Sophie Mercier
email: sophie.mercier@univ-pau.fr

Carmen Sang Üesa

A general multivariate lifetime model with a multivariate additive process as conditional hazard rate increment process

Keywords: Mathematics Subject Classification (2010) MSC 60G51, MSC 60G55, MSC 62H05 Reliability, Multivariate survival function, Multivariate covariate process, Multivariate additive process, Multivariate Lévy process, Poisson random measure

The object of the present paper is the study of the joint lifetime of d components subject to a common stressful external environment. Out of the stressing environment, the components are independent and the lifetime of each component is characterized by its failure (hazard) rate function. The impact of the external environment is modelled through an increase in the individual failure rates of the components. The failure rate increments due to the environment increase over time and they are dependent among components. The evolution of the joint failure rate increments is modelled by a non negative multivariate additive process, which include Lévy processes and non-homogeneous compound Poisson processes, hence encompassing several models from the previous literature. A full form expression is provided for the multivariate survival function with respect to the intensity measure of a general additive process, using the construction of an additive process from a Poisson random measure (or Poisson point process). The results are next specialized to Lévy processes and other additive processes (time-scaled Lévy processes, extended Lévy processes and shock models), thus providing simple and easily computable expressions. All results are provided under the assumption that the additive process has bounded variations, but it is possible to relax this assumption by means of approximation procedures, as is shown for the last model of this paper.

Introduction

Using the vocabulary of reliability theory, the object of the present paper is the study of the joint lifetime of d components subject to a common stressful external environment. Out of the stressing environment, the components are assumed to be independent and the lifetime of each component is characterized by its failure (hazard) rate function.

Most classically, the environment can act through shocks on the components. In that case, each shock may lead to possibly simultaneous failures, also called common cause failures in the reliability literature. Classical models for possibly simultaneous failures are the binomial failure rate (BFR) model [START_REF] Atwood | The binomial failure rate common cause model[END_REF][START_REF] Vesely | Nuclear systems reliability engineering and risk assessment, chapter Estimating common cause failure probabilities in reliability and risk analysis: Marshall-Olkin specializations[END_REF] and the Marshall-Olkin multivariate exponential distribution family [START_REF] Marshall | A multivariate exponential distribution[END_REF]. As noticed by [25, p. 223], "even if common cause failures are caused by a common cause, they do not need to occur at the same time. A rather long time between failures does not necessarily mean that there is no dependency between the failure events". This drawback of the two previous classical models for common cause failures has lead to the development of cumulative shock models, where a shock simultaneously increases some intrinsic characteristics (hazard rate, deterioration level, age, etc.) of the components, leading to some possibly delayed failures among components although due to a common cause (see [START_REF] Mallor | Classification of shock models in system reliability[END_REF] for an overview on shock models and [START_REF] Marshall | Multivariate shock models for distributions with increasing hazard rate average[END_REF] for the study of a particular cumulative shock model).

An approach for a realistic shock model with mixed effect (possible instantaneous failure or sudden increase of a characteristic) has been studied in [START_REF] Cha | On a stochastic survival model for a system under randomly variable environment[END_REF] in the univariate case, where a shock can be fatal to the system with a probability depending on the shock's arrival time, and where a non fatal shock increases the system failure rate of a random increment. This model has been extended to the bivariate case in [START_REF] Mercier | A random shock model with mixed effect, including competing soft and sudden failures, and dependence[END_REF] , where more references on shock models for reliability can also be found. In both [START_REF] Cha | On a stochastic survival model for a system under randomly variable environment[END_REF] and [START_REF] Mercier | A random shock model with mixed effect, including competing soft and sudden failures, and dependence[END_REF], the shocks arrive according to a non-homogeneous Poisson process (N (t)) t≥0 , and the successive increments in the failure rate due to the shocks are assumed to be independent and identically distributed (i.i.d.). The process describing the increment of the failure rates due to the shocks hence appear as a nonhomogeneous compound Poisson process (univariate [START_REF] Cha | On a stochastic survival model for a system under randomly variable environment[END_REF] or bivariate [START_REF] Mercier | A random shock model with mixed effect, including competing soft and sudden failures, and dependence[END_REF]). Given the bivariate non-homogeneous compound Poisson process in [START_REF] Mercier | A random shock model with mixed effect, including competing soft and sudden failures, and dependence[END_REF], the two components are assumed to be conditionally independent, and the conditional failure rate of each component is the sum of its intrinsic failure rate and of the corresponding margin in the compound Poisson process. We then say that the bivariate non-homogeneous compound Poisson process stands for the conditional hazard rate increment process, which appears as a specific type of covariate process, in the vocabulary of statistics literature.

In practice, the impact of the external environment on the components do not always arrive through isolated shocks: in [START_REF] Singpurwalla | Multivariate distributions induced by dynamic environments[END_REF] for instance, the authors consider an extended gamma process as conditional hazard rate increment process for the modeling of a "cumulative effect of the environment" (in a bivariate setting) and in [START_REF] Wenocur | A reliability model based on the gamma process and its analytic theory[END_REF], the authors envision a process constructed as a function of a standard gamma process (in a univariate setting). Extended gamma processes have also been used as prior in a Bayesian context [START_REF] Dykstra | A bayesian nonparametric approach to reliability[END_REF], leading to similar computations as in [START_REF] Singpurwalla | Multivariate distributions induced by dynamic environments[END_REF].

Noticing that non-homogeneous compound Poisson processes and (standard / extended) gamma processes are specific additive processes [START_REF] Sato | Lévy processes and infinitely divisible distributions[END_REF], we here suggest to consider multivariate additive processes as conditional hazard rate increment processes, thus leading to a unified view for different models from the previous literature. Note that this model mostly leads to lifetimes with increasing hazard rates, that is to lifetimes with the Increasing Failure Rate (IFR) property in the vocabulary of reliability theory. This means that with such a model, the components are not supposed to recover from the impact of the environment on their health statement. This IFR property is quite common an assumption in reliability theory since the pioneering work by Barlow & Proschan (see, e.g., [START_REF] Barlow | Mathematical theory of reliability[END_REF]). Indeed, it means that the components are aging over time and that they cannot regenerate themselves (that is out of maintenance actions). This is typically the case of industrial systems, where, for instance, the wear and tear of a wind turbine blade induced by a high wind on some time period cannot be recovered without any repair and induces a failure rate that remains higher from the high wind period. The model hence is well adapted to model multivariate lifetimes in the context of reliability theory, where component lifetimes usually exhibit aging (IFR) properties and where an external environment commonly has a lasting adverse effect on the failure rate.

The point of the paper is the study of the multivariate lifetime induced by the suggested model. The paper is organized as follows: we begin with some technical reminders in Section 2. We next present the model for a general multivariate additive process in Section 3 and provide the expression for the multivariate survival function with respect to the intensity measure of the additive process. The results are specialized in Section 4 to the case of (homogeneous) univariate and multivariate Lévy processes, and in Section 5 to other additives processes (time-scaled Lévy processes, extended Lévy processes and shock models). The paper ends with concluding remarks and perspectives in Section 6.

Technical reminders

Let X = (X (t)) t≥0 be a multivariate non negative additive process, which is assumed to be right-continuous with left-side limits, with no loss of generality. We recall that, following [START_REF] Sato | Lévy processes and infinitely divisible distributions[END_REF]Def. 1.6], the process X is such that:

-X (0) = 0 a.s., -X has independent increments, -X is stochastically continuous.

Based on its non negativity, the process X is known to be (componentwise) non decreasing. In the specific case where the increments of X are homogeneous, that is if the distribution of X (t + s) -X (t) depends only on s for all s,t > 0, then X is a Lévy process.

In all the following, the process X is assumed to have bounded variations, not to be almost surely equal to zero (P (X ̸ = 0) > 0) and to have no drift. Hence X is a pure jump process. We now introduce some notations and make a small review on the jump structure of such multivariate non negative additive processes (without drift). We refer to [START_REF] Sato | Lévy processes and infinitely divisible distributions[END_REF] for more details.

First, let us recall that for each t > 0, the distribution of the random vector X (t) is infinitely divisible and we set ρ t (du) to be its Lévy measure, where ρ t (du) is a (non null) Radon measure concentrated on R d + \ {0} such that

R d + min (∥u∥ ∞ , 1) ρ t (du) < +∞, (1)
based on the bounded variations of X.

The Laplace transform of X(t) can be expressed as:

L X(t) (y) = E e -⟨y,X(t)⟩ = exp - R d + 1 -e -⟨y,x⟩ ρ t (dx)
for all y ∈ R d + . In the specific case where X is a Lévy process, it is enough to specify the Lévy measure ρ 1 of X (1), which is known to control the jump structure of all the process (as ρ t = t ρ 1 for all t > 0). In that case, ρ = ρ 1 is called the Lévy measure of the process X.

Coming back to the general case of an additive process and following the notations of [START_REF] Sato | Lévy processes and infinitely divisible distributions[END_REF]Theorem 19.3], we now define the measure ρ on

F = (0, ∞)× R d + \ {0} by ρ ((0,t] × B) = ρ t (B) for all B ∈ B R d + \ {0} , (2)
which we call intensity measure of X. (In the case of a Lévy process, then ρ (ds, dx) = ds ρ (dx)).

We also set

J (C) = # {s > 0 : (s, ∆ X (s)) ∈ C} for all C ∈ B (F) ,
where ∆ X (s) := X (s) -X (s -) for all s > 0. Setting {T n , n ∈ N} to be the jump times of X (which are know to be almost surely countably many, see [START_REF] Sato | Lévy processes and infinitely divisible distributions[END_REF]Theorem 21.3]), we can also write

J (ds, dx) = ∑ n∈N δ (T n ,∆ X(T n )) (ds, dx)
with probability 1.

With such notations, we know from [START_REF] Sato | Lévy processes and infinitely divisible distributions[END_REF]Theorem 19.3] that:

-J is a Poisson random measure on F with intensity measure ρ; -X (t) = (0,t]×R d + \{0} x J (ds, dx) for all t > 0 almost surely (or to be more specific, there exists a sample space on which this is true).

Hence, any non negative additive process can be written as an integral with respect to a Poisson random measure. This will be the key point to compute the survival function of the multivariate lifetime in the following.

