Jörg Hennig

Marcus Ansorg
email: marcus.ansorg@helmholtz-muenchen.de

Regularity of Cauchy horizons in S 2 × S 1 Gowdy spacetimes

Keywords: numbers: 98.80.Jk, 04.20.Cv, 04.20.Dw

We study general S 2 × S 1 Gowdy models with a regular past Cauchy horizon and prove that a second (future) Cauchy horizon exists, provided that a particular conserved quantity J is not zero. We derive an explicit expression for the metric form on the future Cauchy horizon in terms of the initial data on the past horizon and conclude the universal relation A p A f = (8πJ) 2 where A p and A f are the areas of past and future Cauchy horizon respectively.

Introduction

The well-known singularity theorems by Hawking and Penrose [START_REF] Hawking | The large scale structure of spacetime[END_REF] show that cosmological solutions to the Einstein equations generally contain singularities. As discussed by Clarke [START_REF] Clarke | The analysis of space-time singularities[END_REF] (see also [START_REF] Ellis | [END_REF] for a comprehensive overview) there are two types of singularities: (i) Curvature singularities, for which components of the Riemann tensor or its kth derivatives are irregular (e.g. unbounded), and (ii) quasiregular singularities, which are associated with peculiarities in the topology of space-time (e.g. the vertex of a cone), although the local geometry is well behaved. In addition, the curvature singularities are divided up into scalar singularities (for which some curvature invariants are badly behaved) and nonscalar singularities (for which arbitrarily large or irregular tidal forces occur). The singularity theorems mentioned above provide, however, in general no information about the specific type of singularity -they make statements solely about causal geodesic incompleteness. This lack of knowledge concerning the specific nature of the singular structure is the reason for many open outstanding problems in general relativity, including the strong cosmic censorship conjecture and the BKL conjecture (see [START_REF] Andersson | The global existence problem in general relativity The Einstein equations and the large scale behavior of gravitational fields: 50 years of the Cauchy problem in General Relativity[END_REF] for an overview).

A major motivation for the study of Gowdy spacetimes as relatively simple, but non-trivial inhomogeneous cosmological models results from the desire to understand the mathematical and physical properties of such cosmological singularities. The Gowdy cosmologies, first studied in [17,18], are characterized by an Abelian isometry group U (1) × U (1) with spacelike group orbits, i.e. these spacetimes possess two associated spacelike and commuting Killing vector fields ξ and η. Moreover, the definition of Gowdy spacetimes includes that the twist constants αβγδ ξ α η β ∇ γ ξ δ and αβγδ ξ α η β ∇ γ η δ (which are constant as a consequence of the field equations) are zero †.

For compact, connected, orientable and smooth three manifolds, the corresponding spatial topology must be either T 3 , S 3 , S 2 × S 1 or L(p, q), cf. [18] (see also [26,[START_REF] Neumann | 3-dimensional G-manifolds with 2-dimensional orbits[END_REF][START_REF] Fischer | The theory of superspace Relativity[END_REF]). Note that the universal cover of the lens space L(p, q) is S 3 and hence this case need not be treated separately, see references in [10].

In the T 3 -case, global existence in time with respect to the areal foliation time t was proved by Moncrief [25]. Moreover, he has shown that the trace of the second fundamental form blows up uniformly on the hypersurfaces t = constant in the limit t → 0. As a consequence, the solutions do not permit a globally hyperbolic extension beyond the time t = 0. However, to date it has not been clarified whether the solutions are extendible (as non-globally hyperbolic C 2 -solutions) or are generically subject to curvature singularities at t = 0.

Although global existence of solutions inside the "Gowdy square" (i.e. for 0 < t < π, cf. Fig. 1 below) was shown by Chruściel for S 2 × S 1 and S 3 topology, see Thm. 6.3 in [10], it is still an open question whether globally hyperbolic extensions beyond the hypersurfaces t = 0 or t = π exist. It is expected that these hypersurfaces contain either curvature singularities or Cauchy horizons; the theorem in [10] however does not in fact exclude the possibility that these are merely coordinate singularities.

For polarized Gowdy models, where the Killing vector fields can be chosen to be orthogonal everywhere, the nature of the singularities for all possible spatial topologies has been studied in [22,11]. In particular, strong cosmic censorship and a version of the BKL conjecture have been proved. Investigations of singularities in the unpolarized case for T 3 topology can be found in [START_REF] Berger | [END_REF]24,33,34].

For unpolarized S 3 or S 2 × S 1 Gowdy spacetimes not many results on singularities (strong cosmic censorship, BKL conjecture, Gowdy spikes) are known. Particular singular solutions have been constructed with Fuchsian techniques in [35]. Moreover, numerical studies indicate that the behavior near singularities and the appearance of spikes are similar to the T 3 -case [START_REF] Garfinkle | [END_REF]6,7].

In this paper, we study general (unpolarized or polarized) S 2 × S 1 Gowdy models with a regular Cauchy horizon (with S 2 × S 1 topology) at t = 0 (cf. Fig. 1) ‡ and assume that the spacetime is regular (precise regularity requirements are given below) at this horizon as well as in a neighborhood. As mentioned above, a theorem by Chruściel [10] implies then that the metric is regular for all t < π, i.e. excluding only the future hypersurface t = π. With the methods utilized in this paper we are able to provide † The assumption of vanishing twist constants is non-trivial only in the case of spatial T 3 topology. Note that in spatial S 3 or S 2 × S 1 topology there are specific axes on which one of the Killing vectors vanishes identically, which leads to vanishing twist constants. ‡ Without loss of generality we choose a past Cauchy horizon H p . the missing piece, i.e. we prove that under our regularity assumptions the existence of a regular second (future) Cauchy horizon H f (at t = π) is implied, provided that a particular conserved quantity J is not zero §. Moreover, we derive an explicit expression for the metric form on the future Cauchy horizon in terms of the initial data on the past horizon. From this explicit formula, the universal relation A p A f = (8πJ) 2 between the areas A p , A f of past and future Cauchy horizons and the above mentioned conserved quantity J can be concluded.

The proofs of these statements can be found by relating any S 2 × S 1 Gowdy model to a corresponding axisymmetric and stationary black hole solution (with possibly nonpure vacuum exterior, e.g. with surrounding matter), considered between outer event and inner Cauchy horizon. Note that the region between these horizons is regular hyperbolic, i.e. the Einstein equations are hyperbolic PDEs in an appropriate gauge with coordinates adapted to the Killing vectors, see [START_REF] Ansorg | [END_REF]3,23] . (The Kerr metric is an explicitly known solution of these PDEs, see [START_REF] Obregón | [END_REF].) ¶ As a consequence, the results on the regularity of the interior of such black holes and existence of regular Cauchy horizons inside the black holes obtained in [START_REF] Ansorg | [END_REF]3,23] can be carried over to Gowdy spacetimes.

The results in [START_REF] Ansorg | [END_REF] were found by utilizing a particular soliton method -the so-called Bäcklund transformation. Making use of the theorem by Chruściel mentioned earlier, it was possible to show that a regular Cauchy horizon inside the black hole always exists, provided that the angular momentum of the black hole does not vanish. (The above quantity J is the Gowdy counterpart of the angular momentum.)

In [3,23] these results have been generalized to the case in which an additional Maxwell field is considered. The corresponding technique, that is the inverse scattering method, again comes from soliton theory and permits the reconstruction of the field quantities along the entire boundary of the Gowdy square. Hereby, an associated linear matrix problem is analyzed, whose integrability conditions are equivalent to the nonlinear field equations in axisymmetry and stationarity. Note that in this article we restrict ourselves to the pure Einstein case (without Maxwell field) and refer the reader to [3,23] for results valid in full Einstein-Maxwell theory.

We start by introducing appropriate coordinates, adapted to the description of regular axes and Cauchy horizons at the boundaries of the Gowdy square, see Sec. 2. Moreover, we revisit the complex Ernst formulation of the field equations and corresponding boundary conditions and introduce the conserved quantity J in question. In this formulation we can translate the results of [START_REF] Ansorg | [END_REF]3,23] and obtain the metric on § As we will see in Sec. 2.3, the conserved quantity J vanishes in polarized Gowdy models.

The interior of axisymmetric and stationary black hole solutions is non-compact and has spatial S 2 × R topology. Here the R-factor is generated by a subgroup of the symmetry group corresponding to one of the Killing fields. Therefore, it is possible to factor out a discrete subgroup such that S 2 × S 1 topology is achieved. ¶ Another interesting example of a spacetime with a region isometric to Kerr is the Chandrasekhar and Xanthopoulos solution [START_REF] Chandrasekhar | [END_REF] which describes colliding plane waves. It turns out that the region of interaction of the two waves is an alternative interpretation of a part of the Kerr spacetime region between event horizon and Cauchy horizon, cf. [START_REF] Griffiths | Colliding plane waves in General Relativity[END_REF][START_REF] Helliwell | [END_REF].

the future Cauchy horizon in terms of initial data on the past horizon, see Sec. 3. As another consequence we arrive at the above equation relating A p , A f and J, see Sec. 4. Finally, in Sec. 5 we conclude with a discussion of our results.

Coordinates and Einstein equations

Coordinate system, Einstein equations and regularity requirements

We introduce suitable coordinates and metric functions by adopting our notation from [START_REF] Garfinkle | [END_REF]. Accordingly, we write the Gowdy line element in the form

ds 2 = e M (-dt 2 + dθ 2 ) + sin t sin θ e L (dϕ + Qdδ) 2 + e -L dδ 2 , ( 1
)
where the metric functions M, L and Q depend on t and θ alone. In these coordinates, the two Killing vectors are given by

η = ∂ ∂ϕ , ξ = ∂ ∂δ . ( 2
)
As mentioned in Sec. 1, any S 2 × S 1 Gowdy model can be related to the spacetime portion between outer event and inner Cauchy horizon of an appropriate axisymmetric and stationary black hole solution. Black hole spacetimes of this kind have been studied by Carter [START_REF] Carter | Black hole equilibrium states, Black holes[END_REF] and Bardeen [START_REF] Bardeen | Rapidly rotating stars, disks, and black holes, Black holes[END_REF]. Among other issues they discussed conditions for regular horizons. In this paper we adopt their regularity arguments for our study of Gowdy spacetimes. Accordingly we rewrite the line element (1) in the form

ds 2 = e M (-dt 2 + dθ 2 ) + e u sin 2 θ(dϕ + Qdδ) 2 + e -u sin 2 t dδ 2 (3)
where

u = ln sin t -ln sin θ + L. ( 4
)
Now, at a regular horizon (clear statements about the type of regularity follow below) the metric functions M, Q and u are regular, meaning that L possesses a specific irregular behavior there. + At this point, some remarks about the specific regularity requirements needed in our investigation are necessary. A crucial role is played by a theorem of Chruściel (Theorem 6.3 in [10]) which provides us with the essential regularity information valid in the interior of the Gowdy square. In this theorem it is assumed that initial data are given on an interior Cauchy slice, described by t = constant = t 0 , 0 < t 0 < π. These data are supposed to consist of (i) metric potentials that are H k -functions of θ and (ii) first time derivatives that are H k-1 -functions of θ (with k ≥ 3). Here H k denotes the Sobolev space W k,2 that contains all functions for which both the function and its weak derivatives up to the order k are in L 2 . With these assumptions the theorem by + We achieve the form of the line element used in [START_REF] Ansorg | [END_REF]3,23] from (3) by introducing the Boyer-Lindquisttype coordinates (R, θ, ϕ, t) with R := r h cos t, t := δ/(2r h ), r h = constant, and the metric functions μ := e M , û := e u , ω := -2r h Q. Since the potentials μ > 0, û > 0 and ω are regular at the axes and at the Cauchy horizon (cf. [START_REF] Bardeen | Rapidly rotating stars, disks, and black holes, Black holes[END_REF]), we see that M , u and Q are regular as well.

