 = q-1 r,s=1 2q-1-2(r∨s) a=0 c q,r,s,a I 4q-2-2r-2s-2a A (f ⊗ r f )(x, •) ⊗ a (f ⊗ s f )(x, •) dµ(x) ,
implying in turn

E D 2 F ⊗ 1 DF 4 c q-1 r,s=1 2q-1-2(r∨s) a=0 A (f ⊗ r f )(x, •) ⊗ a (f ⊗ s f )(x, •) dµ(x) 2 c q-1 r,s=1 2q-1-2(r∨s) a=0 A (f ⊗ r f )(x, •) ⊗ a (f ⊗ s f )(x, •) dµ(x) 2 c q-1 r,s=1 A (f ⊗ r f )(x, •) (f ⊗ s f )(x, •) dµ(x) 2 c q-1 r=1 A (f ⊗ r f )(x, •) 2 dµ(x) 2 = c q-1 r=1 f ⊗ r f 2 2 c(E[F 4 ] -3) 2 ,
the last inequality following from [24, identities (5.2.5)-(5.2.6)]. On the other hand, we can also write

D 2 F 2 = q 2 (q -1) 2 A 2 I q-2 (f (x, y, •)) 2 dµ(x)dµ(y) = q 2 (q -1) 2 q-2 r=0 r! q -2 r 2 I 2q-4-2r A 2 f (x, y, •) ⊗ r f (x, y, •)dµ(x)dµ(y) = q 2 (q -1) 2 q r=2 (r -2)! q -2 r -2 so that, using moreover that f ⊗ r f 2 f 4 = 1/q! 2 , E[ D 2 F 4 ] = q 4 (q -1) 4 q r=2 (r -2)! 2 q -2 r -2 4 (2q -2r)! f ⊗ r f 2 c.
Finally, recall from [START_REF] Nourdin | Normal Approximations using Malliavin Calculus: from Stein's Method to the Universality[END_REF]Lemma 5.2.4] that E[Σ 2 ]

q-1 3q (E[F 4 ] -3). Hence, by plugging all the previous estimates in (3.31), one finally obtains the desired inequality (1.10). ✷

Proof of Corollary 1.2

As we said in the Introduction, the equivalences between (a), (b), (c), (d) and (e) (provided (1.9) holds true for the latter one), as well as the estimates (1.11) and (1.13), are straightforward consequences of the main results contained in [START_REF] Hu | Convergence of densities of some functionals of Gaussian processes[END_REF], [START_REF] Nourdin | Stein's method on Wiener chaos[END_REF], [START_REF] Nourdin | Entropy and the fourth moment phenomenon[END_REF] and [START_REF] Nualart | Central limit theorems for sequences of multiple stochastic integrals[END_REF]. Note in passing that Shimizu's inequality (1.8) indeed takes place for the random variable we are considering. This is because, if F ∈ H q satisfies E[ DF -4-ε ] < ∞ then, by [START_REF] Nualart | The Malliavin calculus and related topics[END_REF]Proposition 2.1.1], F has a continuous density given by

p F (x) = E[1 {F >x} δ(DF DF -2 )].
Hence, using moreover that E[δ(DF DF -2 )] = 0, one deduces that

p 2 F (x) P (|F | > x)E[δ(DF DF -2 ) 2 ] |x| -r E |F | r E[δ(DF DF -2 ) 2 ],
implying in turn that p(x) = o(|x| -n ) as |x| → ∞.

The equivalencebetween (f ) and (a), provided condition (1.14) is fulfilled, follows immediately from Theorem 1.1 (for one implication) and the fact that bounds on the Fisher information translates directly into bounds on the total variation distance through (1.10) and (1.4) (for the other implication).