Finally, we end this technical section by a summary of our assumptions:

Assumption 1 The process X = (X (t)) t≥0 is a multivariate non negative additive process such that:

-X is right-continuous with left-side limits; -X has bounded variations (which means that (1) is true); -X is not almost surely equal to zero (P (X ̸ = 0) > 0); -X has no drift.

3 The multivariate lifetime model

The dependence model

Let us consider d components. Out of the stressing environment, these components are independent and for i = 1, • • • , d, the intrinsic hazard rate function of the i-th component is h i (t), t ≥ 0 and its cumulative hazard rate function is

H i (t) = t 0 h i (s) ds, t ≥ 0.
The components are in a common stressing environment which increases their hazard rates.

We set X = (X (t)) t≥0 to be a multivariate non negative additive process fulfilling Assumption 1.

For

each i = 1, • • • , d, the conditional hazard rate of the i-th item given X is h X i (t) = h i (t) + X i (t) (3)
for all t > 0.

The random variable X i (t) hence corresponds to the increment in the hazard rate of component i due to the stressing environment. In the following, the process X is called conditional hazard rate increment process.

The corresponding conditional cumulative hazard rate is provided by

H X i (s i ) = H i (s i ) + s i 0 X i (t) dt (4)
for all s i > 0.

Given (X (t)) t≥0 , the components are assumed to be independent. All the dependence between the components hence comes from the fact that the processes

(X i (t)) t≥0 , i = 1, • • • , d are dependent.
We now introduce the multivariate lifetime

τ = (τ 1 , • • • , τ d )
and the corresponding multivariate survival function:

Fτ (s) = P d i=1 {τ i > s i } , for all s = (s 1 , • • • , s d ) ∈ R d + .
The object of the paper is the study of the multivariate lifetime τ.

The multivariate survival function

Our aim here is to see how to compute Fτ (s) in a general setting. For a better understanding, we begin with a preliminary analysis.

Lemma 1 Let X = (X (t)) t≥0 be a multivariate non negative additive process fulfilling Assumption 1. We have

Fτ (s) = e -H(s) G (s) with H (s) = d ∑ i=1 H i (s i ) , (5)
G (s) = E e -∑ d i=1 s i 0 X i (t)dt , (6)
for all s = (s 1 , • • • , s d ) ∈ R d + .
Proof Conditionning by X and based on the conditional independence assumption, we have:

Fτ (s) = P d i=1 {τ i > s i } = E P d i=1 {τ i > s i } |X = E d ∏ i=1 P (τ i > s i |X) .
Now remembering (4), we get:

Fτ (s) = E d ∏ i=1 e -H i (s i ) e -s i 0 X i (t) dt = e -∑ d i=1 H i (s i ) E e -∑ d i=1 s i 0 X i (t) dt ,
which provides the result.

⊓ ⊔

The main point hence is to see how to compute G (s), which is done in the next theorem.

Theorem 2 Let X = (X (t)) t≥0 be a multivariate non negative additive process fulfilling Assumption 1 with intensity measure ρ (dt, dx). Then:

Fτ (s) = e -H(s)-K(s) (7)
for all s = (s

1 , • • • , s d ) ∈ R d
+ where H (s) is defined by (5) and

K (s) = F 1 -e -∑ d i=1 (s i -t) + x i ρ (dt, dx) (8)
with x = (x 1 , • • • , x d ) and F = (0, ∞) × R d + \ {0} in the integral. Proof Let s = (s 1 , • • • , s d ) ∈ R d + .
The point is to compute G (s), as defined by [START_REF] Cha | On a stochastic survival model for a system under randomly variable environment[END_REF]. Let us first note that

s i 0 X i (t) dt = s i 0 (0,t]×R d + \{0}
x i J (dr, dx) dt

= F 1 {r≤s i } s i r dt x i J (dr, dx) = F (s i -r) + x i J (dr, dx) (9)
for all i = 1, • • • , d and all s i ≥ 0.

For each s ∈ R d + , let us set

ϕ s (r, x) = d ∑ i=1 (s i -r) + x i (10)
with x = (x 1 , • • • , x d ) ∈ R d + .
Based on (9), we have:

d ∑ i=1 s i 0 X i (t) dt = F ϕ s (r, x) J (dr, dx)
and using the definition of G, we get G (s) = E e -F ϕ s (r,x) J(dr,dx) . Now our aim is to use Theorem 2.7 from [START_REF] Kyprianou | Introductory lectures on fluctuations of Lévy processes with applications[END_REF] to compute G (s). For that, we have to check that

F min (ϕ s (r, x) , 1) ρ (dr, dx) < ∞.
Let us first note that based on the definition of ϕ s given in [START_REF] Duffie | Affine Processes and Applications in Finance[END_REF], we have ϕ s (r, x) = 0 as soon as r > S = ∥s∥ ∞ . Also

0 ≤ ϕ s (r, x) ≤ d S ∥x∥ ∞ for all r ≤ S and all x = (x 1 , • • • , x d ) ∈ R d + . This provides F min (ϕ s (r, x) , 1) ρ (dr, dx) = (0,S]×R d + \{0}
min (ϕ s (r, x) , 1) ρ (dr, dx)

≤ (0,S]×R d + \{0} min (d S ∥x∥ ∞ , 1) ρ (dr, dx) = R d + \{0} min (d S ∥x∥ ∞ , 1) ρ S (dx)
based on the definition of ρ given in (2). Now, it is easy to check that min (d S ∥x∥ ∞ , 1) ≤ max (d S, 1) min (∥x∥ ∞ , 1) which entails that F min (ϕ s (r, x) , 1) ρ (dr, dx) ≤ max (d S, 1)

R d + \{0}
min (∥x∥ ∞ , 1) ρ S (dx) < +∞ based on the technical condition (1). This allows us to derive that

G (s) = exp - F 1 -e -ϕ s (r,x) ρ (dr, dx) = exp - F 1 -e -∑ d i=1 (s i -r) + x i ρ (dr, dx)
from [START_REF] Kyprianou | Introductory lectures on fluctuations of Lévy processes with applications[END_REF]Theorem 2.7] (where the result is provided for the Fourier transform but the arguments still hold for the Laplace transform), which achieves this proof.

⊓ ⊔ Remark 1 In the univariate case, the previous theorem writes down

Fτ (s) = exp - s 0 h (r) dr - (0,s]×R * + 1 -e -(s-r)x ρ (dr, dx)
which mostly is [17, Theorem 3.1], except from the fact that the terms W i 's of the quoted theorem are not present here and the fact that contrary to the quoted paper, we do not assume ρ (dr, dx) to be of the shape ρ (dr, dx) = dr ρr (dx). Also, the result is given here in a multivariate setting whereas it is given in a univariate setting in [START_REF] Kebir | On hazard rate processes[END_REF]Theorem 3.1].

The remaining of the paper is devoted to the study of the multivariate lifetime, considering different kinds of multivariate additive processes.

Lévy processes as conditional hazard rate increment processes

Let Y = (Y (t)) t≥0 be a multivariate non negative Lévy process, which is assumed to be right-continuous with left-side limits, to have bounded variations and no drift, and to be not almost surely equal to zero. Then Y is a specific additive process fulfilling Assumption 1.

Let ρ be its Lévy measure. The Lévy measure of Y t then is ρ t = t ρ and ρ (ds, du) = ds ρ (du), which implies that Theorem 2 writes down

Fτ (v) = e -H(v)-K(v) (11)
with

K (v) = F 1 -e -∑ d i=1 (v i -s) + x i ds ρ (dx) (12)
for all v = (v 1 , • • • , v d ) ∈ R d + , where we recall that F = (0, ∞) × R d + \ {0}
. For a better understanding, we first explore the univariate case.

Case of a univariate Lévy process

We first give some expressions for the survival and hazard rate fonctions with respect to the Laplace transform of Y (1) for a univariate Lévy process (Y (t)) t≥0 with Lévy measure ρ, where we recall that

L Y (1) (u) = exp - R * + 1 -e -ux ρ (dx) (13)
for all u ≥ 0 (see e.g. [START_REF] Kyprianou | Introductory lectures on fluctuations of Lévy processes with applications[END_REF] ).

Proposition 1 Let Y = (Y (t)) t≥0 be a univariate non negative Lévy process fulfilling Assumption 1, with Lévy measure ρ. Then:

The hazard rate and survival functions of the lifetime τ are provided by

h τ (v) = h (v) -ln L Y (1) (v) , Fτ (v) = e -H(v)-K(v)
for all v ≥ 0, respectively, where

K (v) = - v 0 ln L Y (1) (u) du, ∀v ≥ 0. ( 14
)
Proof Based on (12), we have

K (v) = R + ×R * + 1 -e -(v-s) + x ds ρ (dx) = v 0 R * + 1 -e -(v-s)x ρ (dx) ds = v 0 R * + 1 -e -ux ρ (dx) du = - v 0 ln L Y (1) (u) du
with u = vs in the third line and using (13) for the last line. This provides the result for Fτ (v), from where we easily derive the result for h τ (v), using that

h τ (v) = f τ (v) Fτ (v) = - F′ τ (v) Fτ (v) .

⊓ ⊔

We now provide other expressions for the survival and hazard rate functions of the lifetime τ with respect to both ρ and corresponding tail integral function, that can be of specific interest in the multivariate case later on.

Proposition 2 Let Y = (Y (t)) t≥0 be a univariate non negative Lévy process fulfilling Assumption 1, with Lévy measure ρ. The survival function of the lifetime τ is provided by

Fτ (v) = e -H(v)-K(v) with K (v) = R * + 1 x e -vx + vx -1 ρ (dx) = ∞ 0 U (y) γ v (y) dy, ∀v ≥ 0, where U (y) = (y,∞) ρ (dx) , for all y > 0 ( 15
)
is the tail integral function of ρ and

γ v (y) = 1 y 2 1 -e -vy -vye -vy , ∀y, v > 0. ( 16
)
The corresponding hazard rate function is given by

h τ (v) = h (v) + R * + 1 -e -vx ρ (dx) = h (v) + v ∞ 0 U (y) e -vy dy (17)
for all v > 0.