0 π π θ t A B C D H p H f A 1 A 2 Figure 1.
The Gowdy square. We assume a H 4 -regular metric and H 3 -regular time derivatives on all slices t = constant in a neighborhood (grey region) of the past Cauchy horizon (H p : t = 0) and find by virtue of the results in [START_REF] Ansorg | [END_REF], that then the metric is H 4 regular on all future slices t = constant, 0 ≤ t ≤ π (unless the quantity J introduced in ( 25) is zero). In particular, a H 4 -regular future Cauchy horizon (H f : t = π) exists.

Chruściel guarantees the existence of a unique continuation of the given initial data for which the metric is H k on all future spatial slices t = constant with t 0 < t < π, i.e. only the future boundary t = π of the Gowdy square is excluded. (Note that Theorem 6.3 as formulated in [10] assumes the metric to be smooth. However, this condition can be relaxed considerably to the assumption of H k spaces [12].) Now, for the applicability of our soliton methods it is essential that the metric potentials in (3) possess C 2 -regularity. Therefore, in order to apply both Chruściel's theorem and the soliton methods, we need to require that the metric potentials M, u, Q be H 4 -functions and the time derivatives H 3 -functions of θ on all slices t = constant in a neighborhood of the horizon H p , see Fig. 1. * Then Chruściel's theorem ensures the existence of an H 4 -regular continuation which implies (via Sobolev embeddings and the validity of the Einstein equations) that the metric potentials M, u, Q are C 2 -functions of t and θ for (t, θ) ∈ (0, π) × [0, π], i.e. in the entire Gowdy square with the exception of the two horizons H p (t = 0) and H f (t = π). Now, in accordance with Carter's and Bardeen's arguments concerning regularity at the horizon, we require that this C 2regularity holds also for t = 0, i.e. we assume in this manner a specifically regular past horizon H p .

As mentioned above, these requirements allow us to utilize our soliton methods at H p . Since H p is a degenerate boundary surface of the interior hyperbolic region, the study of the Einstein equations provides us with specific relations that permit the identification of an appropriate set of initial data of the hyperbolic problem at the past Cauchy horizon H p .

For the line element (3), the Einstein equations read as follows:

-u ,tt -cot t u ,t + u ,θθ + cot θ u ,θ = 2 - sin 2 θ sin 2 t e 2u Q 2 ,t -Q 2 ,θ , (5)
-Q ,tt + cot t Q ,t + Q ,θθ + 3 cot θ Q ,θ -2(u ,t Q ,t -u ,θ Q ,θ ) = 0, (6)
-M ,tt + M ,θθ - 1 2 u ,t (u ,t -2 cot t) + 1 2 u ,θ (u ,θ + 2 cot θ) - 1 2 sin 2 θ sin 2 t e 2u Q 2 ,t -Q 2 ,θ = 0. ( 7
)
Alternatively to (7), the metric potential M can also be calculated from the first order field equations

(cos 2 t -cos 2 θ)M ,t = 1 2 e 2u sin 3 θ sin t cos t sin θ(Q 2 ,t + Q 2 ,θ ) -2 sin t cos θ Q ,t Q ,θ + 1 2 sin t sin θ cos t sin θ(u 2 ,t + u 2 ,θ ) -2 sin t cos θ u ,t u ,θ + (2 cos 2 t cos 2 θ -cos 2 t -cos 2 θ)u ,t + 2 sin t cos t sin θ cos θ(u ,θ -tan θ), ( 8
) (cos 2 t -cos 2 θ)M ,θ = - 1 2 e 2u sin 3 θ sin t sin t cos θ(Q 2 ,t + Q 2 ,θ ) -2 cos t sin θ Q ,t Q ,θ - 1 2 sin t sin θ sin t cos θ(u 2 ,t + u 2 ,θ ) -2 cos t sin θ u ,t u ,θ + 2 sin t cos t sin θ cos θ(u ,t + tan t) + (2 cos 2 t cos 2 θ -cos 2 t -cos 2 θ)u ,θ . (9)
These expressions tell us that (see Appendix A for a detailed derivation)

M ,t = Q ,t = u ,t = 0 , Q = Q p = constant , M + u = constant ( 10
)
holds on H p . As the t-derivatives of all metric functions vanish identically at H p , a complete set of initial data at H p consists of

Q = Q p ∈ R, u ∈ H 4 , Q ,tt ∈ H 2 , ( 11
)
where Q ,tt is in H 2 as a consequence of the regularity assumptions discussed above. Note that among the second t-derivatives only Q ,tt can be chosen freely since the values of M ,tt as well as u ,tt are then fixed, as again the study of the field equations ( 5)-( 7) near H p reveals. Similarly, M is also fixed on H p by the choice of the data in (11). It turns out that the constant Q p is a gauge degree of freedom. This results from the fact that the line element (1) is invariant under the coordinate change Σ : (t, θ, ϕ, δ) → Σ : (t, θ, ϕ = ϕ -Ωδ, δ), (12) leading to Q p = Q p +Ω in the new coordinates . We use this freedom in order to exclude two specific values, namely Q p = 0 and Q p = 1/J, where J is the already mentioned conserved quantity that will be introduced in (26). This exclusion becomes necessary since the analysis carried out below breaks down if Q p takes one of these values.

Note that for the corresponding black hole spacetimes, the coordinate change (12) describes a transformation into a rigidly rotating frame of reference (for more details see [START_REF] Ansorg | [END_REF]3,23]).

We note further that as another consequence of our regularity requirements, the following axis condition holds at least in a neighborhood of the points A and B (cf. Fig. 1):

A 1/2 : M = u. ( 13
)
Moreover, at these points A, B we have (see Appendix A)

M A = M B = u A = u B . ( 14
)
Note that solutions which are also C 2 -regular up to and including H f satisfy corresponding conditions at the points C and D.

The Ernst equation

In order to introduce the Ernst formulation of the Einstein equations, we define the complex Ernst potential

E(t, θ) = f (t, θ) + ib(t, θ), ( 15
)
where the real part f is given by

f := -ξ i ξ i = -e -u sin 2 t -Q 2 e u sin 2 θ ( 16
)
and the imaginary part b is defined in terms of a potential a,

a := ξ i η i ξ j ξ j = - Q f e u sin 2 θ, (17)
via

a ,t = 1 f 2 sin t sin θ b ,θ , a ,θ = 1 f 2 sin t sin θ b ,t . ( 18
)
In this formulation, the vacuum Einstein equations are equivalent to the Ernst equation

(E) (-E ,tt -cot t E ,t + E ,θθ + cot θ E ,θ ) = -E 2 ,t + E 2 ,θ , (19)
where (E) denotes the real part of E. As a consequence of [START_REF] Griffiths | Colliding plane waves in General Relativity[END_REF], the integrability condition a ,tθ = a ,θt of the system ( 18) is satisfied such that a may be calculated from ( 18) using E. Moreover, given a and E we can use ( 16) and ( 17) to obtain the metric functions u and Q. Finally, the potential M may be calculated from

M ,t = - f ,t f + 1 2f 2 sin t sin θ cos 2 t -cos 2 θ cos t sin θ f 2 ,t + f 2 ,θ + b 2 ,t + b 2 ,θ -2 sin t cos θ (f ,t f ,θ + b ,t b ,θ ) -4f 2 cos t sin θ , ( 20
)
M ,θ = - f ,θ f - 1 2f 2 sin t sin θ cos 2 t -cos 2 θ sin t cos θ f 2 ,t + f 2 ,θ + b 2 ,t + b 2 ,θ -2 cos t sin θ (f ,t f ,θ + b ,t b ,θ ) -4f 2 cos θ sin t ( 21
)
since the Ernst equation ( 19) also ensures the integrability condition M ,tθ = M ,θt .

As for the potentials introduced in Sec. 2.1 we conclude axis conditions which hold at least in a neighborhood of the points A and B (cf. Fig. 1):

A 1/2 : E ,θ = 0, a = 0. ( 22
)
Moreover, at the points A, B we have f = 0. Again, solutions which are also H 4 -regular on H f satisfy corresponding conditions at the points C and D.

It turns out that initial data E p (θ) ≡ E(0, θ) = f p (θ) + ib p (θ) of the Ernst potential are equivalent to the inital data set consisting of u, Q = Q p , Q ,tt at H p . Both sets are related via

f p = -Q 2 p e u(0,θ) sin 2 θ, ( 23
)
b p = b A + 2Q p (cos θ -1) -Q 2 p θ 0 e 2u(0,θ ) Q ,tt (0, θ ) sin 3 θ dθ , ( 24
)
where b A = b(0, 0) is an arbitrary integration constant.

Conserved quantities

As a consequence of the symmetries of the Gowdy metric, there exist conserved quantities, i.e. integrals with respect to θ that are independent of the coordinate time t. One of them is J, defined by

J := - 1 8 π 0 Q ,t (t, θ) sin t e 2u(t,θ) sin 3 θ dθ = constant. ( 25
)
As for the black hole angular momentum in the corresponding axisymmetric and stationary black hole spacetimes (cf. discussion at the end of Sec. 1), this quantity determines whether or not a regular future Cauchy horizon exists. In fact, it exists if and only if J = 0 holds. Note that J vanishes in polarized Gowdy models, where we have Q ,t ≡ 0. It turns out that J can be read off directly from the Ernst potential and its second θ-derivative at the points A and B on H p (see Fig. 1),

J = - 1 8Q 2 p (b A -b B -4Q p ), Q p = - 1 2 b ,θθ | A ( 26
)
where

b B = b(t = 0, θ = π).
A detailed derivation of these formulae can be found in [23].

Potentials on A 1 , A 2 , and H f

Ernst potential

In the previous sections we have derived a formulation which permits the direct translation to the situation in which the hyperbolic region inside the event horizon of an axisymmetric and stationary black hole (with possibly non-pure vacuum exterior, e.g. with surrounding matter) is considered, as was done in [START_REF] Ansorg | [END_REF]3,23].

In [START_REF] Ansorg | [END_REF] it has been demonstrated that a specific soliton method (the Bäcklund transformation, see Appendix B) can be used to write the Ernst potential E in terms of another Ernst potential E 0 which corresponds to a spacetime without a black hole, but with a completely regular central vacuum region. Interestingly, the potential E 0 = E 0 (t, θ) possesses specific symmetry conditions which translate here into

E 0 (t, 0) = E 0 (0, t) potential at A 1 , E 0 (t, π) = E 0 (0, π -t) potential at A 2 , E 0 (π, θ) = E 0 (0, π -θ) potential at H f .
Hence the potential values at the boundaries A 1 , A 2 and H f are given explicitly in terms of those at H p . Now the Bäcklund transformation carries these dependencies over to the corresponding original Ernst potential E, i.e. we obtain E at A 1 , A 2 and H f completely in terms of the initial data at H p . An alternative approach (see [3,23]) uses the inverse scattering method. In these papers the potentials on A 1 , A 2 and H f were obtained from the investigation of an associated linear matrix problem. The integrability conditions of this matrix problem are equivalent to the non-linear field equations, see Appendix B. We may carry the corresponding procedure over to our considerations of Gowdy spacetimes. Accordingly we are able to perform an explicit integration of the linear problem along the boundaries of the Gowdy square. Since the resulting solution is closely related to the Ernst potential, it provides us with the desired expressions between the metric quantities on the four boundaries of the Gowdy square.