Inequality (1.16) also follows immediately from (1.10). ✷

4 Proof of Theorem 1.3

Preparation to the proof

For F n as in the statement, recall from [START_REF] Biermé | Optimal Berry-Esseen rates on the Wiener space: the barrier of third and fourth cumulants[END_REF][START_REF] Nourdin | The optimal fourth moment theorem[END_REF] that their exist c, C > 0 independent of n such that, for all n large enough,

E[F 4 n ] -3 C ×    n -1 if 0 < H < 5 8 n -1 log 3 n if H = 5 8 n 8H-6 if 5 8 < H < 3 4 12 and d T V (F n , N) c n -1 if H > 5 8
. Assume for an instant that (1.14) has been checked. Then, as far as the upper (resp. lower) bound is concerned, the desired conclusion directly follows from (1.15) and (1.16) (from (1.4) and (1.7), respectively).

So, in order to complete the proof of Theorem 1.3, it remains to check that (1.9) holds true.

Checking (1.9)

For simplicity, throughout all the proof, we write B instead of B H to indicate the fractional Brownian motion of index H ∈ (0, 1) we are dealing with. We know that B has an integral representation of the form

B t = t 0 K 1 (t, s)dW s , t 0, (4.32)
where {W t , t ≥ 0} is a standard Brownian motion. By convention we will assume that

K 1 (t, s) = 0 if t ≤ s. Also, we set ∆K 1 (t, s) = K 1 (t, s) -K 1 (t -1, s).
Recall the definition (1.21) of ξ k and the definition (1.22) of F n . We claim that, for any p ≥ 1, there exist n 0 such that

sup n≥n 0 E( DF n -p ) < ∞. (4.33)
Note that (4.33) implies (1.14). The proof of our claim (4.33) is based on the following approach. First we will derive a lower bound for DF n 2 (in distribution) denoted by B n and defined in (4.34), which can be expressed as the sum of the square norms of n Gaussian random variables. Then, we fix an integer N ≥ 1 and we decompose B n into the sum of N blocks B i n , i = 0, . . . , N -1 of size [n/N]. The basic inequality (4.35) reduces the problem to estimate negative moments of order p N of each block B i n , provided these blocks are independent. These negative moments can be estimated by the Carbery-Wright inequality if N is large enough. Actually, the blocks are not independent, but we can control the conditional expectation of each block given the previous ones, using the properties of the fractional Brownian motion. Then, it suffices to show that these conditional expectations do not vanish as n tends to infinity (condition (4.40)), which is done in the two final steps of the proof.

The random variables ξ k form a centered stationary Gaussian sequence with covariance

ρ(k) = E[ξ r ξ r+k ] = 1 2 |k + 1| 2H + |k -1| 2H -2|k| 2H .
We can thus write, with D the Malliavin derivative with respect to B,

DF n 2 = 4 nv n n j,k=1 ξ j ξ k ρ(j -k).
Suppose that { ξ j , 1 ≤ j ≤ n} is an independent copy of the sequence {ξ j , 1 ≤ j ≤ n}. Then,

DF n 2 = 4 nv n E   n j=1 ξ j ξ j 2   ,
with obvious notations. The sequence ξ j can be chosen of the form ξ j = B j -B j-1 , where B is a fractional Brownian motion of Hurst index H which is independent of B. We know that B has a representation of the form (different in nature from (4.32))

B t = κ H t 0 (t -s) H-1 2 d W s + Z t ,
where W is a standard Brownian motion (independent of W ), Z is a process independent of W (and of W ) and κ H is a constant only depending on H. We set K 2 (t) = t H-1 2 if t > 0 and K 2 (t) = 0 if t ≤ 0, and ∆K 2 (t) = K 2 (t) -K 2 (t -1). With this notation we can write

ξ j = κ H j 0 ∆K 2 (j -s)d W s + Z j -Z j-1 .
As a consequence, and since

v n = 2 n n k,l=1 ρ(k -l) 2 2 j∈Z ρ(j) 2 < ∞ for H ∈ (0, 3 4 ), DF n 2 = c H n E   n j=1 ξ j j 0 ∆K 2 (j -s)d W s 2   = c H n E   n j=1 ξ j j h=1 h h-1 ∆K 2 (j -s)d W s 2   = c H n E   n h=1 h h-1 n j=h ξ j ∆K 2 (j -s) d W s 2   = c H n n h=1 h h-1 n j=h ξ j ∆K 2 (j -s) 2 ds.
Making a change of indices, we obtain

