Zeina Al Masry

Landy Rabehasaina

Ghislain Verdier

Change-level detection for Lévy subordinators

Keywords: Change detection, Lévy process, subordinator, CUSUM, sequential testing

published or not. The documents may come

Introduction

Let X = (X t ) t≥0 be a continuous stochastic process. More precisely, we here assume that X is a monotone increasing Lévy process prior to hitting m 0 , then it is another increasing Lévy process once this threshold is crossed. The process X = (X t ) t≥0 is observed on-line and m 0 ∈ [0, +∞] is unknown. In this context, it naturally appears the need to detect the change, if it took place. One could think about processes that describe gradual deterioration due to continuous use such as erosion, corrosion, concrete creep, crack propagation (see [START_REF] Van Noortwijk | A survey of the application of gamma processes in maintenance[END_REF] in the case of a gamma process). A unit system could have an accelerating degradation whenever its degradation level crosses m 0 . This latter is usually unknown and an abrupt change in the degradation could be one of the causes that complicate the determination of the failure time of the system. Recall that industries seek to maintain their equipment's available while minimizing their total maintenance cost including the unavailability cost. For that purpose, many maintenance policies have been proposed in the literature. The maintenance decision could be based on many characteristics such as the reliability level, the cost function, the remaining useful life or the change time. We here are interested in the change time characteristic.

Recall that the theory of change detection consists in developing tools to detect the change as soon as possible and by taking into consideration the false alarm constraints. Many works in the literature have studied the online change detection for continuous-time stochastic processes which are Lévy processes. In [START_REF] Davis | A note on the Poisson disorder problem[END_REF] and [START_REF] Peskir | Solving the Poisson disorder problem[END_REF], the authors studied the problem Preprint submitted to Stochastic Processes and their Applications of Poisson disorder problem which seeks to get a stopping time which is close to the time of disorder (or change-point) when the intensity of an observed Poisson process changes from one value to another one at a certain (unknown) time. A revisited version of the latter work has been proposed by [START_REF] Bayraktar | Quickest detection of a minimum of two Poisson disorder times[END_REF] by providing a complete solution of the Poisson disorder problem. Recently, [START_REF] Krawiec | Quickest drift change detection in Lévy-type force of mortality model[END_REF] proposed to solve the quickest drift change detection problem for a Lévy process under the Bayesian set-up by assuming an exponential a priori distribution of the change point. Furthermore, the authors in [START_REF] Fouladirad | Condition-based maintenance for a system subject to a nonhomogeneous wear process with a wear rate transition[END_REF][START_REF] Fouladirad | On-line change detection and condition-based maintenance for systems with unknown deterioration parameters[END_REF][START_REF] Fouladirad | On the use of on-line detection for maintenance of gradually deteriorating systems[END_REF] have studied the online change detection for a gamma process in the framework of the condition-based maintenance strategy. They used the classical CUSUM rule to determine the change time. As for [START_REF] Figueroa-López | Change-point detection for Lévy processes[END_REF], they considered the change detection problem for continuous-time Lévy processes by approximating an adapted sequence of change-point problems and where the optimality of a CUSUM rule is shown. To sum up, in the previous works, the proposed techniques were based on an a priori distribution for the change-time or a deterministic unknown change-time.

The aim of this paper is to propose a detection level rule to ensure a quick detection when the degradation of a unit system crosses m 0 while minimizing the false alarm rate. For that purpose, we consider a procedure inspired from a CUSUM detection rule applied to intervals between jumps larger than some given constant, rather than on the increments of the process as it is the case in the usual setup. Moreover, in the classic methodology of on-line change detection, the change is related to the temporal aspect. As a natural consequence, the performance criteria of the change detection rules result from the mean time between false alarms when there is no change and the mean time before the detection of a change. These quantities are named the Average Run Lengths (and denoted respectively ARL ∞ and ARL 0 ). However, in our case, the change is no more related to a temporal aspect but rather on what we could call a spatial aspect: the change takes place when the system reaches a given level m 0 . Consequently, we here consider, instead of the Average Run Length, a kind of Average Run Level (ARLev) criterion for the evaluation of the detection rules. Roughly speaking, we are interested by the accumulated level since the change rather than the delay before the detection. To motivate this approach, if we consider again the context of the accumulative deterioration of a system, one can imagine that the level of the accumulated degradation since the change occurred is just as important as the delay for detection (even if, obviously, the two are related).

The remainder of the paper is as follows. Section 2 provides a quick presentation of the CUSUM procedure. The proposed methodology along with the main results are presented in Section 3. In Section 4, the proofs of the main results are exposed. Finally, numerical results are given in Section 5. In particular, the present detection procedure is compared to a naive CUSUM approach and is shown to perform better on some examples.

A review of the CUSUM procedure for discrete time observed sequences

In the classic online change-point detection problem, it is assumed that a sequence of i.i.d. random variables Z 1 , Z 2 , . . . , with probability density function (pdf) f 1 , is observed sequentially, until a change occurs at an unknown instant denoted K ∈ N. After the change, the observations Z K , Z K+1 , . . . are again i.i.d. but with a pdf f 2 such that f 2 = f 1 . K is called the change time. We can write:

Z k = Z 1 k 1 [k<K] + Z 2 k 1 [k≥K] , k ∈ N, (1)
where (Z 1 n ) n∈N and (Z 2 n ) n∈N are two i.i.d. sequences with respectively common pdf f 1 and f 2 .

Whenever a new observation is collected, a decision must be made: either there is no evidence against the hypothesis of no change and the system is declared under control and the monitoring continues, waiting for the next observation, or there is clear evidence that a change occurred in the past and an alarm is issued resulting in the monitoring to stop. This decision is taken from a change-detection rule, whose aim is to detect as soon as possible the change from the two operating modes, guaranteeing a low false alarm rate.

The unknown change time K can be either deterministic or random. In the latter case, K is supposed to be a random variable which can be dependent on the observations or completely independent of the observations. The change-point detection rules then rely on a prior distribution, that means a sequence of probabilities π n = P [K = n|Z 1 , . . . , Z n ] for n ∈ N. In this bayesian setting, [START_REF] Shiryaev | On optimum methods in quickest detection problems[END_REF] obtains the asymptotic optimality of a rule based on a likelihood ratio in the case where the prior distribution of the change time is supposed to be geometric and independent of the observations. Since then, its works were extended to non-independent observations, to more general prior distributions and considering several optimality criteria (see for example [START_REF] Tartakovsky | Asymptotic optimality in bayesian changepoint detection problems under global false alarm probability constraint[END_REF], [START_REF] Tartakovsky | State-of-the-art in bayesian changepoint detection[END_REF] or even [START_REF] Tartakovsky | General asymptotic bayesian theory of quickest change detection[END_REF] for an overview). Up to our knowledge, in all the previous papers, a prior distribution (usually the geometric distribution) on the change time should be assumed. As for our case, the procedure is different since we are here interested in the level detection time. As a consequence, the latter cannot have an imposed prior distribution.

In the non-Bayesian setting, the most popular change-detection rule is probably the CUSUM, initially proposed by [START_REF] Page | Continuous inspection schemes[END_REF] in 1954. The CUSUM consists in constructing some likelihood ratio between two hypothesis:

• H 1 : K = +∞ • H 2 : K < ∞
At time n, the CUSUM statistic is defined by

g n = max 1≤k≤n n i=k log f 2 (Z i ) f 1 (Z i ) . (2)
The CUSUM stopping rule is then defined by

τ CU SU M = inf{n ≥ 1 : g n ≥ γ} (3)
where γ > 0 refers to a given threshold. It is standard, as γ > 0, that the test statistic in the definition (3) of τ CU SU M can be written, equivalently, in the following recursive form:

g n+1 = g n + log f 2 (Z n+1 ) f 1 (Z n+1 ) + (4)
with g 0 = 0 and a + = max(a, 0), see [15, (2.2.9) p.38], or in the following form

g n = max 1≤k≤n n i=k log f 2 (Z i ) f 1 (Z i ) ∨ 0 (5)
which will be used throughout the paper. Let us define the set F of all monitoring schemes, i.e. of all stopping times adapted to the filtration induced by (Z i ) i∈N . For all positive real number h, let us define F h by

F h = T ∈ F : E 1 T := E (∞) (T ) ≥ h , (6)
i.e. the set of all monitoring schemes such that the mean time before a false alarm is larger than h. Let also the worst mean delay for detection be defined as

E 2 T = sup K≥1 ess supE (K) [(T -K + 1) + |Z 1 , . . . , Z K-1 ]. (7)
In Equations ( 6) and ( 7), E (∞) and E (K) respectively refer to the expectation with respect to the probability distribution P (∞) of no change and probability distribution P (K) of a change at time K.

In [START_REF] Lorden | Procedures for reacting to a change in distribution[END_REF], the asymptotic optimality of the CUSUM is obtained by showing that

E 1 τ CU SU M ≥ exp(γ), (8)
meaning that τ CU SU M in (3) belongs to F h by considering γ = log h, and that

E 2 τ CU SU M ∼ log h I as h → ∞ (9)
∼ inf

T ∈F h E 2 T as h → ∞, (10)
where I refers to the Kullback-Leibler (KL) distance between the two distributions with respective densities f 1 and f 2 , which is defined as follows

I = KL(f 2 ||f 1 ) = log f 2 (x) f 1 (x) f 2 (x)dx. (11)
Among the most significant extended results of [START_REF] Lorden | Procedures for reacting to a change in distribution[END_REF] on the CUSUM rule, we may cite [START_REF] Moustakides | Optimal stopping times for detecting changes in distributions[END_REF] who obtains the optimality of the CUSUM for a fixed value for h (i.e. a non asymptotic optimality result) and [START_REF] Lai | Information bounds and quick detection of parameter changes in stochastic systems[END_REF] who considers dependent observations, and also several other optimality criteria.