Note that in both approaches the axes A 1 and A 2 are considered first. Starting at H p and using the theorem by Chruściel [10], which ensures H 4 -regularity of the metric inside the Gowdy square (i.e. excluding only H f ), we derive first the Ernst potentials at A 1 and A 2 in terms of the values at H p . It turns out that for J = 0 these formulae can be extended continuously to the points C and D at which A 1 and A 2 meet H f (cf. Fig. 1). Moreover, with the values at C and D it is possible to proceed to H f , and in this way we eventually find an Ernst potential which is continuous along the entire boundary of the Gowdy square. As the theorem by Chruściel ensures unique solvability of the Einstein equations inside the Gowdy square, we conclude that the H 4 -regularity of the Ernst potential holds up to and including H f which therefore turns out to be an H 4 -regular future Cauchy horizon.

The resulting expressions of the Ernst potentials at the boundaries A 1 , A 2 and H f read

A 1 : E 1 (x) := E(t = arccos x, θ = 0) = i[b A -2Q p (x -1)]E p (x) + b 2 A E p (x) -i[b A + 2Q p (x -1)] , ( 27
)
A 2 : E 2 (x) := E(t = arccos(-x), θ = π) = i[b B -2Q p (x + 1)]E p (x) + b 2 B E p (x) -i[b B + 2Q p (x + 1)] , ( 28
)
H f : E f (x) := E(t = π, θ = arccos(-x)) = a 1 (x)E p (x) + a 2 (x) b 1 (x)E p (x) + b 2 (x) , ( 29
)
where

E p (x) := E(t = 0, θ = arccos x) ( 30
)
denotes the Ernst potential on H p and a 1 , a 2 , b 1 , and b 2 in (29) are polynomials in x, defined by

a 1 = i[16Q 2 p (1 -x 2 ) + 8Q p (b A (x + 1) + b B (x -1)) + (b A -b B )(b A (x -1) 2 -b B (x + 1) 2 )], (31)
a 2 = 8Q p [b 2 A (x + 1) + b 2 B (x -1)] -4b A b B (b A -b B )x, (32)
b 1 = 4(4Q p + b B -b A )x, (33)
b 2 = i[4Q p (1 -x 2 ) -b A (1 + x) 2 + b B (1 -x) 2 ](4Q p + b B -b A ). (34)
A discussion of [START_REF] Neugebauer | Gravitostatics and rotating bodies[END_REF] shows that E f is indeed always regular provided that the black hole angular momentum does not vanish, which in turn means that J = 0, cf. (25). In order to prove this statement, we first note that both numerator and denominator on the right hand side of ( 29) are completely regular functions in terms of x, since a 1 , a 2 , b 1 , b 2 are polynomials in x and the initial function E p is regular by assumption. Hence, an irregular behavior of the potential E f could only be caused by a zero of the denominator. Consequently, we investigate whether the equation

b 1 (x)E p (x) + b 2 (x) = 0 (35)
has solutions x ∈ [-1, 1]. The real part of ( 35) is given by

4x(4Q p + b B -b A )f p (x) = 0. (36)
Using ( 23) and ( 26) together with our gauge Q p = 0 and the assumption J = 0 we find that (36) has exactly the three zeros, x = -1, x = 0 and x = 1 (corresponding to θ = π, θ = π/2 and θ = 0). Now, for x = 0 the imaginary part of (35) does not vanish, whereas for x = ±1 it does. Thus we find that the only zeros of the denominator in ( 29) are located at the two axes (x = ±1). As a matter of fact, the regular numerator of ( 29) also vanishes at x = ±1, as can be derived in a similar manner. Consequently, we study the behavior of E f for x = ±1 in terms of the rule by L'Hôpital. As both numerator and denominator in (29) have non-vanishing values of the derivative with respect to x for x = ±1, we conclude that the Ernst potential is regular everywhere whenever J = 0 holds. Consider now the limit J → 0 for which the expression

4Q p + b B -b A
vanishes, cf. (26). As this term appears as a factor in both b 1 and b 2 (cf. ( 33),( 34)), we find that the denominator in (29) vanishes identically. The numerator, however, remains non-zero, which means that Ernst potential diverges on the entire future boundary t = π, 0 ≤ θ ≤ π. We conclude that H f becomes singular in the limit J → 0. This divergent behavior of the Ernst potential corresponds to the formation of a (scalar) curvature singularity at H f . In order to illustrate this property, we calculate the Kretschmann scalar at the point C on H f (see Fig. 1). Using the axis conditions discussed in Sec. 2 and the Einstein equations, we obtain

R ijkl R ijkl | C = 12 e -2u (1 + 2u ,tt ) 2 -Q 2 ,tt C . ( 37
)
In terms of the Ernst potential, this expression reads (cf. Eq. (40) below)

R ijkl R ijkl | C = 1 3 (f ,tt + f ,tttt ) 2 -(b ,tt + b ,tttt ) 2 C . ( 38
)
Now we can use (27) to derive a formula that contains only the initial data on the past horizon H p . Together with (26) we get

R ijkl R ijkl | C = - 3 256Q 8 p J 6 (16Q 4 p J 2 -4b ,xx Q 2 p J -f 2 ,x ) 2 -16Q 4 p J 2 (f ,xx -2f ,x ) 2 B , ( 39
)
where x = cos θ. Note that the numerator is well-defined and bounded for our H 4regular metric, a fact which is ensured by the validity of the Einstein equations near H p . Equation (39) indicates that the Kretschmann scalar diverges as J -6 in the limit J → 0. In fact, as we choose Q p = 0 (see Sec. 2.1), and furthermore f ,x = 0 holds (because 2πf ,x = -Q 2 p A p where A p , 0 < A p < ∞, is the horizon area of H p , see Sec. 4), we conclude that f 4 ,x is the dominating term in the numerator of (39) for sufficiently small J. Hence the Kretschmann scalar indeed diverges as J -6 in the limit J → 0.

Metric potentials

From the Ernst potentials 27), (28), [START_REF] Neugebauer | Gravitostatics and rotating bodies[END_REF] we may calculate the metric potentials M, Q and u on the boundaries of the Gowdy square. Using ( 16), ( 17), ( 18), (10), [START_REF] Clarke | The analysis of space-time singularities[END_REF], [START_REF] Ellis | [END_REF] we obtain

E 1 = f 1 + ib 1 , E 2 = f 2 + ib 2 , E f = f f + ib f in (
A 1 : e M 1 = e u 1 = - sin 2 t f 1 , Q 1 = b 1,t 2 sin t , ( 40
)
A 2 : e M 2 = e u 2 = - sin 2 t f 2 , Q 2 = - b 2,t 2 sin t , ( 41
)
H f : e M f = - f 2 ,θθ | C 4Q 2 f sin 2 θ f f , Q = Q f , e u f = - f f Q 2 f sin 2 θ , ( 42
)
where

Q f = b A -b B + 4Q p b A -b B -4Q p Q p . ( 43
)
Note that Q f = 0 in our gauge (cf. ( 26)):

b A -b B + 4Q p = (b A -b B -4Q p ) + 8Q p = -8Q 2 p J + Q p = 8Q p (1 -JQ p ) = 0
, in accordance with the discussion in Sec. 2.1 where the gauge freedom was used to assure 0 = Q p = 1/J. Furthermore, using ( 27)-( 29) and our regularity assumptions for the initial data, it is straightforward to show that (-sin 2 t/f 1 ), (-sin 2 t/f 2 ) and (-sin 2 θ/f f ) are regular and positive functions on the entire boundaries A 1 , A 2 and H f , respectively. Moreover, the terms (b 1,t / sin t) and (b 2,t / sin t) are regular on A 1 and A 2 , respectively. Consequently, the above boundary values for the metric potentials M, u and Q are regular, too.

A universal formula for the horizon areas

In [START_REF] Ansorg | [END_REF] a relation between the black hole angular momentum and the two horizon areas of the outer event and inner Cauchy horizons was found. This relation emerged from the explicit expressions of the inner Cauchy horizon potentials in terms of those at the event horizon. Translated to the case of general S 2 × S 1 Gowdy spacetimes, this relation is given by

A p A f = (8πJ) 2 (44)
where the areas A p and A f of the Cauchy horizons H p and H f are defined as integrals over the horizons (in a slice δ = constant),

A p/f = S 2 √ g θθ g ϕϕ dθdϕ = 2π π 0 e M +u 2 | H p/f sin θ dθ = 4πe u | A/C . ( 45
)

Discussion

In this paper we have analyzed general S 2 × S 1 Gowdy models with a past Cauchy horizon H p . As any such spacetime can be related to a corresponding axisymmetric and stationary black hole solution, considered between outer event and inner Cauchy horizons, the results on the regularity of the interior of such black holes (obtained in [START_REF] Ansorg | [END_REF]3,23]) can be carried over to the Gowdy spacetimes treated here. In particular, specific soliton methods have proved to be useful, (i) the Bäcklund transformation and (ii) the inverse scattering method. Both methods imply explicit expressions for the metric potentials on the boundaries A 25)) does not vanish. In the limit J → 0, the future Cauchy horizon transforms into a curvature singularity. Moreover, for J = 0 the universal relation

A p A f = (8πJ) 2 (46)
holds, where A p and A f denote the areas of past and future Cauchy horizons.

Remark. Note that the statements in Thm. 1 can be generalized to S 2 × S 1 Gowdy spacetimes with additional electromagnetic fields, see [3,23]. The proof utilizes a more general linear matrix problem in which the Maxwell field is incorporated. Again the corresponding integrability conditions are equivalent to the coupled system of field equations that describe the Einstein-Maxwell field in electrovacuum with two Killing vectors (associated to the two Gowdy symmetries). It turns out that apart from J a second conserved quantity Q becomes relevant. The corresponding counterpart of this quantity in Einstein-Maxwell black hole spacetimes describes the electric charge of the black hole. For Gowdy spacetimes we conclude that a regular future Cauchy horizon exists if and only if J and Q do not vanish simultaneously. Moreover, we find that Eq. ( 46) generalizes to

A p A f = (8πJ) 2 + (4πQ 2 ) 2 .
With the above theorem we provide a long outstanding result on the existence of a regular future Cauchy horizon in S 2 × S 1 Gowdy spacetimes. We note that the soliton methods being utilized in order to derive our conclusions are not widely used in previous studies of this kind. Therefore we believe that these techniques might enhance further investigations in the realm of Gowdy cosmologies. inverse scattering method. Both methods make use of the following linear matrix problem (see [27,28]), which read in our coordinates as follows: where T n is a matrix polynomial in λ of degree n. From Φ, determined via (B.6), one can finally calculate the corresponding new Ernst potential E, see [START_REF] Neugebauer | Gravitostatics and rotating bodies[END_REF]. Note that our specific expressions for the metric at the future Cauchy horizon H f in Gowdy spacetimes can be obtained by considering the particular case of a twofold Bäcklund transformation (n = 2), for which the new Ernst potential E reads

Φ ,x = B x 0 0 A x + λ 0 B x A x
E = [
with α i ᾱi = 1,
where λ 1 := λ(x, y, K = -1), λ 2 := λ(x, y, K = 1).