Proof The first expression of h τ (v) is a direct consequence of Proposition 1 and (13), from where we can also write

K (v) = R * + v 0 1 -e -ux du ρ (dx) (18)
= R * + 1 x e -vx + vx -1 ρ (dx) ,
which provides the first expression for K (v).

Using that

1 -e -ux = x 0 ue -yu dy, (19)
we can also write [START_REF] Kyprianou | Introductory lectures on fluctuations of Lévy processes with applications[END_REF] as

K (v) = (0,v)×(0,∞) x 0 ue -yu dy du ρ (dx) .
Based on Fubini's theorem for the first and third lines, we get:

K (v) = (0,v)×(0,∞) (y,∞) ρ (dx) ue -uy du dy = (0,v)×(0,∞) U (y) ue -uy du dy = R * + U (y) v 0 ue -uy du dy = R * + U (y) 1 y 2 1 -e -vy -vye -vy dy (20)
which is the second expression for K (v).

Using again [START_REF] Mallor | Classification of shock models in system reliability[END_REF], we also have

R * + 1 -e -vx ρ (dx) = R * + x 0 v e -vy dy ρ (dx) = v R * + (y,∞) ρ (dx) e -vy dy = v R * + U (y) e -vy dy,
which provides the second expression for h τ (v) (which can also be obtained through derivation of Fτ (v)).

⊓ ⊔

The two previous propositions provide us with several expressions for Fτ (v) and h τ (v) (and hence also for the p.d.f. f τ (v) = h τ (v) Fτ (v)), which allows to chose the most appropriate one, according to the context.

Remark 2

The first expression of the hazard rate function in Proposition 2 was already given in [START_REF] Kebir | On hazard rate processes[END_REF]Corollary 3.3], as well as [START_REF] Kyprianou | Introductory lectures on fluctuations of Lévy processes with applications[END_REF] in [17, Corollary 3.2] (considering a = 0 in the quoted paper and omitting the terms h (t) and H (t)). [START_REF] Cont | Financial modelling with jump processes[END_REF]Proposition 3.13] for their expression with respect to ρ), it is easy to check that

Remark 3 Assuming m 1 = E (Y (1)) = R * + x ρ (dx) and m 2 = var (Y (1)) = R * + x 2 ρ (dx) to be finite (see
h τ (v) = v→0 + h (v) + R * + vx + x 2 O v 2 ρ (dx) = v→0 + h (v) + vm 1 + m 2 O v 2 .
Hence, the increment in the hazard rate due to the stressing environment mostly is a linear function of v when v is small. The result of the previous proposition is now applied to the case of a homogeneous gamma process, which is a Lévy process such that Y (t) ∼ Γ (at, b) with a, b > 0, where

f Γ (at,b) (y) = b at Γ (at) y at-1 e -by 1 R + (y) for all t > 0, with E (Y (t)) = at/b, var (Y (t)) = at/b 2 , Lévy measure ρ Γ (a,b) (dx) = ae -bx /x dx and Laplace transform L Y (1) (v) = b b + v a , ∀v ≥ 0 (21)
(see e.g. [1, page 6]).

Corollary 1 (Univariate homogeneous gamma process) Let (Y (t)) t≥0 be a univariate homogeneous gamma process with Y (t) ∼ Γ (at, b). Then

Fτ (v) = e -H(v)+at b b + v a(b+v) , h τ (v) = h (v) + a ln b + v b .
Proof The expression for h τ (v) is directly obtained from Proposition 1 and (21).

The expression of K (v) in the same proposition provides

K (v) = v 0 a ln b + u b du = -av -ln b b + v a(b+v)
and the result for Fτ (v) .

⊓ ⊔

We now look at the case of an inverse gaussian process (Y t ) t≥0 , which is a Lévy process such that Y (t) ∼ IG (at, b) with a, b > 0, where

f IG(at,b) (x) = at (2πx 3 ) 1 2 exp - b 2 2 x + abt - (at) 2 2 1 x for all t > 0, with E (Y (t)) = at/b, var (Y (t)) = at/b 3 , Lévy measure ρ IG(a,b) (dx) = a √ 2πx 3 exp - b 2 2 x 1 R * + (x) dx
and Laplace transform

L Y (1) (v) = exp -a b 2 + 2v -b , ∀v ≥ 0 (22)
(see e.g. [15, pages 595 and 658]).

Corollary 2 (Inverse gaussian process) Let (Y (t)) t≥0 be an inverse gaussian process with Y (t) ∼ IG (at, b). Then

Fτ (v) = exp -H (v) -a 1 3 b 2 + 2v 3 2 -b 3 -bv h τ (v) = h (v) + a b 2 + 2v -b .
Proof The expression for h τ (t) is directly obtained from Proposition 1 and (22). The expression of K (v) in the same proposition gives Remark 4 Any (homogenous) Lévy process has a linear trend in the sense that E (Y (t)) = Ct for some C > 0. However, it can be seen from the previous examples that the increment in the unconditional hazard rate due to the stressing environment can have much less impact than a linear function, as it asymptotically behaves like a logarithm function in the case of a gamma process and as a square root function in the case of an inverse gaussian process.

K (v) = v 0 a b 2 + 2u -b du = a √ b 2 +2v b (x -b) x dx = a 1 3 b 2 + 2v
The increment in the unconditional hazard rate is plotted in Figure 1 for the case of a gamma and an inverse gaussian increment process which share the same mean function E (Y (t)) = at/b with a = 1 and b = 2 (left), b = 0.25 (right). As can be seen, the increment in the unconditional hazard rate can be larger when t is small for a gamma process than for a inverse gaussian process but after a while, it becomes larger for the inverse gaussian process (in coherence with the asymptotic behaviour).

Case of a multivariate Lévy process

We now come to the multi-dimensional case, where we recall that ρ (dt, dx) = dt ρ (dy), and where the Laplace transform of Y (1) is provided by

L Y(1) (v) = exp - R d + \{0} 1 -e -∑ d i=1 v i x i ρ (dy) (23)
for all v = (v 1 , • • • , v d ) ∈ R d + .
This allows to provide new expressions for Fτ (v) and h τ (v) in this specific setting.

Proposition 3 Let Y = (Y (t)) t≥0 be a multivariate non negative Lévy process fulfilling Assumption 1. We have

Fτ (s) = e -H(s)-K(s) (24) for all s = (s 1 , • • • , s d ) ∈ R d
+ where H (s) is defined by (5) and

K (s) = - ∞ 0 ln L Y(1) (s 1 -t) + , • • • , (s d -t) + dt for all s = (s 1 , • • • , s d ) ∈ R d + .
Proof The result is a direct consequence from (12) and (23).

⊓ ⊔

A common way to model a multivariate non negative Lévy process is to construct it through superposition (see [START_REF] Barndorff-Nielsen | Multivariate subordination, self-decomposability and stability[END_REF]). We next provide the result for the bivariate case. Corollary 3 Let (Z i (t)) t≥0 , i = 1, 2, 3 be three independent univariate non negative Lévy processes fulfilling Assumption 1. We set

Y 1 (t) = Z 1 (t) + Z 3 (t) Y 2 (t) = Z 2 (t) + Z 3 (t) (25)
for all t ≥ 0. Then Y = (Y (t)) t≥0 be a bivariate non negative Lévy process and the survival function is provided by [START_REF] Nelsen | An introduction to copulas[END_REF] with

K (s) = - s 1 0 ln L Z 1 (1) (t) dt - s 2 0 ln L Z 2 (1) (t) dt - max(s 1 ,s 2 ) 0 ln L Z 3 (1) (s 1 -t) + + (s 2 -t) + λ (t) dt.
Proof We have

L Y(1) (s 1 , s 2 ) = E e -s 1 Y 1 (1)-s 2 Y 2 (1) = E e -s 1 Z 1 (1)-s 2 Z 2 (1)-(s 1 +s 2 )Z 3 (1) = L Z 1 (1) (s 1 ) L Z 2 (1) (s 2 ) L Z 3 (1) (s 1 + s 2 )
due to the independence assumption. Hence

K (s) = - ∞ 0 ln L Z 1 (1) (s 1 -t) + dt - ∞ 0 ln L Z 2 (1) (s 2 -t) + dt - ∞ 0 ln L Z 3 (1) (s 1 -t) + + (s 2 -t) + dt,
which easily provides the result.

⊓ ⊔

The previous corollary is next specialized to the case of gamma processes.

Corollary 4 Let (Z i (t)) t≥0 , i = 1, 2, 3 be three independent univariate gamma processes, such that Z i (t) ∼ Γ (a i t, b i ) with a i , b i > 0, i = 1, 2, 3, and let (Y i (t)) t≥0 , i = 1, 2 be constructed through [START_REF] Rausand | System Reliability Theory: Models, Statistical Methods, and Applications[END_REF]. Then

Fτ (s) = e -H(s)+a 1 s 1 +a 2 s 2 +a 3 max(s 1 ,s 2 ) b 1 b 1 + s 1 a 1 (b 1 +s 1 ) b 2 b 2 + s 2 a 2 (b 2 +s 2 ) (26) × b a 3 (b 3 +max(s 1 ,s 2 )) 3 (b 3 + |s 2 -s 1 |) a 3 (b 3 +|s 2 -s 1 |)/2 (b 3 + s 1 + s 2 ) a 3 (b 3 +s 1 +s 2 )/2 for all s = (s 1 , s 2 ) ∈ R 2 + .
Proof Based on the result for univariate gamma processes given in Corollary 1, the only point is to compute

I (s) = max(s 1 ,s 2 ) 0 ln L Z 3 (1) (s 1 -t) + + (s 2 -t) + λ (t) dt.
Assume that s 1 < s 2 and let us write

I (s) = I 1 (s) + I 2 (s)
with

I 1 (s) = s 1 0 ln L Z 3 (1) (s 1 + s 2 -2t) dt = a 3 s 1 0 ln b 3 b 3 + s 1 + s 2 -2t dt = a 3 s 1 + ln b s 1 3 (b 3 -s 1 + s 2 ) (b 3 -s 1 +s 2 )/2 (b 3 + s 1 + s 2 ) (b 3 +s 1 +s 2 )/2
and

I 2 (s) = s 2 s 1 ln L Z 3 (1) (s 2 -t) dt = a 3 s 2 s 1 ln b 3 b 3 + s 2 -t dt = a 3 (s 2 -s 1 ) + ln b 3 b 3 + s 2 -s 1 (b 3 +s 2 -s 1 )
. This provides

I (s) = a 3 s 2 + ln b a 3 (b 3 +s 2 ) 3 (b 3 -s 1 + s 2 ) a 3 (b 3 +s 2 -s 1 )/2 (b 3 + s 1 + s 2 ) a 3 (b 3 +s 1 +s 2 )/2
Similar results can be obtained in the case s 2 ≥ s 1 , and the two cases are next synthesized in [START_REF] Sato | Lévy processes and infinitely divisible distributions[END_REF].

