D Armentano

J-M Azaïs

F Dalmao
email: fdalmao@unorte.edu.uy

J R León

E Mordecki
email: mordecki@cmat.edu.uy.

* Armentano

Dalmao

On the finiteness of the moments of the measure of level sets of random fields

Keywords: AMS2000 Classifications: Primary 60G60 and Secondary 60G15 Moments of measure of level sets, Kac-Rice formula, Crofton formula, shot noise process

General conditions on smooth real valued random fields are given to ensure the finiteness of the moments of the measure of their level sets. As a by product, a new generalized Kac-Rice formula (KRF) for the expectation of the measure of these level sets in the one-dimensional case is obtained when the second moment can be uniformly bounded. The conditions involve (i) the differentiability of the trajectories up to a certain order k, (ii) the finiteness of the moments of the k-th partial derivatives of the field up to another order and (iii) the boundedness of the field's joint density and some of its derivatives. Particular attention is given to the shot noise processes and fields. Other applications include stationary Gaussian processes, Chi-square processes and regularized diffusion processes.

Introduction

Level sets of random fields play a key role in several branches of mathematics such as random algebraic geometry, probability, and mathematical physics. The focus is on different geometric characteristics of the level sets, typically their geometric measure. Depending on the dimensions, these characteristics can be the number of crossings of a stochastic process, the length of level curves of a random field, etc. Since the actual distribution of these random variables is usually out of reach, it is natural to study their moments and asymptotic distributions.

The present work concerns with the following two issues: (i) assessing the finiteness of the moments and (ii) their computation or estimation. For (ii) the main tool is the Kac-Rice formula (KRF for short) which gives the expected value and the higher order moments of the measure of level sets of smooth random fields. These two issues are highly connected, for instance:

-The KRF of order one is, in general, valid under conditions that imply (without further hypotheses) that the expectation is finite. Furthermore, in some cases, necessary and sufficient conditions for the finiteness of the expectation of the measure of level sets can be obtained from the KRF (see [START_REF] Armentano | Conditions for the finiteness of the moments of the volume of level sets[END_REF] and the references therein).

-For stationary Gaussian random fields, the KRF of order two can be used to obtain the finiteness of the second moment of the measure of level sets as in the works [START_REF] Geman | On the Variance of the Number of Zeros of a Stationary Gaussian Process Ann[END_REF], [START_REF] Kratz | On the second moment of the number of crossings by a stationary Gaussian process[END_REF] and [START_REF] Azais | Necessary and sufficient conditions for the finiteness of the second moment of the measure of level sets[END_REF]. Some complicated study has been performed for higher moments in [START_REF] Belyaev | On the number of intersections of a level by a Gaussian stochastic process[END_REF]. In other cases, the calculations are intractable.

-In the other direction, explained in section 7, the finiteness of the second moment is a tool to establish the validity of the KRF.

The seminal works of this field are that of Kac [19] and Rice [START_REF] Rice | Mathematical analysis of random noise II[END_REF]. The KRFs were first established for Gaussian stochastic processes profiting from the fact that the Gaussian framework allows not only to obtain conditions under which the formulas are valid but also permits some explicit computations. Adler in [START_REF] Adler | The geometry of random fields[END_REF] obtained KRF for Gaussian fields. Recent works of [START_REF] Angst | On the absolute continuity of the random nodal volumes[END_REF] and [START_REF] Armentano | Conditions for the finiteness of the moments of the volume of level sets[END_REF] are also worth mentioning, as they show the finiteness of the moments of the measure of nodal sets of a real-valued Gaussian stationary field. For a panoramic and contemporary view of these matters, we refer to the books [START_REF] Adler | Random fields and geometry[END_REF], [START_REF] Azaïs | Level sets and extrema of random processes and fields[END_REF] and [START_REF] Berzin | Kac-Rice formulas for random fields and theirs applications in: random geometry, roots of random polynomials and some engineering problems[END_REF]. Mainly motivated by the applications, there has been an interest in studying such formulas for non-Gaussian processes. The first successful attempt was that of Marcus [START_REF] Marcus | Level crossings of a stochastic process with absolutely continuous sample paths[END_REF] who provided a formula for the expected number of crossings of a process whose trajectories are absolutely continuous. Concerning finiteness of moments Nualart & Wschebor [START_REF] Nualart | Intégration par parties dans l'espace de Wiener et approximation du temps local[END_REF], by using properties of the trajectories of regular processes, show that the expectation of the number of crossings and its moments of order greater than one can be bounded. This result is based on the idea that a "nice function" cannot have too many zeros. Unfortunately, the proof heavily relies on the intermediate value theorem, and for this reason, it applies only to stochastic processes. Wschebor in [START_REF] Wschebor | Surfaces aléatoires, Mesure géométrique des ensembles de niveau[END_REF] establishes the KRF for the measure of the level sets of fields (not necessarily Gaussian). However, it is important to point out that the hypotheses of these formulas are difficult to check. An important exception is when the field is a nonlinear transformation of a Gaussian one, as is the case of χ 2 , t, or F random fields. The books [START_REF] Adler | Random fields and geometry[END_REF] and [START_REF] Azaïs | Level sets and extrema of random processes and fields[END_REF] contain a comprehensive update of these subjects. The papers [START_REF] Biermé | A Fourier approach for the level crossings of shot noise processes with jumps[END_REF], [START_REF] Borokov | On Rice formula for stationary multivariate piecewise smooth process[END_REF] and [START_REF] Dalmao | Rice formula for processes with jumps and applications[END_REF] consider crossings for discontinuous processes, and the two last works include KRFs. Moreover, Biermé & Desolneux (in [START_REF] Biermé | Crossings of smooth shot noise processes[END_REF] and [START_REF] Biermé | On the perimeter of excursion sets of shot noise random fields[END_REF]) studied crossings problems and KRF for shot noise processes.

Within the applications of the KRF we mention the random sea surface modeling, and the articles [START_REF] Machado | Wave statistics in non-linear random seas[END_REF], [START_REF] Podgorki | Slepian noise approach for Gaussian and Laplace moving average processes[END_REF], [START_REF] Baxevani | Sample path asymmetries in non-Gaussian random seas[END_REF] and [START_REF] Podgorki | Envelope crossing distribution for Gaussian fields[END_REF] contain Gaussian and general KRFs. Worsley computes the expectations of some level sets characteristic in the context of medical image processing [START_REF] Worsley | Estimating the number of peaks in a random field using the Hadwiger characteristic of excursion sets, with applications to medical images[END_REF] and in astrophysics [START_REF] Worsley | Boundary corrections for the expected Euler characteristic of excursion sets of random fields, with an application to astrophysics[END_REF]. Other applications and KRFs for fields can be found in the recent monograph [START_REF] Berzin | Kac-Rice formulas for random fields and theirs applications in: random geometry, roots of random polynomials and some engineering problems[END_REF].

The present paper considers first finiteness of moments which remains an open problem, excepting the stationary Gaussian case limited to the first two moments.

It can be a first step to establish a speed of convergence in the "ergodic" case when we observe the random field over an increasing set of parameter.

In the particular case of η-dependent random processes (a random field X is η-dependent if X(t), X(s) are independent whenever |t -s| > η), the finiteness of the second moment gives directly a central limit theorem.

This finiteness gives also a central limit theorem in the case of increasing number of independent observations of the random field.

The main result in dimension one is given in section 2. Section 3 considers the application of the previous result to different examples with new results, even in the case of Gaussian processes, or χ 2processes. Section 4 contains the study of shot noise processes, this theme constitutes, together with shot noise random fields, the main application of our results. In section 5, the previous result is extended to level sets of co-dimension one that corresponds to random fields from R d and the sphere S d to R. Section 6 deals with shot noise random fields. Finally, non-Gaussian KRFs are in section 7. All results presented in the examples are new but, except for the toy Example 1, impossible it is to know if they are optimal or not.

Processes from R to R

The basic idea of this section is the following (see the details below): Let f (•) be defined as, say, [0, 1]. Assume that the k-th derivative of f is bounded by M and that f has

k zeros on [0, 1]. Then, f satisfies |f (1/2)| ≤ M (1/2) k k! .
Now, if we replace the function f by the paths of a random process X that admits a density at t = 1/2 and we assume that this density is bounded by C. Then, the probability of the event

{|X(1/2)| ≤ M (1/2) k k! } is smaller than 2CM (1/2) k k! .
Theorem 2.1 is just a systematic exploitation of this method with some generalization because we consider the joint density of X(t) and some derivatives.

Assume that X = {X(t) : t ∈ R} is a real valued stochastic process with smooth paths. Define the number of crossings through level u by the process X over the finite time interval I ⊂ R by

N u = N u (I) := #{t ∈ I : X(t) = u}.
Let |I| denote the length of I and let Ī be the middle point of I.

As usual, we interpret the zeroth derivative X (0) as X itself. Thus, we have the following result.

Theorem 2.1. Consider u, X, I and N u as above. Assume that X satisfies (H1) The sample paths of X are C k (I) for some k ≥ 2.

(H2) For some m = 1, 2, . . . , there exists a constant D m such that

E |X (k) | m ∞ ≤ D m .
(H3) There exist 0 ≤ h ≤ k and a constant C > 0 such that the joint density of X(t), X (t), . . . , X (h) (t) is bounded by C uniformly in t ∈ I, and on a neighborhood of (u, 0, . . . , 0).

Then, for p = 1, 2, . . . , such that

p < k - h 2 - 1 1 + h 1 m + 1 1 + h -1 , (1)
the p-th moment of the number of crossings N u is finite. If in addition we assume (H4) the density in (H3) is bounded by C uniformly in t ∈ I and

(u 1 , . . . , u h+1 ) ∈ R h+1 . Then E(N p u ) ≤ (k -1) p + D m • E α,k,p + C|I| (h+1)(k-h/2) • D α,k,h,p , where α is any real number such that p m < α < k -h 2 -1+p 1+h , E α,k,p = p(k -1) p-1 ∞ a=1 (a + 1) p-1 a mα and D α,k,h,p = 2 (h+1)(1+h/2-k) k! • (k -h)! ∞ a=1 (a + 1) p-1 a 1-(h+1)(k-h/2-α) .
Remark 2.2 (Large m). The limit case, when m can be chosen arbitrarily large (as is the case for Gaussian processes), corresponds to α close to zero. The limit condition (1) then becomes

p < k - h 2 (h + 1) -1.
Remark 2.3 (Large k). When the process X has C ∞ (I) paths, if (H2) holds true for m = 1 with arbitrary k, and (H3) holds true for some h = 0, 1, . . . , then, all the moments of N u are finite.