In our second approach, the inverse scattering method, the linear problem (B.1) is integrated along the boundaries of the Gowdy square. It turns out that explicit formulae can be found and that, moreover, the resulting solution must be continuous at this boundary (provided that the solution is regular at H f , which is true for J = 0, see discussion in Sec. 3). In this way we find the expressions that constitute the statements of this paper.

5 )

 5 = Φ(x, y, K) is a 2 × 2 matrix pseudopotential depending on the coordinatesx = cos(t -θ), y = cos(t + θ) (B.2)as well as on the spectral parameter K ∈ C. The function λ is defined asλ(x, y, K) = K -y K -x . (B.3)For fixed values x, y, the equation (B.3) describes a mapping C → C, K → λ from a two-sheeted Riemann surface (K-plane) onto the complex λ-plane. In the K-plane the two K-sheets are connected at the branch pointsK 1 = x (λ = ∞), K 2 = y (λ = 0). (B.4)Examining the integrability conditions Φ ,xy = Φ ,yx yields, on the one hand, that the quantities A x , A y , B x and B y are given in terms of a single complex 'Ernst' potentialE = f + ib,On the other hand, the integrability conditions Φ ,xy = Φ ,yx tell us that this potential E satisfies the Ernst equation[START_REF] Griffiths | Colliding plane waves in General Relativity[END_REF]. Conversely, any solution E to the Ernst equation implies the existence of an associated matrix Φ which obeys the above linear matrix equations (B.1) where the functions A x , A y , B x and B y follow from (B.5). Now, with a Bäcklund transformation a new potential E can be constructed from a previously known one E 0 . Starting from E 0 and the corresponding matrix function Φ 0 , we consider transformations of the form BT n : Φ 0 → Φ = T n Φ 0 , n ∈ N even, (B.6)

  1 , A 2 , H f of the Gowdy square in terms of the initial values at H p . Moreover we obtain statements on existence and regularity of a future Cauchy horizon as well as a universal relation for the horizon areas. These results are summarized in the following. Consider an S 2 × S1 Gowdy spacetime with a past Cauchy horizon H p , where the metric potentials M, u and Q appearing in the line element (3) are H 4functions and the time derivatives H 3 -functions of the adapted coordinate θ on all slices t = constant in a closed neighborhood N := [0, t 0 ]×[0, π], t 0 ∈ (0, π), of H p . In addition, suppose M, Q, u ∈ C 2 (N ). Then this spacetime possesses an H 4 -regular future Cauchy horizon H f if and only if the conserved quantity J (cf. (

	Theorem 1.

  α 1 (cos t + cos θ) -α 2 (cos tcos θ)] E 0 + 2 Ē0 αHere, α 1 and α 2 are solutions of the Riccati equationsα i,x = -(λ i α 2 i + α i )

			E 0,x 2f 0	+ (α i + λ i )	Ē0,x 2f 0	,		(B.8)
	α i,y = -	1 λ i	α 2 i + α i	E 0,y 2f 0	+ α i +	1 λ i	Ē0,y 2f 0	, i = 1, 2,	(B.9)

1 (cos t + cos θ) -α 2 (cos tcos θ) -2 .

(B.7)

* In[START_REF] Ansorg | [END_REF]3,23], the much stronger assumption was made that the metric functions be analytic in an exterior neighborhood of the black hole's event horizon. This stronger requirement was necessary to conclude that the metric is also regular (in fact analytic) in an interior vicinity of the event horizon, a requirement needed for applying Chruściel's theorem.

Acknowledgments

We would like to thank Florian Beyer and Piotr T. Chruściel for many valuable discussions and John Head for commenting on the manuscript. This work was supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative Research Centre SFB/TR7 "Gravitational wave astronomy".

Appendix A. Derivation of initial and boundary conditions

We provide a derivation of the initial and boundary conditions through a thorough study of the Einstein equations near the past Cauchy horizon H p , i.e. the initial surface t = 0. First multiply the field equation ( 6) with sin t and consider subsequently the limit t → 0. Taking our regularity assumptions into account (cf. discusscion in Sec. 2.1), we arrive at

With the result (A.1), we study the limit t → 0 of (6) in terms of the rule by L'Hôpital and find

Evaluation at θ = 0 shows that the constant vanishes, leading to

Next multiply Eq. ( 5) with sin t and study the limit t → 0. With (A.1) and (A.3) we obtain

Using the previous results, we derive from (8) in the limit t → 0

On the other hand, (9) leads to

Similarly, we study the Einstein equations on the axis, i.e. in the limit sin θ → 0. Multiplication of Eqs. ( 5) and ( 6) with sin θ leads for sin θ → 0 to u ,θ = 0, Q ,θ = 0 for sin θ = 0.

(A.7)

From ( 8) and ( 9) we obtain

As a consequence of an axis regularity condition that excludes the appearance of struts or knots along the axis (see [START_REF] Stephani | Exact solutions of Einstein's field equations[END_REF] for details), it turns out that the constant in (A.8) vanishes. Hence we get M = u for sin θ = 0. (A.9)

At the points A and B (see Fig. 1), (A.9) means that M A = u A and M B = u B , and with M A + u A = M B + u B (cf. (A.6)) we conclude

(A.10)

Finally, we derive

by multiplying the Ernst equation [START_REF] Griffiths | Colliding plane waves in General Relativity[END_REF] with sin θ and considering the limit sin θ → 0. Moreover, it follows from the definition (17) of the potential a that a = 0 for sin θ = 0. (A.12)

Appendix B. The linear problem and Bäcklund transformations

In this appendix we briefly discuss the mathematical structure of the Ernst equation [START_REF] Griffiths | Colliding plane waves in General Relativity[END_REF] which permits the application of so-called soliton methods. More details can be found in [START_REF] Ansorg | [END_REF]3,23]. For a sophisticated introduction to soliton methods for the axisymmetric and stationary Einstein equations we refer the reader to [START_REF] Neugebauer | Gravitostatics and rotating bodies[END_REF]. There are two soliton methods which lie at the heart of the treatment of S 2 × S 1 Gowdy spacetimes pursued in this paper: (i) the Bäcklund transformation and (ii) the


L Gardes

A Guillou

A Schorgen

Estimating the conditional tail index by integrating a kernel conditional quantile estimator

Keywords: Heavy-tailed distribution, Covariates, Kernel estimator, Asymptotic normality AMS Subject Classifications: 62G32, 62G05, 62E20

This paper deals with the estimation of an extreme value index of a heavy-tailed distribution in the presence of covariates. A class of estimators is proposed in this context and its asymptotic normality established under mild regularity conditions. These estimators are functions of a kernel conditional quantile estimator depending on some tuning parameters. The finite sample properties of our estimators are illustrated on a small simulation study.

Introduction

Tail index estimation has been extensively studied in the literature and several estimators proposed. The most popular estimator was proposed by [START_REF] Hill | A simple general approach to inference about the tail of a distribution[END_REF] in the context of heavytailed distributions. [START_REF] Diebolt | Bias-reduced estimators of the Weibull tail-coefficient[END_REF] considered the case of Weibull-tail distributions and the general case was studied by [START_REF] Dekkers | A moment estimation for the index of an extremevalue distribution[END_REF]. The aim of this paper is to extend this framework to the case where some covariate x is recorded simultaneously with the variable of interest Y . In this case, the tail index depends on the covariate x and is referred in the following to as the conditional tail index. Such a context has been already considered by [START_REF] Smith | Extreme value analysis of environmental time series: an application to trend detection in ground-level ozone (with discussion)[END_REF] or [START_REF] Davison | Models for exceedances over high thresholds[END_REF] who proposed parametric models, while semi-parametric approaches has been used in [START_REF] Hall | Nonparametric analysis of temporal trend when fitting parametric models to extreme-value data[END_REF] or [START_REF] Beirlant | Regression with response distributions of Pareto-type[END_REF]. Also, fully non-parametric methods can be used for instance based on splines (see Chavez-Demoulin and [START_REF] Chavez-Demoulin | Generalized additive modelling of sample extremes[END_REF] or local polynomials (see [START_REF] Davison | Local likelihood smoothing of sample extremes[END_REF]. These latter estimators have been then extended by [START_REF] Beirlant | Regression with response distributions of Pareto-type[END_REF] in case of multidimensional covariates. More recently, [START_REF] Gardes | Conditional extremes from heavy-tailed distributions: an application to the estimation of extreme rainfall return levels[END_REF] have addressed the problem of the estimation of a conditional extreme quantile via a nearest neighbour approach. We propose here new conditional extreme value index estimators based on a class of functions satisfying some mild conditions. We only assume that the conditional distribution of Y given x is heavy-tailed whereas no parametric assumption on the covariates x is made. The conditional distribution of Y given x is then of polynomial-type, with a rate of convergence driven by the conditional tail index.

The remainder of this paper is organized as follows. Our class of estimators is given in Section 2. Their asymptotic normality is established in Section 3. In Section 4, a sub-class of estimators is studied and their finite sample properties are illustrated in Section 5. All the proofs are postponed to Section 6.

A class of conditional tail index estimators

Let E be a metric space associated to a distance d. For x ∈ E, denote by F (., x) (resp. q(., x)) the conditional distribution function (resp. the conditional quantile function) of Y given x. We assume that for all x ∈ E, F (y, x) = y -1/γ(x) L(y, x),

(1)

or, equivalently, for α ∈ (0, 1),

q(α, x) = F ← (α, x) = α -γ(x) ℓ(α -1 , x),
where F ← (α, x) = inf{t, F (t, x) ≤ α} denotes the generalized inverse of the conditional survival function and, for x fixed, L(., x) and ℓ(., x) are slowly varying functions, that is for all λ > 0,

lim y→∞ L(λy, x) L(y, x) = lim y→∞ ℓ(λy, x) ℓ(y, x) = 1.
Here, γ(.) is a unknown positive function of the covariate referred in the following to as the conditional tail index. For a given x ∈ E, our aim is to propose an estimator of γ(x).

Suppose that we have at our disposal pairs (Y1, x1), ..., (Yn, xn) of independent observations from model (1) where the design points x1, . . . , xn are assumed to be non random. We propose to estimate the conditional tail index as a function of the kernel conditional quantile estimator b qn(α, x)

:= b F ← n (α, x), with b F n(y, x) = n X i=1 H " d(x, xi) h1,n « K " y -Yi h2,n « , n X i=1 H " d(x, xi) h1,n « , (2)
where h1,n and h2,n are non random positive sequences. This kernel estimator was defined for instance in [START_REF] Ferraty | Nonparametric Functional Data Analysis: Theory and Practice[END_REF]. The function H(.) is an asymmetrical kernel with support in [0, 1] and the function K(.) is defined by

K(v) = Z ∞ v g(s)ds,
where g(.) is a bounded probability function with support included in [-1, 1]. Our class of conditional tail index estimators is given by b

γ(un,x, x) = Z un,x 0 Ψ(α, un,x, x) log b qn(α, x)dα, (3)
where for all u ∈ (0, 1), Ψ(., u, x) is a non-null continuous function in L1(0, 1) such that R u 0 Ψ(α, u, x)dα = 0 and un,x is a positive sequence. The sequence un,x is introduced in order to select only the largest observation to estimate γ(x). Its choice is thus very important in practice. Note that the estimator b γ(un,x, x) depends only on the Yi's for which the corresponding xi's belong to the ball B(x, h1,n) = {t ∈ E; d(t, x) ≤ h1,n}. The smoothness of the function b F n(., x) is also controlled by the bandwidth h2,n. Finally, note that b F n(y, x) ∼ Fn(y, x) as h2,n → 0, where

Fn(y, x) = n X i=1 H " d(x, xi) h1,n « 1l {Y i >y} , n X i=1 H " d(x, xi) h1,n « ,
is the empirical estimator of the conditional survival function. The same result is established for the associated conditional quantile in Lemma 1. Note also that in the special case where H(.) is the indicator function, Fn(., x) leads to the estimator proposed in [START_REF] Gardes | A moving window approach for nonparametric estimation of the conditional tail index[END_REF].