⊓ ⊔

Another common way to model the dependence between the marginal processes in a Lévy process is to use a Lévy copula. We limit the study to the bidimensional case for sake of simpicity, but the results could clearly be extended. Following [START_REF] Cont | Financial modelling with jump processes[END_REF][START_REF] Kallsen | Characterization of dependence of multidimensional Lévy processes using Lévy copulas[END_REF], we recall that a (two-dimensional non-negative) Lévy-copula is any 2-increasing function

C : [0, ∞] 2 → [0, ∞] such that C (x, ∞) = C (∞, x) = x (uniform margins) and C (0, x) = C (x, 0) = 0 for all x ∈ [0,
+∞] (grounded function). Sklar's Theorem for Lévy copulas says that, starting from a Lévy process Y = (Y (t)) t≥0 , there exists a Lévy-copula C such that

U (y 1 , y 2 ) = C (U 1 (y 1 ) ,U 2 (y 2 )) (27)
for all (y 1 , y 2 ) ∈ [0, ∞] 2 , where, for i = 1, 2, the function U i is the tail integral function of the Lévy measure ρ i of (Y i (t)) t≥0 defined in [START_REF] Jeanblanc | Mathematical methods for financial markets[END_REF] and where U is the tail integral function of ρ, with

U (y) = U (y 1 , y 2 ) = (y 1 ,+∞)×(y 2 ,+∞) ρ (dy)
for all y= (y 1 , y 2 ) ∈ R 2 + \ {(0, 0)}, U (y 1 , ∞) = U (∞, y 2 ) = 0 and U (0, 0) = +∞. Recall also that the Lévy copula is unique as soon as the tail integral functions U i (y i ), i = 1, 2 are continuous.

In case of a Lévy process with dependence between margins modelled by a Lévy copula, we need to express the survival function with respect to the tail integral function U, in order to be able to compute it. This is the main object of the following theorem.

Theorem 3 Let Y = (Y (t)) t≥0 be a bivariate non negative Lévy process fulfilling Assumption 1 with C as Lévy copula C and U i as univariate tail integral function of

ρ i , i = 1, 2. Then Fτ (v) = exp R 2 + C (U 1 (u 1 ) ,U 2 (u 2 )) γ v (u) du 1 du 2 Fτ 1 (v 1 ) Fτ 2 (v 2 ) (28)
with

γ v (u) = min(v 1 ,v 2 ) 0 (v 1 -s) (v 2 -s) e -(v 1 -s)u 1 -(v 2 -s)u 2 ds (29) = |v 2 -v 1 | + 2 u 1 + u 2 e -u 1 (v 1 -v 2 ) + -u 2 (v 2 -v 1 ) + (u 1 + u 2 ) 2 -v 1 v 2 + v 1 + v 2 u 1 + u 2 + 2 (u 1 + u 2 ) 2 e -(u 1 v 1 +u 2 v 2 ) u 1 + u 2 for all u, v ∈ R 2 + .
Proof Starting again from [START_REF] Dykstra | A bayesian nonparametric approach to reliability[END_REF], the point is to compute K (v) with

K (v) = F 1 -e -∑ 2 i=1 (v i -s) + y i ds ρ (dy) where v = (v 1 , v 2 ) ∈ R 2 + . Assume first that v 1 ≤ v 2 . Then, we have K (v) = R 2 + \{0} v 1 0 1 -e -(v 1 -s) y 1 -(v 2 -s) y 2 + v 2 v 1 1 -e -(v 2 -s) y 2 ds ρ (dy) = v 1 0 R 2 + \{0} 1 -e -(v 1 -s) y 1 -(v 2 -s) y 2 ρ (dy) ds + v 2 v 1 R 2 + \{0} 1 -e -(v 2 -s) y 2 ρ (dy) ds. (30)
Let us write

1 -e -(v 1 -s) y 1 -(v 2 -s) y 2 = -1 -e -(v 1 -s)y 1 1 -e -(v 2 -s)y 2 + 1 -e -(v 2 -s)y 2 + 1 -e -(v 1 -s)y 1
in the first integral. We obtain:

K (v) = - v 1 0 R 2 + \{0}
1e -(v 1 -s)y 1 1e -(v 2 -s)y 2 ρ (dy) ds

+ v 1 0 R 2 + \{0}
1e -(v 2 -s)y 2 ρ (dy) ds

+ v 1 0 R 2 + \{0}
1e -(v 1 -s)y 1 ρ (dy) ds

+ v 2 v 1 R 2 + \{0} 1 -e -(v 2 -s) y 2 ρ (dy) ds = - v 1 0 R 2 + \{0} 1 -e -(v 1 -s)y 1 1 -e -(v 2 -s)y 2 ρ (dy) ds (31)
+ v 1 0 R * + 1 -e -(v 1 -s)y 1 ρ 1 (dy 1 ) ds + v 2 0 R * + 1 -e -(v 2 -s) y 2 ρ 2 (dy 2 ) ds
Looking at the case v 2 ≥ v 1 provides a similar result, where the upper bound v 1 in the integral in (31) is substituted by v 2 .

Hence, for all v 1 , v 2 > 0, we have

K (v) = -J (v) + J 1 (v 1 ) + J 2 (v 2 )
where

J i (v i ) = v i 0 R * + 1 -e -(v i -s) y i ρ i (dy i ) ds, i = 1, 2,
have already been computed in the univariate case and where

J (v) = min(v 1 ,v 2 ) 0 R 2 + \{0} 1 -e -(v 1 -s)y 1 1 -e -(v 2 -s)y 2 ρ (dy) ds. Writing 1 -e -(v i -s)y i = y i 0 (v i -s) e -(v i -s)u i du i for i = 1, 2, we get J (v) = min(v 1 ,v 2 ) 0 R 2 + \{0} y 1 0 (v 1 -s) e -(v 1 -s)u 1 du 1 × y 2 0 (v 2 -s) e -(v 2 -s)u 2 du 2 ρ (dy) ds = min(v 1 ,v 2 ) 0 R 2 + (u 1 ,+∞)×(u 2 ,+∞) ρ (dy) (v 1 -s) (v 2 -s) e -(v 1 -s)u 1 -(v 2 -s)u 2 ds du 1 du 2 = R 2 + U (u) γ v (u) du 1 du 2 with γ v (u) = min(v 1 ,v 2 ) 0 (v 1 -s) (v 2 -s) e -(v 1 -s)u 1 -(v 2 -s)u 2 ds = e -u 1 v 1 -u 2 v 2 min(v 1 ,v 2 ) 0 (v 1 -s) (v 2 -s) e s(u 1 +u 2 ) ds.
A double integration by parts provides

γ v (u) = e -u 1 v 1 -u 2 v 2 u 1 + u 2 e (u 1 +u 2 ) min(v 1 ,v 2 ) 1 u 1 + u 2 (v 1 + v 2 -2 min (v 1 , v 2 )) + 2 (u 1 + u 2 ) 2 -v 1 v 2 - v 1 + v 2 u 1 + u 2 - 2 (u 1 + u 2 ) 2 = v 1 + v 2 -2 min (v 1 , v 2 ) + 2 u 1 + u 2 e -u 1 v 1 -u 2 v 2 +(u 1 +u 2 ) min(v 1 ,v 2 ) (u 1 + u 2 ) 2 -v 1 v 2 + v 1 + v 2 u 1 + u 2 + 2 (u 1 + u 2 ) 2 e -(u 1 v 1 +u 2 v 2 ) u 1 + u 2 = |v 2 -v 1 | + 2 u 1 + u 2 e -u 1 (v 1 -v 2 ) + -u 2 (v 2 -v 1 ) + (u 1 + u 2 ) 2 -v 1 v 2 + v 1 + v 2 u 1 + u 2 + 2 (u 1 + u 2 ) 2 e -(u 1 v 1 +u 2 v 2 ) u 1 + u 2
Using Proposition 2 for J 1 (v 1 ) and J 2 (v 2 ), we obtain

K (v) = - R 2 + U (u) γ v (u) du 1 du 2 + R * + U 1 (u 1 ) γ v 1 (u 1 ) du 1 + R * + U 2 (u 2 ) γ v 2 (u 2 ) du 2 ,
which allows to conclude, due to [START_REF] Singpurwalla | Multivariate distributions induced by dynamic environments[END_REF].

⊓ ⊔

Remark 5 Based on (29), it is clear that γ v (u) ≥ 0 for all u, and hence we can see that [START_REF] Vesely | Nuclear systems reliability engineering and risk assessment, chapter Estimating common cause failure probabilities in reliability and risk analysis: Marshall-Olkin specializations[END_REF]. This shows that τ has the positive quadrant dependence property, which entails that higher values for τ 1 leads to higher values for τ 2 (and conversely), see e.g. [START_REF] Nelsen | An introduction to copulas[END_REF] for more details on this notion. This behaviour is coherent with what could be expected in such a context (see e.g. [START_REF] Mercier | A bivariate failure time model with random shocks and mixed effects[END_REF]). Example 1 We here consider a Clayton-Lévy copula

Fτ (v) ≥ Fτ 1 (v 1 ) Fτ 2 (v 2 ) from (
C Θ (v 1 , v 2 ) = v -θ 1 + v -θ 2 -1 θ
with θ > 0, where we recall that the dependence increases with θ (independence: θ → 0 + ; complete dependence: θ → ∞), see [START_REF] Cont | Financial modelling with jump processes[END_REF] for more details. We also take Y i (t) ∼ Γ (t, 1) for i = 1, 2, with

U i (u i ) = ∞ u i e -x x dx = Ei (x) , ∀u i > 0,
where Ei is the exponential integral function. We set Fτ ⊥ and Fτ ∥ to be the survival functions when (Y 1 (t)) t≥0 and (Y 2 (t)) t≥0 are independent and completely dependent, respectively. The functions Fτ -Fτ ⊥ and Fτ ∥ -Fτ are plotted in Figure 2 (left and right plots, respectively) for θ = 1. We can see that both functions remain non negative, which is coherent with the previous remark.