Remark 2.4 (h = 0). The case h = 0 and m = 1 corresponds to (3.23) in pg. 82 in [START_REF] Azaïs | Level sets and extrema of random processes and fields[END_REF], obtained in Lemma 1.2 in [START_REF] Nualart | Intégration par parties dans l'espace de Wiener et approximation du temps local[END_REF]. When h = 0 and m > 1 Theorem 2.1 assures that the pth moment is finite for

p < m m + 1 (k -1).
Before proving the theorem we establish three preliminary lemmas. Recall that f (0) = f .

|f ( Ī) -u| ≤ |f (k) | ∞ k! |I| 2 k , |f ( Ī)| ≤ |f (k) | ∞ (k -1)! |I| 2 k-1 , . . . , |f (h) ( Ī)| ≤ |f (k) | ∞ (k -h)! |I| 2 k-h .
Proof. The proof is based on the Lagrange remainder form for polynomial interpolation. That is, let g : I → R be C k , t 1 , . . . , t k ∈ I and P be the only polynomial of degree k -1 such that g(t j ) = P (t j ), j = 1, . . . , k. Then, for t ∈ I we have

g(t) -P (t) = 1 k! k j=1 (t -t j ) • g (k) (ξ),
for some ξ such that min{t 1 , . . . , t k , t} < ξ < max{t 1 , . . . , t k , t}, see Lemma 5.2, p. 135 in [START_REF] Azaïs | Level sets and extrema of random processes and fields[END_REF]. Now, for g = f -u, t = Ī and using as t 1 , . . . , t k the roots of f -u, we get P = 0 and so the first inequality follows. The proofs of the other inequalities follow in a similar way.

Lemma 2.6. Let Z be a random variable taking non negative integer values. Then,

E(Z p ) = ∞ =1 ( p -( -1) p ) P(Z ≥ ) ≤ p ∞ =1 p-1 P(Z ≥ ).
Proof. Use Fubini's Theorem and the convexity of the function x → x p . Lemma 2.7. Assume that conditions (H1), (H2), (H3) and (H4) of Theorem 2.1 hold true. Let ≥ k and define a and r such that = a(k -1) + r with 1 ≤ r ≤ k -1. Then, for B > 0, we have:

P(N u ≥ , |X (k) | ∞ ≤ B) ≤ aC(2B) h+1 h i=0 |I| 2a k-i 1 (k -i)! .
where C is the bound in (H3). When (H4) does not hold, the inequality is true for sufficiently large only.

Proof. We divide I into a equal subintervals I 1 , . . . I a , s.t. at least one of them has k zeros. After this, we use an union bound:

P(N u ≥ , |X (k) | ∞ ≤ B) = P(N u ≥ a(k -1) + r, |X (k) | ∞ ≤ B) ≤ P   a j=1 {N u (I j ) ≥ k}, |X (k) | ∞ ≤ B   ≤ a j=1 P |X( Īj ) -u| ≤ B k! |I| 2a k , h i=1 |X (i) ( Īj )| ≤ B (k -i)! |I| 2a k-i ≤ aC h i=0 2B (k -i)! |I| 2a k-i
, obtaining the result. Of course, when h = 0, the intersection in the equation above must be absent.

Proof of Theorem 2.1. Assume (H4) for the moment From Lemma 2.6, we have

E(N p u ) ≤ (k -1) p + ∞ =k p p-1 P(N u ≥ ). (2)
In order to bound the summands in the r.h.s. of (2), for given consider a and r as in Lemma 2.7, and α > 0 to be defined later. We use the following bound:

P(N u ≥ ) ≤ P(N u ≥ , |X (k) | ∞ ≤ a α ) + P(|X (k) | ∞ > a α ).
Using now Lemma 2.7 for the first summand and Chebyshev's inequality for the second, we obtain

E(N p u ) ≤ (k -1) p + p(k -1) p-1 ∞ a=1 (a + 1) p-1 P(N u ≥ ) ≤ (k -1) p + p(k -1) p-1 C2 h+1 |I| 2 (h+1)(k-h/2) h i=0 1 (k -i)! . ∞ a=1 (a + 1) p-1 a (h+1)(α+h/2-k)+1 +D m ∞ a=1 (a + 1) p-1 a -mα .
The conditions for the simultaneous convergence of the two series above are

p m < α < k - h 2 - 1 + p 1 + h .
Under condition [START_REF] Adler | The geometry of random fields[END_REF], there always exists an adequate α, concluding the proof of Theorem 2.1. When (H4) does not hold, the calculation above is true only for (or a) sufficiently large, and this does not change the condition of convergence of the series.

Examples

Example 3.1 (Conditional sine-cosine process). We begin with a toy example. Consider a random variable ω such that E (|ω| M ) < ∞ and E (|ω| M +1 ) = ∞. This variable is the random frequency of a process X = {X(t) : 0 ≤ t ≤ 2π} that we construct as

X(t) = ξ 1 sin ωt + ξ 2 cos ωt,
where ξ 1 and ξ 2 are two standard normal independent random variables, also independent from ω. Conditionally to the value of ω, the process X is a Gaussian sine-cosine process. It is clear that the density of X(t) is standard normal for each t. Nevertheless, it should be observed that X(t) is, in general, not a Gaussian process. The number N 0 of roots of X(t) on [0, 2π] coincides with the number of roots of the process

X(t) = ξ 1 ξ 2 1 + ξ 2 2 sin ωt + ξ 2 ξ 2 1 + ξ 2 2 cos ωt = cos(ωt -θ),
where θ = arctan(ξ 1 /ξ 2 ) is defined a.s. Therefore, N 0 satisfies |N 0 -2ω| ≤ 2. This means that for the considered process, the maximal finite moment of N 0 is M . To apply Theorem 2.1, we compute

|X (k) | ∞ ≤ ω k ξ 2 1 + ξ 2 2 .
Note that ξ 2 1 + ξ 2 2 follows a Rayleigh's distribution. Then, based on the independence of the three random variables above, we have

E |X (k) | m ∞ ≤ H m E(ω mk ),
where H m is the finite moment of the Rayleigh distribution. So, we can apply Theorem 2.1 with h = 0, k = M and m = 1. Assuring that the moments of order p ≤ (M -1)/2 are finite. This result should be compared with the true result which is M .

Example 3.2 (Stationary Gaussian process). Consider a stationary Gaussian process with spectral measure supported in a set with an accumulation point. By Ex. 3.4 in [START_REF] Azaïs | Level sets and extrema of random processes and fields[END_REF] we know that for all h X(t), X (t), . . . , X (h) (t) admit a joint density. By Borell-Sudakov-Tsirelson Theorem (see, for example Th. 2.8 in [START_REF] Azaïs | Level sets and extrema of random processes and fields[END_REF]), for all m:

E |X (k) | m ∞ ≤ +∞.
It is then possible to take h = k and m arbitrarily large. Thus, Formula (1) gives a finite moment of order

p≤ k(k + 1) 2 -2.
To visualize it, see Table 1.

k 2 3 4 p 1 2,3,4 5,6,7,8 Table 1: If X is C k , following Theorem 2.1 we obtain that E(N p u ) is finite for values of k and p above.
This result improves the previous one obtained in [START_REF] Nualart | Intégration par parties dans l'espace de Wiener et approximation du temps local[END_REF] that is recovered in the case h = 0 and m = 1.

Example 3.3 (Chi-square process). Consider a Chi-square process Y = {Y (t) : t ∈ R} given by

Y (t) = |X(t)| 2 = n i=1 X i (t) 2 ,
where X = {X(t) : t ∈ R} with X(t) = (X 1 (t), . . . , X n (t)) is an R n valued random process whose coordinates are n independent copies of a stationary Gaussian process with C k (R) paths and variance 1. First, observe that the case n = 1 admits an ad-hoc treatment. If n = 1 for u = 0 we have

N Y 0 [0, T ] = N X 0 [0, T ], and for u > 0 we have N Y u [0, T ] = N X √ u [0, T ] + N X - √ u [0, T ].
For n ≥ 2, the level u = 0 is not interesting, as for this level the process is differentiable and non-negative and has a bounded density, so Bulinskaya Lemma (Prop. 1.20 in [START_REF] Azaïs | Level sets and extrema of random processes and fields[END_REF]) gives that a.s. the number of crossings is zero. Consequently, we consider n ≥ 2 and u > 0. Observe that, excluding the uninteresting case where λ 2 = Var X (t) = 0, the joint density of the random variables X(t), X (t) is bounded, as they conform a pair of independent Gaussian random variables.

It is not difficult to check the hypothesis of Theorem 2.1 in two different situations: n = 2 and n ≥ 3. It is direct to see that Y has C k paths and that |Y | ∞ has moments of every order when considering a finite interval I ⊂ R. Observe now that Y (0) has a χ 2 (n) density that is bounded for n ≥ 2. We can then apply Theorem 2.1 with the given k, arbitrary m, and h = 0. Based on (1), we obtain the finiteness of the moments of the crossings, for u > 0, of order

p = k -2,
which is relevant only in the case k ≥ 3. A more refined analysis can be carried out. Regarding the derivative, we have

Y (t) = 2 n i=1 X i (t)X i (t).
We

P(Y ∈ B 1 , Y ∈ B 2 ) = R n dx B1 p Y |X=x (y)1 {Y ∈B2} p X (x)dy ≤ (Const) R n |B 1 |Y -1/2 1 {Y ∈B2} p X (x)dx = (Const)|B 1 | B2 y -1/2 p Y (y)dy = (Const)|B 1 | B2 y n/2-3/2 e -y/2 dy, (3)
where p X (.) and p Y (.) are the density functions of X and Y , respectively. Note that we used the explicit expression of the density of a χ 2 (n) distribution. When n ≥ 3, the integrand in (3) is bounded yielding that

P (Y ∈ B 1 , Y ∈ B 2 ) ≤ (Const)|B 1 ||B 2 |.
This proves that the process satisfies the hypotheses of Theorem 2.1 with h = 1. The conclusion is that for n ≥ 3, the number of crossings of Y has a finite moment of order p = 2k -3.

Example 3.4 (Regularized processes). In this example, we consider the number of crossings with a level u of a regularized diffusion. We depart from a diffusion X = {X(t) : t ≥ 0} defined as the solution of the stochastic differential equation

X(t) = x 0 + t 0 b(s, X(s))ds + t 0 σ(s, X(s))dW (s), t ≥ 0.
Here, W = {W (s) : s ≥ 0} is a standard Brownian motion, x 0 ∈ R, and b, σ : [0, ∞) × R → R are Lipschitz w.r.t. the second variable, that is, for T > 0 there exists K T > 0 such that for x, y ∈ R and 0 ≤ s ≤ T it holds that

|b(s, x) -b(s, y)| + |σ(s, x) -σ(s, y)| ≤ K T |x -y|.
Assume also the linear growth condition

|b(s, x)| + |σ(s, x)| ≤ K T (1 + |x|).
Then, the above equation has a unique strong solution that satisfies

sup 0≤t≤T E(X(t) 2 ) ≤ H T < ∞.
for a constant H T (see Theorem 7.1 in [START_REF] Borodin | Stochastic processes[END_REF]). The regularized diffusion is defined by

X Ψ (t) = (Ψ * X)(t),
where Ψ : R → [0, ∞) is a C ∞ -function with support contained in [-1, 1] that integrates to one. Thus, X Ψ is obtained from X by path-wise convolution with Ψ. We consider two cases:

Case I. We assume that b = 0 and σ : R + × R → R is strictly positive and C 3 .