Main result

In this section, we give the useful notations and assumptions in order to establish the asymptotic normality of our estimators.

(A.1) The slowly varying function ℓ(., x) is normalized.

Assumption (A.1) is equivalent to suppose that for all y > 1,

ℓ(y, x) = c(x) exp Z y 1 ∆(v, x) v dv,
where c(x) > 0 and ∆(v, x) → 0 as v → ∞.

(A.

2) The function |∆(., x)| is ultimately decreasing.

The largest oscillation of the log-quantile function with respect to its second variable is defined for all a ∈ (0, 1/2) as:

ωn(a) = sup ˛l og q(α, x) q(α, x ′ ) ˛, α ∈ (a, 1 -a), (x, x ′ ) ∈ B(x, h1,n) × B(x, h1,n) ff .
(A.

3) The function |∆(., x)| is regularly varying with index ρ(x) < 0, that is for all λ > 0, |∆(λy, x)|/|∆(y, x)| → λ ρ(x) as y → ∞. Moreover ∆(., x) is assumed to have a constant sign at infinity.

Conditions (A.1) and (A.3) imply that for all v > 0,

log ℓ(vy, x) ℓ(y, x) = ∆(y, x) v ρ(x) -1 ρ(x) (1 + o(1)) as y → ∞,
which is the so-called second order condition classically used to establish the asymptotic normality of tail index estimators. Note that the second order parameter ρ(x) controls the rate of convergence of ℓ(λy, x)/ℓ(y, x) to 1. In particular if ρ(x) is close to 0, this convergence is slow and thus the estimation of the conditional tail index is difficult. Let mx = mn,x be the number of covariates in the ball B(x, h1,n). The covariates belonging to the ball B(x, h1,n) are denoted by {x * i , i = 1, . . . , mx} and the associated observations by {Y * i , i = 1, . . . , mx}. The order statistics are denoted by Y * 1,mx ≤ . . . ≤ Y * mx,mx and the corresponding covariates by {x *

(1) , . . . , x * (mx) }. We introduce the set {β0, . . . , βm x } where β0 = 0, βm x = 1 and, for i = 1, . . . , mx -1,

βi = mx X j=mx-i+1 H " d(x, x * (j) ) h1,n « , mx X j=1 H " d(x, x * (j) ) h1,n « . (4)
The next two conditions are related to the kernel function H(.) and the function Ψ(.). There are similar in spirit to the ones used by [START_REF] Beirlant | On exponential representations of log-spacings of extreme order statistics[END_REF]. Let kx = kn,x be a sequence of integers such that 1 < kx < mx.

(B.1) There exists two positive functions f1(., x) and f2(., x) such that for all i = 1, . . . , kx, f1

" i kx , x « ≤ kx i Z β i 0 Ψ(α, β kx , x)dα ≤ f2 " i kx , x « , satisfying 0 < Z 1 0 f 2 1 (α, x)dα,
and for some ξ > 0,

Z 1 0 f 2+ξ 2 (α, x)dα < ∞. (B.
2) There exists a positive function f3(., x) such that for i = 1, . . . , kx kx

˛Z β i β i-1 Ψ(α, β kx , x)dα ˛≤ f3 " i kx , x « , with Z 1 0 f3(α, x) max(log(1/α), 1)dα < ∞.
We are now in position to state our main result. We consider the case where the threshold un,x in (3) is taken equal to β kx . Thus, the choice of the sequence un,x ∈ (0, 1) is replaced by the choice of a sequence of integers kx ∈ {2, . . . , mx -1}.

Theorem 1. Suppose (A.1), (A.2), (A.3), (B.1) and

(B.2) hold. If kx → ∞, mx/kx → ∞, k 1/2 x ∆ " mx kx , x « → λ(x) ∈ R, k 1/2 x h2,n q(kx/mx, x) → 0 and k 2 x ωn(m -(1+δ) x ) → 0, for some δ > 0 then σn,x " b γ(β kx , x) -γ(x) -γ(x)b1,n(x) -∆ " mx kx , x « b2,n(x) « , converges in distribution to a N (0, γ 2 (x)) random variable with σ 2 n,x = kx X i=1 " 1 i Z β i 0 Ψ(α, β kx , x)dα « 2 ! -1 b1,n(x) = kx X i=1 " 1 i Z β i 0 Ψ(α, β kx , x)dα « -1 b2,n(x) = kx X i=1 " 1 i Z β i 0 Ψ(α, β kx , x)dα « " i kx + 1 « -ρ(x)
.

The asymptotic bias is divided into two terms. The first one (b1,n(x)) only depends on the functions Ψ(.) and H(.). The second term (b2,n(x)) depends on the unknown second order parameter ρ(x). Minimizing this second bias term requires the estimation of ρ(x) (we refer to Cairo and Gomes (2002) for estimators of the second order parameter in the unconditional case). Note that, in order to cancel the first bias term b1,n(x), one can define the following estimator :

b γ * (β kx , x) = b γ(β kx , x) , kx X i=1 " 1 i Z β i 0 Ψ(α, β kx , x)dα « . (5)
The asymptotic normality of this estimator is a straighforward consequence of Theorem 1.

Corollary 1. Under the assumptions of Theorem 1,

σ * n,x " b γ * (β kx , x) -γ(x) -∆ " mx kx , x « b * 2,n (x) « , converges in distribution to a N (0, γ 2 (x)) random variable with (σ * n,x ) 2 = kx X i=1 1 i Z β i 0 Ψ(α, β kx , x)dα ! 2 , kx X i=1 " 1 i Z β i 0 Ψ(α, β kx , x)dα « 2 b * 2,n (x) = kx X i=1 " 1 i Z β i 0 Ψ(α, β kx , x)dα « " i kx + 1 « -ρ(x) , kx X i=1 " 1 i Z β i 0 Ψ(α, β kx , x)dα « .

A particular case

In order to simplify the assumptions of Theorem 1, we consider the case where the function H(.) is given by H

(x) = 1l {x∈[0,1]} . (6)
In this situation, βi = i/mx for all i = 0, . . . , mx. Assumptions (B.1) and (B.2) are replaced by the following assumptions:

(C.1) Let u ∈ (0, 1). There exists a decreasing function Φ(., x) such that for all β ∈ (0, u] and for some ξ > 0:

u β Z β 0 Ψ(α, u, x)dα = Φ(β/u, x)
with

Z 1 0 Φ(α, x)dα = 1 and Z 1 0 Φ 2+ξ (α, x)dα < ∞.
(C.2) There exists a positive function f4(., x) such that for i = 1, . . . , kx

˛iΦ " i kx , x « -(i -1)Φ " i -1 kx , x «˛≤ f4 " i kx , x « , with Z 1 0 f4(α, x) max(log(1/α), 1)dα < ∞.
We denote by b γ(kx/mx, x) the estimators obtained by taking the function H(.) as in ( 6). The following result is a consequence of Theorem 1:

Corollary 2. Suppose (A.1), (A.2), (A.3), (C.1) and (C.2) hold. If kx → ∞, mx/kx → ∞, k 1/2 x Z 1/kx 0 Φ(α, x)dα → 0, k 1/2 x ∆ " mx kx , x « → λ(x) ∈ R, k 1/2 x h2,n q(kx/mx, x) → 0 and k 2 x ωn(m -(1+δ) x ) → 0,
for some δ > 0 then

k 1/2 x " b γ(kx/mx, x) -γ(x) -∆ " mx kx , x « AB(Φ, ρ(x), x)
« converges to a N (0, γ 2 (x)AV(Φ, x)) random variable with:

AB(Φ, ρ(x), x) = Z 1 0 Φ(α, x)α -ρ(x) dα and AV(Φ, x) = Z 1 0 Φ 2 (α, x)dα.
As an example of functions satisfying the conditions of Corollary 2, one can mention for all α < u:

Ψ θ (α, u, x) := 8 > > > > > > > > < > > > > > > > > : (θ + 1) 2 θu θ+1 " u θ θ + 1 -α θ « if θ > 0 1 u " log " u α " - 1
" if θ = 0 1 u if θ = ∞.
Note that Ψ0(., ., .) and Ψ∞(., ., .) correspond to the limiting cases of Ψ θ (., ., .).

Denote by γθ (x) the estimator obtained by taking the function H(.) as in (6) and by using the function Ψ θ (., u, x). Its asymptotic normality is a direct consequence of Corollary 2.

Corollary 3. Suppose (A.1), (A.2), (A.3) hold. If kx → ∞, mx/kx → ∞, k 1/2 x ∆ " mx kx , x « → λ(x) ∈ R, k 1/2 x h2,n q(kx/mx, x) → 0 and k 2 x ωn(m -(1+δ) x ) → 0,
for some δ > 0 then

k 1/2 x " γθ (x) -γ(x) -∆ " mx kx , x « θ + 1 (1 -ρ(x))(1 + θ -ρ(x)) « converges to a N (0, 2γ 2 (x)(θ + 1)/(1 + 2θ)) random variable.
Note that when θ = 0, AB(Φ0, ρ(x), x) = 1/(1ρ(x)) 2 and AV(Φ0, x) = 2 which are the same values than the ones obtain for the conditional Zipf estimator defined in [START_REF] Gardes | A moving window approach for nonparametric estimation of the conditional tail index[END_REF]. This estimator is an extension in the conditional framework of the Zipf estimator introduced by [START_REF] Kratz | The QQ-estimator and heavy tails[END_REF] and [START_REF] Schultze | On least squares estimates of an exponential tail coefficient[END_REF]. When θ = ∞, AB(Φ∞, ρ(x), x) = 1/(1ρ(x)) and AV(Φ∞, x) = 1, which correspond to the asymptotic bias and variance of the conditional Hill estimator proposed again in [START_REF] Gardes | A moving window approach for nonparametric estimation of the conditional tail index[END_REF].

To choose the value of θ, one can use for instance the criterion introduced in Gardes and Girard (2010). More precisely, we introduce a modified version of the Asymptotic Mean Squared Error (AMSE) as follows:

AM SE * := ∆ 2 " mx kx , x « Z 0 -∞ AB 2 (Φ θ , ρ, x)dρ + γ 2 (x)AV(Φ θ , x) kx = π(θ) kx 8 < : kx∆ 2 " mx kx , x " AV(Φ θ , x) + γ 2 (x) R 0 -∞ AB 2 (Φ θ , ρ, x)dρ 9 = ; ≤ C(x) π(θ) kx
where C(x) is a constant independent of θ and

π(θ) = Z 0 -∞ " θ + 1 (1 -ρ)(1 + θ -ρ) « 2 dρ × 2(1 + θ) 1 + 2θ = 2 " θ + 1 θ « 3 1 1 + 2θ (θ -2 log(1 + θ) -1/(1 + θ) + 1).
Therefore, a good choice of θ could be to take θπ minimizing the upper bound π(θ). Numerical optimization leads to θπ ≈ 0.6833.