5 Other additive processes as conditional hazard rate increment processes

Time-scaled Lévy process

We begin with a first extension of the previous section considering time-scaled Lévy processes, which are known from [START_REF] Sato | Lévy processes and infinitely divisible distributions[END_REF], to be additive processes.

Definition 1 Let Y = (Y (t)) t≥0 be a multivariate non negative Lévy process and let ρ be its Lévy measure. Let λ : R + -→ R + be a continuous function such that λ (t) > 0 for all t > 0 and Λ (t) = t 0 λ (s) ds < +∞ for all t > 0, and lim t→∞ Λ (t) = ∞. The process X = (X (t)) t≥0 with

X (t) = Y (Λ (t)) for all t ≥ 0 (32)
is called a time-scaled Lévy process with time-scaling function Λ and Lévy measure ρ.

We first look at the intensity measure of X (for which we could not find any reference so that a proof is provided).

Lemma 2

The intensity measure of the time-scaled Lévy process X with time-scaling function Λ and Lévy measure ρ is:

ρ (ds, du) = λ (s) ds ρ (du) .
Proof Let J X and J Y be the Poisson random measures with intensity measures ρ (ds, du) and ρY (ds, du), respectively.

For each B ∈ B R d + \ {0} and each t > 0, we have:

ρ ((0,t] × B) = E [J X ((0,t] × B)] = E [# {s > 0 : s ∈ (0,t] and ∆ X (s) ∈ B}] where ∆ X (s) = X (s) -X (s -) = Y (Λ (s)) -Y Λ (s) -= Y (Λ (s)) -Y (Λ (s -)) = ∆ Y (Λ (s)) (by continuity of Λ (•)).
As Λ (•) is a one-to-one continuous function from (0,t] to (0,Λ (t)], we now have:

ρ ((0,t] × B) = E [# {Λ (s) > 0 : Λ (s) ∈ (0,Λ (t)] and ∆ Y (Λ (s)) ∈ B}] = E [# {u > 0 : u ∈ (0,Λ (t)] and ∆ Y (u) ∈ B}] setting u = Λ (s). Hence: ρ ((0,t] × B) = E [J Y ((0,Λ (t)] × B)] = ρY ((0,Λ (t)] × B) = Λ (t) ρ Y (B) ,
because ρY (ds, du) = ds ρ Y (du), which provides the result.

⊓ ⊔

Based on Theorem 2, and using similar arguments as for Proposition 3, Corollary 3 and Theorem 3, it is now straightforward to derive the following results in case of a time-changed Lévy process.

Corollary 5 Let X be a time-scaled Lévy process with time-scaling function Λ and Lévy measure ρ, such that the underlying homogeneous Lévy process Y (see (32)) fulfills Assumption 1. Then we have the following results:

1. The process X fulfills Assumption 1 and

Fτ (s) = e -H(s)-K(s) with K (s) = F 1 -e -∑ d i=1 (s i -t) + x i λ (t) dt ρ (dx) = - ∞ 0 ln L Y(1) (s 1 -t) + , • • • , (s d -t) + λ (t) dt for all s = (s 1 , • • • , s d ) ∈ R d + .
2. In the bivariate case, if Y is constructed through superposition as in Corollary 3, we have

K (s) = - s 1 0 ln L Z 1 (1) (s 1 -t) λ (t) dt - s 2 0 ln L Z 2 (1) (s 2 -t) λ (t) dt - max(s 1 ,s 2 ) 0 ln L Z 3 (1) (s 1 -t) + + (s 2 -t) + λ (t) dt 3.
Assuming the dependence in Y to be modelled by a Lévy copula C with U i (s i ) the tail integral functions of ρ i , i = 1, 2, we have

Fτ (s) = exp R 2 + C (U 1 (s 1 ) ,U 2 (s 2 )) γ v,λ (s) ds 1 ds 2 Fτ 1 (v 1 ) Fτ 2 (v 2 ) with γ v,λ (s) = min(v 1 ,v 2 ) 0 (v 1 -t) (v 2 -t) e -(v 1 -t)s 1 -(v 2 -t)s 2 λ (t) dt for all s = (s 1 , • • • , s d ) ∈ R d + .
Remark 6 Remark 5 remains valid here and τ exhibits the positive quadrant dependence property.

As an example, we now look at the case of a univariate non-homogeneous gamma process.

Example 2 Let (Y (t)) t≥0 be a univariate homogeneous gamma process with Y (t) ∼ Γ (t, b), where b > 0. Let X (t) = Y (Λ (t)) for all t > 0. Then, (X (t)) t≥0 is a univariate non-homogeneous gamma process with X (t) ∼ Γ (Λ (t) , b) and

Fτ (s) = e -H(s) exp - s 0 λ (t) ln b + s -t b dt , h τ (s) = h (s) + a s 0 λ (t) b + s -t dt.
The increment in the unconditional hazard rate is plotted in Figure 3 for the case of a non homogenous gamma conditional increment process with a = b = 1 and λ (s) = s β with β = -0.5, β = 0 and β = 1, which leads to Λ (s) = s 0.5 /0.5 (concave function), Λ (s) = s and Λ (s) = s 2 /2 (convex function), and for λ (s) = e s . We can see that for small s, the largest influence of the stressing environment is obtained in the concave case whereas for large s, it increases with β and the largest hazard rate is obtained for λ (s) = e s . Fig. 3 The increment in the unconditional failure rate for a non-homogeneous gamma increment process, with a = b = 1 and λ (s) = s β with β = -0.5, β = 0 and β = 1, and λ (s) = e s .

Extended time-scaled Lévy process

Let Y = (Y (t)) t≥0 be a multivariate time-scaled Lévy process with time-scaling function Λ and Lévy measure ρ as in Subsection 4. We here consider a process X = (X (t)) t≥0 which is constructed as a stochastic integral with respect to Y, with

X (t) = (0,t] b (u) dY (u) = (0,t] b 1 (u) dY 1 (u) , • • • , (0,t] b d (u) dY d (u) (33)
for all t ≥ 0, where b

(u) = (b 1 (u) , • • • , b d (u)) for all u ≥ 0 is a componentwise positive and continuous function on R * + d .

Definition 2

The process X = (X (t)) t≥0 constructed through (33) is said to be an extended time-scaled Lévy process with parameter (b, ρ,Λ ).

Remark 7

The name "extended Lévy process" comes from the extended gamma process, which is defined as an integral with respect to a gamma process in several papers such as [START_REF] Al Masry | Approximate simulation techniques and distribution of an extended gamma process[END_REF][START_REF] Dykstra | A bayesian nonparametric approach to reliability[END_REF][START_REF] Guida | A time-discrete extended gamma process for time-dependent degradation phenomena[END_REF]. Note that in [START_REF] ¸inlar | On a generalization of gamma processes[END_REF], such processes are called local gamma processes.

Lemma 3 Let X = (X (t)) t≥0 be a multivariate extended time-scaled Lévy process with parameter (b, ρ,Λ ) such that the underlying Lévy process Y (as in (33)) fulfills Assumption 1 and such that

(0,t] b i (u) λ (u) du < ∞ (34)
for all i ∈ {1, • • • , d} and all t > 0. Then X is an additive process wich fulfills Assumption 1, and its intensity measure is

ρ (ds, du) = λ (s) ds ρϕ -1 b(s) (du)
where

ϕ b(s) (x) = (b 1 (s) x 1 , • • • , b d (s) x d ) (35)
for all s > 0 and all x ∈ R d + , and where ρϕ -1 b(s) stands for the push-forward measure of ρ by the function ϕ b(s) .

In the specific case where ρ (dx) = ρ (x) dx, then

ρ (ds, du) = λ (s) ds ρ u 1 b 1 (s) , • • • , u d b d (s) 1 b 1 (s) × • • • × b d (s) du.
Proof Remembering that

Y (t) = (0,t]×R d + \{0} y J Y (ds, dy) ,
where J Y is a Poisson random measure on F with intensity measure ρY (ds, du) = λ (s) ds ρ (du), we can write:

X (t) = R * + ×R d + \{0} 1 (0,t] (s) (b 1 (s) y 1 , • • • , b d (s) y d ) J Y (ds, dy) for all t ≥ 0. For m ≥ 1, 0 ≤ t 1 < • • • < t m and 1 ≤ i ≤ m -1, we have X (t i+1 ) -X (t i ) = R * + ×R d + \{0} 1 (t i ,t i+1 ] (s) (b 1 (s) y 1 , • • • , b d (s) y d ) J Y (ds, dy) ,
which shows that the random vectors

X (t i+1 ) -X (t i ), i = 1, • • • , m -1 are inde- pendent, because the Borel sets (t i ,t i+1 ] × R k + are disjoint.
Hence, (X t ) t≥0 is a process with independent increments.

For each 1 ≤ j ≤ d, t > 0, 0 ≤ s ≤ 1, we have

X j (t + s) -X j (t) = (t,t+s] b j (u) dY j (u) ≤ sup u∈[t,t+1] b j (u) × (Y j (t + s) -Y j (t)) .
Based on the continuity of b j , the supremum in the right side is finite. The stochastic continuity of (X j (t)) t≥0 for each 1 ≤ j ≤ d now is a direct consequence of the same property for Y, and hence, X is stochastically continuous. Finally, (X t ) t≥0 is a rightcontinuous process with left-side limits by construction (and X 0 = 0 a.s.). Hence (X t ) t≥0 is an additive process. Now, our aim is to use Proposition 19.5 from [START_REF] Sato | Lévy processes and infinitely divisible distributions[END_REF] to derive the Laplace transform of X (t). For that, we have to boil down to a Poisson random measure with a finite intensity measure. For ε > 0 and t > 0, we set J t,ε Y (ds, dy) to be the trace of J Y (ds, dy) on (0,t] × (ε, ∞) d \ {0} , which is known to be a Poisson random measure with intensity measure ρt,ε

Y (ds, du) = 1 (0,t] (s) 1 (ε,∞) d \{0} (u) λ (s) ds ρ (du) .
Based on the bounded variations of Y (that is (1) for Y) and the assumption on

λ , we have ρt,ε Y (0,t] × (ε, ∞) d < ∞.
We set

X ε (t) = R * + ×R d + \{0} 1 (0,t] (s) (b 1 (s) y 1 , • • • , b d (s) y d ) J t,ε Y (ds, dy) = R * + ×R d + \{0} 1 (0,t] (s) ϕ b(s) (y) J t,ε Y (ds, dy) ,
where ϕ b(s) is defined in (35).