Case II. The volatility σ is strictly positive and C 3 as in Case I, and there exist constants B T ,c T and

C T such that |b(s, x)| ≤ B T , 0 < c T ≤ σ(s, x) ≤ C T for all 0 ≤ s ≤ T and x ∈ R.
We start with Case I. With regards to the hypotheses of Theorem 2.1, we have that X Ψ is C ∞ since it inherits the regularity of Ψ. Furthermore, for a large enough (depending on Ψ) the random variable X Ψ (t) has a uniformly bounded density on an interval [a, T ] for all finite T , see Lemma 3.1 in [START_REF] Nualart | Intégration par parties dans l'espace de Wiener et approximation du temps local[END_REF]. Besides, a direct computation gives

|X (h) Ψ | ∞ ≤ c • |X| ∞ , with c = |Ψ (h) (u)|du
E(|X| 2 ∞ ) ≤ C 1 E([X] T ),
with [X] T the quadratic variation of X on [0, T ]. Now, by Theorem II-29 in Protter [START_REF] Protter | Stochastic integration and differential equations[END_REF], we have

[X] T = T 0 σ(s, X(s))dW (s) T = T 0 σ(s, X(s)) 2 ds ≤ K 2 T T 0 (1 + |X(s)|) 2 ds ≤ 2K 2 T T 0 (1 + |X(s)| 2 )ds.
Taking expectations, we have

E([X] T ) ≤ 2K 2 T T 0 (1 + E(X(s)) 2 )ds ≤ 2K 2 T T (1 + H T ) < ∞.
Hence, by Theorem 2.1 with h = 0, m = 2 and arbitrary k, we obtain that

E(N p u ) < ∞,
for all p, as obtained in [START_REF] Nualart | Intégration par parties dans l'espace de Wiener et approximation du temps local[END_REF].

For Case II, we apply Girsanov's theorem [14, Th. 10.1-10.2]. We then have two SDE

dX 0 (t) = σ(t, X 0 (t))dW (t) dX(t) = b(t, X(t))dt + σ(t, X(t))dW (t).
Consider the process density:

ρ(t) = exp t 0 b(s, X(s)) σ(s, X(s)) dW (s) - 1 2 t 0 b(s, X(s)) σ(s, X(s)) 2 ds .
Then, Girsanov's Theorem states that for an arbitrary function on the trajectories of the process

F : C([0, T ], R) → R we have E (F (X)) = E (ρ(T )F (X 0 )) .
If F is the p-power of the number of crossings on [0, T ], we have

E (N u (I, X) p ) = E (ρ(T )N u (I, X 0 ) p ) ≤ E ρ(T ) 2 1/2 E N u (I, X 0 ) 2p 1/2 ,
applying the Hölder's inequality (X and X 0 are defined above). To bound the first expectation in the r.h.s. consider

ρ(t) 2 = exp 2 t 0 b(s, X(s)) σ(s, X(s)) dW (s) -2 t 0 b(s, X(s)) σ(s, X(s)) 2 ds exp t 0 b(s, X(s)) σ(s, X(s)) 2 ds
Then,

ρ(t) 2 ≤ exp 2 t 0 b(s, X(s)) σ(s, X(s)) dW (s) -2 t 0 b(s, X(s)) σ(s, X(s)) 2 ds exp B T c T 2 T ,
and since the first exponential is a martingale, we get

Eρ(T ) 2 ≤ exp B T c T 2 T < ∞.
As, for all p, E N u (I, X 0 ) 2p is finite by Case I, the finiteness of E (N u (I, X) p ) follows.

Shot noise processes

The stationary shot noise process X = {X(t) : t ∈ R d } is defined by

X(t) = i β i g(t -τ i ), (4)
where (β i ) is a sequence of i.i.d. random, the "impulse", variables, (τ i ) is a Poisson field on R d with constant intensity λ, and g : R d → R is some function called the kernel function. Following [START_REF] Biermé | Crossings of smooth shot noise processes[END_REF], we assume that β 1 is an integrable random variable and that g ∈ L 1 (R d ). This ensures the a.s. convergence of the series in [START_REF] Armentano | Conditions for the finiteness of the moments of the volume of level sets[END_REF]. A key issue to apply Theorem 2.1 is the verification of condition (H4) with h = 0, which requires the boundedness of a density. This is a delicate issue as was previously noticed by Biermé and Desolneux in [START_REF] Biermé | Crossings of smooth shot noise processes[END_REF] (see also [START_REF] Biermé | Regularity and Crossings of Shot Noise Processes[END_REF]), that is the main reference of this section from where we borrow the presentation and notations. We consider below d = 1, the case d > 1 (as well as shot noise random fields defined on the sphere) will be considered in Section 6. We begin by specializing Theorem 2.1 to the present situation.

Corollary 4.1. Consider a stationary shot noise process (4) satisfying condition (H1) for some k ≥ 1, (H2) for some m ≥ 1 and k above, and (H3) for h = 0. Then,

E(N p u ) < ∞ for p < m m + 1 (k -1). ( 5
)
The rest of the section is devoted to obtain sufficient conditions to verify this corollary. The differentiability of the sample paths follows directly from the differentiability of the kernel g:

X (k) (t) = i β i g (k) (t -τ i ),
provided that the k-th derivative of g is integrable, i.e. g (k) ∈ L 1 (R).

Boundedness of moments (H2)

. We now give conditions on the impulse and the kernel in order to verify (H2) for given k and m. For simplicity of exposition we assume that I = [-1, 1] and the general case can be treated in the same way. We consider the partition of the real line I 1 = I = [-1, 1] and

I n = [-n, -n + 1) ∪ (n -1, n] (n ≥ 2)
. In this way we can write

X (k) (t) = ∞ n=1 i:τi∈In β i g (k) (t -τ i ).
D k = ∞ n=1 d k,n < ∞. (6)
Then, the condition (H2) holds true for m and k as above, i.e.

E |X (k) | m ∞ < ∞.
Proof. We have

|X (k) | ∞ = max -1≤t≤1 |X (k) (t)| ≤ max -1≤t≤1 ∞ n=1 i:τi∈In |β i ||g (k) (t -τ i )| ≤ ∞ n=1 i:τi∈In |β i | max -1≤t≤1 |g (k) (t -τ i )| ≤ ∞ n=1 i:τi∈In |β i | sup -1≤t≤1,s∈In |g (k) (t -s)| = ∞ n=1 d k,n Z n ,
where Z n := τi∈In |β i |. As the sets I n are disjoint and have the same length, the random variables Z n are i.i.d. Each one has a compound Poisson distribution. As E(|β 1 | m ) < ∞, we obtain E(Z m 1 ) < ∞. We now use Jensen's inequality, based on the convergence in [START_REF] Azaïs | Level sets and extrema of random processes and fields[END_REF], to obtain the following bound:

∞ n=1 d k,n Z n m = D m k ∞ n=1 d k,n D k Z n m ≤ D m-1 k ∞ n=1 d k,n Z m n .
Now, as the random variables Z m n are i.i.d. and have finite moments,

E |X (k) | m ∞ ≤ D m-1 k ∞ n=1 d k,n E (Z m n ) = D m k E (Z m 1 ) < ∞,
it concludes that condition (H2) holds true for k and m.

Remark 4.3. Given a kernel g (k) : R → R we define by

G (k) : [0, ∞) → [0, ∞) the monotone hull of max(|g (k) (t)|, |g (k) (-t)|), t ≥ 0, as the smallest non-increasing function that dominates |g (k) (t)| and |g (k) (-t)| for t ≥ 0. More precisely G (k) (t) = sup s>t (|g (k) (s)|, |g (k) (-s)|).
Then, condition ( 6) is implied by the integrability of G (k) . Then, when the kernel g (k) decreases monotonously for large positive and large negative values, condition (6) follows automatically from the integrability of the kernel g (k) .

Boundedness of the density (H4).

In [START_REF] Biermé | Regularity and Crossings of Shot Noise Processes[END_REF] Section 3.2, it is shown that when β 1 = 1 a.s., in the two following particular situations the stationary shot noise process has a bounded density:

(a) The kernel is g(t) = e -t 1 {t≥0} , and the intensity λ > 1.

(b) The kernel satisfies g(t) = t -α for t ≥ A for some A > 0 and α > 1/2.

We present below a generalization of the results of [START_REF] Biermé | Crossings of smooth shot noise processes[END_REF] that constitutes one of the contributions of the present paper. (B2) The same as (B1) replacing g(x) by g(-x) and g + * (t) by

g - * (t) := inf 0≤s≤t |g (-s)|, (t ≥ 0),
Then X(0) has a bounded density.

Note that Proposition 4.4 implies (H4) with h = 0. The proof requires the following simple result, that has a direct proof.

X(0) = i : τi>0 β i g(-τ i ) + i : τi≤0 β i g(-τ i ) =: X + (0) + X -(0),
as X + (0) and X -(0) are independent, it is enough to see that X + (0) has a bounded density. Define by T 1 and T 2 the first two positive occurrences of the Poisson process. Condition on T 2 and apply (b) in Lemma 4.5 to obtain that β 1 g(-T 1 ) has a conditional density bounded by BE(1/|g(-T 1 )| | T 2 ). Applying now (a) in Lemma 4.5, we obtain that the sum X + (0) has a conditional density, denote it by f X + (0)|T2 (x) with the same bound. Finally, integrating

f X + (0) (x) = E(f X + (0)|T2 (x)) ≤ E(BE(1/|g(-T 1 )| | T 2 )) = BE(1/|g(-T 1 )|),
concluding the proof in this case.

Let us consider now case (B1), the proof in the case (B2) is similar. We see first that, conditional on T 2 , the random variable g(-T 1 ) has a bounded density. In fact, by the change-of-variable formula

f g(-T1)|T2 (u) = 1 |g (g -1 (u))| 1 T 2 1 {g -1 (u)≤T2} ≤ 1 T 2 g + * (T 2 )
.

By (b) in Lemma 4.5 the product β 1 g(T 1 ) has a conditional density bounded by

1 T 2 g + * (T 2 ) E(1/|β 1 |). (7)
Then, by conditional independence and (a) in Lemma 4.5, the sum X + (0) has a conditional density with the same bound. Finally, as the density of T 2 is λ 2 t 2 e -λt2 , integrating the bound (7), we obtain that

f X + (0) (u) = E(f X + (0)|T2 (u)) ≤ ∞ 0 1 t 2 g + * (t 2 ) E(1/|β 1 |)λ 2 t 2 e -λt2 dt 2 = λE(1/|β 1 |)E(1/g + * (T )).
This concludes the proof of the proposition giving the respective bounds.