A small simulation study

In this section, we illustrate the efficiency of our estimator γθ (.) proposed in Section 4 on a small simulation study. Keeping the notations of Sections 3 and 4 in mind, we recall the expression of our estimator

γθ (x) = Z kx/mx 0 Ψ θ (α, kx/mx, x) log b qn(α, x)dα. (7)
The asymmetric kernel H(.) is taken as the indicator function ( 6) and the integrated kernel K(.) is chosen as the integrated biweight kernel:

K(x) = Z ∞ x 15 16 (1 -s 2 ) 2 1l {s∈[-1,1]} ds.
Concerning the parameter θ, it will be fixed at the value θπ = 0.6833 as explained in Section 4. Moreover, we assume that the conditional survival function of Y given the covariate x is a Burr-type distribution, that is

F (y, x) = " 1 + y -ρ/γ(x) " 1/ρ , (8)
where ρ < 0 is the second order parameter defined in condition (A.3). In this particular setting, the slowly varying function ℓ(., x) and ∆(., x) are given by

ℓ(z, x) = (1 -z ρ ) -γ(x)/ρ , ∆(z, x) = γ(x) z ρ 1 -z ρ , with z > 1.
Concerning the conditional tail index, as in [START_REF] Daouia | Kernel estimators of extreme level curves, Test, to appear[END_REF], we choose

γ(x) = 1 2 " 1 10 + sin(πx) « 11 10 - 1 2 exp -64 " x - 1 2 « 2 !! . ( 9
)
In order to proceed to our simulations, the next step is to select kx and the two bandwidths h1,n and h2,n of kernels H(.) and K(.), i.e to find their optimal values according to some criterion.

5.1 Optimization of (m x , k x ) -AMSE minimization

In our case, H(.) is the indicator function. Therefore the estimator only depends on the number of points in the ball B(x, h1,n). Thus to find the optimal numbers (mx, kx), we minimize:

AM SE(mx, kx) = 2(θπ + 1) 2θπ + 1 γ 2 (x) kx + » θπ + 1 (1 -ρ)(1 + θπ -ρ) ∆ " mx kx , x «-2
with respect to these parameters. Now, concerning the two unknown quantities γ(x) and ∆(mx/kx, x), they will be estimated using an exponential regression model (see ( 18) in the proof of Theorem 1), taking ρ(x) = ρ := -1 as a canonical value. This choice is usual (see for instance [START_REF] Feuerverger | Estimating a tail exponent by modelling departure from a Pareto distribution[END_REF] or [START_REF] Beirlant | Tail index estimation and an exponential regression model[END_REF]) and leads to a least squares estimator for γ(x) and ∆( mx kx , x), denoted respectively by b γLS(x) and b ∆LS( mx kx , x). Therefore, the AMSE can be estimated by

AM SE(mx, kx) = 2(θπ + 1) 2θπ + 1 b γ 2 LS (x) kx + » θπ + 1 2(θπ + 2) b ∆LS " mx kx , x «-2
from which we deduce our criterion of selection for (mx, kx) :

( b mx, b kx) = arg min mx,kx
AM SE(mx, kx).

Note that the AMSE does not depend on the bandwidth parameter h2,n. Therefore we need another criterion to find its optimal value.

Optimization of h 2,n -Subsample division

The task is now to find an optimal value for the bandwidth h2,n of the integrated kernel K(.).

To this end, in the spirit of [START_REF] Ferraty | Nonparametric Functional Data Analysis: Theory and Practice[END_REF], we propose to split our sample into two random subsamples I1 and I2 of the same sample size. Then, we fix the covariate on x0 (for instance taken in I1) and find the nearest neighbour x * 0 of x0 in I2. Finally, we calculate the estimation γ(1) θπ (x0) of γ(x0) from I1 and γ(2) θπ (x * 0 ) from I2, using the optimal value for (mx, kx) as described in Section 5.2. The resulting optimization criterion for h2,n can then be proposed :

b h2,n = arg min

h 2,n |γ (1) θπ (x0) - γ(2) θπ (x * 0 )|.

Graphs and computational issues

We simulate 100 replications of n = 5 000 pairs (Yi, xi), i = 1, ..., n where xi = i/n and the distribution of Y given x is the Burr-type distribution (8). The conditional quantile estimator b qn(α, x) is obtained by dichotomous inversion of the conditional empirical survival function defined in (2). For a fixed value of the covariate x, we perform the optimizations of (mx, kx) and h2,n using grids:

• the number mx of neighbours of x belongs to a grid from 2% to 10% of the sample size, i.e from mx = 100 to 500 in our case; to avoid too small or too large values for kx, this parameter has been taken in the interval (0.15 × mx; 0.85 × mx);

• h2,n goes from 0.05 to 1 by step of 0.05.

Finally, two values of ρ were used: ρ = -1 and -0.8. Note that as usual in the absence of covariates, the closer ρ is to 0, the more difficult is the estimation of γ(.).

Figure 1 below illustrates the procedure presented in Section 5.1 for x = 0.4. In panel (a), we plot the estimated and theoretical AMSE for one realization. The boxplot of b kx for the 100 replications is given in panel (b). As observed, b k0.4 is very close with small variability to the optimal value (k opt 0.4 ) which minimizes the AMSE. Now, we come back to our original problem, that is the estimation of γ(.). Again we simulate 100 replications from the Burr-type distribution (8). The L 2 -errors between b γ(.) and γ(.) are computed. The median, 10% and 90% quantiles of these errors are represented in Figure 2. From these plots, we observe that our estimator performs quite well with a deterioration in the estimation of γ(.) if |ρ| decreases.

Proofs

Preliminary results

The difference between the estimators b qn(., x) and F ← n (., x) of the conditional quantile is controlled by the following lemma:

Lemma 1. For all x ∈ E, sup α∈[0,1) ˛b qn(α, x) -F ← n (α, x) ˛≤ h2,n.
Proof -The empirical estimator of the survival function can be rewritten as:

Fn(y, x) = 8 < : β0 = 0 if y ≥ Y * mx,mx , βj if y ∈ [Y * mx-j,mx , Y * mx-j+1,mx ), βm x = 1 if y < Y * 1,mx ,
where βj, j = 0, . . . , mx are defined in (4). It is then easy to check that:

F ← n (α, x) = mx-1 X i=0 Y * mx-i,mx 1l {α∈[β i ,β i+1 )} . ( 10
)
Let j ∈ {0, . . . , mx -1}. Since K(u) = 1 for all u ≤ -1 and K(u) = 0 for all u ≥ 1, we have for all y ≥ Y * mx-j,mx + h2,n that

K " y -Y * i,mx h2,n « = 0 for all i ∈ {1, . . . , mx -j}.
Furthermore, for all y ≤ Y * mx-j,mx -h2,n,

K " y -Y * i,mx
h2,n « = 1 for all i ∈ {mxj, . . . , mx}.

Thus, for all y ≥ Y * mx-j,mx + h2,n, b F n(y, x) ≤ βj and for all y ≤ Y * mx-j,mx -h2,n, b

F n(y, x) ≥ βj+1. Hence, for all α ∈ [βj, βj+1), b F ← n (α, x) ∈ (Y * mx-j,mx -h2,n, Y * mx-j,mx + h2,n). (11)
Equations ( 10) and ( 11) conclude the proof.

The following lemma establish the rate of convergence of Riemann sums.

Lemma 2. Let ϕ(.) be an integrable function defined on (0, 1].

i)

lim N →∞ 1 N N X i=1 ϕ " i N « = Z 1 0 ϕ(x)dx.
ii) Furthermore, if ϕ(.) is a decreasing function, we have for all

N ∈ N * , 0 ≤ Z 1 0 ϕ(x)dx - 1 N N X i=1 ϕ " i N « ≤ Z 1/N 0 ϕ(x)dx - ϕ(1) N .
Proof -The proof of part i) is straightforward. Concerning ii), remark that, since ϕ is decreasing,

Z 1 0 ϕ(x)dx = N X i=1 Z i/N (i-1)/N ϕ(x)dx ≥ 1 N N X i=1 ϕ " i N « . ( 12
) Next, Z 1 1/N ϕ(x)dx = N -1 X i=1 Z (i+1)/N i/N ϕ(x)dx ≤ 1 N N -1 X i=1 ϕ " i N « . (13)
Equations ( 12) and ( 13) conclude the proof.

Let us introduce the rescaled log-spacings:

Zi,m x (x) = i ˘log(Y * mx-i+1,mx ) -log(Y * mx-i,mx ) ¯, i = 1, . . . , mx -1.
The next result provides an expansion of the conditional tail index estimator b γ(β kx , x) where β kx is defined in (4).

Lemma 3. Assume that conditions (A.1), (A.2) hold. If kx → ∞, mx/kx → ∞ and k 2

x ωn(m

-(1+δ) x ) → 0 for some δ > 0 then, b γ(β kx , x) = kx X i=1 " 1 i Z β i 0 Ψ(α, β kx , x)dα « Zi,m x (x) + O P " h2,n q(kx/mx, x) « . Proof of Lemma 3 -From Lemma 1, sup α∈[0,β kx ) ˛b qn(α, x) F ← n (α, x) -1 ˛≤ h2,n Y * mx-kx,mx , (14)
since, for all α ∈ [0,

β kx ), F ← n (α, x) ≥ F ← n (β kx , x) = Y * mx-kx,mx
. From Gardes and Girard (2010, Lemma 1), there exists a sequence of event An with P(An) → 1 as n → ∞ such that for some δ > 0,

log Y * mx-kx,mx |An d = log q(U kx+1,mx , x) + O P " ωn(m -(1+δ) x ) "˛A n, (15)
where Ui, i = 1, . . . , mx are independent standard uniform variables, U1,m x ≤ . . . ≤ Um x ,mx are the associated order statistics. Hence, for all ε > 0,

P "˛l og Y * mx-kx,mx -log q " kx mx , x «˛> ε « =: T1,n + T2,n, where
T1,n := P " ˛log Y * mx-kx,mx -log q " kx mx , x «˛> ε ˛An « P(An)
and

T2,n := P "˛l og Y * mx-kx,mx -log q " kx mx , x «˛> ε ∩ A C n « .
Clearly from (15),

T1,n d = P " ˛log q(U kx+1,mx , x) q (kx/mx, x) + O P " ωn(m -(1+δ) x ) " ˛> ε ˛An « P(An).
Using the fact that for a positive sequence kx such that kx → ∞ and mx/kx → ∞, U kx+1,mx P ∼ kx/mx and since q(., x) is a regularly varying function we have T1,n → 0 using the fact that ωn(m

-(1+δ) x ) → 0. Furthermore, T2,n ≤ P(A C n ) → 0 since P(An) → 1.
Hence, for all ε > 0,

P "˛l og Y * mx-kx,mx -log q " kx mx , x «˛> ε « → 0,
and thus ( 14) implies that sup α∈[0,β kx )

˛b qn(α, x)

F ← n (α, x) -1 ˛= O P " h2,n q(kx/mx, x) « . ( 16
) Now remark that b γ(β kx , x) = Z β kx 0 Ψ(α, β kx , x) log F ← n (α, x)dα + Z β kx 0 Ψ(α, β kx , x) log b qn(α, x) F ← n (α, x) dα.
From ( 16) and since Ψ(., u, x) ∈ L1(0, 1),

˛Z β kx 0 Ψ(α, β kx , x) log b qn(α, x) F ← n (α, x) dα ˛= O P " h2,n q(kx/mx, x) « . (17)
Furthermore, from (10) and since Z β kx 0 Ψ(α, β kx , x)dα = 0, we have

Z β kx 0 Ψ(α, β kx , x) log F ← n (α, x)dα = kx X i=1 Z β i β i-1 Ψ(α, β kx , x)dα ! log Y * mx-i+1,mx = kx X i=1 " 1 i Z β i 0 Ψ(α, β kx , x)dα « Zi,m x (x)
which achieves the proof.