Based on Proposition 19.5 from [START_REF] Sato | Lévy processes and infinitely divisible distributions[END_REF] (written for Fourier transform), the Laplace transform of

X ε (t) is E e -⟨u,X ε (t)⟩ = exp - R d + \{0} 1 -e -⟨u,x⟩ ρt,ε Y ϕ -1 t,b (dx)
for all u ∈ R d + and t > 0, where ϕ t,b (s, x) = 1 (0,t] (s) ϕ b(s) (x). This means that the Lévy mesure of the random vector

X ε (t) is ρt,ε Y ϕ -1 t,b . Then, the intensity measure ρ of X is such that ρ ((0,t] × B) = lim ε→0 + ρt,ε Y ϕ -1 t,b ((0,t] × B) = lim ε→0 + ρY (s, x) ∈ (0,t] × (ε, ∞) d \ {0} : (b 1 (s) x 1 , • • • , b d (s) x d ) ∈ B = ρY (s, x) ∈ (0,t] × R d + \ {0} : (b 1 (s) x 1 , • • • , b d (s) x d ) ∈ B = (0,t] ρ x ∈ R d + \ {0} : ϕ b(s) (x) ∈ B λ (s) ds = (0,t] ρϕ -1 b(s) (B) λ (s) ds for all B ∈ B R d + \ {0}
. This provides the expression for ρ (ds, du). In the specific case where ρ (dx) = ρ (x) dx, then

ρϕ -1 b(s) (B) = R d + \{0} 1 B ((b 1 (s) x 1 , • • • , b d (s) x d )) ρ (x) dx = R d + \{0} 1 B (u) ρ u 1 b 1 (s) , • • • , u d b d (s) 1 b 1 (s) × • • • × b d (s) du with u = (b 1 (s) x 1 , • • • , b 2 (s) x d ) .
We finally check that X fulfills Assumption 1, where the only thing remaining to prove is that (1) is true. We have:

R d + min (∥u∥ ∞ , 1) ρ t (du) = (0,t]×R d + min (∥u∥ ∞ , 1) ρ (dt, du) = (0,t]×R d + min ϕ b(t) (u) ∞ , 1 λ (t) dt ρ (du) with ϕ b(t) (u) ∞ = ∥(b 1 (t) u 1 , • • • , b d (u) u d )∥ ∞ ≤ ∥b (t)∥ ∞ ∥u∥ ∞ and min ϕ b(t) (u) ∞ , 1 ≤ min (∥b (t)∥ ∞ ∥u∥ ∞ , 1) ≤ max (∥b (t)∥ ∞ , 1) min (∥u∥ ∞ , 1) . Hence R d + min (∥u∥ ∞ , 1) ρ t (du) ≤ (0,t] max (∥b (t)∥ ∞ , 1) λ (t) dt R d + min (∥u∥ ∞ , 1) ρ (du) ≤ (0,t] 1 + d ∑ i=1 b i (t) λ (t) dt R d + min (∥u∥ ∞ , 1) ρ (du) = Λ (t) + d ∑ i=1 (0,t] b i (t) λ (t) dt R d + min (∥u∥ ∞ , 1) ρ (du) < +∞
based on (34) and Assumption (1) for Y, which achives this proof.

⊓ ⊔

We next provide several expressions for Fτ (s) in the following corollary. The first expression is a direct consequence of Lemma 3 and Theorem 2, as

R d + \{0} φ (u) ρϕ -1 b(s) (du) = R d + \{0} φ ϕ b(s) (u) ρ (du)
for any non negative Borel function φ . Similar arguments as for Proposition 3, Corollary 3 and Theorem 3 can be used for the other expressions.

Corollary 6 Let X = (X (t)) t≥0 be an extended time-scaled Lévy process with parameter (b, ρ,Λ ) such that the underlying Lévy process Y (see (33)) fulfills Assumption 1 and (34). Considering X as conditional hazard rate increment process, we get the following results:

1. We have

Fτ (s) = e -H(s)-K(s) (36) with K (s) = F 1 -e -∑ d i=1 (s i -t) + b i (t)x i λ (t) dt ρ (dx) = - ∞ 0 ln L Y(1) (s 1 -t) + b 1 (t) , • • • , (s d -t) + b d (t) λ (t) dt for all s = (s 1 , • • • , s d ) ∈ R d
+ where H (s) is defined in (5). 2. In the bivariate case, assuming Y to be constructed through superposition as in Corollary 3, we have

K (s) = - s 1 0 ln L Z 1 (1) ((s 1 -t) b 1 (t)) λ (t) dt - s 2 0 ln L Z 2 (1) ((s 2 -t) b 2 (t)) λ (t) dt - max(s 1 ,s 2 ) 0 ln L Z 3 (1) (s 1 -t) + b 1 (t) + (s 2 -t) + b 2 (t) λ (t) dt.
3. Finally, assuming the dependence in Y to be modelled by a Lévy copula C with U i (s i ) the tail integral functions of ρ i , i = 1, 2, we have

Fτ (s) = exp R 2 + C (U 1 (s 1 ) ,U 2 (s 2 )) γ v,λ ,b (s) ds 1 ds 2 Fτ 1 (v 1 ) Fτ 2 (v 2 ) with γ v,λ ,b (s) = min(v 1 ,v 2 ) 0 b 1 (t) b 2 (t) (v 1 -t) (v 2 -t) e -(v 1 -t)b 1 (t)s 1 -(v 2 -t)b 2 (t)s 2 λ (t) dt for all s = (s 1 , • • • , s d ) ∈ R d + .
Remark 8 Remark 5 remains valid and τ exhibits the positive quadrant dependence property.

As an example, we now look at the case of a bivariate extended time-scaled gamma process constructed though superposition.

Example 3 Let Z i (t) ∼ Γ (a i t, 1) with a i > 0, i = 1, 2, 3, and let (Y i (t)) t≥0 , i = 1, 2 be constructed through [START_REF] Rausand | System Reliability Theory: Models, Statistical Methods, and Applications[END_REF].

Then Fτ i (s) = exp -H i (s) -(a i + a 3 ) s 0 λ (t) ln (1 + (s -t) b i (t)) dt
for i = 1, 2 and s > 0, and

K (s) = a 1 s 1 0 λ (t) ln (1 + (s 1 -t) b 1 (t)) dt + a 2 s 2 0 λ (t) ln (1 + (s 2 -t) b 2 (t)) dt + a 3 max(s 1 ,s 2 ) 0 λ (t) ln 1 + (s 1 -t) + b 1 (t) + (s 2 -t) + b 2 (t) dt for all s = (s 1 , s 2 ) ∈ R 2
+ , which provides the bivariate survival function through (36).

Remark 9 Considering H (s) = 0 for all s ∈ R 2 + , a 1 = 0 = a 2 = 0 and a 3 = 1, we get

Fτ (s) = exp - max(s 1 ,s 2 ) 0 λ (t) ln 1 + (s 1 -t) + b 1 (t) + (s 2 -t) + b 2 (t) dt = exp s 1 0 λ (t) ln (1 + (s 1 -t) b 1 (t) + (s 2 -t) b 2 (t)) dt × exp s 2 s 1 λ (t) ln (1 + (s 2 -t) b 2 (t)) dt
Comparing such results with Equations (2.2) and (2.3) from [START_REF] Singpurwalla | Multivariate distributions induced by dynamic environments[END_REF], even if it is not exactly the same model, it seems strange that s 1 -t and s 2 -t do not intervene in these two equations (just as st intervenes in [START_REF] Kebir | On hazard rate processes[END_REF]Theorem 3.1] for instance, providing a similar result as in the present paper).

Shock models

We finally explore the case where the impact of the stressing environment arrives through shocks according to a non homogenous Poisson process, and may have different impacts on the components. We begin with a general result, that we next decline through different cases.

Proposition 4 Let (N t ) t≥0 be a Poisson process on R + with intensity function λ (•) and points {T n , n ∈ N * }, and let Q (t, du) be a transition probability kernel from (R + , B (R + )) to R k + , B R k + . We set (U n ) n∈N * to be a sequence of non negative k-dimensional random vectors such that given {T n , n ∈ N * }, the U n 's, n = 1, 2, • • • are conditionaly independent with conditional distribution Q (T n , du). Finally let h :

R + × R k + , B R + × R k + -→ R d + , B R d + be a measurable function with h (t, u) = (h 1 (t, u) , • • • , h d (t, u))
for all t ≥ 0 and u ∈R k + , such that the function

t -→ (0,t]×R k + ∥h (v, u)∥ ∞ λ (v) dv Q (v, du) (37)
is continuous on R * + and such that

(0,t]×R k + min (∥h (v, u)∥ ∞ , 1) λ (v) dv Q (v, du) < ∞ (38)
for all t > 0.

Let us set:

X (t) = N t ∑ n=1 h (T n , U n ) = ∞ ∑ n=1 1 (0,t] (T n ) h (T n , U n ) for all t > 0.
Then (X t ) t≥0 is an additive process with intensity measure

ρ (ds, dx) = λ (s) ds Q h(s,•) (s, dx) ,
where, for each s, Q h(s,•) (s, dx) stands for the push-forward measure of Q (s, du) by the function h (s, •).