Corollary 4.6. Consider a shot noise process with a C ∞ (R) kernel g such that for every k ≥ 1

g (k) (t) c k (t)e αt as t → -∞ and g (k) (t) C k (t)e -αt as t → +∞, ( 8
)
where means equivalence and c k , C k are polynomials of an arbitrary degree. Assume furthermore that λ > α and E(|β 1 |) < ∞. If in addition either (i) β 1 has bounded density or

(ii) E(1/|β 1 |) < ∞ and g, or g(-x
), has a strictly negative derivative on (0, +∞), then E(N p u ) < ∞, for any p ≥ 1.

Proof. We apply Corollary 4.1 with m = 1 and arbitrarily large k. In view of ( 5), we derive the result for arbitrary p. The kernel is differentiable for any k. The relation ( 8) ensures condition [START_REF] Azaïs | Level sets and extrema of random processes and fields[END_REF], as it gives the integrability of the derivatives of any order of the kernel. It remains to see that the density of X(0) is bounded, and this follows in case (i) from the fact that λ > α giving E(1/|g(-T 1 )|) < ∞, and the boundedness of the density follows by (A) in Proposition 4.4. In case of (ii), we apply (B) in Proposition 4.4. In this way we conclude the proof.

5 Random fields from R d and S d to R

We begin with the case when the domain is R d . Consider a real valued random field X = {X(t) : t ∈ R d } and define, for a given u ∈ R, the level set C u restricted to D a (the closed ball with radius a centered at the origin) by the formula

C u = C u (D a ) := {t ∈ D a : X(t) = u}.
Observe that under regularity conditions, the level set C u is almost surely a manifold of co-dimension one. In fact, Theorem 1 in [START_REF] Belyaev | On the number of intersections of a level by a Gaussian stochastic process[END_REF] (see [START_REF] Wschebor | Surfaces aléatoires, Mesure géométrique des ensembles de niveau[END_REF] for a proof) states that {t ∈ D a : X(t) = u, ∇X(t) is singular} is a.s. empty and, provided the a.s. absence of critical points, the Implicit Function Theorem gives a local chart a.s. The aim of the first part of this section is to generalize Theorem 2.1 into this framework.

To this end, we compute the (d -1)-dimensional Hausdorff measure of a Borel set B⊂ D a , based on Crofton's formula ( [START_REF] Morgan | Geometric measure theory[END_REF] p. 31):

H d-1 (B) = c d-1 (a) v∈S d-1 y∈v ⊥ ∩Da # {B ∩ v,y } dv ⊥ a (y) dS d-1 (v). (9)
Here dS d-1 is the uniform probability on the sphere S d-1 , dv ⊥ a is the uniform probability on v ⊥ ∩ D a , and v,y is the affine linear space {y + tv : t ∈ R}. The constant c d-1 (a) can be easily computed in the particular case of the boundary of D a , namely S d-1 a := {t ∈ R d : |t| = a}, yielding,

c d-1 (a) = 1 2 H d-1 (S d-1 a ) = π d/2 Γ(d/2) a d-1 . ( 10
)
Remark 5.1. When B ⊂ D a is a codimension one smooth submanifold H d-1 (B) coincides with the (induced) Riemannian measure.

In view of [START_REF] Berzin | Kac-Rice formulas for random fields and theirs applications in: random geometry, roots of random polynomials and some engineering problems[END_REF], to obtain the finiteness of the moments of H d-1 (C u ), the idea is to give conditions on X that ensure that its restriction to an arbitrary line in R d verifies the hypothesis of Theorem 2.1 for some values of m, h and k, as stated in the following result. (H2 ) For some m = 1, 2, . . . there exists a constant D m such that

max |v|=1 E ∂ k X ∂v k m ∞ ≤ D m ,
where ∂ k ∂v k denotes the k-th directional derivative w.r.t. the vector v. (H3 ) For some 0 ≤ h ≤ k there exists a constant C > 0 such that the joint density of

X(t), ∂X(t) ∂v , . . . , ∂ h X(t) ∂v h is bounded by C uniformly in t ∈ D a , v ∈ S d-1
, and (u 1 , . . . , u h+1 ) ∈ R h+1 .

Then, for p ≥ 1 satisfying (1), the p-th moment of the measure of the level set H d-1 (C u ) is finite and is bounded by

(c d-1 (a)) p (k -1) p + D m • E α,k,p + C|2a| (h+1)(k-h/2) • D α,k,h,p ,
where α and the coefficients E α,k,p and D α,k,h,p are as in Theorem 2.1.

Proof of Theorem 5.2. We first apply Jensen's inequality in [START_REF] Berzin | Kac-Rice formulas for random fields and theirs applications in: random geometry, roots of random polynomials and some engineering problems[END_REF]:

(H d-1 (C u )) p = c d-1 (a) v∈S d-1 y∈v ⊥ ∩Da # {C u ∩ v,y } dS d-1 (v) dv ⊥ a (y) p ≤ (c d-1 (a)) p v∈S d-1 y∈v ⊥ ∩Da (# {C u ∩ v,y }) p dS d-1 (v) dv ⊥ a (y).
Now, take expectation and apply Tonelli's Theorem,

E (H d-1 (C u )) p ≤ (c d-1 (a)) p v∈S d-1 y∈v ⊥ ∩Da E (# {C u ∩ v,y }) p dS d-1 (v) dv ⊥ a (y).
Let us apply Theorem 2.1 to the expectation inside the integral. It is clear that the bound is maximal when the interval is maximal, and it corresponds to y = 0. So,

E (H d-1 (C u )) p ≤ (c d-1 (a)) p (k -1) p + D m • E α,k,p + C|2a| (h+1)(k-h/2) • D α,k,h,p .
The expectation in the r.h.s. above is finite due to Theorem 2.1.

An easy consequence is

Corollary 5.3. Assume that the random field in Theorem 5.2 is stationary Gaussian with a spectral density. Then, the moment of order p of the Hausdorff measure of C u is finite with p = k(

Proof. The only point to check is that, if we limit the parameter set to a line, the obtained Gaussian stationary process has a density so it satisfy the conditions of Example 3.2.

Let us mention that the case p = 2 and the case of arbitrary p were considered in [START_REF] Azais | Necessary and sufficient conditions for the finiteness of the second moment of the measure of level sets[END_REF] and in [START_REF] Armentano | Conditions for the finiteness of the moments of the volume of level sets[END_REF] respectively.

We move to random fields defined on a sphere. Without loss of generality, we can assume that the sphere is

S d = {x ∈ R d+1 : |x| = 1}.
Given t ∈ S d and a tangent vector v ∈ T t S d , we denote by ∂ k X(t) ∂v k the k-th order derivative of X along the sphere.

For the definition of the integral geometric measure on homogeneous spaces, we refer [START_REF] Santaló | Integral Geometry and Geometric Probability[END_REF] or [START_REF] Howard | The kinematic formula in Riemannian homogeneous spaces[END_REF]. In this case, if M ⊂ S d is a co-dimension one regular set, the Crofton's formula reads

H d-1 (M) = β 2,d+1 E∈G 2,d+1 #(M ∩ E) dG 2,d+1 (E). (11)
Here, G 2,d+1 denotes the Grassmanian of 2-dimensional subspaces of R d+1 , and the integral is with respect to the induced Haar measure as homogeneous space of the orthogonal group of R d+1 . This is the unique probability measure that is invariant under the action of this group.

Remark 5.4. It is easy to see that the given probability measure on G 2,d+1 can be generated by the span of two independent standard Gaussian vectors on R d+1 .

Remark 5.5. Note that E ∩ S d is a great circle for every E ∈ G 2,d+1 .

The constant β 2,d+1 can be computed in the same fashion as in [START_REF] Biermé | A Fourier approach for the level crossings of shot noise processes with jumps[END_REF]. In this case, we have

β 2,d+1 = π d/2 Γ(d/2)
.

Theorem 5.6. Consider a real valued random field X = {X(t) : t ∈ S d }. Assume that:

(H1 ) X is C k (S d ) for a given k.

(H2 ) there exist m ∈ N and a constant D m such that

E max t∈S d ,v∈TtS d ,|v|=1 ∂ k X(t) ∂v k m ≤ D m . (H3 ) Let 0 ≤ h ≤ k.
Assume that there exists a constant C > 0 such that the joint density of

X(t), ∂X(t) ∂v , . . . , ∂ h X(t) ∂v h is bounded by C uniformly in t ∈ S d and v ∈ T t S d , |v| = 1, and for (u 1 , . . . , u h+1 ) ∈ R h+1 .
Then, for p = 1, 2, . . . , satisfying (1) the p-th moment of the measure of the level curve H d-1 (C u ) is finite and bounded by

β p 2,d+1 (k -1) p + D m • E α,k,p + C|2π| (h+1)(k-h/2) • D α,k,h,p ,
where α and the coefficients E α,k,p , D α,k,h,p are given in Theorem 2.1.

Proof. From Remark 5.5 and ( 11) applied to M = C u , we observe that

#(C u ∩ E) = #{t ∈ E ∩ S d : X(t) = u}
is the number of crossings of a process defined on a great circle of radius 1 in such a way that Theorem 2.1 can be applied. Now, the proof follows the same lines as proof of Theorem 5.2.

Shot noise random field

In this section, we first consider a shot noise defined in R d and then on the sphere S d . Consider a shot noise random field X : R d → R defined by ( 4), where we now consider the case d > 1. We are interested in the finiteness of high-order moments of the measure of level sets of this differentiable random field which is restricted to the closed centered ball D a . For the computation of expectations of excursion sets of shot noise random fields with realizations with bounded variation (possibly discontinuous), see [START_REF] Biermé | On the perimeter of excursion sets of shot noise random fields[END_REF].

The following result is a direct application of Theorem 5.2.

Corollary 6.1. Consider a shot noise random field (4) satisfying condition (H1 ) for some k ≥ 1, (H2 ) for some m ≥ 1 and k above, and (H3 ) for h = 0. Then,

E(H d-1 (C u (D a )) p ) < ∞ (12)
for p < m m + 1 (k -1).

In the following paragraphs, we obtain sufficient conditions to verify this corollary. The differentiability of the trajectories follows directly from the differentiability of the kernel g, see [START_REF] Biermé | On the perimeter of excursion sets of shot noise random fields[END_REF]. Consider a multiindex α = (α 1 , . . . , α d ) of non-negative integers such that |α| = i α i = k and the usual notation for the partial derivatives. Then,

∂ k X(t) ∂t α = i β i ∂ k g(t) ∂t α (t -τ i ),
is a shot noise provided that the α-th derivative of g is integrable.

Boundedness of moments (H2 ). For simplicity of exposition, we take a = 1, i.e. the unitary ball D 1 . Define the partition of R d given by A 1 = D 1 and

{A n = D rn \ D rn-1 : n = 2, 3, . . . },
where the sequence (r n ) has r 1 = 1 and is such that the volume of each set A n equals the volume of A 1 .