Proofs of the main results

Proof of Theorem 1 -From Lemma 1 in [START_REF] Gardes | Conditional extremes from heavy-tailed distributions: an application to the estimation of extreme rainfall return levels[END_REF], there exists a sequence of event An with P(An) → 1 as n → ∞ such that {Zi,m x (x), i = 1, . . . , kx|An} has the same distribution as  i log q(Ui,m x , x) q(Ui+1,m x , x)

+ iO P " ωn(m -(1+δ) x ) " , i = 1, . . . , kx ˛An ff .
Denoting by

e γ(β kx , x) = kx X i=1 " 1 i Z β i 0 Ψ(α, β kx , x)dα « Zi,m x (x),
we thus have that e γ(β kx , x)|An has the same distribution as

kx X i=1 " 1 i Z β i 0 Ψ(α, β kx , x)dα « i log q(Ui,m x , x) q(Ui+1,m x , x) + O P " kxωn(m -(1+δ)
x ) " (b1,n(x) + 1) ˛An.

Let us introduce the following notation ∆x = ∆ (mx/kx, x). From Gardes and Girard (2010, Lemma 4), to show that σn,x(e γ(β kx , x)-γ(x)(b1,n(x)+1)-∆xb2,n(x)) converges to a N (0, γ 2 (x)) random variable, it is sufficient to prove that σn,x(T1,n-γ(x)(b1,n(x)+1)-∆xb2,n(x))+σn,xT2,n converges to a N (0, γ 2 (x)) random variable, where

T1,n = kx X i=1 " 1 i Z β i 0 Ψ(α, β kx , x)dα « i log q(Ui,m x , x) q(Ui+1,m x , x) ,
and T2,n = O P " kxωn(m -(1+δ)

x ) " (b1,n(x) + 1).

From Beirlant et al. (2002, Theorem 2.1), {i(log q(Ui,m x , x)-log q(Ui+1,m x , x)), i = 1, . . . , kx} has the same distribution as ( γ(x) + ∆x

" i kx + 1 « -ρ(x) ! Fi + νi,n(x) + o P (∆x), i = 1, . . . , kx ) , (18)
where F1, . . . , F kx are independent standard exponential random variables and, uniformly in i = 1, . . . , kx,

kx X j=i |νj,n(x)| j = o P " ∆x max " log kx + 1 i , 1 «« . ( 19
)
Thus, σn,x(T1,nγ(x)(b1,n(x) + 1) -∆xb2,n(x)) has the same distribution as σn,x (γ(x)T1,1,n + ∆xT1,2,n + T1,3,n + (b1,n(x) + 1)o P (∆x)) , with

T1,1,n = kx X i=1 " 1 i Z β i 0 Ψ(α, β kx , x)dα « (Fi -1) T1,2,n = kx X i=1 " 1 i Z β i 0 Ψ(α, β kx , x)dα « " i kx + 1 « -ρ(x) (Fi -1) T1,3,n = kx X i=1 " 1 i Z β i 0 Ψ(α, β kx , x)dα « νi,n(x).
Let us first consider the term T1,1,n. From Lyapunov's Theorem, if for some ξ > 0,

kx X i=1 " 1 i Z β i 0 Ψ(α, β kx , x)dα « 2 ! -(1+ξ/2) kx X i=1 ˛1 i Z β i 0 Ψ(α, β kx , x)dα ˛2+ξ → 0, (20) then
, σn,xT1,1,n d → N (0, 1). ( 21
) From condition (B.1), kx X i=1 " 1 i Z β i 0 Ψ(α, β kx , x)dα « 2 ! -(1+ξ/2) kx X i=1 ˛1 i Z β i 0 Ψ(α, β kx , x)dα ˛2+ξ ≤ k -ξ/2 x 1 kx kx X i=1 f 2 1 " i kx , x « ! -(1+ξ/2) 1 kx kx X i=1 f 2+ξ 2 " i kx , x « ! .
Using Lemma 2 i) and condition (B.1), it then easy to check that condition (20) is satisfied. Now, let us focus on the term T1,2,n. We have E(T1,2,n) = 0 and

Var(T1,2,n) = 1 k 2 x kx X i=1 " kx i Z β i 0 Ψ(α, β kx , x)dα « 2 " i kx + 1 « -2ρ(x) = O(k -1 x ),
from Lemma 2 i) and condition (B.1). Thus, T1,2,n = o P (1). ( 22) Furthermore,

T1,3,n = kx X i=1 " 1 i Z β i 0 Ψ(α, β kx , x)dα « νi,n(x) = kx X i=1 i X j=1 Z β j β j-1 Ψ(α, β kx , x)dα ! νi,n(x) i = kx X j=1 Z β j β j-1 Ψ(α, β kx , x)dα kx X i=j νi,n(x) i .
Thus, from (B.2), (19) and using Lemma 2 i)

|T1,3,n| = ∆xo P kx X j=1 ˛Z β j β j-1 Ψ(α, β kx , x)dα ˛max " log kx + 1 j , 1 « ! = ∆xo P 1 kx kx X j=1 f3 " j kx , x « max " log kx + 1 j , 1 « ! = o P (∆x). (23)
Finally, using (B.1) and Lemma 2 i), b1,n(x) + 1 = O(1) and thus

σn,xT2,n = O P " k 3/2 x ωn(m -(1+δ) x " = o P (1). (24)
As a first conclusion, ( 21) to ( 24) entail that σn,x(e γ(β kx , x)γ(x)(b1,n(x) + 1) -∆xb2,n(x)) converges to a N (0, γ 2 (x)). The conclusion of the proof is then straighforward using Lemma 3 and the fact that under (B.1) and Lemma 2 i), σn,xO P " h2,n q(kx/mx, x)

« = O P k 1/2
x h2,n q(kx/mx, x)

! = o P (1). ( 25
)
Proof of Corollary 2 -It is quite easy to check that the assumptions of Corollary 2 imply the ones of Theorem 1. Thus, it is sufficient to prove the three following statements:

θ + 1 θ (1 -α θ ) if θ > 0 -log(α) if θ = 0 1 if θ = ∞,
for α < 1. Note that Φ0(., .) and Φ∞(., .) correspond to the limiting cases of Φ θ (., .).

Concerning the condition (C.2), the mean value theorem leads to :

˛iΦ θ " i kx , x « -(i -1)Φ θ " i -1 kx , x
«˛≤ sup

α∈I i ˛θ + 1 θ (1 -α θ ) -α θ (1 + θ) ≤ sup α∈I i θ + 1 θ (1 + α θ (θ -1)),
where Ii = [(i -1)/kx, i/kx]. Clearly, if θ ≥ 1, for all i = 1, . . . , kx,

˛iΦ θ " i kx , x « -(i -1)Φ θ " i -1 kx , x «˛≤ θ + 1 θ 1 + " i kx « θ (θ -1) ! . ( 29
) If θ < 1, then for i = 1, ˛iΦ θ " i kx , x « -(i -1)Φ θ " i -1 kx , x «˛= Φ θ " 1 kx , x « , (30)
and for i = 2, . . . , kx,

˛iΦ θ " i kx , x « -(i -1)Φ θ " i -1 kx , x «˛≤ θ + 1 θ 1 + " i -1 kx « θ (θ -1) ! ≤ θ + 1 θ 1 + " i 2kx

  is a direct consequence of Lemma 2 i) and condition (C.1). Using Lemma 2 ii) we have that b1,n(x) = o(k (26), σn,xb1,n(x) = o(1). Finally, (28) is straightforward using Lemma 2 i). This achieves the proof of Corollary 2.Proof of Corollary 3 -It is sufficient to prove that the function Ψ θ (., u, x) satisfies conditions (C.1) and (C.2). Clearly, (C.1) is satisfied with Φ θ (α, x)

  1)(1 + (θ -1)α θ )/θ if θ ≥ 1, (θ + 1)(1 + (θ -1)(α/2) θ )/θ if θ < 1.Since f4(., x) satisfies the condition in (C.2), the proof is complete.

Figure 1 :

 1 Figure 1: (a) Theoretical (-) and estimated (• • •) AMSE for one realization of the Burr-type distribution (8) with ρ = -1 and γ(0.4) defined in (9). The vertical lines correspond to the optimal values k opt 0.4 and b k0.4 ; (b) Boxplot of b k0.4 for the 100 replications. The horizontal line is the optimal value k opt 0.4 .

Figure 2 :

 2 Figure 2: Estimation of γ(.): Median (full line), 10% (dashed line) and 90% quantiles (dotted line) of the L 2 -errors and true function (-) based on 100 replications from the Burr-type distribution (8) with γ(.) defined in (9) and (a) ρ = -1 or (b) ρ = -0.8.


R Pierre

Bertrand

Mehdi Fhima
email: mehdi.fhima@math.univ-bpclermont.fr

Arnaud Guillin
email: arnaud.guillin@math.univ-bpclermont.fr

Fast change point analysis on the Hurst index of piecewise fractional Brownian motion

Keywords: Détection de ruptures, Dérivée Filtrée, mouvement Brownien fractionnaire par morceaux, paramètre de Hurst, Satistique de Bernouili des accroissements Change point analysis, Filtered derivative with p-value method, Hurst parameter, Increment Bernoulli Statistic, piecewise fractional Brownian motion

. La statistique sous-jacente de la technologie FDpV est un nouvel estimateur de l'indice de Hurst, appelé statistique de Bernouilli des accroissements (IBS). À la fois les méthodes FDpV et IBS ont une complexité linéaire par rapport à la taille de la série d'observation, aussi bien en temps de calcul que pour la mémoire, donc également leur combinaison.

Introduction

Recent measurement methods allow us to record and to stock large data sets, so called "the data deluge". For instance, today technology allows recording of heartbeat series during 24 hours leading to data sets of size n ≥ 100, 000, and very high frequency (VHF) financial series leads to data size n ≥ 40, 000. Tomorrow, many other series will be recorded at VHF leading to millions of data.

Large or huge series with small meshes of time can be described as continuous time processes observed at discrete times. Such a stochastic process X belongs to a certain class of model, that is X ∈ M = {X θ , θ ∈ Θ}, where Θ is a subset of R d and d is the dimension of the model. The structural parameter θ is believed to provide relevant information on the system which generates the series, and statisticians have to estimate it.