Description of the model and main results

Let us now consider an increasing stochastic process X = (X t ) t≥0 which behaves as one of two given processes X 1 or X 2 , depending on whether it is below or above a certain (unknown) threshold m 0 ∈ [0, +∞]. More precisely, both processes X 1 = (X 1 t ) t≥0 and X 2 = (X 2 t ) t≥0 are Lévy processes with respective characteristic exponents

ψ j (θ) = ia j θ + R (1 -exp(iθx) + iθx1 [|x|<1] )Q j (dx)
where θ ∈ R and Q j (.), j = 1, 2 refers to the Lévy measure that verifies E[exp(iθX j t )] = exp(-tψ j (θ)), ∀t ≥ 0, see [START_REF] Kyprianou | Introductory lectures on fluctuations of Lévy processes with applications[END_REF]Section 1.1 p.4]. Let us consider the following assumptions

(A 1 ) Q j (-∞, 0) = 0, d j = -(a j + (0,1) xQ j (dx)) = 0 and (0,∞) (1 ∧ x)Q j (dx) < ∞, j = 1,
(A 2 ) (0,∞) x 2 Q j (dx) < ∞ (this implies that (0,∞) xQ j (dx) < ∞ based on (A 1 )),
(A 3 ) Q j ((0, +∞)) = +∞ for j = 1 or j = 2, and possibly both at the same time,

(A 4 ) lim inf →0 Q2 ( ) Q1 ( ) -1 > 0, where Q j ( ) = Q j (( , +∞)) for j = 1, 2.
These assumptions may be interpreted as follows. (A 3 ) means that at least one of the two processes (X 1 or X 2 ) admits infinitely many small jumps in a finite time interval. Assumption (A 4 ) is used to discriminate between X 1 and X 2 . Indeed, Q j ( ) represents the average number of jumps greater than by unit of time [START_REF] Cinlar | Introduction to stochastic processes[END_REF]Remark 1.16 p.318]. Intuitively speaking, this assumption says that the order of magnitude of the number of jumps larger than is different when tends to 0. Assumptions (A 1 ), (A 2 ) and (A 4 ) are supposed to hold throughout the paper. In the sequel, the study of the proposed detection level rule will be done when Assumption (A 3 ) holds but also when it does not hold, i.e. when X 1 and X 2 are both Compound Poisson processes. Also, both processes X 1 and X 2 being subordinators (see (A 1 )), we recall that they may be characterized by their Laplace exponent φ j (θ) , instead of the characteristic exponent ψ j (θ), j = 1, 2, that verify E[exp(-αX j t )] = exp(-tφ j (α)) for all t ≥ 0 and α ≥ 0. Since the subordinators are driftless processes, those Laplace exponent have here the simple expression

φ j (α) = (0,+∞) (1 -e -αx )Q j (dx), α ≥ 0, j = 1, 2, see [21, Section 2.2 page 9].
The latter processes allow for the process X to be now written as

X t = X 1 t 1 [t≤τm 0 ] + (X 2 t-τm 0 + X 1 τm 0 )1 [t>τm 0 ] , ∀t ≥ 0, (12)
with the crossing time of level m 0 of the process X 1 defined as

τ m0 = inf{t ≥ 0|X 1 t ≥ m 0 }, (13)
where m 0 ∈ [0, ∞] is unknown. Moreover, following [19, Section 2.4, p.44], and because X is driftless, it will be convenient to note that, thanks to (A 2 ) and more precisely to the fact that (0,∞) xQ j (dx) is finite, each process X j , j = 1, 2, may be expressed in function of a Poisson random measure N j on ([0, +∞) × (0, +∞), B([0, +∞)) × B((0, +∞)), η j ) as follows

X j t = [0,t] (0,+∞) xN j (ds × dx), t ≥ 0, (14)
where η j refers to a measure on ([0, +∞) × (0, +∞), B([0, +∞)) × B((0, +∞)) given by η j (ds × dx) = ds Q j (dx). Throughout the paper, E m0 (.) refers to the expectation under the assumption that the process X change its behavior when it exceeds m 0 and E ∞ (.) to the expectation where m 0 = ∞, i.e. when there is no change in the behavior of X, so that X t = X 1 t for all t ≥ 0, see [START_REF] Tartakovsky | State-of-the-art in bayesian changepoint detection[END_REF]. The aim of the paper is to determine, by sequentially observing a sample path t ≥ 0 → X t , a detection rule which achieves the two following goals:

1. To guarantee a quick detection of the crossing of level m 0 if it occurs, such that the overshoot of the underlying process at the time of detection is not too large with respect to the fixed threshold m 0 .

2. To provide a low false alarm rate if there is no change, that means m 0 = +∞.

This will be achieved by proving the forthcoming main Theorems 2 and 3. Theorem 2 holds when (A 3 ) is satisfied and Theorem 3 holds when (A 3 ) is not satisfied.

Construction of the -detection rule

Let > 0 and (T i ) i∈N the sequence constructed from the observed trajectory X t , t ≥ 0, in the following way:

T 0 = 0, T i+1 = inf{t > T i | ∆X t = X t -X t-> }, i ≥ 0. ( 15
)
Thus, the T i , i ∈ N, are the successive times when an observed jump of the process X is larger than a given . The corresponding inter-arrival times are then defined by:

η i = T i -T i-1 , i ≥ 1. ( 16
)
Intuitively, due to the Lévy nature of the processes X 1 and X 2 , η i seems to be exponentially distributed with parameter Q1 ( ) if τ m0 > T i or parameter Q2 ( ) if τ m0 < T i-1 . In other words, η i looks either distributed as E( Q1 ( )) or E( Q2 ( )) whether we are before or after having crossed the threshold. Although this latter statement is not rigorous and is only intuitively correct, this however motivates the use of the following CUSUM type rule. More precisely, we introduce the CUSUM statistic (G η n ) n∈N associated to the sequence (η i ) i∈N as in (4) by G η 0 = 0 and

G η n+1 = G η n + φ (η n+1 ) + , n ≥ 0 (17)
where

x ∈ [0, +∞) → φ (x) := log Q2 ( ) Q1 ( ) + (-Q2 ( ) + Q1 ( ))x := a + b x (18)
is the logarithm of the likelihood ratio of the exponential distributions with respective parameters Q2 ( ) and Q1 ( ).

Remark 1. One can verify easily that φ (.) is linear, and is increasing (resp. decreasing) when Q2 ( ) ≤ Q1 ( ) (resp. ≥).

Finally, we set the associated stopping rule related to a threshold γ( ) (which will be made explicit later on), by

τ ,η CU SU M := inf{n ≥ 0| G η n ≥ γ( )}, (19)
and the associated "pseudo-level":

M = τ ,η CU SU M i=1 ∆X T i = τ ,η CU SU M i=1 X T i -X T i -. (20)
The idea behind the above construction is that the smaller is, the closer M is from X at the detection time defined by

d = τ ,η CU SU M i=1 η i . (21)
In this case, we will denote by L the detection level defined as follows:

L = X d . (22)
Figure 1 displays all the quantities above that we have used for the construction of the detection rule. One should note the importance of comparing the proposed model with the classical one described in Section 2 where the model features an unknown (but not random) instant change of the process behavior. In our case, the process behavior changes once it exceeds a certain threshold. The main idea of this paper is to exchange the role of time and space, so that the analog of time detection τ CU SU M in (3) is the level detection L in [START_REF] Shaked | Stochastic orders[END_REF].

t τm 0 Xt X 1 t X 2 t-τm 0 + X 1

Main results

The three following theorems are the main contributions of the paper.

Theorem 1. Assume that (A 1 ) -(A 4 ) hold and define the CUSUM statistics τ ,η CU SU M in (19) with γ( ) := log h( ) and

h( ) := [ Q2 ( )I ] 2 if Q2 (0) = +∞, [ Q1 ( )] β if Q2 (0) < +∞ (and consequently Q1 (0) = +∞), ( 23
)
where β > 2 is arbitrary, and

I = log Q2 ( ) Q1 ( ) -1 + Q1 ( ) Q2 ( ) , ∀ > 0. ( 24
)
Then, for all > 0, there exist two positive quantities c 1 and c 2 such that the following properties are satisfied:

(P 1 ) E m0 ([M -m 0 ] + ) ≤ c 1 , for all > 0, (P 2 ) E ∞ (M ) ≥ c 2 , for all > 0, (P 3 ) c 1 = o(c 2 ) and lim →0 c 2 = +∞.
In the case where the following assumption holds

lim sup →0 Q1 ( ) Q2 ( ) < ∞, (25)
then (P 3 ) can be made more precise as

(P 3 ) lim sup →0 c 1 < ∞ and lim →0 c 2 = +∞.
Theorem 2. Under the assumptions of Theorem 1, it holds that

lim →0 E m0 (L -M ) = 0, ∀m 0 ≥ 0, ( 26
)
and the properties (P 1 ), (P 2 ), (P 3 ) and (P 3 ) are still valid by substituting M by L .

Remark 2. Note that E m0 ([Lm 0 ] + ) corresponds to the mean overshoot of the process X at detection time above the threshold m 0 . In a degradation context, this corresponds to the mean degradation above m 0 . E ∞ (L ) is the mean detection level when there is no regime change, which will be denoted as the Average Run Level (ARLev ∞ ). This latter quantity is the analog of the Average Run Length (ARL) in the usual temporal context for detection rules. Properties (P 1 ) and (P 2 ) in Theorem 2, applied to L , combined with property (P 3 ) ensure that, as tends to 0, the mean overshoot of the process is negligible against the mean level when there is no regime change E ∞ (L ). Even better, (P 3 ) guarantees that the mean overshoot is bounded, which is useful in practical situations. Moreover, in some specific cases (such as when X 1 is a gamma process or an inverse Gaussian process), it can be easily verified from the proofs of Theorems 1 and 2 that the quantities c 1 and c 2 could be explicitly expressed for a fixed .

When the assumption (A 3 ) is not satisfied, the processes X 1 and X 2 are compound Poisson processes. In that case, the detection procedure is slightly different. The jump and interarrival times of the process X are defined as

T 0 := 0, T i+1 := inf{t > T i | ∆X t = X t -X t-> 0}, i ≥ 0, η i := T i -T i-1 , i ≥ 1.
The sequence (T i ) i∈N corresponds to the jump times of X 1 when T i is less than τ m0 (i.e. before the process crossed the level m 0 ), and to the jump times of X 2 when T i is larger than τ m0 . In other word, this corresponds to (T i ) i∈N in (15) with = 0. We aim here at devising a "classical" CUSUM rule to the sequence (η i ) i∈N defined above. More precisely we let the associated CUSUM statistic (G n ) n∈N , defined in [START_REF] Moustakides | Optimal stopping times for detecting changes in distributions[END_REF], with G 0 = 0 and

G n+1 = (G n + φ 0 (η n+1 )) + , n ≥ 0,
where we recall from ( 18) that φ 0 (x) = log Q2 (0) Q1 (0) + (-Q2 (0) + Q1 (0))x, and

τ η CU SU M := inf{n ≥ 0| G n ≥ log h}, h > 1. (27)
Finally, the associated level is defined as

L = τ η CU SU M i=1 ∆X Ti = τ η CU SU M i=1 X Ti -X Ti -. (28)
The equivalent of Theorems 1 and 2 in the present case is as follows.

Theorem 3. Let us suppose that Assumptions (A 1 ) -(A 2 ) hold and that X 1 and X 2 are compound Poisson processes with finite different intensities Q1 (0) and Q2 (0). Then there exists c h 1 and c h 2 such that the following properties are satisfied

(P 0 1 ) E m0 ([L -m 0 ] + ) ≤ c h 1 for all h > 1, (P 0 2 ) E ∞ (L) ≥ c h 2 for all h > 1, (P 4 ) c h 1 = o(c h 2 ) as h → +∞ and lim h→+∞ c h 2 = +∞.
Remark 3. Similarly to Remark 2, Theorem 3 identifies the detection level by detecting a change in the respective intensities. When the intensities are equal, then the jumps necessarily have distinct distributions whenever the process X t is less or larger than m 0 . An intuitive solution for detecting the change would be to perform a CUSUM rule on the successive jump sizes of the process. However, theoretical results on the detection level with this latter solution seem to be difficult to prove, and this case remains an open problem.