In this way, we can write

∂ k X ∂v k (t) = ∞ n=1 i:τi∈An β i ∂ k g ∂v k (t -τ i ).
We now present a result useful to verify the moment condition (H2 ). The proof follows the same lines as that of Proposition 4.2 in dimension one, and is omitted.

∂ k g ∂v k (t -s) .
Then, the condition (H2 ) holds true for m and k as above, i.e.

sup |v|=1 E ∂ k X ∂v k m ∞ < ∞.
Boundedness of the density (H3 ). Introduce the volume of the d-dimensional ball of radius r by

V (r) = k d r d , k d = π d/2 /Γ(1 + d/2).
1 |∇g(t)| dH d-1 (t) satisfies E(G * (|T 1 |)) < ∞.
Then X(0) has a bounded density.

The following auxiliary result is needed in the proof under (B). Its proof is elementary and thus omitted.

f |T1| (x) = dλk d e -λk d x d x d-1 , f |T2| (x) = dλ 2 k 2 d e -λk d x d x 2d-1 .
Proof of Proposition 6.3. This proof follows the same lines as those of the proof of Proposition 4.4. To prove case (A), we write

X(0) = β 1 g(-T 1 ) + ∞ i=2 β i g(-T i ).
Applying (b) in Lemma 4.5, we obtain that β 1 g(-T 1 ) has a conditional density bounded by BE(1/|g(-T 1 )| | T 2 ). Applying now (a) in Lemma 4.5, by conditional independence we obtain that the sum X(0) has a conditional density: f X(0)|T2 (x) with the same bound. Finally, integrating

f X(0) (x) = E(f X(0)|T2 (x)) ≤ E(BE(1/|g(-T 1 )| | T 2 )) = BE(1/|g(-T 1 )|),
concluding the proof in this case. Let us consider now case (B). Applying the co-area formula we see that, conditional on T 2 , the random variable g(-T 1 ) has a bounded density:

f g(-T1)|T2 (u) = t∈g -1 (u)∩D |T 2 | 1 |∇g(t)| 1 k d |T 2 | d H d-1 (dt) ≤ G * (|T 2 |) k d |T 2 | d .
By (b) in Lemma 4.5, the product β 1 g(-T 1 ) has a conditional density bounded by

G * (|T 2 |) k d |T 2 | d E(1/|β 1 |). (13)
Then, by conditional independence and (a) in Lemma 4.5, the sum X(0) has a conditional density with the same bound [START_REF] Biermé | On the perimeter of excursion sets of shot noise random fields[END_REF]. Integrating the bound [START_REF] Biermé | On the perimeter of excursion sets of shot noise random fields[END_REF] with the density of |T 2 | in Lemma 6.4, we obtain that

f X(0) (x) = E(f X(0)|T2 (x)) ≤ E(1/|β 1 |) k d ∞ 0 G * (r) r d e -λk d r d λ 2 dk 2 d r 2d-1 dr = dk d λ 2 E(1/|β 1 |) ∞ 0 G * (r)e -λk d r d r d-1 dr = λE(1/|β 1 |)E(G * (|T 1 |)),
which is finite because of our hypotheses, concluding the proof. Let us check the finiteness of the moments [START_REF] Biermé | Regularity and Crossings of Shot Noise Processes[END_REF]. It is straightforward to verify (H1 ) and (H2 ). To see (H3 ) assume first that β 1 has a bounded density. Then, as g is radial, we have

E 1 g(-T 1 ) = E 1 g(|T 1 |) = ∞ 0 1 g(r) f |T1| (r)dr < ∞, (14)
due to the form of the kernel, Lemma 6.4, and condition (i). Assume now that E(1/|β 1 |) < ∞. The sup in the function G * (r) in (B) in Proposition 6.3 is attained when u = e -r 2q , giving G * (r) = e r 2q r d-2q . A computation similar to [START_REF] Borodin | Stochastic processes[END_REF] gives EG * (|T 1 |) < ∞. In conclusion, the hypotheses of Corollary 6.1 are valid with arbitrary k, obtaining that the shot noise random field (4) verifies [START_REF] Biermé | Regularity and Crossings of Shot Noise Processes[END_REF] with arbitrary p. Now, we move to the shot noise random field defined on the sphere. Let P be the standard Poisson field on the unit sphere S d of R d+1 . We realize P as a sequence T 1 , T 2 , . . . , T N of uniformly distributed random points on S d where N is a Poisson random variable, all variables being independent.

Let g : [0, π) → R be a C ∞ function. The shot noise process is defined by

X(t) = N i=1 β i g(dist 2 (t, T i )), t ∈ S d , ( 15
)
where dist is the geodesic distance, and the β i 's are i.i.d. with distribution F . We assume the following conditions:

(H4) β 1 g(dist 2 (t, T )) admits a bounded density, where T has a uniform distribution on the sphere.

(H5) The distribution F has moments of any order.

Note that (H4) is rather weak, it is met, for example, if β 1 has a bounded density and g is strictly positive on [0, π] (apply Lemma 4.5). We have the following result.

Proposition 6.6. Consider a shot noise random field defined by [START_REF] Borokov | On Rice formula for stationary multivariate piecewise smooth process[END_REF] and assume (H4)-(H5). Then, for every level u = 0, for every compact set W and for every integer p

E H p d-1 (C u ∩ W ) < ∞.
The proof follows from Theorem 5.6.

Differentiability (H1 ) and Boundedness of moments (H2 ). We now check (H1 ) and (H2 ) for arbitrary k and m.

Let v be a norm 1 vector orthogonal to t ∈ S d . Since the number of realizations of P is almost surely finite, for any k, the derivative of X along the sphere at t ∈ S d in the direction of v is given by

∂ k X(t) ∂v k = N i=1 β i d k dz k g(dist 2 (γ t,v (z), T i )) z=0
, where γ t,v (z) is a C ∞ arc-length parametrization of a great circle through t in the direction of v such that γ t,v (0) = t. Then, X is C k for all k, and by compactness of the sphere S d we obtain

∂ k X(t) ∂v k ≤ (Const) N i=1 β i .
The condition (H5) implies that the compound Poisson distribution of N i=1 β i admits moments of every order giving the desired result.

Boundedness of the density (H3 ) with h = 0. For simplicity, we consider the hypothesis (H3 ) with h = 0 and study the marginal density of X(t). Lemma 6.7. For every t ∈ S d the distribution of X(t) is the sum of one atom at zero and a defective probability with bounded density.

Proof. We consider the distribution of X(t) conditional to P(S d ) = k in the case k > 0. Under that condition it is well known that the T i 's, i = 1 . . . , k, are i.i.d. with uniform distribution on S d , so we write X(t) = β 1 g(dist 2 (t, T 1 )) + k i=2 β i g(dist 2 (t, T i )).

The terms in this sum are independent and, because of (H4), the first term has a bounded density. By convolution it is the same for the conditional distribution of X(t) which admits a density bounded by the same constant. Since this bounds does not depend on k, it is also a bound for the density of X(t) conditional to P(S d ) > 0. Obviously, when P(S d ) = 0, X(t) = 0 which gives the atom at zero.

Non Gaussian Kac-Rice Formula

Gaussian KRFs are valid under weak and simple conditions and a comprehensive reference is the book [START_REF] Azaïs | Level sets and extrema of random processes and fields[END_REF] that treats all the relevant dimensions, i.e. random fields X from R D to R d with d ≤ D. These formulas give the expectation or the higher moments of the H D-d Hausdorff measure of C u (H) = {t ∈ H : X(t) = u}, that is the level set restricted to a compact set H ⊂ R D . Though the proofs use basically the change-ofvariable formula (or its generalization: the co-area formula) and have nothing to do with Gaussianity, its generalization to non-Gaussian cases encounters difficulties in defining properly the quantities involved in the formulas.

For instance, in the simplest case, the KRF for the expectation when D = d = 1 formally reads for a compact interval I:

E(N u (I)) = I E |X (t)| X(t) = u p X(t) (u)dt. ( 16
)
In the non-Gaussian case, the conditional expectation is defined only for almost every level u. As a consequence, the punctual values of the r.h.s. of ( 16) are not defined unless some kind of continuity is established. This is why the non-Gaussian KRF requires complicated conditions. See [START_REF] Marcus | Level crossings of a stochastic process with absolutely continuous sample paths[END_REF] and [START_REF] Azaïs | Level sets and extrema of random processes and fields[END_REF] for the case D = d = 1; for the case D = d > 1 the only reference is [START_REF] Adler | Random fields and geometry[END_REF]. To our knowledge, in the non-Gaussian case, with the exception of the complicated treatment in [START_REF] Wschebor | Surfaces aléatoires, Mesure géométrique des ensembles de niveau[END_REF], there exists no proof of the KRF for the case D > d.

Often, the process or the random field has C k paths with k "large" and the conditions can be drastically simplified. This is the object of this section. Note that in its full generality, the statement of the KRF cannot be stated in the classical form.

Our first main result is the following: Theorem 7.1. Let us consider a real-valued process X = {X(t), t ∈ I}, where I is a bounded interval of R. Assume that there exist u ∈ R and > 0 such that:

Proof. By (b) there exists a finite K such that

E(N 2 v ) ≤ K,
for all |v -u| < . Our conditions imply that, with probability 1, the process X(t) cannot take the value u at the two extremities of I. In addition by the Bulinskaya Lemma (Prop. 1.20 in [START_REF] Azaïs | Level sets and extrema of random processes and fields[END_REF]) there are, with probability 1, no extremes at the level u. Thus, the Kac Lemma (Lemma 3.1 in [START_REF] Azaïs | Level sets and extrema of random processes and fields[END_REF]) yields By the area formula (Prop 6.1 in [START_REF] Azaïs | Level sets and extrema of random processes and fields[END_REF])

N u = lim
N δ u = 1 2δ u+δ u-δ N v dv.
This, associated to the Jensen inequality, yields

E (N δ u ) 2 ≤ K.
This implies in turn that the family N δ u is uniformly integrable. As a consequence, N δ u converges also in L 1 , yielding E(N u ) = lim δ→0 E(N δ u ).

Making explicit the r.h.s. above we get directly [START_REF] Geman | On the Variance of the Number of Zeros of a Stationary Gaussian Process Ann[END_REF]. Under the integrability of X (t), the conditional expectation E |X (t)| X(t) is well defined giving [START_REF] Howard | The kinematic formula in Riemannian homogeneous spaces[END_REF].

Lemma 2 . 5 .

 25 Consider a function f : I → R of class C k (I) for an interval I and k ≥ 1. If f -u has k roots in I and 0 ≤h ≤ k, we have

  and the infinity norm is taken on the interval [a, T ]. Hence, it suffices to bound the moments of |X| ∞ . Now, the Burkholder-Davis-Gundy inequality [27, Th.48] gives

Proposition 4 . 2 .

 42 Consider the shot noise in[START_REF] Armentano | Conditions for the finiteness of the moments of the volume of level sets[END_REF] with impulse such that E(|β 1 | m ) < ∞ for given m ≥ 1. Define d k,n = sup t∈I1,s∈In |g (k) (t -s)|,for k ≥ 1, and assume that

Proposition 4 . 4 .