DF n 2 ≥ A n := c H n n h=1 n-h+1 n-h h j=1 ξ n-j+1 ∆K 2 (n -j + 1 -s) 2 ds.
The sequences {ξ j , 1 ≤ j ≤ n} and {ξ n-j+1 , 1 ≤ j ≤ n} have the same law, so A n has the same law as With the change of variable s → n + 1 -s, we get

B n := c H n n h=1 n-h+1 n-h h j=1 ξ j ∆K 2 (n -j + 1 -s)
B n = c H n n h=1 h+1 h h j=1 ξ j ∆K 2 (s -j) 2 ds.
Fix an integer N ≥ 1 and let M = [n/N] be the integer part of n/N. Then, n ≥ NM. As a consequence,

B n ≥ c H n N -1 i=0 (i+1)M h=iM +1 h+1 h h j=1 ξ j ∆K 2 (s -j) 2 ds.
Set

B i n = c H n (i+1)M h=iM +1 h+1 h h j=1 ξ j ∆K 2 (s -j) 2
ds, i = 0, . . . , N -1.

We are going to use the estimate

(B n ) -p ≤ N -1 i=0 (B i n ) -p N . (4.35)
Consider again the representation of the sequence ξ as stochastic integrals with respect to a Brownian motion W , and denote by {F W t } the filtration generated by the Brownian motion W . Then,

E[(B n ) -p ] ≤ E N -1 i=0 (B i n ) -p N = E E[(B N -1 n ) -p N |F W (N -1)M ] N -2 i=0 (B i n ) -p N . ( 4

.36)

Let us estimate the conditional expectation appearing in the right-hand side. In the same spirit that (1.19), it is immediate that

E[(B N -1 n ) -p N |F W (N -1)M ] ≤ 1 + p N 1 0 P B N -1 n x|F W (N -1)M x -p N -1 dx.
By Carbery-Wright's inequality (2.30) with d = 2,

P B N -1 n x|F W (N -1)M ≤ c √ x E B N -1 n |F W (N -1)M -1 d . (4.37) The conditional expectation E B N -1 n |F W (N -1)M
is given by

E B N -1 n |F W (N -1)M = c H n N M h=(N -1)M +1 h+1 h E   h j=1 ξ j ∆K 2 (s -j) 2 |F W (N -1)M   ds. (4.38)
Taking into account that the sequence ξ j is Gaussian, the conditional expectation appearing in the above equation can be bounded below by the conditional variance which is not random. More precisely,

E   h j=1 ξ j ∆K 2 (s -j) 2 |F W (N -1)M   ≥ Var h j=1 ξ j ∆K 2 (s -j) F W (N -1)M = E     h j=(N -1)M +1 j (N -1)M ∆K 1 (j, u)dW u ∆K 2 (s -j)   2   . ( 4
i = 0, . . . , N -1, lim inf M →∞ 1 M (i+1)M h=iM +1 h+1 h E   h j=iM +1 j iM ∆K 1 (j, u)dW u ∆K 2 (s -j) 2   ds > 0. (4.40)
Proof of (4.40). First we compute the expectation in (4.40):

E   h j=iM +1 j iM ∆K 1 (j, u)dW u ∆K 2 (s -j) 2   = h j,k=iM +1 ∆K 2 (s -j)∆K 2 (s -k) j∧k iM ∆K 1 (j, u)∆K 1 (k, u)du. Set β M j,k = j∧k iM ∆K 1 (j, u)∆K 1 (k, u)du.
We can write, exchanging the order of the summation