Example: Gamma Processes

Setting A : R + → R + to be a measurable, increasing and right-continuous function with A(0) = 0 and b > 0, let us recall that a standard (non homogeneous) gamma process Y = (Y t ) t≥0 , with A(.) as shape function and b as scale parameter (denoted by Y ∼ Γ(A(.), b)), is a stochastic process with independent, non-negative and gamma distributed increments such that Y 0 = 0 almost surely. The pdf of an increment Y t -Y s (with 0 < s < t) is given by

f (x) = b A(t)-A(s) Γ(A(t) -A(s))
x A(t)-A(s)-1 exp(-bx), ∀x ≥ 0.

Gamma processes are largely used in reliability, notably to model the cumulative deterioration of a system (see [START_REF] Van Noortwijk | A survey of the application of gamma processes in maintenance[END_REF] for an overview).

When the shape function is linear, i.e. A(t) = γt, the gamma process is said to be homogeneous. A homogeneous gamma process is a subordinator.

We consider the case in which X 1 and X 2 are two homogeneous gamma processes, so that

X 1 ∼ Γ(γ 1 ., b 1 ) and X 2 ∼ Γ(γ 2 ., b 2 )
where b 1 and b 2 refer to the scale parameters of X 1 and X 2 , respectively.

The Lévy measures of the process X 1 and X 2 are given by:

Q j (dx) = γ i 1 x e -bj x dx, for j = 1, 2.
If γ 1 = γ 2 , the change consists in a modification in the mean rate of degradation or the variability of the system considered. One can check that the latter condition is equivalent to the assumption (A 4 ) and that, when b 1 = b 2 = b, an explicit form of φ can be obtained as follows, for i ≥ 1:

φ (η i ) = log γ 2 γ 1 + (-γ 2 + γ 1 )g(0, b )η i ,
where g(., .) is the upper gamma incomplete function.

Proof of the main results

The idea of the proof of Theorems 1 and 2 is to approximate the original process X by a somewhat simpler jump process X of which behavior changes too when it crosses m 0 and such that X t converges towards X t point-wise as → 0. The construction of this process X is given in Subsection 4.1. The change detection for the approximating process is easier to deal with. The point of Section 4.2 is to study some properties of the stopping procedure related to X and τ ,η CU SU M defined in [START_REF] Kyprianou | Introductory lectures on fluctuations of Lévy processes with applications[END_REF].

4.1. The approximating process and its associated CUSUM statistics Construction of X Let > 0. The idea here is to approximate X j , j = 1, 2, by X j, which is given by ( 14) by removing the jumps less or equal to . Consequently, similarly to the representation ( 14), we can express X j, as follows

X j, t = [0,t] ( ,+∞) xN j (ds × dx), t ≥ 0. (29)
Moreover, it is standard that X j, t is a compound Poisson process (see [START_REF] Kyprianou | Introductory lectures on fluctuations of Lévy processes with applications[END_REF]Lemma 2.8,p.44]) that can be written as

X j, t = N j, X (t) i=1 ∆ j, i , ∀t ≥ 0,
with underlying Poisson process N j, X = (N j, X (t)) t≥0 , intensity Q j ( ) and jumping times denoted by (T j, i ) i≥0 . The distribution of the associated increments ∆

j, i = X j, T j, i -X j, T j, i-1 is
given by 1

Q j ( ) Q j (dx)1 [x> ] . (30)
In the following, we will let ∆ j, , j = 1, 2, be generic random variables with same distribution as the ∆ j, i , i ∈ N. Once the processes X j, are constructed, then the process X can be defined in a similar way of Equation ( 12)

X t = X 1, t 1 [t≤τ m 0 ] + (X 2, t-τ m 0 + X 1, τ m 0 )1 [t>τ m 0 ] (31)
with the first crossing time of level m 0 of process X 1, defined as

τ m0 = inf{t ≥ 0|X 1, t ≥ m 0 }. ( 32
)
At this point, we may note that the pseudo-level in (20) may be conveniently expressed as

M = X d (33)
where we recall that d is the detection time defined by [START_REF] Doney | Fluctuation theory for Lévy processes[END_REF].

CUSUM statistics associated to X

We then define the jumping times of the process X by

U i = T 1, i 1 [i≤N 1, ] + (T 2, i-N 1, + T 1, N 1, )1 [i>N 1, ] , i ≥ 0, (34)
where

N 1, = inf n ∈ N n i=1 ∆ 1, i ≥ m 0 . (35)
Note that N 1, refers to the index of the jumping time where X 1, exceeds m 0 , i.e. that

N 1, = inf{i ∈ N|T 1, i ≥ τ m0 }.
We also define the index of the first jumping time of the process X 1, following τ m0 by

D = inf{i ∈ N|T 1, i ≥ τ m0 }. ( 36
)
We define the inter-arrival times of the jump processes X j, and X as

η j, i = T j, i -T j, i-1 , j = 1, 2, i ≥ 1, δ i = U i -U i-1 , i ≥ 1.
Thus, (η i ) i∈N defined in ( 16) is associated to the observed process X, (η j, i ) i∈N referred to the inter-arrival times of the process N j, X , j = 1, 2, and (δ i ) i∈N is related to the constructed process X . A trajectory of X and X as well as the corresponding stopping times τ m0 and τ m0 on crossing level m 0 are illustrated in Figure 2. An illustration of the construction of the above sequences is given in Figure 3. As in [START_REF] Moustakides | Optimal stopping times for detecting changes in distributions[END_REF] for the sequence (η i ) i∈N , let us now introduce the CUSUM statistic (G δ n ) n∈N associated to the sequence (δ i ) i∈N by G δ 0 = 0 and

G δ n+1 = G δ n + φ (δ n+1 ) + , n ≥ 0, (37)
and the resulting stopping rule related to the threshold γ( )

τ ,δ CU SU M := inf{n ≥ 0| G δ n ≥ γ( )}. ( 38
)
Outline of Theorems 1, 2, 3 proofs The proofs of Theorems 1, 2 have different methodologies than that of Theorem 3. More precisely, the sketch of the proof of Theorems 1, 2 is the following:

1. We perform a detection of the random index N 1, defined in (35) through the CUSUM procedure τ ,δ CU SU M . 2. We prove that τ ,η CU SU M (that depends on the observed process X) is stochastically smaller than τ ,δ CU SU M (that depends on the unobserved process X ). This part is one of the main difficulty of the proof and is the object of the forthcoming Proposition 1 in Section 4.2. 3. Since X t point-wise converges to X t as → 0, we argue that the so-called pseudo-level M and detection level L defined respectively in ( 20) and [START_REF] Shaked | Stochastic orders[END_REF], which depend on the observed inter-arrival times (η i ) i∈N and the associated jumps (∆X T i ) i∈N , satisfy the properties (P 1 ), (P 2 ) and (P 3 ) or (P 3 ).

The proof of Theorem 3 is on the other hand simpler, as we let = 0 i.e. we directly deal with the observed process X and construct consequently the CUSUM statistic that detects the index N 1, .

t m0 τm 0 X ǫ t τ ǫ m0 = U ǫ N 1,ǫ = T 1,ǫ N 1,ǫ U ǫ D ǫ -1 U ǫ D ǫ Xt X 1,ǫ t X 2,ǫ t-τ ǫ m0 + X 1,ǫ τ ǫ m0 X 1 t X 2 t-τm 0 + X 2 τm 0 Figure 2:
Trajectories for X and X and switching times on reaching level m 0 .

U ǫ D ǫ -1 U ǫ D ǫ τ ǫ m0 = U ǫ N 1,ǫ τ m0 δ ǫ N 1ǫ = η 2,ǫ 1 δ ǫ D ǫ -1 = η 1,ǫ D ǫ -1 δ ǫ D ǫ = η 1,ǫ D ǫ δ ǫ D ǫ +1 = η 1,ǫ D ǫ +1 δ ǫ N 1,ǫ +1 = η 2,ǫ 2 η 2,ǫ 1 η 2,ǫ 2 η ǫ D ǫ η ǫ D ǫ -1 = η 1,ǫ D ǫ -1 η ǫ D ǫ +1 = η 2,ǫ 2 η 2,ǫ 3 η ǫ D ǫ +2 = η 2,ǫ 3 η 1,ǫ 0 Figure 3: Illustration of definitions for the sequences (η i ) i∈N , (δ i ) i∈N and (η j, i ) i∈N , j = 1, 2.
Recall that (η i ) i∈N is related to the observed process X and (δ i ) i∈N is related to the unobserved process X .

Intermediary results

Firstly, one should note that, in view of the explicit expression (5) for the CUSUM statistic, the following equalities hold

G δ n < γ( ) = max 1≤k≤n n i=k φ (δ i ) ∨ 0 < γ( ) , n ∈ N, (39)
{G η n < γ( )} = max 1≤k≤n n i=k φ (η i ) ∨ 0 < γ( ) , n ∈ N. ( 40
)
One can easily prove by induction on n ≥ m the following relation between

G δ n and G δ m G δ n =   max m+1≤j≤n n r=j φ (δ r )   ∨ 0 ∨ G δ m + n r=m+1 φ (δ r ) . (41)
One should note that τ ,η CU SU M , which is the CUSUM stopping time based on the observed path X is based on the sequence (η i ) i∈N . Nevertheless, we can see on Figure 3 that the particular time interval η D causes a problem since its distribution is unknown because the regime change does not occur necessarily after a jump of size larger than (i.e. at time U D ), unlike τ ,δ CU SU M . The following results, that will be used in the proof of Theorem 1, shows that τ ,δ CU SU M is larger than τ ,η CU SU M in some sense. Throughout the following sections and for the sake of readability, we denote E m0 and P m0 by E and P, respectively, when m 0 is finite and when no confusion is possible. Let us recall that, given X and Y two random variables, X is said to be stochastically smaller than Y , denoted X ≤ st Y (see [22, Chapter 1, p.3]) if, for all x ∈ R:

P(Y ≥ x) ≥ P(X ≥ x). Proposition 1. It holds that τ ,η CU SU M | N 1, = p ≤ st τ ,δ CU SU M | N 1, = p for all p ∈ N * .
Proof. Setting p ∈ N * , we have to prove the following inequality

P(τ ,δ CU SU M ≥ n, N 1, = p) ≥ P(τ ,η CU SU M ≥ n, N 1, = p) (42)
for all n ∈ N. We set throughout the proof

G := G η D -1 + b (τ m0 -U D -1 ) (43)
with b defined in [START_REF] Lai | Information bounds and quick detection of parameter changes in stochastic systems[END_REF]. By construction of the process X , one has G η i = G δ i for all i = 1, . . . , D -1 (see Figure 3), so that G η D and G δ D may be expressed as

G η D = G + φ (η 2, 1 ) + , (44)
G δ D = G + φ (η 1, 0 ) + , η 1, 0 := U D -τ m0 . (45)
A crucial remark is that, since X 1, has independent increments, η 1, 0 defined above is independent from τ m0 , G δ D -1 and U i , i = 1, . . . , D -1, and hence is in particular independent from G . Also, since X 1, and X 2, are independent processes, η 2, 1 is independent from G and is E( Q2 ( )) distributed. In fact, it will be proved later on that the r.v. η 1, 0 is E( Q1 ( )) distributed. Hence, the statement that we want to prove is justified by the following heuristic argument: from (44) and (45), it is clear that both CUSUM statistics G δ n and G η n coincide up to index D -1. Starting from index D , these two quantities may then be seen as two CUSUM statistics starting from G , with increments respectively given by φ (η 1, 0 ), φ (η 1, D +1 ), . . . , φ (η 1, N -1 ) (cf. Figure 3) which are negative in expectation then switch to φ (η 2, 1 ), φ (η 2, 2 ), . . ., which are positive in expectation, and φ (η 2, 1 ), φ (η 2, 2 ), . . ., which are positive in expectation. Consequently, (G η n ) n∈N will tend to hit the threshold γ( ) before (G δ n ) n∈N . The proof is decomposed into several steps as follows.