 44 Consider a shot noise process (4) with a differentiable kernel. Denote by T an exponential random variable with parameter λ. Assume that either (A) The impulse β 1 has a density bounded by B and either E(1/|g(-T )|) < ∞ or E(1/|g(T )|) < ∞. (B1) Assume that E(1/|β 1 |) < ∞. Assume further that g(x) has a strictly negative derivative in (0, +∞), and the function g + * (t) := inf 0≤s≤t |g (s)|, (t ≥ 0), satisfies E(1/g + * (T )) < ∞.

Lemma 4 . 5 .

 45 Consider two independent random variables X and Y , where X has a density bounded by B. (a) Then, the sum X + Y has a density bounded by B. (b) If E(1/|Y |) < ∞, the product XY has a density bounded by BE(1/|Y |). Proof of Proposition 4.4. Consider (A), assuming that E(1/|g(-T )|) < ∞ (the other case is analogous). Based on Lemma 4.5 and the decomposition

Theorem 5 . 2 .

 52 Consider a real valued random field X = {X(t) : t ∈ D a }. Assume that X satisfies (H1 ) The sample paths of X are C k (D a ), k ≥ 2.

Proposition 6 . 2 .

 62 Consider a shot noise with impulse s.t. E(|β 1 | m ) < ∞ for given m, and kernel s.t. D k = ∞ n=1 d k,n < ∞, for given k, with d k,n = sup |v|=1 sup t∈A1,s∈An

Proposition 6 . 3 .

 63 Consider a shot noise random field (4) with a differentiable kernel. Denote by T 1 the occurrence of the Poisson field closest to the origin with the Euclidean distance (a.s. defined). Assume that either (A) The impulse β 1 has a density bounded by B and E(1/|g(-T 1 )|) < ∞. (B) E(1/|β 1 |) < ∞ and the function G * (r) := sup u∈R t∈g -1 (u)∩Br

Lemma 6 . 4 .

 64 Denote by (T n ) the occurrence of the Poisson field ordered by the Euclidean distance to the origin. Then, the random variables |T 1 | and |T 2 | have the following densities:

Example 6 . 5 .

 65 Consider a shot noise with a radial kernel of the form g(t) = e -|t| 2q (q = 1, 2, . . . ), and impulse s.t. E(|β 1 |) < ∞. Assume further that (i) 2q < d, or 2q = d and λk d > 1, (ii) β 1 has a bounded density or E(1/|β 1 |) < ∞.

( a ) 17 )

 a17 The sample paths of X are C 1 (I).(b) sup |v-u|< E(N v (I) 2 ) < ∞.(c) The density of X(t) at point v is uniformly bounded for t ∈ I and for |v -u| < .Then,E(N u (I)) = lim δ→0 1 2δ I E |X (t)|1 |X(t)-u|≤δ dt. (Suppose in addition that X (t) has a finite expectation for every t ∈ I. Define for |v -u| < R(v) :=I E |X (t)| X(t) = v p X(t) (v)dt.Then, (17) can be rewritten asE(N u (I)) = lim

  (t)|1 {|X(t)-u|≤δ} dt.

  see that conditional to X(t), the random variable Y (t) has a Gaussian distribution with variance 4λ 2 Y (t). Its conditional density p Y |X=x is bounded by

(Const) Y (t) -1/2 , where (Const) denotes a meaningless constant whose value may change from line to line. Let now B 1 and B 2 be two Borel sets of R with respective measures |B 1 | and |B 2 |. Writing Y and Y for Y (t) and Y (t)

Remark 7.2. Analogous to the situation in Theorem 7.1, the finiteness of the second moment of the level set can be obtained using Theorem 5.2.

Example 7.3 (The case D = d = 1). Suppose that X(t) satisfies the conditions of Theorem 2.1 with and k, m, h such that

In most of the cases, it is very easy to see that the process also satisfies all the conditions of Theorem 7.1, although this fact is not an exact consequence. This is the case for all the examples considered in Section 3. Of course, for Gaussian processes and χ 2 processes, the validity of Rice formula has been known for a long time.


Chapter 11

Juggling between Social Obligations and Personal Benefit in Western Côte d'Ivoire

How Do Ex-combatants Spend Their Cash Allowance?

Magali Chelpi-den Hamer

The use of cash transfers in aid programming has received growing support in recent years despite the fear of misuse and loss of control. There is one sector in particular in which its use has rarely been questioned: in processes dealing with the disarmament, demobilization, and reintegration of ex-combatants (commonly known as DDR). The use of cash has become increasingly widespread in this field, yet few grounded studies systematically document its diversity of impacts. Cash transfers are sometimes used in the disarmament stage to "buy back" weapons, in the hope of reducing the number of guns and ammunition in circulation, and they are also sometimes used in the reinsertion/reintegration stage, with the aim to facilitate excombatants' return to civilian life. What lies behind these practices is the assumption that giving money to ex-combatants can have a positive and direct effect on their lives and will prevent them from taking up weapons again. Based on a careful examination of over one hundred individual testimonies collected in western Côte d 'Ivoire between December 2006 and July 2007, this chapter examines how the financial incentives distributed in the summer of 2006 to progovernmental demobilized militias were spent, providing an empirical base on the subject.

Cash Transfers in the Ivorian DDR: Process and Specificities in a Nutshell

The Ivorian DDR process planned to use cash transfers during the reinsertion stage with the intended aim of facilitating ex-combatants' transitioning from soldiering to civilian life. The process was designed in line with what was agreed upon in Linas-Marcoussis in 2003, which was several times reiterated in follow-up agreements. At the time of my fieldwork, the National DDR Commission (CNDDR) was the governmental institution mandated to implement the Ivorian DDR program with the help of the United Nations peacekeeping mission. [START_REF]For detailed information on the process[END_REF] Official parties were included in the process (which consisted of the Ivorian army and recognized rebel groups), as well as paramilitary and militia groups, which had been active on both sides. Inclusion in the DDR process was based on lists of recruits provided by the respective chiefs of staff, checked by the CNDDR. In line with the national plan, demobilized combatants were entitled to a financial package of 499,500 CFA francs (equivalent to €760 / $940 U.S.), locally designated as filet de sécurité (this term literally means "net of safety" and is the term locally used in official wording). The financial package was scheduled to be disbursed in three installments: 25 percent paid upon demobilization, 25 percent paid forty-five days later, and 50 percent paid ninety days after demobilization. It was supposed to cover for ex-combatants' basic expenses for the first six months following their demobilization, helping them face the immediate challenges encountered after their return into their community of origin.

In the summer of 2006 (July-August), after a series of political stalls and false starts, progovernmental militia members started an official demobilization process that included cash transfers. [START_REF]At the time of doing fieldwork[END_REF] The cash was handed over in three installments, as initially planned (only the third installment was delayed), in Duékoué, western Côte d'Ivoire, under the supervision of the CNDDR and the UN peacekeeping mission. A peculiarity of the Ivorian situation was that the majority of pro-governmental militias undergoing the process were local recruits and had already spontaneously demobilized in 2004-2005 and resumed personal activity. The bulk of the weapons had in fact been collected by their leaders during the course of 2005, when the military situation had stabilized, for storage in local arsenals, ready for a potential redistribution should the situation deteriorate or for a hand-over in case of sustainable peace. Right before the start of the militias' demobilization in the summer of 2006, militia leaders had surrendered batches of weapons to peacekeeping forces as a sign of goodwill. Most of the weapons collected during the summer demobilization operation, however, were defective, and the ratio of combatant per surrendered weapon/ammo was also very low. This situation led the DDR division of the United Nations Operation in Côte d'Ivoire to suspend the demobilization operation in the beginning of August. Nevertheless, 981 militia members had already been registered and had received financial compensation. Out of those, five hundred could enter a follow-up reinsertion program, run by the German organization GTZ-IS, [START_REF][END_REF] to benefit from individual advice and socioeconomic support. This chapter specifically analyzes the use of cash of these low-ranked demobilized militia members one year after receiving their financial compensation. One hundred individuals were interviewed using two GTZ-IS reinsertion centers as premises for conducting the meetings.

Respondents were not observed in situ, when they received their cash transfers, or when they spent them; instead, they were asked to reconstruct their perceptions of past events by taking their life story as a point of departure.

How Were the Cash Allowances Spent?

Among the interviewed recruits, genuine belief that they had earned the right to compensation was a recurring litany throughout the discussions. "We lost five years of our time." "We've been delayed." There was something intrinsic to these statements, namely, the expression of a certain conception of fairness and the implicit claim that financial compensation was due, regardless of the outcome of the conflict. The majority of militia members emphasized the fact that, despite having been called to fight for their country, they had gained little from it, and they had mainly lost their time in the movement instead of being able to work elsewhere. They were therefore entitled to a proper reward for the services rendered. Some threatened to return to violence if not satisfied, but such claims were mostly rhetorical since most of them had already put down their guns and were no longer armed.

While most low-ranked recruits saw financial compensation as a back-payment for their services, military leaders, and especially the high-ranking ones, had a different perspective: they mainly used it as a carrot-an incentive-to retain their troops, especially after it was institutionalized in the national DDR plan. Indeed, the prospect of getting no financial reward at all would probably have led many recruits to abandon their ranks after a certain time, especially among the western militias, who were in this respect much freer than their rebel counterparts.

But the prospect of being on the official demobilization list, hence the prospect of potentially being eligible to receive a cash allowance, had the reverse effect of maintaining recruits' loyalty.

As already noted above, the financial incentive distributed in the summer of 2006 to the 981 militia members was dispatched in three installments: the first installment was supplied in July 2006, right after their official demobilization (125,000 CFA francs); 4 the second installment (same amount) was dispatched mid-September; and the third installment (249,500 CFA francs) 5 was distributed at the end of November.

The three interview fragments presented below show different examples of use of the financial incentives received:

The first installment, I shared with my family and my in-laws. I kept a little, but not much. Most of the money was used to pay miscellaneous expenses. The second installment, I gave 50,000 to my wife for her small business. I also set up a cabine [local phone booth] for my little brother but he screwed everything up. I used the last installment to invest in my own business, a maquis [local eating place] and to improve my home.

I gave the money to my family, and I also paid for my uncle's medical expenses.

I also gave petty cash to friends I had in the militia group, who were not yet demobilized. I also bought a plot of land in Guiglo, near the timber industry for 50,000 CFA francs. I am currently building my own house. I purchased wood from the timber company, to make charcoal, and I opened a bank account at the local post. I improved my home. I set up a water tap for 58,000 CFA francs and I purchased an electricity counter at 72,000 CFA francs. I helped my brother start his own hevea field and I gave him 125,000 CFA francs. I also gave petty cash to my family.