(i+1)M h=iM +1 h j,k=iM +1 h+1 h ∆K 2 (s -j)∆K 2 (s -k)ds β M j,k = (i+1)M j,k=iM +1 (i+1)M h=j∨k h+1 h ∆K 2 (s -j)∆K 2 (s -k)ds β M j,k = (i+1)M j,k=iM +1 α M j,k β M j,k , where α M j,k = (i+1)M +1 j∨k ∆K 2 (s -j)∆K 2 (s -k)ds.
Then we are interested in the liminf, as M tends to infinity, of

1 M (i+1)M j,k=iM +1 α M j,k β M j,k .
We make now the change of indices j → j -iM and k → k -iM and we obtain the expression

Ψ M := 1 M M j,k=1 α M j,k β M j,k , where
β M j,k = (j+iM )∧(k+iM ) iM ∆K 1 (j + iM, u)∆K 1 (k + iM, u)du = j∧k 0 ∆K 1 (j + iM, u + iM)∆K 1 (k + iM, u + iM)du, and
α M j,k = M +1 j∨k ∆K 2 (s -j)∆K 2 (s -k)ds.
Step 1: Case H > 1 2 . In this case ∆K 1 and ∆K 2 are nonnegative. On the other hand, by [30, (5.10)], we have

∂K 1 ∂t (t, s) = c H (t/s) H-1 2 (t -s) H-3 2 ,
where c H =

H(2H-1) β(2-2H,H-1/2) . Therefore, assuming j ≥ k ≥ 2, β M j,k ≥ k-1 1 ∆K 1 (j + iM, u + iM)∆K 1 (k + iM, u + iM)du ≥ c 2 H k-1 1 j+iM j+iM -1 x u + iM H-1 2 (x -u -iM) H-3 2 dx × k+iM k+iM -1 y u + iM H-1 2 (y -u -iM) H-3 2 dy du ≥ c 2 H k-1 1 (j + iM -1)(k + iM -1) (u + iM) 2 H-1 2 (j -u) H-3 2 (k -u) H-3 2 du.
The term (j+iM -1)(k+iM -1)

(u+iM ) 2 is lower bounded by 1. Therefore

β M j,k ≥ c 2 H k-1 1 (j -u) H-3 2 (k -u) H-3 2 du = c 2 H k-1 1 (x + j -k) H-3 2 x H-3 2 dx ≥ c 2 H k-1 1 (x + j -k) 2H-3 dx = C (j -k + 1) 2H-2 -(j -1) 2H-2 .
By similar arguments we obtain, assuming again j ≥ k ≥ 2, α M j,k ≥ terms, we can get a lower bound as in the case H > 1 2 . When j = k, the integrands in the definition of β M j,k and α M j,k are nonnegative. On the other hand, for j ≥ k + 1, we can write In this way we obtain the decomposition

α M j,k = α M j,k,1 + α M j,k,2 , where α M j,k,1 = j+1 j (s -j) H-1 2 (s -k) H-1 2 -(s -k -1) H-1 2 ds ≤ 0,
Ψ M = 1 M M j=1 α M j,j β M j,j + 2 M 1≤k≤j-1≤M -1 α M j,k,1 + α M j,k,2 β M j,k,1 + β M j,k,2 = Ψ 1 M + Ψ 2 M + Ψ 3 M , where
Ψ 1 M = 1 M M j=1 α M j,j β M j,j + 2 M 1≤k≤j-1≤M -1 α M j,k,1 β M j,k,1 + α M j,k,2 β M j,k,2 , Ψ 2 M = 2 M 1≤k≤j-1≤M -1 α M j,k,1 β M j,k,2 ,
and

Ψ 3 M = 2 M 1≤k≤j-1≤M -1 α M j,k,2 β M j,k,1 .
The term Ψ 1 M is nonnegative and it can be bounded below as follows

Ψ 1 M ≥ 2 M 2≤k≤j-1≤M -1 M +1 j+2 ∆K 2 (s -j)∆K 2 (s -k)ds × k-1 1
∆K 1 (j + iM, u + iM)∆K 1 (k + iM, u + iM)du .