Step 1: Decomposition of the event {τ ,δ CU SU M ≥ n} Let us now decompose the left-hand side of (42) as

P(τ ,δ CU SU M ≥ n, N 1, = p) = P(τ ,δ CU SU M ≥ n, n ≥ D , N 1, = p) + P(τ ,δ CU SU M ≥ n, n < D , N 1, = p) (46)
and consider separately the two terms in the right-hand side of (46). We start by considering

P(τ ,δ CU SU M ≥ n, n < D , N 1, = p). One has τ ,δ CU SU M ≥ n, n < D = G δ k < γ( ), k = 1, . . . , n -1, n < D = {G η k < γ( ), k = 1, . . . , n -1, n < D } = {τ ,η CU SU M ≥ n, n < D } ,
as indeed, for all i = 1, . . . , D -1, δ i = η i = η 1, i and using (39) and (40). Hence we have the equality

P(τ ,δ CU SU M ≥ n, n < D , N 1, = p) = P(τ ,η CU SU M ≥ n, n < D , N 1, = p). ( 47
)
The main bulk of the proof concerns the first term in the right-hand side of (46). Conditioning on the crossing time τ m0 , the occurrences of the jump times of X prior to τ m0 as well as D , and since N 1, ≥ D , we obtain that

P(τ ,δ CU SU M ≥ n, n ≥ D , N 1, = p) = ∞ t=0 n∧p d=1 t1≤...≤t d-1 ≤t P(τ ,δ CU SU M ≥ n, τ m0 ∈ dt, D = d, N 1, = p, T 1, i ∈ dt i , i = 1, . . . , d-1). (48)
To avoid cumbersome notation, we define

A := {τ m0 ∈ dt, D = d, N 1, = p, T 1, i ∈ dt i , i = 1, . . . , d -1} = {τ m0 ∈ dt, T 1, d-1 < t, T 1, d ≥ t, N 1, = p, T 1, i ∈ dt i , i = 1, . . . , d -1} (49)
where the last equality comes from the definition (36) of D . We now consider the integrand on the RHS of (48). The rest of the proof is dedicated to prove that P(τ ,δ CU SU M ≥ n, A) ≥ P(τ ,η CU SU M ≥ n, A). To this end, we first observe, according to (38), that

{τ ,δ CU SU M ≥ n, n ≥ D = d} = {G δ k < γ( ), k = 1, . . . , n -1, n ≥ D = d} = {G δ k < γ( ), k = 1, . . . , d -1} {G δ d < γ( )} {G δ k < γ( ), k = d + 1, . . . , n -1} {D = d}. ( 50
)
We now consider each event on the RHS of the above equality intersected with A. Thanks to the explicit expression (5) of the CUSUM statistics as well as (39), the first event may be expressed as

{G δ k < γ( ), k = 1, . . . , d -1} A = max 1≤k≤d-1 max 1≤j≤k k i=j φ (t i -t i-1 ) < γ( ) A.
Then, thanks to (45), the second event can be written as

{G δ d < γ( )} A = (f d-1 (t 1 , . . . , t d-1 , t) + φ (η 1, 0 )) + < γ( ) A
where

f d-1 : (t 1 , . . . , t d-1 , t) → f d-1 (t 1 , . . . , t d-1 , t) := max 1≤k≤d-1 d-1 i=k φ (t i -t i-1 ) ∨ 0 + b (t -t d-1 ),
and we recall that η 1, 0 := U D -τ m0 refers to the residual time before a jump larger than after that the process X crosses the threshold m 0 (see Figure 3). Finally, the third event of (50) intersected with A can be written as follows thanks to (41):

{G δ k < γ( ), k = d + 1, . . . , n} A = max d+1≤k≤n     max d+1≤j≤k k r=j φ (1 [r≤p] η 1, r + 1 [r>p] η 2, r-p )   ∨ 0 ∨ (f d-1 (t 1 , . . . , t d-1 , t) + φ (η 1, 0 )) + + k r=d+1 φ (1 [r≤p] η 1, r + 1 [r>p] η 2, r-p ) < γ( ) A.
Hence, we obtain

P(τ ,δ CU SU M ≥ n, n ≥ D , A) = P(τ ,δ CU SU M ≥ n, A) = P v(t 1 , . . . , t d-1 ) < γ( ), Ψ(η 1, 0 , (η 1, j ) d+1≤j≤p , (η 2, i ) 1≤i≤n-p )) < γ( ), A = 1 v(t 1 , . . . , t d-1 ) < γ( ) P Ψ(η 1, 0 , (η 1, j ) d+1≤j≤p , (η 2, i ) 1≤i≤n-p )) < γ( ), A (51)
where we define the following two functions

v(t 1 , . . . , t d-1 ) = max 1≤k≤d-1    max 1≤j≤k k i=j φ (t i -t i-1 )    ∨ 0, Ψ : (z 1 , . . . , z n-d+1 ) → max (f d-1 (t 1 , . . . , t d-1 , t)+φ (z 1 )) + , max d+1≤k≤n     max d+1≤j≤k k r=j φ (z r-d+1 )   ∨ 0 ∨ (f d-1 (t 1 , . . . , t d-1 , t) + φ (z 1 )) + + k r=d+1 φ (z r-d+1 ) . (52)
Note that Ψ depends on (fixed) t 1 , . . . , t d-1 , t. For the sake of clarity, this dependence is not mentioned, as there is no possible ambiguity in the following.

Step 2: General properties of η 1, 0

Let us now argument the fact that η 1, 0 is independent from A and is distributed as

E( Q1 ( )) i.e. P(η 1, 0 ≥ x, A) = exp(-Q1 ( )x)P(A), ∀x ≥ 0. ( 53
)
Recalling the definition of η 1, 0 in (45), we have η 1, 0 = U D -τ m0 which can be written as η 1, 0 = T 1, D -τ m0 , since N 1, ≥ D and because of the definition of U i in (34). Additionally, the definition of D in (36) implies that {N 1, X (t) = d -1} = {D = d} on τ m0 ∈ dt. Then, we have for x ≥ 0,

{η 1, 0 ≥ x, τ m0 ∈ dt, D = d} = {T 1, d -t ≥ x, τ m0 ∈ dt, N 1, X (t) = d -1} = {N 1, X (t + x) -N 1, X (t) = 0, τ m0 ∈ dt, N 1, X (t) = d -1}. (54) Since {N 1, = p} = p i=1 ∆ 1, i ≥ m 0 > p-1 i=1 ∆ 1,
i and thanks to the definition (49) of the event A, the left-hand side of (53) thus reads

P(η 1, 0 ≥ x, A) = P N 1, X (t + x) -N 1, X (t) = 0, τ m0 ∈ dt, N 1, X (t) = d -1, p i=1 ∆ 1, i ≥ m 0 > p-1 i=1 ∆ 1, i , T 1, i ∈ dt i , i = 1, . . . , d -1 ,
which can be written in integral form, with respect to the first d -1 inter-arrival times

∆ 1, i , . . . , ∆ 1, d-1 , as follows z1,...,z d-1 ≥0 P N 1, X (t + x) -N 1, X (t) = 0, τ m0 ∈ dt, N 1, X (t) = d -1, d-1 i=1 z i + p i=N 1, X (t)+1 ∆ 1, i ≥ m 0 > d-1 i=1 z i + p-1 i=N 1, X (t)+1 ∆ 1, i , T 1, i ∈ dt i , ∆ 1, i ∈ dz i , i = 1, . . . , d-1 . (55)
We now observe for fixed z 1 , . . . , z d-1 , that the event {τ m0 ∈ dt, N 1,

X (t) = d -1, T 1, i ∈ dt i , ∆ 1, i ∈ dz i , i = 1, . . . , d -1} depends on X 1 s for s ∈ [0, t], whereas {N 1, X (t + x) - N 1, X (t) = 0, d-1 i=1 z i + p i=N 1, X (t)+1 ∆ 1, i ≥ m 0 > d-1 i=1 z i + p i=N 1, X (t)+1 ∆ 1,
i } depends on the increments ∆X 1 s for s ≥ t. Since X 1 is a Lévy process, its increments after time t are independent from its history up to time t, we hence deduce that (55) can be written as

z1,...,z d-1 ≥0 P N 1, X (t + x) -N 1, X (t) = 0, d-1 i=1 z i + p i=N 1, X (t)+1 ∆ 1, i ≥ m 0 > d-1 i=1 z i + p-1 i=N 1, X (t)+1 ∆ 1, i × P τ m0 ∈ dt, N 1, X (t) = d -1, T 1, i ∈ dt i , ∆ 1, i ∈ dz i , i = 1, . . . , d -1 . (56)
Since N 1, X is a Poisson process, the random variables N 1, X (t + x) -N 1, X (t), N 1, X (t) are independent. Additionally, since X 1, is a compound Poisson process, the jump times are independent from the jump size (∆ 1, i ) i=1,...,p (see Section 4.1). Consequently, (56) can be written as

z1,...,z d-1 ≥0 P N 1, X (t + x) -N 1, X (t) = 0 × P d-1 i=1 z i + p i=N 1, X (t)+1 ∆ 1, i ≥ m 0 > d-1 i=1 z i + p-1 i=N 1, X (t)+1 ∆ 1, i × P τ m0 ∈ dt, N 1, X (t) = d -1, T 1, i ∈ dt i , ∆ 1, i ∈ dz i , i = 1, . . . , d -1 . ( 57
)
Since the intensity of the Poisson process N 1, X is Q1 ( ), we have P N 1, X (t + x) -N 1, X (t) = 0 = e -Q1 ( )x , so that we may write (57) as

e -Q1 ( )x z1,...,z d-1 ≥0 P d-1 i=1 z i + p i=N 1, X (t)+1 ∆ 1, i ≥ m 0 > d-1 i=1 z i + p-1 i=N 1, X (t)+1 ∆ 1, i × P τ m0 ∈ dt, N 1, X (t) = d -1, T 1, i ∈ dt i , ∆ 1, i ∈ dz i , i = 1, . . . , d -1
which is the right-hand side of (53). Similarly, one can easily argument that the vector (η 1, 0 , (η 1, j ) d+1≤j≤p , (η 2, i ) 1≤i≤n-p ) is independent from A and has independent components. Finally, the independence of processes X 1 and X 2 immediately implies that A is independent from (η 2, 1 , . . . , (η 2, i ) n-d+1 ).