As seen in the testimonies, respondents used their safety net very differently, constantly juggling between social obligations and personal benefit. A rough categorization of expenses is presented below. The findings are grouped into five categories of expenses: (1) the reimbursement of creditors, (2) response to familial demands, (3) investments, (4) dealing with social events (such as medical expenses, funerals, home improvements), and (5) reward to the military leader.

Reimbursing Creditors

Reimbursing creditors was not an uncommon answer, and many respondents said that upon receipt of the cash, one of the first things they did was to pay the debts they-or their familyhad accumulated during the war period. One respondent had accumulated as much as 325,000 CFA francs in debt since the beginning of the war. He had not paid his house rent for several months (which had put him 25,000 CFA francs in debt every month), not to mention the unpaid utilities bills (electricity/water) that had accumulated over the years. For him, it was important to pay his creditors first since he feared they would have gone to the police to file a formal complaint when they found out that he had received a cash allowance (the official demobilization process had been widely advertised in the local medias so it was difficult to hide having received something). Another respondent explained that even if he had slept most of the war in military camps, he had kept renting a home in town for his wife. Several people mentioned having to buy food on credit during the war because even if they could usually eat in the military camps, their dependents were usually not entitled to free food, so they had to contract loans to have them eat.

One respondent contracted a loan to bury his mother while he was in the armed group. His dead parent had remained in the mortuary for a month (which is billed 5,000 CFA francs a day), the coffin cost 50,000 CFA francs, and he had to borrow to pay for the stay of his relatives who had come to attend the funeral. Most of his cash allowance was therefore used to clear this debt.

Responding to Familial Demands

The relationships respondents had with their family after receiving their cash entitlement were quite complex, and the way they managed these ranged from strategies of avoidance to strategies of resignation. How to best manage the burden eventually appeared to be the most standard concern. Several respondents reported difficulties in avoiding relatives' requests. As this interview fragment tells, "People know when you get your cash. The following morning, you see all your relatives in a row in front of your door and you have to give them something. You give 5,000 to your aunt, 5,000 to your cousin, 5,000 to your other cousin, etc." One respondent succeeded in bypassing such an issue by putting all his cash in the local bank and by returning empty-handed to his village. Even when cornered, he was essentially incapable of giving cash to anyone.

A few respondents mentioned having loaned money to a sibling (up to 100,000 CFA francs). While most loans were not yet repaid at the time of the interviews, some had already been bitterly abused. One respondent mentioned he had given his complete first installment to his family, "to be left in peace." He used to be employed as a katakata driver before the war (a katakata is a sort of bush tractor that transports all kinds of merchandise), and he wanted to save the second and third installment of his cash allowance to buy a katakata for himself. Given the second-hand cost of such a vehicle (1.5 million CFA francs), one of his brothers convinced him to entrust him with his money. The plan was to travel to Abidjan to ask their eldest brother to participate in the purchase by paying the remaining million. What eventually happened is that the entrusted brother eventually usurped all the money and claimed to the former militiaman that he had been robbed on the way. But regardless of these cases of peer abuse, a recurring argument in favor of family support is gratefulness. Close to the front line, villages often played an important role in supplying food to combatants, and complex mechanisms of money collection occurred between armed groups, villagers, and the educated elite in Abidjan who were native to these war-affected areas (the cadres villageois). One interview fragment summarized it very well: "You have to be grateful and reward those who fed you during the war." Several respondents therefore felt obliged to reward their benefactors, and these included close relatives who participated in the war effort and who supported them during difficult times, including the times spent in the armed group. Rewards to the spouse and to direct parents (father/mother) were rather standard patterns, and several stories were heard of respondents setting up a farm for their father, or helping their wife to develop her own business.

Investing

Several respondents mentioned that it was not until the last installment that they could do something productive with their money. Western Côte d'Ivoire is particularly conducive to agriculture; some also invested in wood and bought loads from the nearby timber industry to turn into charcoal. Others entered the growing sector of hevea cultivation or expanded the plantation they already had (this sector was especially in vogue in 2006-2007, during the period of fieldwork). Respondents who were already engaged in cultivating hevea before the war bought new plants, and those who were new to the activity purchased land, cleared a field, and/or joined the myriad of private and local humanitarian projects that were then offering incentives to cultivate hevea in the Moyen-Cavally region.

One respondent used his last installment to purchase one hectare of hevea for 180,000 CFA francs. 6 He had used the previous two transfers to give petty cash to his relatives (wife, brothers, and sister), to pay for his son's school boarding expenses (35,000 CFA francs), to buy food (three bags of rice at 36,000 CFA francs), and to purchase agricultural equipment (a pulverizer at 48,500 CFA francs). Another respondent used his third transfer to enter a local project, the Nouvelles Plantations Hévéïcoles de l'ouest Montagneux de la Côte d'Ivoire, to benefit from free seedlings. When we met, he had just started contracting workers to clear two hectares of forest to start a hevea plantation.

But respondents did not solely invest in agricultural activities. One used his safety net to purchase a sewing machine, a Singer-à-tête-noire (75,000 CFA francs). He was planning to return to tailoring, an activity he was doing before the war, and to run his own workshop.

Another had to wait for the third installment to restock his shop with 200,000 CFA francs worth of new merchandise. Another invested in a chicken farm. He bought poultry, food, and vaccines, and resumed his prewar activity. Another used most of his safety net to open a local restaurant.

He spent 300,000 CFA francs to purchase a freezer, several chairs, a few tables, the first stock of drinks, and to cover the various costs related to the installation (restoration, rent, and a security deposit). Other types of investments included partial payment to an auto school (in the prospect of getting a driving license to be able to work as a taxi/truck/minivan driver), paying city fees for setting up a market stall or a small business, paying intermediaries to find a job, paying registration fees for being authorized to take national civil service exams, and paying bribes and fees to be allowed to take the ninth-and twelfth-grade exams in the hope that this would open more doors (having these diplomas allows the person to sit civil service entrance exams).

Social Uses

Improving one's home, buying a plot of land (when not for agricultural purposes), building a house, purchasing basic furniture, buying clothes, marrying, or rewarding old men who provided mystic protection during the war (gris-gris), all these different uses were cited as common expenses. Being further recognized, within the household and/or the community of belonging, as capable of doing so was also sending important social messages. Notably, a certain number of respondents took advantage of their financial safety net to leave the family home and to settle independently: "Before the war, I used to sleep at my parents. With the safety net, I detached myself. I built a 2-chambres-salon." [START_REF]2-chambres-salon is the local term to designate a small house with two bedrooms and a living room[END_REF] One respondent used his second installment to have his identity papers drawn up. In a country where the lack of documentation often impedes free circulation, this is worth noting. Many respondents also mentioned having spent substantial sums on medical expenses right after receipt of their cash entitlement, either for themselves or for close relatives. Usually, such a use was a major expense, and there was rarely anything left. It is difficult to label such types of expenses unproductive even if, indeed, the money was eventually wasted on social matters. When a serious disease hit them (or their family), respondents preferred to refer to the period as "bad timing."

Perhaps the best statement to illustrate how most respondents felt upon receipt of their cash allowance is the following: "When you are not paid for years, you live one day after the other. This cash we got, we could do nothing good with it." [START_REF]Quand vous êtes resté quelque part sans salaire pendant des années, tu vis au jour le jour. Cet argent, on ne pouvait rien faire avec[END_REF] That partly explains why a lot of respondents spent their money quickly, with relatively insignificant personal benefit. Many respondents were also not happy with the fact that the safety net had come in three installments.

They would have preferred to receive everything in one go, as smaller cash amounts were easily wasted in their views, in particular by responding to familial demands and by dealing with social events (such as medical expenses, funerals, and home improvements). 9 Payment by installment nevertheless continues to remain the preferred international approach for fear of misuse and mismanagement, while, paradoxically, western Côte d'Ivoire is known for seasonal farming activities, with local people relatively used to circulation of large amounts of cash in one go.

Allocating Money to War Chiefs: Racket or Reward?

Allocating part of the financial safety net to war chiefs-the "chiefs' share," as Olivier de Sardan and Hamani label it, in their chapter on Niger (chapter 12 in this volume)-was also a common feature in western Côte d'Ivoire, which raises the question of the extent to which it should be interpreted as a racket or a reward. If we look at interview fragments, there is plenty of anecdotal evidence suggesting that extortion was real. One account is particularly enlightening: "It was serious in Duékoué. [START_REF]protected compound, but as soon as people got out, they were prone to abuse[END_REF] You were obliged to give. If you did not give, you could not get out. The first time, my leader took 30,000 out of my installment as recognition payment. The next two times, he did not get anything. As I was walking out of the compound with the cash in my pockets, he asked me, but I lied to him. I told him that I was still expecting my money and that I just wanted to get out for a drink." Retaliation rarely followed such avoidance strategies, and once the immediate threat had passed, there was usually no follow-up action on the part of war chiefs. Other respondents were less resourceful and could not avoid being shaken down after having received each installment. [START_REF]One respondent could not avoid giving half of his safety net[END_REF] If some militia leaders seemed more prone to extorting from recruits, some were also more magnanimous than others, and who really benefited from this money eventually remained vague in many situations. The distinction between war leaders and war leaders' envoys was often blurred on the ground and in the respondents' discourse, and it was not easy to distinguish between organized racketeering orchestrated by militia leaders and simple robbery by higherranked recruits who were taking advantage of their position to abuse their peers. Most accounts showed, however, that rather than giving cash directly to their chiefs, recruits were more likely to pay an intermediary, especially if the person was known to have a close relationship with their leader (some even signed receipts!). In addition, there was a certain group dynamic, and after the first demobilized recruits had given money to their chiefs, the others were likely to do the same.

Rewarding war chiefs thus resembled a racket in some cases, but that was not the only pattern, and the informal back payment to warlords varied considerably from one individual to another. Several respondents pointed out that they were not forced to give money. There were even accounts of militia leaders gathering recruits before the first installment was made and telling them that they were not obliged to make any financial contributions. 12 Some recruits beside did not perceive it as coercion. One respondent reported that he had rewarded his chief by paying for several rounds of drinks, and another militia member mentioned that although his chief had mentioned that he did not want anything in return, he was happy to give him 20,000 out of his last installment. Allocating money to war chiefs therefore seemed strongly connected to social obligation. One has to bear in mind that the majority of recruits were grateful to have been put on the official demobilization list by their militia leaders, which turned out to be the key to being eligible for financial compensation.

A last feature to take into account is also to whom recruits wanted to give. Militias also have their hierarchy, and while some felt more inclined to give to their general leader, others preferred rewarding a war chief who was closer to them in the battlefield and to whom they related more. A few respondents who fought in advanced positions reported being keen on rewarding the chiefs of their section. In their view, he had succeeded in the most important thing: he had preserved their lives during combat. One female recruit liberally praised her local commander: "I really say thank you to Colonel T. because he took good care of us. We did not lose anyone in our group. He watched our back. When he knew the day was not good, he would tell us not to move. That's the way we worked." The social value of reward associated with mystic beliefs must therefore not be downplayed when examining the relationship between war chiefs and recruits.