Taking (X t ) t≥0 as conditional hazard rate increment process, we get

Fτ (v) = e -H(v) exp - F 1 -e -∑ d i=1 (v i -s) + h i (t,x) λ (s) ds Q (s, dx) for all v = (v 1 , • • • , v d ) ∈ R d + .
Proof Based on [8, Theorem 3.2 page 264], we know that {(T n , U n ) , n ∈ N * } are the points of a Poisson random measure Ĵ (ds, du) with intensity measure ρ (ds, du) = λ (s) ds Q (s, du). Then, we may write

X (t) = ∞ ∑ n=1 1 (0,t] (T n ) h (T n , U n ) = R + ×R k + 1 (0,t] (s) h (s, u) Ĵ (ds, du)
for all t ≥ 0. The fact that X has independent increments can be proved in a similar way as in Lemma 3. Also, (X (t)) t≥0 is a right-continuous process with left-side limits by construction (and X (0) = 0 a.s.) and the stochastic continuity of (X (t)) t≥0 is a direct consequence of (37), as

P (∥X (v + s) -X (v)∥ ∞ > ε) ≤ 1 ε E (∥X (v + s) -X (v)∥ ∞ ) ≤ 1 ε (v,v+s]×R k + ∥h (t, u)∥ ∞ λ (t) dt Q (t, du)
for all ε, v, s > 0. Hence (X (t)) t≥0 is an additive process. Besides, we have E e -⟨y,X(t)⟩ = E e -y,

R + ×R k + 1 (0,t] (s)h(s,u) J(ds,du) = exp - R + ×R k + 1 -e -⟨y,1 (0,t] (s)h(s,u)⟩ λ (s) ds Q (s, du) = exp - t 0 R k + 1 -e -⟨y,x⟩ Q h(s,•) (s, dx) λ (s) ds for all y ∈ R d + , where Q h(s,•) (s, dx)
is defined in the proposition. This can be written E e -⟨y,X(t)⟩ = exp and next the expression for Fτ (v), based on Theorem 2. Finally, Assumption (1) for X is a direct consequence of (38), which ends this proof.

⊓ ⊔

In the specific case where the shocks induce i.i.d. (multivariate) increments U j , j = 1, 2, • • • in the hazard rates of the components, we obtain the following result.

Corollary 7 (Non-homogeneous compound Poisson process) Let X t be a multivariate compound Poisson process (CPP) of the shape

X t = N t ∑ j=1 U j ,
where (N t ) t≥0 is a Poisson process with intensity function λ (•), independent of the U j 's, which are i.i.d. d-dimensional non negative random variables such that E (U 1 ) is finite with µ (du) as common distribution. Then

Fτ (v) = e -H(v) exp - F 1 -e -∑ d i=1 (v i -s) + x (i) λ (s) ds µ (du) , for all v = (v 1 , • • • , v d ) ∈ R d + .
Proof With the notations of Proposition 4, we have

h (t, u) = u, Q (t, du) = µ (du) .
Using the fact that E (U 1 ) is finite, it is easy to check that (37) and (38) are true, and next derive the expression for Fτ (v).

⊓ ⊔

In Proposition 4, all the dependence between the τ i 's, i = 1, • • • , d, comes from the dependence between the U (i) j 's with

U j = U (1) j , • • • ,U (d) j
for all j ≥ 1. We next envision possible additional dependence between components due to possibly simultaneous fatal shocks, with a time-dependent impact of the shocks on the components. A fatal shock on a component is modelled through an infinite increment in its hazard rate at the time of the shock. More specifically, we assume that by a shock at time T n and for each subset of components C ⊂ {1, • • • , d}, there is a probability a C (T n ) that:

the shock is fatal to all components in C, the shock induces an increment in the hazard rate function of all other components (out of C).

Corollary 8 (Possibly fatal shocks and time-dependent impact of a shock) Let (N t ) t≥0 be a Poisson process on R + with intensity function λ (•) and points {T n , n ∈ N * }, and let Q (t, du) be a transition probability kernel from (R + , B (R + )) to Rd + , B Rd + , with R+ = R + ∪{+∞}. We set (V n ) n∈N * to be a sequence of non negative d-dimensional random vectors such that given {T n , n ∈ N * }, the V n 's, n = 1, 2, • • • are conditionaly independent with conditional distribution Q (T n , du) and

Q (t, du) = ∑ C⊂{1,••• ,d} a C (t) ν C (du)
where:

-the summation is taken over all subsets C of {1, • • • , d}; -for each t > 0, {a C (t) ,C ⊂ {1, • • • , d}} is a set of non negative real numbers such that ∑ C⊂{1,••• ,d} a C (t) = 1;
because the S i 's are known when X is known. Using the conditional independence for the intrinsic lifetimes (that is the U i 's when i ∈ F and the τ i 's when i / ∈ F), we get

E d ∏ i=1 1 {τ i >v i } X = ∏ i∈F 1 {S i >v i } ∏ i∈F E 1 {U i >v i } X ∏ i̸ ∈F E 1 {τ i >v i } X = ∏ i∈F E 1 {S i >v i } 1 {U i >v i } X ∏ i̸ ∈F E 1 {τ i >v i } X = d ∏ i=1 E 1 {τ i >v i } X for all v ∈ R d + .
Hence, the conditional independence still holds and we can use Lemma 1 for the process X. This provides

Fτ (v) = e -H(v) E e -∑ d i=1 v i 0 X i (t) dt .

Now, let us introduce the event

A v in which no fatal shock is induced in any component i during [0, v i ], with i = 1, . . . , d. Note that if a fatal shock is induced in a component i for a given T n < v i , then v i 0 X i (t) dt = ∞, so that E e -∑ d i=1 v i 0 X i (t)dt 1 Āv = 0. Hence Fτ (v) = e -H(v) E e -∑ d i=1 v i 0 X i (t)dt 1 A v . (39)
Our aim now is to compute this expression using an approximation procedure. To this end, for a given M > 0, let us define an approximate process (X M t ) t≥0 in which the measure ν C is changed into

ν M C (du) = ∏ i∈C δ M (du i ) × µ C (du C) ,
keeping the remaining elements. That is, each time a fatal shock is induced in the original process X, its hazard rate is increased of an amount M in X M (where M can be arbitrarily big).

Let FM τ be the multivariate survival function of the approximate process. Our aim is to show that lim

M→∞ FM τ = Fτ . (40)
Now, for each t > 0, let R t = t -T N t be the time elapsed at time t since the last arrival in the non-homogeneous Poisson process. Observe that on the set Āv and provided that component i has a fatal shock before v i , we have

v i 0 X M i (t) dt ≥ MR v i and therefore lim M→∞ E e -∑ d i=1 v i 0 X M i (t)dt 1 Āv ≤ lim M→∞ E e -MR v i 1 Āv = 0.
Thus, taking into account the fact that on the set A we have

X M i (t) = X i (t), 0 ≤ t ≤ v i , we now have lim M→∞ E e -∑ d i=1 v i 0 X M i (t)dt = lim M→∞ E e -∑ d i=1 v i 0 X M i (t)dt 1 A v = E e -∑ d i=1 v i 0 X i (t)dt 1 A v .
Using Lemma 1 for the approximate process X M and (39), we obtain that lim

M→∞ FM τ (v) = lim M→∞ e -H(v) E e -∑ d i=1 v i 0 X M i (t)dt = e -H(v) E e -∑ d i=1 v i 0 X i (t)dt 1 A v = Fτ (v) ,
thus showing (40). Now, applying Proposition 4 to the approximate process, we have

FM τ (v) = e -H(v)-K M (v) , (41)
with

K M (v) = F 1 -e -∑ d i=1 (v i -s) + x i λ (s) ds Q M (s, dx) for all v = (v 1 , • • • , v d ) ∈ R d + such that v 0 = 0 < v 1 < • • • < v d . Noting that the integrand is null when s > v d , we get K M (v) = v d 0 R d + Q M (s, dx) λ (s) ds - v d 0 R d + e -∑ d i=1 (v i -s) + x i Q M (s, dx) λ (s) ds = Λ (v d ) - d ∑ j=1 K M j (v) , (42)
where

K M j (v) = v j v j-1 R d + e -∑ d i=1 (v i -s) + x i Q M (s, dx) λ (s) ds = v j v j-1 R d + e -∑ d i= j (v i -s)x i Q M (

Concluding remarks and perspectives

We have here provided a new model for a multivariate lifetime, which encompasses many models from the previous literature and allows several types of dependence between components: simultaneous failures, simultaneous increase of the individual failure rates of the components and some dependence between the simultaneous individual failure rates increments.

Even if the results have been provided separately for the different hazard rate increment processes envisioned in the paper (Lévy processes with extensions and shock processes), the results could easily be extended to the case of an additive process of the shape Z t = X t + Y t , ∀t ≥ 0, where (X t ) t≥0 and (Y t ) t≥0 are independent additive processes studied in the paper, which allows to still enlarge the results.

In the same way, most results have been provided under the assumption that the additive process has bounded variations, but, as we have seen in the last model, it is possible to relax this assumption by means of an approximation procedure, which allows to enlarge the possible scope in another way.

Finally, note that for application purpose, statistical estimation procedures remain to be developed, which could be the study of a future work. Typically, such procedures would highly depend on the available data and specifically on the possibility to monitor the additive process or not, in addition to the observation of lifetimes.

As noted in the introduction, the model developped in the paper mostly leads to lifetimes with the Increasing Failure Rate (IFR) property, that is with a lasting effect of an adverse environment on the failure rate. This is for instance well adapted to the reliability field (and also to some biostatistics models). However, it could be questionable in other contexts. In that case, as suggested by a first referee, it could be interesting to consider the impact of the additive process on the cumulated hazard rate function instead of an impact on the hazard rate function. This would provide a very interesting alternate model, which could allow for non monotonic hazard rates. A second referee pointed out another possibility with a similar purpose by considering an affine Markov process as conditional hazard rate increment process. This would allow to enlarge the results of the paper as additive processes are specific affine Markov processes, please see [START_REF] Duffie | Affine Processes and Applications in Finance[END_REF][START_REF] Eberlein | Mathematical Finance[END_REF][START_REF] Filipović | Time-inhomogeneous affine processes[END_REF] for more details on affine Markov processes.

These two pists provide very promising alternate models, which could allow for a non monotonic conditional hazard rate increment process. Their mathematical tractability remains to be studied.

3 2 -b 3 -bv with x = √ b 2 +

 32 2u in the second line, which provides the result for Fτ (v).