A slightly different approach is based on change point analysis: The structural parameter θ is assumed to be piecewise constant with an unknown configuration of change τ . In this framework, the first task of statisticians is the estimation of the location of the change points and the second one could be the estimation of the structural parameters between change points. There is a huge literature on change point analysis and model selection since the fifties, see e.g. [START_REF] Basseville | Detection of abrupt changes: theory and application[END_REF], [START_REF] Brodsky | Nonparametric methods in change-point problems[END_REF], [START_REF] Csörgo | Limit Theorem in Change-Point Analysis[END_REF], [START_REF] Birgé | Minimal penalties for Gaussian model selection[END_REF] or [START_REF] Bertrand | Off-line detection of multiple change points by the Filtered Derivative with p-Value method, to appear in Sequential Analysis[END_REF] and the references therein. However, most of the studies are devoted to change on the mean, on the variance or on the regression parameters. But relevant informations are also provided by the time dependence structure of the process, see e.g. [START_REF] Ayache | Discretization error of wavelet coefficient for fractal like process[END_REF] and [START_REF] Khalfa | Heart Rate Regulation processed through wavelet analysis and change detection[END_REF]. Fractional Brownian motion (fBm) is a paradigmatic example of such process, indeed fBm is a zero mean Gaussian process depending on two parameters: The Hurst index H linked to the time structure and a scale parameter σ.

In this presentation, we consider a simple model, that is a process X which is a piecewise fBm with an unknown configuration of changes. Moreover, we set us in the frame of huge datasets, and we focus our attention on time and memory complexity. These two reasons have lead us to propose a new change point procedure for detection of change on the Hurst index for a piecewise fBm. Our new procedure is the combination, on the one hand of the FDpV method, introduced in Bertrand et al. (2011) for fast and light detection of change on the mean, variance or regression parameter, and on the other hand of the Increment Bernoulli Statistic (IBS) a new estimator for Hurst index, which is a variation on the Increment Ratio Statistic (IRS) estimator introduced in [START_REF] Bardet | Measuring Roughness of Random Paths by Increment Ratios[END_REF].

The rest of this paper is organized as follows: At first, in Section 1, we define our model of piecewise fBm. Next, in Section 2, we introduce a new fast and robust estimator of the Hurst index of fBm, namely the Increment Bernoulli Statistic. Then, in Section 3, we describe the transposition of the FDp-V method to Hurst index.

Our model

We observe a process X at the discrete and regularly spaced time t i = i/n, where i = 0, . . . , n. We assume the existence of a segmentation τ = (τ k ) k=0,...,K+1 , with 0 = τ 0 < τ 1 < • • • < τ K < τ K+1 = n, such that the restriction of the process X on each interval (τ k , τ k+1 ) for k = 0, . . . , K is a fBm with Hurst index H k and scale parameter σ k . The integer K corresponds to the number of change points and (K + 1) to the number of segments. Stress that K can be zero and in this case the process X is a fBm.

Let us precise that in our roadmap the process X should be almost surely with continuous paths. For this reason, the so-called piecewise fBm can not be defined by plugging a piecewise Hurst index into one of the representations of the fBm. Indeed, by doing so, the process X would almost surely have discontinuity at at each change point on Hurst index, and method for detecting change on the mean will also be efficient for detecting change on Hurst index. Let us refer to Taqqu and Samorodinitsky (1994) as a reference book on the different representations of fBm and to [START_REF] Ayache | Multifractional process with random exponent[END_REF] for the construction of multi-fBm by plugging a continuous time varying Hurst index into one of the fBm representations. A rather complicated solution to avoid the drawback of pathwise discontinuities due to Hurst index discontinuities, was proposed in [START_REF] Benassi | Identification of the Hurst exponent of a Step Multifractional Brownian motion[END_REF] and cosigned by the first author. However, as point out by Antoine Ayache during private conversations held in 2004, for statistical applications, it suffices to cancel the discontinuity by adding a correction term. The same solution is also adopted in [START_REF] Bardet | Detecting abrupt changes of the long-range dependence or the self-similarity of a Gaussian process[END_REF] .

The model having being specified, we are concerned with change point analysis on the Hurst parameter, where the number of change K is unknown. There are few references on this problem. To our best knowledge, the only reference are [START_REF] Benassi | Identification of the Hurst exponent of a Step Multifractional Brownian motion[END_REF] and [START_REF] Bardet | Detecting abrupt changes of the long-range dependence or the self-similarity of a Gaussian process[END_REF].

Increment Bernoulli Statistic for fBm

In this section, we investigate the properties of a new estimator of the Hurst parameter of fBm, namely the Increment Bernoulli Statistic (IBS). IBS is a variation on IRS which has been introduced by [START_REF] Surgailis | The increment ratio statistic[END_REF] and applied to fBm by [START_REF] Bardet | Measuring Roughness of Random Paths by Increment Ratios[END_REF]. Both IRS and IBS are fast and robust estimator of the Hurst index. By fast we mean estimator with linear time complexity, and by robust we mean estimator with invariant scaling property. The choice of the IBS instead of the IRS is motivated by the fact that the IBS is bit less expensive, in terms of time complexity, than the IRS.

In the next section, the IBS is used as the underlying estimator for the FDpV method, see [START_REF] Antoch | Procedures for the detection of multiple changes in series of independent observations[END_REF]. For this reason, we define IBS for a every process X, even if we apply it to fBm in this section. Let X be a process observed at a family of discrete times t k , we define the second order increments by ∆X(t k ) := X(t k+2 ) -2X(t k+1 ) + X(t k ).

Then, the Increment Bernoulli Statistic (IBS) is based on the comparison of the signs of consecutive second order increments. The results of these comparisons will be equal to 1 if the consecutive second order increments have the same sign, and 0 otherwise. Hence, this explains the name of our new estimator, that is to say: Increment Bernoulli Statistic (IBS) which is given by

IBS n (X) = 1 n -2 n-3 k=0 ψ (∆X(t k ), ∆X(t k+1 ))
where ψ(•, •) is described as follows ψ(x, y) = 0 if sign(x) = sign(y) and 0 otherwise, where sign(z) denotes the sign of z. Let us remark that IBS is scale invariant: Indeed, since ψ(σx, σy) = ψ(x, y)

for σ > 0, then IBS n (σX) = IBS n (X).
When X is a fBm, that is X = B H with a Hurst index H ∈ (0, 1), then the IBS converges in distribution to a continuous monotonic increasing function Λ(H) defined as follows

Λ(H) := Λ 0 (ρ(H)) Λ 0 (r) := 1 π arccos(-r) ρ(H) = -3 2H + 2 2H+2 -7 8 -2 2H+1 -1
where ρ(H) ∈ (-1, 1) represents the correlation between two successive second order increments. The graph of Λ(H) is given in Figure 1. Then, due to the fact that Λ(•) is a reversible function, it is easy to deduce an estimator of the Hurst parameter H given by H n = Λ -1 (IBS n (B H )). Furthermore, we note that the function φ(•,

•) = ψ(•, •) -Λ(H)

Filtered Derivative with p-Value method

In this section, we describe the Filtered Derivative with p-value method (FDp-V). First, we define the Filtered Derivative function. Next, we describe the two steps of the FDp-V method:

Step 1 is based on Filtered Derivative and select the potential change points, whereas Step 2 calculate the p-value associated to each potential change point, for disentangling right change points and false alarms. At first, we note that Λ(H) is a continuous monotonic increasing function of H, see Figure 1. So, the detecting of change points on the Hurst parameter H is equivalent to detecting change points on Λ(H). Consequently, the estimator IBS n (B H ) of the parameter Λ(H) is used as the underlying estimator for the FDpV method. We reefer to [START_REF] Bertrand | Off-line detection of multiple change points by the Filtered Derivative with p-Value method, to appear in Sequential Analysis[END_REF] and [START_REF] Bertrand | Filtered Derivative with p-Value Method for Multiple Change-Points Detection[END_REF] for the introduction of FDpV technology and its numerical efficiency. Let us stress that the choice of the direct estimator H n = Λ -1 (IBS n (B H )) as the underlying estimator for the FDpV would be more expensive in term of numerical complexity.

Filtered Derivative function

Let X be a piecewise fBm observed at a family of discrete times t j = j/n, for j = 0, . . . , n. The Filtered Derivative for IBS is defined as the difference between the estimators of the parameter Λ(H) computed on two sliding windows respectively at the right and at the left of the index k, both of size A, that is specified by the following function

D(k, A) = IBS (X, k, A) -IBS (X, k -A, A) for k ∈ [A, n -A] (1)
where

IBS (X, k, A) = A -1 k+A j=k+1 ψ (∆X(t k ), ∆X(t k+1 ))
is an estimator of Λ(H) on the sliding box [k + 1, k + A]. It is easy to see that the Filtered Derivative function D is computed by recurrence with linear time and memory complexity.

Eventually, this method consists on filtering data by computing the estimators of the parameter Λ(H) before applying a discrete derivation. This construction explains the name given by [START_REF] Benveniste | Detection of abrupt changes in signals and dynamical systems: some statistical aspects, Analysis and optimization of systems[END_REF], the so-called Filtered Derivative method.

Step 1: Detection of potential change points

In order to detect the potential change points, we test the null hypothesis (H 0 ) of no change in the Hurst parameter H against the alternative hypothesis (H 1 ) indicating the existence of at least one change point (H 1 ) : There is an integer

K ∈ N * and 0 = τ 0 < τ 1 < • • • < τ K < τ K+1 = n such that H 1 = • • • = H τ 1 = H τ 1 +1 = • • • = H τ 2 • • • = H τ K +1 = • • • = H τ K+1 .
where H j = H τ k is the value of the Hurst parameter at t j ∈ [τ k-1 /n, τ k /n). Now, we fix a probability of type I error at level p * 1 , and we determine the corresponding critical value C 1 given by

P max k∈[A:n-A] |D(k, A)| > C 1 |H 0 is true = p * 1 .
Of course, such a probability is usually not available, so that we only consider the asymptotic distribution of the maximum of |D|. Then, the change points τ k is selected as a potential change point if its local maxima satisfy |D( τ k , A)| > C 1 . We remark through the graph of the function |D| that there are not only the "right hats" (surrounded in green in Figure 2) which gives the right change points, but also false alarms (surrounded in black in Figure 2). Consequently, we have introduced another step in order to keep just the right change points.

Step

Conclusion

In conclusion, it appears that the combination of the FDpV and the IBS methods provides a fast (time) and cheap (memory) algorithm to the detection of change points on the Hurst parameter of piecewise fBm. So, this algorithm is adapted to segment random signals with large datasets. In future work, we will develop the FDpV + IBS method in order to detect abrupt changes on parameters of real data drawn from financial and physiological domains.

Figure 1 :

 1 Figure 1: The graph of Λ(H).

2 :

 2 Elimination of false alarmsThe list of potential change points ( τ 1 , . . . , τ Kmax ) obtained at step 1 contains right change points but also false alarms. In the second step, a test is carried out to remove the false alarms from the list of change points found at step 1. More precisely, for all potential change point τ k , we test whether the Hurst parameter is the same on the two successive intervals ( τ k-1 /n, τ k /n) and ( τ k /n, τ k+1 /n), or not. Formally, for all 1 ≤ k ≤ K max , we apply the following hypothesis testing(H 0,k ) : H k = H k+1 versus (H 1,k ) : H k = H k+1 ,where H k is the value of H on the segment ( τ k-1 /n, τ k /n). By using this second test, we calculate new p-values ( p 1 , . . . , p Kmax ) associated respectively to each potential change points ( τ 1 , . . . , τ Kmax ). Then, we only keep the change points which have a p-value smaller than a critical level denoted p * 2 . By doing so, we obtain a subset τ 1 , . . . , τ K of the first list which represents the estimators of the change points in the Hurst parameter of mBm.

Figure 2 :

 2 Figure 2: Detection of potential change points. Above: Simulated piecewise fBm with five change points in the Hurst parameter. Below: Filtered Derivative function -D-.