Has the Spending of Cash Allowances Facilitated Social Acceptance?

Given the well-known controversies associated with using cash transfers in the reinsertion stage of ex-combatants, it is worth questioning the extent to which the financial safety net has been helpful in securing acceptance by the community with whom ex-combatants associated, and whether it played a role in facilitating their transition to civilian life. Examining these questions is particularly relevant since in the western Ivorian context, most low-ranked militias had returned home several months/years before being officially demobilized; hence they eventually faced immediate reinsertion issues upon return without any financial safety net then to help them cope. Return to their home mostly occurred in 2004-2005, after the end of open conflict (2002-2003) and after the end of the encampment period in the military settings of Blolequin, Toulepleu, Zéo, and Zagné (the location where they were based depended on the armed group in which they were integrated). As noted above, the first time any financial compensation was disbursed was in the summer of 2006. Accounts widely varied per respondent. The ones who chose to remain in Guiglo, in the militia leaders' compounds or nearby, were clearly waiting for the official disarmament to start and for the supply of the several-times-announced financial benefits. Some reported having experienced no problems at all upon return and being warmly welcomed by their families: "My folks were told I was dead. They were quite happy to see me back." Others had never severed contacts with their parents during their time in the armed group, so their return was expected. Many local recruits, in fact, appointed near their place of residence, did not need to be resocialized after their military experience since their actual bonds with their prewar network had never been cut.

But how smooth was their return to civilian life? Accounts were rather mixed on this question, and there was no single pattern. While reinsertion problems directly experienced after having demobilized oneself were not often spontaneously mentioned (and unfortunately not systematically probed during interviews), several things came up during the discussions. Shortly after their return, a certain number of militia recruits were feared by the people with whom they usually associated, especially those known to have fought with Liberian mercenaries. The account of this female recruit is particularly enlightening:

In the beginning, I was scaring everyone. Even my cousins were afraid. Even my mother. When I was angry, I often noticed that people acted different. They were all scared of me. But I said no. What I did [during the war], it is past. But even my friends were scared, and they were saying, "She went to war, she's going to kill you at night." It did not feel good. So I moved on elsewhere for some time. I spent two months in Abidjan. When I came back to Guiglo, I started to sell alloco [fried banana] in front of the Becanti. People were coming to see me out of curiosity, also soldiers. Some were surprised, "Eh, you are here now!" Some did not even want to eat alloco, they just came to see me. It took some time, but people eventually saw me differently. I had not changed, but their perception did.

They saw that I behaved well, and that I did not look for arguments with anybody. Now everything is okay.

Like her, several respondents mentioned having felt the need to work on their image shortly after returning to their prewar lives and having to emphasize their nonviolent attitude. If some respondents could resume their prewar activities relatively quickly, others reported encountering difficulties. Again, there was a multiplicity of patterns, which mostly depended on respondents' individual attitudes, characteristics, and social networks. One respondent could not go back to his previous work because he had a bad reputation. He shared his point of view: "Many things happened during the war. I did not know the face of all rebels, so when the chief suspected someone and said, 'This person is a rebel, he has to be killed,' we obeyed. But this man might have family in Guiglo; and now if I approach someone for work, these people can tell him that I'm not a good person and that it is better to avoid me." In contrast, well-known fighters had no problems resuming their prewar jobs. One respondent who had fought in the front line with the Liberians resumed upon return his work as a building painter, and even if he mentioned getting fewer contracts than before, the main cause was the general decrease in the local economic activity due to the displacement of most of the local middle class, not the fear of dealing with him.

Has financial compensation facilitated social acceptance then? For most demobilized militia recruits, the financial safety net was distributed years after their return to their community. This raises doubts about the extent to which cash allowances facilitated social acceptance. In addition, not every combatant benefited from them. As already mentioned, the demobilization program of the summer of 2006 was only partial and only targeted 981 lowranked militia recruits. Not everyone appeared on the list for inclusion, and in a single villageeven in a single family-some received the cash allowance and others did not, even if they also participated in the war effort. Selection appeared quite arbitrary in some cases, and perhaps the most interesting feature is that those who did not receive financial compensation and who were locally known to have participated in the war effort were mocked by their peers. A posteriori, going to war and getting nothing out of it was perceived locally as ridiculous and as a waste of time. There was, however, a certain tolerance with respect to this unfairness, and although the few militia recruits I interviewed who did not receive compensation indeed expressed their frustration, they seemed to accept their situation.

Conclusion

Respondents used their safety net very differently, constantly juggling between social obligations and personal benefit. Out of the five broad categories of expenses identified-( 1) reimbursing creditors, ( 2) responding to familial demands, (3) investing in one's own business, (4) dealing with social events (such as medical expenses, funerals, home improvements), and (5) allocating money to war chiefs-three would not have existed if no cash allowance had been given to excombatants. Creditors would not have rushed to ask for their full payback and would have continued to display the same attitude as toward their other debtors; relatives would not have lined up in front of respondents' doors; and war chiefs and undemobilized friends would not have had a share.

For the two other categories (investing and dealing with social events), the financial compensation was useful but rather limited in time and scope. When discussing investments, a recurring point that came up from the interviews was that the financial safety net should have been given in one go instead of in three installments. Relevant investments usually involve substantial sums of money, and small amounts are easily wasted on day-to-day expenses. With respect to social events, one important contribution that this extra cash has made was in allowing young men to emancipate, by being able to leave the family home and/or by becoming a shortterm provider for their close family.

Given the fact that most local recruits who benefited from DDR-related cash transfers had in fact spontaneously demobilized years before the start of the official process, the financial safety net received had a limited impact on their immediate post-return phase, which is probably when social acceptance was the most challenging. If cash transfers have the potential to curb excombatants' dependencies on their direct support structures when they return to their homes, it is likely in this case that ex-combatants' spouses and families were socially and economically affected by their return, more or less adversely depending on individual situations. This case shows little evidence that individuals used the money unwisely on the so-called "temptation goods," such as drinking alcohol, buying drugs, or gambling. After receiving their safety net, ex-recruits faced a certain number of socioeconomic demands (from creditors, family, and acquaintances made during the war) to which they had to respond, and the range of responses greatly varied from one individual to another. In some instances, the extra cash has helped them face a certain number of social events (such as medical expenses, school fees, or costs associated with marriage, a funeral, or a newborn child), and sometimes it has enabled them to partially invest in a small business. The use of cash transfers in the reinsertion phase has indeed meant a breath a fresh air for ex-combatants, but should not be overrated, as their room to maneuver was eventually limited by the way they individually balanced social obligations with personal benefit. Demobilized militia members did not seem particularly privileged in comparison with the communities they related to, and many ended up probably more exposed to abuse by their direct entourage and by targeted robberies. The government switched focus afterward, away from the militia issue, which obviously had not been solved by the press release, toward the demilitarization of the official parties, which consisted of reducing the size of the Ivorian army while finding ways to incorporate part of the rebels in the new force. In 2008-2009, some disarmament and reintegration initiatives occurred, but they were very localized and small-scale, and in 2010, the DDR process was put to a halt again in view of the resumption of tensions around the presidential elections. After the electoral crisis, demilitarization came back high on the agenda. The modalities of the Ivorian DDR process still largely drew on the procedures developed in the 2000s, but a new institution replaced the CCI and the PNRRC, the ADDR (Autorité pour le désarmement, la démobilisation et la reintegration). If the name change was mostly symbolic, two main adjustments were made.

Postscript

First, a noteworthy increase in terms of financial compensation: while the previous safety net had been set at 499,500 francs CFA, in 2012, it was decided to provide more per ex-combatant (800,000 CFA francs), with flexible disbursement conditions. Second, there was a change in definition of who was considered to be a combatant (hence a change in who had access to financial benefits). Before 2012, the Ivorian DDR process integrated a large diversity of profiles in the definition of who was considered "ex-combatant" (from fighting to support tasks), but after 2012, the definition took a more restrictive turn, and individuals who wanted to be part of the official DDR process had to be identified as fighters by their respective commanders and had to pass a weapons-handling test. With respect to the dismantling of pro-governmental militias, ADDR had planned to process 23,000 individual applications out of a total of 74,000 for the entire process. At the end of January 2015, 46,031 ex-combatants, all categories taken together, had benefited from support, according to ADDR statistics. These successes were only relative though, and the ADDR director was not hiding the persistence of many difficulties. Mistrust of some former militia groups allied with the former regime was one of them, as many were fearing reprisals if they officially integrated the process. Direct observations conducted in Duékoué in May 2015, however, showed that some former pro-Gbagbo militias had been taken into account in the process and had received 800,000 CFA francs in cash transfers (with some in fact having benefited twice, in 2006 and in 2013).

At the time of writing (November 2017), the use of the term "militia" finally seems to

Bibliography

  Ten years after the failed start of the demobilization of pro-governmental militias, it is worth reflecting on how the process evolved. If during the period 2003-2007, DDR progress was very slow to materialize, the dynamics changed in 2007-2009, following the Ouagadougou comprehensive peace agreement. The APO, as it was locally labeled, promoted direct dialogue between the Ivorian government and the rebel forces while reiterating its attachment to the existing National DDR Plan. It created two new structures for dealing with demilitarization issues: the Integrated Command Centre (CCI), which was put in charge of the military component of the process (the disarmament and demobilization phases), and the National Reinsertion and Community Rehabilitation Program (PNRRC), which was empowered to deal with the reinsertion of former combatants. Political appeasement then seemed possible, and a series of symbolic launches of the DDR process were organized in several localities. The dismantling of western militias was quickly announced and scheduled for May 2007. But unlike the 2006 attempt, the 2007 operation was very opaque in terms of cash transfers, and neither the National DDR Commission nor the UN peacekeeping mission were associated with the event. In 2007, the dismantling of militias took the form of a media operation, where 1,026 weapons were collected and handed over to the presidency by several militia leaders in return for a presidential envelope of 280 million CFA, supposed to be distributed to low-ranked recruits. Following that event, an official press release announced the official end of the operation.

  have disappeared in western Côte d'Ivoire, for the best. ADDR officially announced the end of the DDR process in June 2015, and more recently, in June 2017, the UN peacekeeping mission, which was in charge of controlling part of the DDR process, put a term to thirteen years of mandate. Empirical work is still needed, however, to document the impact of the ADDR program and to see-among other things-if the doubling of amount of the cash transfers led to different uses. Magali Chelpi-den Hamer is lecturer/researcher at Aix-Marseille Université and at the Institut des Mondes Africains (IMAF). She holds a Ph.D. in social sciences from the University of Amsterdam and a French doctorate in anthropology from the École des Hautes Études en Sciences Sociales (cum laude). She has extensively published on local processes of militarization and demilitarization, education, and certification of learning in situations of forced displacement in Côte d'Ivoire. Her current interests include transitional justice and the developmenthumanitarian nexus. She has consulted for various institutions (USAID, World Bank, UNICEF, Norwegian Government), and in the early 2000s, she was involved in humanitarian program coordination.