Fig. 1

 1 Fig.1The increment in the unconditional failure rate for a gamma and an inverse gaussian increment process, with a = 1 and b = 2 (left), b = 0.25 (right).

Fig. 2

 2 Fig. 2 The functions Fτ -Fτ ⊥ (left) and Fτ ∥ -Fτ (right). Example 1.

R d + 1 -

 1 e -⟨y,x⟩ ρ t (dx)with ρ t (dx) = t 0 Q h(s,•) (s, dx) λ (s) ds = ρ ((0,t], dx) , which provides ρ (ds, dx) = λ (s) ds Q h(s,•) (s, dx)

∑CCj v j- 1 eC∑C 1 LRemark 10 Example 4 ( 1 0Lexp v 2 v 1 L µ 2

 111041212 s, dx) λ (s) ds d i= j (v i -s)x i ν M C (dx) a C (s) λ (s) ds for 1 ≤ j ≤ d.In the case where { j,• • • , d} ∩C ̸ = ∅, let i C ∈ { j, • • • , d} ∩C. Remembering that the i-th margin of ν M C is δ M , we have: lim M→∞ ⊂{1,••• ,d} such that { j,••• ,d}∩C̸ =∅ d i= j (v i -s)x i ν M C (dx) a C (s) λ (s) ds ≤ lim M→∞ ∑ ⊂{1,••• ,d} such that { j,••• ,d}∩C̸ =∅ v -(v ic -s)M a C (s) λ (s) ds = 0. (43)Hence, we only need to consider the case where { j, • • • , d} ∩C = ∅. Note that in that case, components j, j + 1, • • • , d do not fail, so that the corresponding margin in ν M C coincides with the corresponding marginµ j•••d dx j•••d in the original process. Hence (x 1 ,...x j-1 )∈R j-1 + ν M C (dx) = µ j•••d dx j•••d and ∑ ⊂{1,••• ,d} such that { j,••• ,d}∩C=∅ d i= j (v i -s)x i ν M C (dx) × a C (s) λ (s) ds = ⊂{1,••• ,d} such that { j,••• ,d}∩C=∅ d i= j (v i -s)x i µ j•••d dx j•••d a C (s) λ (s) ds = ∑ C ⊂{1,••• ,d} such that { j,••• ,d}∩C=∅ v j v j-µ j•••d (v js, • • • v ds) a C (s) λ (s) ds.(44)Taking into account (43) and (44), we conclude thatlim M→∞ K M j (v) = ∑ C ⊂{1,••• ,d} such that { j,••• ,d}∩C=∅ v j v j-1 L µ j•••d (v js, • • • v ds) a C (s) λ (s) ds,which, together with (41), (42) and (40) ends this proof. ⊓ ⊔ The previous result is given in the case where v 1 < • • • < v d but all other cases can be easily derived by symmetry. Case d = 2) In the specific case where d = 2, the previous corollary providesFτ (v) = e -H(v)-Λ (v 2 ) exp v µ (v 1s, • • • v 2s) a ∅ (s) λ (s) ds × (v 2s) (a ∅ (s) + a 1 (s)) λ (s) ds for all v = (v 1 , v 2 ) ∈ R 2 + such that v 1 < v 2 .This is the result of [23, Proposition 1], which already enlarged [6,Theorem 1]. Hence the previous corollary enlarges several results from the previous literature.

Acknowledgments. Both authors thank the Editor in Chief and the reviewers for their constructive comments on the model. This has lead us to better motivate the choice of the model and open the path for very interesting alternate models.

is a probability measure on R d + and where µ C (du C) stands for the marginal distribution of µ corresponding to indexes i ∈ C.

as conditional hazard rate increment process, we get

Proof Note firstly that the process X is not of bounded variation as components of V j 's may be infinite, so that we are not in the settings of the previous results. However, it can be seen that Lemma 1 still holds considering X as conditional hazard rate increment process. Indeed, looking at the details of the proof, the only point to check is the fact that the conditional independence assumption between components lifetimes (given X) still holds. As a matter of fact, when X is known, the set F ⊆ {1, . . . d} of components which are subject to an infinite increment in their hazard rate is known, together with the corresponding shock times. For each i ∈ F, let S i be the smallest among these shock times, which is called possibly fatal shock time in the following. Note that these possibly fatal shock times can coincide among components. Note also that a possibly fatal shock on a component may occur after an intrinsic failure of the component (that is a failure due to its failure rate) and then nothing happens at the corresponding shock time. Then, for i ∈ F, the lifetime of component i is the minimum between its possibly fatal shock time S i and its intrisic lifetime (say U i ).

With this in mind, we can write

Conflict of interest statement. On behalf of all authors, the corresponding author states that there is no conflict of interest.


Kariman Abelin-Genevois

Eva Polirststok

Erik Estivalèzes

Jérôme Briot

Annick Sevely

Jérôme Sales De Gauzy

Pascal Swider

Eva Polirsztok

Erik Estivalezes

Jerome Briot

Hydration properties of the lumbar intervertebral discs in AIS after surgical correction:five years follow-up and comparison with an age-matched control group

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  Open Archive TOULOUSE Archive Ouverte (OATAO)OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This is an author-deposited version published in : http://oatao.univ-toulouse.fr/ Eprints ID : 15778 To link to this article : URL: http://link.springer.com/article/10.1007/s11832-015-0637-0 To cite this version : Abelin-Genevois, Kariman and Polirststok, Eva and Estivalèzes, Erik and Briot, Jérôme and Sevely, Annick and Sales de Gauzy, Jérôme and Swider, Pascal Hydration properties of the lumbar intervertebral discs in AIS after surgical correction:five years follow-up and comparison with an age-matched control group. (2015) In: EPOS 34th Annual Meeting, 15 April 2015 -18 April 2015 (Marseille, France).


Eric Kulanthaivelu

GROWTH : THE IMPORTANCE OF

MOTIVATION

Tropical cyclones are arguably one of the most damaging and threatening natural disasters for human systems. Among other examples, the 2005 Hurricane Katrina caused the displacement of approximately 650,000 people and destroyed more than 200,000 homes along the US Gulf Coast. A number of empirical studies have explored the short and long-run economic relationship between tropical cyclones and national growth rates, but no general conclusion can be drawn from them so far. While negative effects are found in samples of exposed countries worldwide, cyclone shocks also show no significant influence in other national-level analyses. This suggests that, beyond inequalities in the exposure to cyclonic risk between countries, there is also inequality regarding these extreme weather events' impacts.

PAPER'S OBJECTIVE AND CONTRIBUTION

This paper explores key dimensions across which one might expect the economic impact of cyclone shocks on economic growth to be heterogeneous. First, since cyclone events are primarily localized along coastal areas, they may only affect a reduced share of national economies. Another aspect to consider is the level of development, as developing countries tend to have agricultural-based economies, they may experience stronger effects from cyclone shocks. In fact, cyclones can trigger a number of catastrophes like floods or landslides, potentially causing losses of crops that spread to the rest of the economy. According to these criteria, the sub-group of Small Islands Developing States (SIDS) is distinguished from the world sample and is specifically analysed. Splitting the initial sample in such a way reveals that in addition to economic development, country size provides another shield against growth losses due to storm shocks.

DATA

The raw data on cyclone events come from the Tropical Cyclone Exposure Database (TCE-DAT;Geiger & al., 2018), which provides consistent pixel-event-level data on cyclone intensity and exposure from 1950 to 2015. Data on precipitation and temperature levels were obtained from the Climatic Research Unit gridded Time Series database (CRU TS; Harris & al., 2020). Information on per capita GDP come from the United Nations Statistics Division (UNSD, 2020).

MÉTHODOLOGY

This study aims to assess the short to long-term impact of cyclone events on per capita GDP growth rate using an exogenous and scale-invariant predictor of cyclone events, adding controls for temperature and precipitation. Panel data regression is used with time and country fixed effects, and the purpose is to study the marginal cumulative effect over 15 years after a cyclone strike. (2021) show that these extreme events are not disruptive enough to be reflected in national economic growth rates in the US or in Hong-Kong respectively. Apart from the geographic dimension, another aspect to consider is the level of development. In the more general framework of natural disasters, developing countries are almost always found to be more adversely affected than advanced economies (Kahn, 2005 ;Noy, 2009 ;Fomby & al., 2013).

RESULTS

Worldwide estimations are led by SIDS, and hence, national growth responses to cyclone events are heterogeneous given the size and the development level of the country. The estimated immediate effect of one additional unit of cyclone exposure is to decrease growth rates in SIDS by -0.016 percentage point, and this negative effect drops to -0.024 percentage point after 15 years. The quasi-constant slope indicates that SIDS manage to avoid an intensification of negative effects, but do not manage to counteract them either. In contrast, point estimates tend to show that non SIDS countries experience an increased growth in the short-run, which plummets in the long-run, but these results are not statistically significant and no conclusion can be drawn from them.

TRANSMISSION CHANNELS AND PUBLIC POLICY IMPLICATIONS

Negative impacts in SIDS appear to be driven by an increased dependence on foreign economic conditions, insufficient reconstruction capacities and difficulties to implement adaptation policies. Policy implications regarding the urge to invest in economic preparedness or early warning systems in SIDS shall emerge from this study.

ROBUSTNESS CHECKS

Results obtained for SIDS are consistent with the neoclassical growth theory in the very short-run as it predicts an immediate negative impact on economic growth. Three core aspects are highlighted by the table below. It confirms the exogeneity of the cyclone indicator as estimates are of same magnitude given 95 % confidence bands. Then, it confirms that lag structures are not particularly influential and short-term impacts are most decisive. Third, the negative sign associated with the lagged value of per capita GDP shows that the economy is still converging towards its steady state level after the cyclone strike.

Figure 1 :

 1 Figure 1: Annual average maximum wind speed (in km/h) from 1950 to 2015 at pixel level for North Atlantic and North East Pacific tropical cyclone basin

  (2014), Hsiang & Jina (2014) or Krichene & al. (2021) find evidence of negative and persistent effects of tropical cyclones on economic growth using universal datasets. In other national level analyses, Strobl (2011) or Zhou & Zhang

Figure 2 :

 2 Figure 2: Measures of the marginal cumulative effects over fifteen years of cyclone events on per capita GDP growth for SIDS (left) and non SIDS (right).