Step 3: End of proof Let us assume Q2 ( ) ≤ Q1 ( ). This implies that (η 1, 0 , (η 1, j ) d+1≤j≤p ) ≤ st (η 2, 1 , . . . , η 2, p-d+1 ). By independence of η 1, 0 , (η 1, j ) j∈N , (η 2, i ) i∈N we thus obtain

(η 1, 0 , (η 1, j ) d+1≤j≤p , (η 2, i ) 1≤i≤n-p )) L = (η 1, 0 , (η 1, j ) d+1≤j≤p , (η 2, i ) p-d+2≤i≤n-d+1 )) ≤ st (η 2, 1 , . . . , η 2, n-d+1 ). ( 58
)
The inequality Q2 ( ) ≤ Q1 ( ) entails that φ defined by ( 18) is increasing (see Remark 1) which in turn implies that Ψ : (z 1 , . . . , z n-d+1 ) → Ψ(z 1 , . . . , z n-d+1 ) (defined in (52)) is an increasing function in each of the variables z i , i = 1, . . . , nd + 1. Hence, the independence of (η 1, 0 , (η 1, j ) d+1≤j≤p , (η 2, i ) 1≤i≤n-p ) from A, argued at the end of Step 2, as well as the stochastic order in (58) (see [22, Theorem 1.A.3 (b), p.6]) imply that (51) is upper bounded as follows

P(τ ,δ CU SU M ≥ n, A) = 1 {v(t1,...,t d-1 )<γ( )} P Ψ(η 1, 0 , (η 1, j ) d+1≤j≤p , (η 2, i ) 1≤i≤n-p )) < γ( ), A = 1 {v(t1,...,t d-1 )<γ( )} P Ψ(η 1, 0 , (η 1, j ) d+1≤j≤p , (η 2, i ) 1≤i≤n-p )) < γ( ) × P(A) ≥ 1 {v(t1,...,t d-1 )<γ( )} P Ψ(η 2, 1 , . . . , η 2, n-d+1 ) < γ( ) × P(A) = P(τ ,η CU SU M ≥ n, A), (59)
where we recall that v(t 1 , . . . , t d-1 ) = max

1≤k≤d-1    max 1≤j≤k k i=j φ (t i -t i-1 )    ∨ 0.
Using the inequality (59), we obtain by integrating in (48) that

P(τ ,δ CU SU M ≥ n, n ≥ D , N 1, = p) ≥ P(τ ,η CU SU M ≥ n, n ≥ D , N 1, = p).
Finally, using ( 46) and (47), we conclude that

P(τ ,δ CU SU M ≥ n, N 1, = p) ≥ P(τ ,η CU SU M ≥ n, N 1, = p) (60)
for all n ∈ N and p ∈ N. If Q2 ( ) > Q1 ( ), one verifies this time that (η 1, 0 , (η 1, j ) d+1≤j≤p ) ≥ st (η 2, 1 , . . . , η 2, p-d+1 ), so that (58) is replaced by

(η 1, 0 , (η 1, j ) d+1≤j≤p , (η 2, i ) 1≤i≤n-p )) ≥ st (η 2, 1 , . . . , η 2, n-d+1 ).
Coupled with the fact that Ψ : (z 1 , . . . , z n-d+1 ) → Ψ(z 1 , . . . , z n-d+1 ) (defined in (52)) is this time a decreasing function in each of the variables z i , i = 1, . . . , nd + 1, one deduces that the (59) as well as the conclusion (60) still hold. This ends the proof.

As explained in the outline of the proof of theorems at the end of Section 4.1, we now find the adequate candidate for detecting the index N 1, , defined in (35), of the jump time where X 1, exceeds m 0 .

Since the inter-arrivals (η 1, i ) i≥0 and (η 2, i ) i≥0 are independent from N 1, , we first observe that

E[(τ ,δ CU SU M -N 1, ) + |N 1, = K] = E (K) [(τ ,δ, CU SU M -K) + ] ( 61
)
where τ ,δ, CU SU M refers to the stopping time of the CUSUM given by (3) associated to the sequence (Z i ) i∈N in (1) such that the sequences (Z 1 i ) i∈N and (Z 2 i ) i∈N are respectively E( Q1 ( )) and E( Q2 ( )) distributed and for a threshold γ( ) = log h( ). Conditioning on Z 1 ,...,Z K results on the following upper bound:

E (K) [(τ ,δ, CU SU M -K) + ] = E (K) (E (K) [(τ ,δ, CU SU M -K) + |Z 1 , . . . , Z K ]) ≤ E (K) sup p≥1 ess supE (p) [(τ ,δ, CU SU M -p) + |Z 1 , . . . , Z p ] = E 2 τ ,δ, CU SU M , (62)
where we recall that E 2 τ ,δ, CU SU M is the worst mean delay defined in [START_REF] Fouladirad | On-line change detection and condition-based maintenance for systems with unknown deterioration parameters[END_REF]. Equation [START_REF] Figueroa-López | Change-point detection for Lévy processes[END_REF] states that

E 2 τ ,δ, CU SU M ∼ log h( ) I as h( ) → ∞ (63)
where h( ) := exp(γ( )) and I refers to the KL distance defined in [START_REF] Tartakovsky | Asymptotic optimality in bayesian changepoint detection problems under global false alarm probability constraint[END_REF] and given by [START_REF] Stein | A note on cumulative sums[END_REF]. At this point, (63) does not provide much information. Indeed, E 2 τ ,δ, CU SU M behaves like log h( ) I as h( ) becomes large. However, this fact is not really useful as we wish to rather know how this quantity behaves when → 0. An upper bound is hence provided in the following lemma:

Lemma 1. (a) For all > 0, the following inequality holds

E 2 τ ,δ, CU SU M ≤ log h( ) + max log Q2 ( ) Q1 ( ) , Q1 ( ) Q2 ( ) -1 I ( 64
)
where I is given by [START_REF] Stein | A note on cumulative sums[END_REF].

(b) Under the assumption (A 4 ), it holds that

lim inf →0 I > 0 and lim sup →0 max log Q2 ( ) Q1 ( ) , Q1 ( ) Q2 ( ) -1 I < ∞.
Proof. We first prove (a). Let τ be the stopping variable of a one-sided sequential probability ratio tests (SPRT) of F 1 = E( Q1 ( )) (associated to a probability measure P 1 and expectation E 1 ) vs F 2 = E( Q2 ( )) (associated to a probability measure P 2 and expectation E 2 ) with likelihood boundary ratio h( ), given by

τ = inf{n ≥ 1|S n ≥ log h( )} ( 65
)
where S n refers to the random walk defined by

S n = n i=1 ξ i = n i=1 log Q2 ( ) Q1 ( ) + (-Q2 ( ) + Q1 ( ))Y i = n i=1 φ (Y i )
and (Y i ) i∈N is i.i.d and follows an exponential distribution with parameter Qj ( ) under P j , j = 1, 2.

We have P 2 (τ < ∞) ≤ h( ) (see [START_REF] Wald | Sequential Analysis[END_REF]), so that [16, (11) Theorem 2] reads here

E 2 τ ,δ, CU SU M ≤ E 2 (τ ). ( 66
)
We know from [START_REF] Wald | Sequential Analysis[END_REF] that E 2 (τ ) is equivalent to log(h( ))/I when h( ) is large, however this information is not satisfactory at this point because we want an estimate for E 2 (τ ) when tends to 0, such that lim →0 h( ) = +∞. Note however that this asymptotic implies that E 2 (τ ) is finite (see [START_REF] Stein | A note on cumulative sums[END_REF]) so that, since τ is a stopping time adapted to the sequence (Y i ) i∈N , Wald's equation (see [START_REF] Ross | Stochastic processes[END_REF]Theorem 3.3.2 p. 105]) reads

E 2 (S τ ) = E 2 (ξ 1 ).E 2 (τ ) = I E 2 (τ ), ( 67
)
so that (66) implies the upper bound

E 2 τ ,δ, CU SU M ≤ E 2 (S τ ) I . ( 68
)
We now observe that,

• if Q2 ( ) ≥ Q1 ( ), then ξ i ≤ log Q2 ( ) Q1 ( ) and hence E 1 (S τ ) ≤ log h( ) + log Q2 ( ) Q1 ( ) ,
• if Q2 ( ) ≤ Q1 ( ), then we have, by the memoryless property of the exponential distribution, that

E 2 (S τ ) = log h( ) + E 2 ((-Q2 ( ) + Q1 ( ))Y i ) = log h( ) + Q1 ( ) Q2 ( ) -1.
In both cases, we have

E 2 (S τ ) ≤ log h( ) + max log Q2 ( ) Q1 ( ) , Q1 ( ) Q2 ( ) -1 , (69)
so that (68) implies (64).

We now come to prove (b). Let us denote

ϕ(x) = -log x + x -1, ∀x > 0, ( 70
)
so that I = ϕ( Q1 ( )/ Q2 ( )) by [START_REF] Stein | A note on cumulative sums[END_REF]. Note that ϕ is a convex function and it admits a unique minimum for x = 1 with ϕ(1) = 0. Assumption (A 4 ) means that Q2 ( ) Q1 ( ) belongs to H := (0, 1 -c) (1 + c, +∞) for some small c ∈ (0, 1), and for ≤ 0 small enough. This implies I ≥ inf x∈H ϕ(x) := d > 0 for ≤ 0 , proving that lim inf →0 I > 0.

Then, we show that lim sup →0 max log

Q2 ( ) Q1 ( ) , Q1 ( ) Q2 ( ) -1 I < ∞.
For that purpose, for ≤ 0 , we consider the following cases:

• If Q2 ( ) Q1 ( ) < 1 -c, then log Q2 ( ) Q1 ( ) < 0 and Q1 ( ) Q2 ( ) -1 > 0. Consequently, we obtain that max log Q2 ( ) Q1 ( ) , Q1 ( ) Q2 ( ) -1 I = Q1 ( ) Q2 ( ) -1 ϕ Q1 ( ) Q2 ( ) ≤ sup x∈(0,1-c) 1/x -1 ϕ(1/x) < +∞. • If Q2 ( ) Q1
Q2 ( ) -1 I = log Q2 ( ) Q1 ( ) ϕ Q1 ( ) Q2 ( ) ≤ sup x>1+c log x ϕ(1/x) < +∞.
This ends the proof of the Lemma.