By the same arguments as in the case H > 1 2 we can show that lim inf M →∞ Ψ 1 M > 0. Therefore, it suffices to show that [(s -j) H-1 2 -(s -j -1)

H-1 2 ][(s -k) H-1 2 -(s -k -1) H-1 2 ]ds = M +1-j 1 [x H-1 2 -(x -1) H-1 2 ][(x + j -k) H-1 2 -(x + j -k -1) H-1 2 ]dx ≤ C + M +1-j 2 [x H-1 2 -(x -1) H-1 2 ][(x + j -k) H-1 2 -(x + j -k -1) H-1 2 ]dx ≤ C + C H M +1-j 2 (x -1) H-3 2 (x + j -k -1) H-3 2 dx ≤ C + C H M +1-j 2 (x -1) 2H-3 dx ≤ C ′ .
On the other hand, we have

| α M j,k,1 | ≤ 1 0 x H-1 2 j -k + x -1) H-1 2 -(j -k + x) H-1 2 dx = 1 3 2 -H 1 0 x H-1 2 j-k j-k-1 (u + x) H-3 2 du dx ≤ 1 ( 3 2 -H)( 1 2 + H) (j -k -1) H-3 2 ,
and

| β M j,k,1 | ≤ k k-1 K 1 (k + iM, u + iM)|∆K 1 (j + iM, u + iM)|du = c H k k-1 K 1 (k + iM, u + iM) j+iM j+iM -1 x u + iM H-1 2 (x -u -iM) H-3 2 dx du ≤ c H j + iM -1 k + iM H-1 2 k k-1 K 1 (k + iM, u + iM) j j-1 (x -u) H-3 2 dx du ≤ C(j -k -1) H-3 2 ,
because j+iM -1 k+iM 1, the integral

∆K 2 1 K 1 ∆K 1

 2111 (s -j)∆K 2 (s -k)ds ≥ 0. Similarly β M j,k = β M j,k,1 + β M j,k,2 , (k + iM, u + iM)∆K 1 (j + iM, u + iM)du ≤ 0, (k + iM, u + iM)∆K 1 (j + iM, u + iM)du ≥ 0.

1 iM∆K 1 iM +k- 1 iM∆K 1 1 iM∆K 1 0 ∆K 1 0 ∆K 1

 1111110101 limits are based on the following estimates. One one hand, β M j,k,2 and α M j,k,2 are uniformly bounded:β M j,k,2 = iM +k-(k + iM, u)∆K 1 (j + iM, u)du ≤ (k + iM, u) 2 du iM +k-(j + iM, u) 2 du ≤ k+iM (k + iM, u) 2 du j+iM (j + iM, u) 2 du = E(|B k+iM -B k+iM -1 | 2 ) E(|B j+iM -B j+iM -1 | 2 ) = 1,

j j- 1 1 K 1

 111 (x -u) H-3 2 dx is bounded by (j -k -1) H-3 2 uniformly in u ∈ [k -1, k], and sup (k + iM, u + iM)du < ∞,as it can be easily checked from the expression of K 1 . Finally,lim M →∞ 1 M 1≤k≤j-2≤M -2 (j -k -1) H-3 2 = 0,which implies both (4.41) and (4.42).

I 2q-2r (f ⊗ r f ),

* I. Nourdin is partially supported by the ANR grant 'Malliavin, Stein and Stochastic Equations with Irregular Coefficients' [ANR-10-BLAN-0121]. † D. Nualart is supported by the NSF grant DMS1208625

other terms in the above expression converge to zero. For instance,

and this last quantity behaves as M 4H-3 , which converges to zero because H < 3 4 . A similar analysis can be done for the other terms.

Step 2: Case H < 1 2 . In this case, see [30, (5.23)], we have that

is negative. Therefore ∆K 1 (j, s) is negative if s < j -1 and positive if j -1 s j. Also,

and ∆K 2 (u) is negative if u 1 and positive if u < 1. Then, it suffices to show that the negative terms do not contribute to the limit, and once we get rid of these negative