Proof of Theorem 1

Theorem 1 aims to study the properties of M defined in [START_REF] Cinlar | Introduction to stochastic processes[END_REF]. The latter can be rewritten as

M = τ ,η CU SU M i=1 ∆ ,η i where ∆ ,η i = ∆ 1, i , i ≤ D -1, ∆ 2, i-D +1 , i ≥ D (71)
where we recall that D is defined in (36).

Proof. Using the basic inequality (a + b) + ≤ a + + b + , we obtain the following upper bound:

E([M -m 0 ] + ) = E     M - N 1, i=1 ∆ ,η i + N 1, i=1 ∆ ,η i - N 1, i=1 ∆ 1, i + N 1, i=1 ∆ 1, i -m 0   +   ≤ E     M - N 1, i=1 ∆ ,η i   +   + E   N 1, i=1 [∆ ,η i -∆ 1, i ] +   + E     N 1, i=1 ∆ 1, i -m 0   +   = A + B + C . ( 72
)
In the above decomposition,

N 1, i=1 ∆ 1, i = X τ m 0
is the level of the process X at the instant where its behavior changes. Thus C refers to the overshoot of the latter process while it crosses m 0 . As for the quantities A and B that involve

N 1, i=1 ∆ ,η
i , it is somehow difficult to provide an intuitive interpretation. The idea is to be able to exploit the fact that τ ,η CU SU M enables to detect the index N 1, where the distribution of the jumps X changes (as mentioned in the outline of the theorems proof in Section 4.1). Firstly, we look at A :

A = E     τ ,η CU SU M i=1 ∆ ,η i - N 1, i=1 ∆ ,η i   +   = E     τ ,η CU SU M i=N 1, +1 ∆ 2, i-D +1   1 {τ ,η CU SU M ≥N 1, +1}   = ∞ d=1 ∞ p=d E     τ ,η CU SU M i=p+1 ∆ 2, i-d+1   1 {τ ,η CU SU M ≥p+1} 1 {D =d,N 1, =p}   . ( 73
)
Since N 1, , D and τ m0 depend on the process X 1 and since τ ,η CU SU M depends on (η 1, i ) i∈N , τ m0 and (η 2, i ) i∈N , then ∆ 2, i-d+1 is independent form N 1, , D and τ ,η CU SU M . Consequently, we can write (73) as follows

A = E ∆ 2, ∞ d=1 ∞ p=d E [τ ,η CU SU M -p] + 1 {D =d,N 1, =p} = E ∆ 2, E τ ,η CU SU M -N 1, + = E ∆ 2, ∞ p=1 E [τ ,η CU SU M -p] + | N 1, = p × P(N 1, = p). ( 74
)
By using Proposition 1, A in (74) can be upper bounded by

A ≤ E ∆ 2, ∞ p=1 E τ ,δ CU SU M -p + N 1, = p × P(N 1, = p). ( 75
)
Additionally, the combination of ( 61) and (62) yields that

E τ ,δ CU SU M -p + N 1, = p ≤ E 2 τ ,δ, CU SU M
from which, thanks to Lemma 1 (a) and ( 75), one can easily obtain the following upper bound for A :

A ≤ E ∆ 2, log h( ) + max log Q2 ( ) Q1 ( ) , Q1 ( ) Q2 ( ) -1 I ∞ p=1 P(N 1, = p) ≤ E ∆ 2, log h( ) + max log Q2 ( ) Q1 ( ) , Q1 ( ) Q2 ( ) -1 I . (76)
Secondly, we deal with the term B . We observe that, from (71

), ∆ ,η i = ∆ 1, i for all i = 1, . . . , D -1, so that B = E   N 1, i=1 [∆ ,η i -∆ 1, i ] +   = E   N 1, i=D [∆ 2, i-D +1 -∆ 1, i ] +   ≤ E   N 1, i=D ∆ 2, i-D +1   ≤ E   N 1, i=1 ∆ 2, i   = E N 1, E ∆ 2, , (77)
the last equality stemming from the independence between N 1, and the ∆ 2, i , i ∈ N. By Lorden's inequality (see e.g. [26, Proposition 6.2 p.160]), we have

E N 1, -1 ≤ m 0 E (∆ 1, ) + E ∆ 1, 2 (E [∆ 1, ]) 2 , (78)
which, plugged into (77), yields

B ≤ m 0 E ∆ 2, E [∆ 1, ] + E ∆ 1, 2 (E [∆ 1, ]) 2 E ∆ 2, + E ∆ 2, . (79)
Finally, we end up by looking to the term C . Since

N 1, i=1 ∆ 1,
i is larger than m 0 , it simplifies as

C = E   N 1, i=1 ∆ 1, i -m 0   .
Wald's equality (see [25, Theorem 3.3.2 p. 105]) and (78) provide

C = E N 1, × E[∆ 1, i ] -m 0 ≤ E ∆ 1, 2 E [∆ 1, ] + E(∆ 1, ). (80)
Gathering (76), ( 79) and (80), we obtain from (72) that E([M -m 0 ] + ) ≤ c 1 and then (P 1 ) is satisfied, where

c 1 := 6 i=1 T i ( ), T 1 ( ) := E ∆ 2, log h( ) + max log Q2 ( ) Q1 ( ) , Q1 ( ) Q2 ( ) -1 I , T 2 ( ) := m 0 E ∆ 2, E [∆ 1, ] , T 3 ( ) := E ∆ 1, 2 (E [∆ 1, ]) 2 E ∆ 2, , T 4 ( ) := E ∆ 2, T 5 ( ) := E ∆ 1, 2 E [∆ 1, ] , T 6 ( ) := E ∆ 1, .
Let us now prove (P 2 ). For that purpose , we set

c 2 := h( )E ∆ 1, (81)
where we recall that h( ) is defined by [START_REF] Wald | Sequential Analysis[END_REF]. When m 0 = +∞ then the process X is equal to X 1 , τ ,δ CU SU M = τ ,η CU SU M , and the expression of M in (71) is simplified to

M = τ ,δ CU SU M i=1 ∆ 1, i .
The independence of τ ,δ CU SU M from the sequence (∆

1, i ) i∈N yields E ∞ (M ) = E ∞   τ ,δ CU SU M i=1 ∆ 1, i   = E ∞ (τ ,δ CU SU M )E(∆ 1, ) ≥ h( )E(∆ 1, ) = c 2 (82)
by [START_REF] Fouladirad | On the use of on-line detection for maintenance of gradually deteriorating systems[END_REF]. Finally, we prove the last property (P 3 ) or (P 3 ). First, the expression (30) of the distribution of ∆ j, , j = 1, 2, and in view of assumption (A 2 ), yields the following estimates for their first and second order moments:

E[∆ j, ] = 1 Qj ( ) ( ,∞) xQ j (dx) ∼ →0 1 Qj ( ) (0,∞) xQ j (dx), E[(∆ j, ) 2 ] = 1 Qj ( ) ( ,∞) x 2 Q j (dx) ∼ →0 1 Qj ( ) (0,∞) x 2 Q j (dx). (83)
We then consider the different cases given in the statement of the Theorem. In each case, we consider the behaviour as → 0 of each term T i ( ), i = 1, . . . 6 in the definition of c 1 . We note that for i = 4, 5, 6, it holds that lim sup →0 T i ( ) is finite, so that we will mainly focus on the terms T 1 ( ), T 2 ( ) and T 3 ( ) in the following. ) -→ 0 as → 0. Hence, thanks to the bound (b) in Lemma 1, we obtain that lim sup

→0 T 1 ( ) ≤ lim sup →0 E[∆ 2, ] log h( ) I + lim sup →0 E[∆ 2, ] max log Q2 ( ) Q1 ( ) , Q1 ( ) Q2 ( ) -1 I = 0,
as indeed Q2 (0) = +∞ and so lim →0 E[∆ 2, ] = 0. We check easily from (83) that T 2 ( ) and T 3 ( ) are O Q1 ( ) Q2 ( ) , hence lim sup →0 T 2 ( ) and lim sup →0 T 3 ( ) are finite under the present assumption. Thus, we have lim sup →0 c 1 < ∞. As for c 2 , we write from (81) that

c 2 = [ Q2 ( )I ] 2 1 Q1 ( ) ( ,∞) xQ 1 (dx) = Q2 ( )[I ] 2 Q2 ( ) Q1 ( ) ( ,∞) xQ 1 (dx), so that lim inf →0 c 2 ≥ lim inf →0 Q2 ( ) lim inf →0 [I ] 2 lim inf →0 Q2 ( ) Q1 ( ) lim inf →0 ( ,∞) xQ 1 (dx) = lim inf →0 Q2 ( ) lim inf →0 [I ] 2 lim inf →0 Q2 ( ) Q1 ( ) (0,∞) xQ 1 (dx) = +∞
as we used the property that lim sup →0 Q1 ( ) Q2 ( ) < ∞ implies that lim inf →0 Q2 ( ) Q1 ( ) > 0 as well as Lemma 1 (b). Finally, we verified that (P 3 ) holds.

Case 2: Q2 (0) = +∞ and lim sup →0

Q1 ( ) Q2 ( ) = ∞.
In this case, h( ) has the same form as in Case 1. We note that this case necessarily implies that Q1 (0) = +∞. As in the previous case, the property lim sup →0 T 1 ( ) < ∞ can be proved similarly. As to T 2 ( ) and T 3 ( ), both of those terms are O Q1 ( ) Q2 ( ) , hence we have

c 1 = O Q1 ( ) Q2 ( ) . (84)
Contrary to the Case 1, lim sup →0 T 2 ( ) and lim sup →0 T 3 ( ) are infinite in the present case. To prove that lim →0 c 2 = +∞, we recall that I = ϕ( Q1 ( ) Q2 ( ) ) where ϕ is defined in (70). We note that lim x→∞ ϕ(x)/x = 1, which implies in particular that ϕ(x) ≥ x/2 for x ≥ K large enough. Recalling that I ≥ d > 0 for ≤ 0 small enough (see proof of Lemma 1), we then have, for ≤ 0 ,

c 2 =        Q2 ( ) ϕ Q1 ( ) Q2 ( ) Q1 ( ) Q2 ( ) I ( ,∞) xQ 1 (dx) ≥ Q2 ( ) 1 2 d ( 0,∞) xQ 1 (dx) if Q1 ( ) Q2 ( ) ≥ K, Q2 ( ) Q2 ( ) Q1 ( ) [I ] 2 ( ,∞) xQ 1 (dx) ≥ Q2 ( ) 1 K d 2 ( 0,∞) xQ 1 (dx) if Q1 ( ) Q2 ( ) < K, so that, all in all, we have c 2 ≥ Q2 ( ) min( 1 2 d, 1 K d 2 ) ( 0 ,∞) xQ 1 (dx) -→ +∞ as → 0. Furthermore, the definition (81) of c 2 combined with (84) implies c 1 c 2 = O 1 [ Q2 ( )I ] 2 [ Q1 ( )] 2 Q2 ( )
. Now, remembering that ϕ(x) ≥ x/2 for x ≥ K, we have, for small enough

1 [ Q2 ( )I ] 2 [ Q1 ( )] 2 Q2 ( ) ≤ 1 Q2 ( )d 2 K 2 1 [ Q1 ( )/ Q2 ( )≤K] + 4 Q2 ( ) 1 [ Q1 ( )/ Q2 ( )>K] ,
so that lim sup →0

1 [ Q2 ( )I ] 2 [ Q1 ( )] 2 Q2 ( ) ≤ lim sup →0 1 Q2 ( )d 2 K 2 + lim sup →0 4
Q2 ( ) = 0. This entails that c 1 = o(c 2 ). Hence (P 3 ) holds.

Case 3: Q1 (0) = +∞ and Q2 (0) < ∞.

In this case, h( ) = [ Q1 ( )] β . Let us first note that, since ϕ(x)

∼ x as x → ∞, I = ϕ Q1 ( ) Q2 ( ) ∼ Q1 ( ) Q2 (
0) as → 0. Hence, log h( )/I is equivalent to β Q2 (0) log Q1 ( )/ Q1 ( ) -→ 0 as → 0. Furthermore, we have from Lemma 1 that

max log Q2 ( ) Q1 ( ) , Q1 ( ) Q2 ( ) -1 I < ∞ as → 0.
Both facts, along with the boundedness of E[∆ 2, ], implies that lim sup →0 T 1 ( ) < +∞. Again, we check easily from (83) that T 2 ( ) and T 3 ( ) are O( Q1 ( )) as → 0. All in all, we have that c 1 is a O( Q1 ( )). Hence

c 1 c 2 = O Q1 ( ) [ Q1 ( )] β / Q1 ( ) = O [ Q1 ( )] 2-β -→ 0, → 0.
Furthermore, it can be easily verified that lim →0 c 2 = +∞ and consequently (P 3 ) holds.

Proof of Theorem 2

We denote throughout this section the processes X j, -= (X j, - t ) t≥0 , j = 1, 2 as well as X -= (X - t ) t≥0 obtained from X j , j = 1, 2 and X by discarding the jumps of height larger than . Using the Poisson random measure N j introduced in Section 3, and with a similar expression as those in ( 14) and ( 29) for X j t and X j, t , this reads

X j, - t := [0,t] (0, ] xN j (ds × dx) = s≤t ∆X j s 1 [∆X j s ≤ ] , j = 1, 2, (85)
X - t := s≤t ∆X s 1 [∆Xs≤ ] = X t -X t , (86)
so that one has, similarly to (12):

X - t = X 1, - t 1 [t≤τm 0 ] + (X 1, - τm 0 + X 2, - t-τm 0 )1 [t>τm 0 ] . (87)
Thus, one can point out that the difference between the detection level L and the pseudolevel M expressed in [START_REF] Shaked | Stochastic orders[END_REF] and (33) respectively, is given by

L -M = X - d
where we recall that d is the detection time defined by [START_REF] Doney | Fluctuation theory for Lévy processes[END_REF]. Indeed, L is the detection level, i.e. the level of the process X at time d , and M is the sum of all jumps larger than of the process X t between t = 0 and t = d . So that, by distinguishing the two cases (d ≤ τ m0 and d > τ m0 ), this difference may also be written thanks to (87) as

0 ≤ L -M = X 1, - d ∧τm 0 + X 2, - d -d ∧τm 0 . ( 88
)
We next show that the expectation of each term on the right-hand side of (88) tends to 0 when → 0. Before showing the latter, we start by mentioning an important result related to martingales associated to the processes X 1 and X 2 . Note first that (0, ) xQ j (dx), j = 1, 2 is finite for small enough (see Assumption (A 1 )). Hence the following process

[0,t] (0, ] xN j (ds × dx) -t (0, ) xQ j (dx) t≥0 = X j, - t -t (0, ) xQ j (dx) t≥0 (89)
is a martingale adapted to X j , j = 1, 2. This fact may be verified by applying [START_REF] Kyprianou | Introductory lectures on fluctuations of Lévy processes with applications[END_REF]Corollary 4.6 p.97 ] with function φ used in that reference given by φ(s, x) := x1 (0, ) (x).

Term X 1, - d ∧τm 0

We first observe that, (X 1, - t ) t≥0 in (86) is increasing so that

X 1, - d ∧τm 0 ≤ X 1, - τm 0 .
Furthermore, for m 0 < +∞, τ m0 in ( 13) is a stopping time adapted to X 1 of finite expectation, so that according to Doob's optional stopping time theorem (see [27, p.4]) applied to the martingale (89) we have for all N ∈ N that

E X 1, - τm 0 ∧N = E(τ m0 ∧ N ) × (0, ) xQ 1 (dx),
which, by letting N → +∞ and using the monotone convergence theorem, yields

E X 1, - τm 0 = E(τ m0 ).
(0, )

xQ 1 (dx) (90)
which tends to 0 as → 0. Hence, we deduce that

X 1, - d ∧τm 0 L 1 -→ 0 as → 0. ( 91
)
Term X 2, - d -d ∧τm 0
Let us introduce the process (V i ) i∈N as a time shifted version of the CUSUM statistic (G η i ) i∈N as V 0 = G defined in (43) and V n = G η D +n-1 for n ≥ 1, so that (V i ) i∈N satisfies from (44), [START_REF] Moustakides | Optimal stopping times for detecting changes in distributions[END_REF] as well as the relation η D +n = η 2, n+1 (see Figure 3 for the illustration of this latter fact) the recursive equation

V n+1 = V n + φ (η 2, n+1 ) + , n ≥ 0 V 0 = G . ( 92
)
Let us associate to this CUSUM statistic the corresponding first passage time above the threshold γ( ) = log h( ) given by

τ V CU SU M := inf{n ≥ 0| V n ≥ log h( )} if τ ,η CU SU M ≥ D , 0 otherwise.
We observe in particular, thanks to the relation between the process (V i ) i∈N and (G η i ) i∈N , that their corresponding CUSUM statistics verify τ V CU SU M = τ ,η CU SU M -D when τ ,η CU SU M ≥ D . We also define the random walk (S

(2) n ) n≥0 by S (2) n = n k=1 φ (η 2, k )
, where φ is defined in [START_REF] Lai | Information bounds and quick detection of parameter changes in stochastic systems[END_REF], with first passage time above the threshold log h( ) given by τ S (2) := inf{n ≥ 0| S (2) n ≥ log h( )}.

It can be easily verified by induction that V n ≥ S

n for all n, hence the corresponding first passage time verifies τ V CU SU M ≤ τ S (2) . Recalling the properties η D +n-1 = η 2, n for n ≥ 1 and τ V CU SU M = τ ,η CU SU M -D when τ ,η CU SU M ≥ D , this results, thanks to the definition of d in [START_REF] Doney | Fluctuation theory for Lévy processes[END_REF], in

d -d ∧ τ m0 = τ V CU SU M k=1 η 2, k ≤ τ S (2) k=1 η 2, k := d ,2 . ( 94
)
which in turns results in

X 2, - d -d ∧τm 0 ≤ X 2, - d ,2 . (95)
Since d ,2 is a stopping time adapted to X 2 , a similar martingale argument to the one leading to (90) leads to the following

E X 2, - d ,2 = E(d ,2 ) × (0, ) xQ 2 (dx). ( 96
)
We then proceed to study the behaviour of E(d ,2 ) as → 0. Following (94) and by using Wald's equation (see [START_REF] Ross | Stochastic processes[END_REF]Theorem 3.3.2 p. 105])), we obtain

E(d ,2 ) = E τ S (2) × E(η 2, ). (97)
One can easily see the following equality

E τ S (2) = E 2 (τ )
with the right-sided term which was defined in the proof of Lemma 1, with τ defined by (65). By using (67) and ( 69), we consequently obtain

E τ S (2) ≤ log h( ) + max log Q2 ( ) Q1 ( ) , Q1 ( ) Q2 ( ) -1 I .
Let us suppose that Q2 (0) = ∞ (see [START_REF] Wald | Sequential Analysis[END_REF]), so that the threshold in ( 93) is given by log h( ) where h( ) = [ Q2 ( )I ] 2 . Multiplying both sides of the above inequality by E(η 2, ) = O 1/ Q2 ( ) , we thus get from (97) and (96), that (98) Assumption (A 2 ) ensures that (0, ) xQ 2 (dx) tends to 0 as tends to 0, and Lemma 1 (b) ensures that max log Q2 ( ) Q1 ( ) , Q1 ( ) Q2 ( ) -1 /I is bounded when tends to 0. Consequently, the term (98) tends to 0 when tends to 0.

E X 2, -
Finally, this implies thanks to (95) that

X 2, - d -d ∧τm 0 L1 -→ 0, → 0. ( 99
)
One can easily show the same results when Q2 (0) < ∞ and h( ) = [ Q1 ( )] β (see [START_REF] Wald | Sequential Analysis[END_REF]).

End of proof

Combining (88), ( 91) and (99), we then deduce [START_REF] Asmussen | Applied probability and queues[END_REF]. By observing that

E([L -m 0 ] + ) ≤ E(L -M ) + E([M -m 0 ] + ),
and by recalling the form of c 1 in the proof of Theorem 1, the property (P 1 ) holds by replacing the quantity c 1 by E(L -M ) + c 1 . As for the property (P 2 ), it holds because the following holds

E ∞ (L ) ≥ E ∞ (M ) ≥ c 2 ,
where the last inequality stems from Theorem 1. Finally, one can verify that (P 3 ) (or (P 3 ) when ( 25) is satisfied) holds because of (26).

Proof of Theorem 3

The idea of the proof is to take = 0 in the construction of the approximated process defined in (31), so that both processes X and X now coincide. The situation is hence less complicated than in Section 4.1 as here τ m0 is equal to T D 0 for some (random) index D = D 0 defined in (36) with = 0. Likewise, when = 0, the index N 1, coincides with D 0 , the crossing time τ m0 is now equal to τ m0 and the quantity η 1, 0 defined in (45) with = 0 is equal to 0. We may then drop the dependence in = 0 now and denote by (∆ 1 i ) i∈N and (∆ 2 i ) i∈N the respective jumps of the compound Poisson processes X 1 and X 2 and M as the pseudo level. We note that, now that = 0 and X = X 0 , the pseudo-level M is now the same as L defined in [START_REF] Qiu | Distribution-free multivariate process control based on log-linear modeling[END_REF], which is nicely illustrated in Figure 1. This latter quantity may also be expressed as

L = τ η CU SU M i=1 ∆ η i where ∆ η i = ∆ 1 i , i ≤ D 0 , ∆ 2 i-D 0 +1 , i > D 0 . (100)
As in (72), we upper bound the expected delay as

E([L -m 0 ] + ) = E([M -m 0 ] + ) ≤ E     M - D 0 i=1 ∆ η i   +   + E     D 0 i=1 ∆ 1 i -m 0   +   = A 0 (h) + C 0 := c h 1 (101)
where h > 1 is the threshold for the CUSUM rule defined in [START_REF] Bertoin | Lévy processes[END_REF]. Similarly to (73) and (74), one obtains

A 0 (h) = E(∆ 2 ) E [τ η CU SU M -D 0 ] + . (102)
Moreover, Lemma 1 is still valid when = 0 by substituting h( ) by h, so that one can easily derive that, similarly to (75),

A 0 (h) ≤ E(∆ 2 ) log h + max log Q2 (0) Q1 (0) , Q1 (0) Q2 (0) -1 I 0 . ( 103
) with I 0 = log Q2 (0) Q1 (0) -1 + Q1 (0)
Q2 (0) . As for C 0 , a similar analysis as in (80) yields that

C 0 = E D 0 .E[∆ 1 i ] -m 0 ≤ E ∆ 1 2 E [∆ 1 ] + E(∆ 1 ). (104)
Thus, (P 0 1 ) is satisfied. Now, we turn to prove (P 0 2 ). For that purpose, one gets similarly to (82) that

E ∞ (L) = E ∞ (τ η CU SU M ) E(∆ 1 ) ≥ h E(∆ 1 ) := c h 2
with E(∆ 1 ) a finite quantity thanks to Assumption (A 1 ) as well as Q1 (0) < +∞. Finally, since c h 1 = O(log h) as h → ∞ because of (103) and (104), one easily verifies that (P 4 ) holds. This ends the proof of Theorem 3.

Numerical illustrations on case study: standard gamma processes with level switching

As previously explained, the process X defined as in equation ( 12) changes from X 1 (called in the sequel Regime 1) to X 2 (called in the sequel Regime 2) when it crosses the level m 0 . In this section, we consider more specifically gamma processes with corresponding shape functions A i : t → γ i t, i = 1, 2 and scale parameter b = 1. That means

X i = (X i t ) t≥0 with X i t ∼ Γ(γ 1 t, 1), i = 1, 2.
The objective of this experiment is to illustrate the -detection rule [START_REF] Kyprianou | Introductory lectures on fluctuations of Lévy processes with applications[END_REF] proposed in this paper and to compare its performances with those of the classical CUSUM rule, applied on the increments (X ti+1 -X ti ) i∈N of the process X, for some temporal discretization (t i ) i∈N of [0, +∞) with constant size t i+1 -t i = s > 0. Indeed, conditionally to the change time, one may think that the increments of the process are independent and gamma distributed random variables Γ(γ 1 s, 1) or Γ(γ 2 s, 1) before and after the regime change which may lead us to believe that a classic CUSUM rule is well adapted. However, not only this latter fact is not true, but in our case, the change time is not deterministic, but random; Worse, it is a stopping time that depends on the trajectory X. Moreover the crossing of level m 0 can happen between two "inspections" t i and the distribution of the resulting increment is therefore unknown (neither Γ(γ 1 s, 1) or Γ(γ 2 s, 1)). Consequently, the particular situation considered in this paper does not correspond to the conditions for applying the classic CUSUM rule. Figure 4 illustrates a sample path of the process X with threshold m 0 = 10 as well as the trajectories for the -detection rule and classical CUSUM statistics.

The process X (top of Figure 4) is supposed to be observed continuously. To apply the classic CUSUM rule (bottom of Figure 4), and as explained earlier on, a time-discretization step-size s is considered, so that t i = is, i ∈ N. For each increment X is -X (i-1)s , i = 1, 2, . . . , a likelihood ratio is computed between Regime 1 and Regime 2 by noting that, intuitively, X is -X (i-1)s is distributed according to a gamma Γ(γ 1 s, 1) as long as is < τ m0 , and to a gamma Γ(γ 2 s, 1) as soon as (i -1)s > τ m0 , with τ m0 the crossing time of level m 0 defined in [START_REF] Tartakovsky | General asymptotic bayesian theory of quickest change detection[END_REF]. The likelihood ratios can then be used sequentially to compute the test statistic (4) of the CUSUM rule. The implementation of the -detection rule (middle of Figure 4) is different. By construction, a new increment of the rule is computed as soon as the process, which is a pure-jump process, observed a jump greater than a fixed value . The number of increments of the -detection rule is therefore random and depends also on the functioning mode: in the gamma case considered in this section (see Section 3.3), the expected delay before a new increment is equal to:

E[η i ] = 1 Qi ( ) = 1 γ i +∞ 1 y e -y dy , (105)
for i = 1 or 2 depending on whether the process is under Regime 1 or Regime 2.

For each of the two change detection rules, it is necessary to determine a threshold γ( ) (for the -detection rule) or γ(s) (for the classic CUSUM rule). In a classic way in statistical process control, the threshold is chosen in practice such that the false alarm "rate" can be controlled. More precisely, the threshold γ( ) of the -detection rule is chosen such that the Average Run Level under Regime 1 E ∞ (L ) (and denoted ARLev ∞ , see Remark 2) is equal to a specific value chosen here equal to 30. It is obtained empirically by a dichotomous approach on γ( ) based on the test statistics of 10000 trajectories of X simulated under Regime 1 (by adapting Algorithm 1 of [START_REF] Qiu | Distribution-free multivariate process control based on log-linear modeling[END_REF] to our context). The same approach is followed for γ(s).

As expected, the test statistic in Figure 4 is close to zero when the system is under Regime 1 and it increases as soon as the system is under Regime 2. Indeed, the logarithm of the likelihood ratio [START_REF] Lai | Information bounds and quick detection of parameter changes in stochastic systems[END_REF] tends to be negative before the change and positive after the change. The regime change is therefore quickly detected.

To go further, the performances of the -detection and classic CUSUM rules are compared. For the same ARLev ∞ , fixed to the value 30, and leading to the determination of the threshold γ( ) and γ(s), we compare the mean overshoot of the process X at detection time above the threshold m 0 , defined by E m0 ([Lm 0 ] + ) for the -detection rule (see again Remark 2). In the sequel, we use m 0 = 0 for the mean overshoot, that means that the system is under Regime 2 at the beginning of the monitoring. It allows to obtain what is called, in the control chart community, the zero-state ARL (as opposed to the steady-state ARL with a non-null m 0 ). The corresponding quantity will be denoted ARLev 0 . Obviously, for the same ARLev ∞ , the best rule is the one with the smallest ARLev 0 . These ARLev will be obtained by Monte Carlo simulations.

For the classic CUSUM, and since we only need the increment values for this approach, the gamma processes are simulated by the "increment" approach (see [START_REF] Van Noortwijk | A survey of the application of gamma processes in maintenance[END_REF]), which is an exact and efficient simulation technique. For the -detection rule, and since it is necessary to observe the jumps of the process, we used a series representation of the gamma process (see [START_REF] Rosiński | Series representations of Lévy processes from the perspective of point processes[END_REF]Section 6], or [START_REF] Al Masry | Approximate simulation techniques and distribution of an extended gamma process[END_REF]Proposition 2]) and simulate the processes with the rejection approach (with B = 30 for the truncation of the series, see [START_REF] Al Masry | Approximate simulation techniques and distribution of an extended gamma process[END_REF]Algorithm 2]).

To make the comparison between the classic CUSUM and the -detection rule relevant, the number of increments of a run must be approximately the same. Remind that the number of increments of the -detection rule is random (of which expectation is given by (105)). For a value s, we then determine the value of which gives, in mean, the same number of increments. For one value for s, we obtain two values for , 1 and 2 depending on whether the process is under Regime 1 or Regime 2. The -detection rule was then applied with two configurations: either we used the greater value for (that means we consider the least advantageous case for the -detection rule), or we used the mean between the two epsilon values.

The Monte Carlo approximations of the ARLev are based on 10000 repetitions. We use for the simulations of the gamma processes, γ 1 = 1 and several values for γ 2 (1.1, 1.2 and 1.5). The time-discretization step-size s for the classic CUSUM is varying from 2 to 0.05. For each value, 1 and 2 are computed, and then is chosen. For example, if s = 0.5, that means that we consider two increments per unit of time for the classic CUSUM and γ 2 = 1.1, then we have to use 1 = 8.2 × 10 -2 to obtain, for the -detection rule, two increments per unit of time in mean when the system is under Regime 1 and 2 = 10 -1 when the system is under Regime 2. We then test two values for : = 10 -1 which corresponds to the "worst-case scenario" and = 9.1 × 10 -2 , the "medium scenario". The results are presented, for the classic CUSUM, in Table 1 and for the -detection rule, in Table 2 for the "worst-case scenario" and Table 3 for the "medium scenario". In each table are reported the threshold (γ(s) for the classic CUSUM and γ( ) for the -detection rule) of the rules leading to an empirical ARlev ∞ close to 30 when the system is under Regime 1, the corresponding observed ARLev ∞ and the empirical standard deviation of the Run Level SDRLev ∞ . For the comparison of the rules when the system is under Regime 2, we find in Table 1 to 3, the time is high (a small s for the classic CUSUM and a small for the -detection rule), more efficient is the detection rule. It is an illustration of the results obtained in the previous sections, which show that the -detection rule is efficient as tends to 0. Another interesting point is that, for a comparable number of increments per unit of time, the -detection rule gives better results (a smaller ARLev 0 ) than the the classic CUSUM, even in the "worst-case scenario", as soon as is sufficiently small ( ≤ 0.1). The difference is even more pronounced when the classic CUSUM is compared to the medium scenario.

2 Figure 1 :

 21 Figure 1: Detection time and level

  ( ) > 1 + c, then log

Case 1 :

 1 Q2 (0) = +∞ and lim sup →0 Q1 ( ) Q2 ( ) < ∞. In this case, we have h( ) = [ Q2 ( )I ] 2 and thus E[∆ 2, ] log h( ) I = O( log( Q2 ( )I ) Q2 ( )I

Figure 4 :

 4 Figure4: Top: Process X with a change from a gamma process Γ(t, 1) to a gamma process Γ(1.5t, 1) at level m 0 = 10 ; Middle: The corresponding -detection rule statistic (plain line) with = 7.2 × 10 -4 for a theoretical Average Run Level (E∞(L )) equal to 30, leading to a threshold equal to γ( ) = 2.599 (dotted line) ; Bottom: The classical CUSUM statistic (plain line) with s = 0.1 for a theoretical Average Run Level equal to 30, leading to a threshold equal to γ(s) = 2.963 (dotted line).

Table 1 :

 1 Empirical Average Run Levels for the classic CUSUM for a change from a gamma process Γ(t, 1) to a gamma process Γ(γ 2 t, 1) obtain from 10000 repetitions.

	s	γ 2	γ(s)	ARLev∞	SDRLev∞	ARLev 0	SDRLev 0