Local Asymptotic Normality Complexity Arising in a Parametric Statistical Lévy Model

Jedidi, Wissem

doi:https://doi.org/10.1155/2021/3143324

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Complexity Arising in Financial Modelling and its Applications

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Research Article | Open Access

Volume 2021 | Article ID 3143324 | https://doi.org/10.1155/2021/3143324

Local Asymptotic Normality Complexity Arising in a Parametric Statistical Lévy Model

Wissem Jedidi¹

Academic Editor: Ning Cai

Received27 Aug 2021

Accepted09 Nov 2021

Published29 Dec 2021

Abstract

We consider statistical experiments associated with a Lévy process observed along a deterministic scheme . We assume that under a probability , the r.v. , has a probability density function , which is regular enough relative to a parameter . We prove that the sequence of the associated statistical models has the LAN property at each , and we investigate the case when is the product of an unknown parameter by another Lévy process with known characteristics. We illustrate the last results by the case where is attracted by a stable process.

1. Introduction

This work is a part of an ambitious program consisting in the estimation of the parameter intervening in the stochastic differential equation driven by a known Lévy process :

These kinds of models are motivated by mathematical finance problems ([1]). In this context, the property of local asymptotic normality property (LAN) has become an important issue [2]. The LAN property is described as follows: a sequence of families of probabilities indexed by an open set is said to have the LAN property at each point with speed , if the sequence of probabilities localized around ,converges, in the sense of weak convergence of the associated likelihood processes, to a Gaussian shift ; see Section 2 for a precise definition. The LAN property allows to recover the so-called asymptotic Fisher information quantity . This quantity is crucial in any estimation procedure, since provides the lower bound of the variance of any estimator of .

The LAN property was investigated by Akritas [3] in models associated with Lévy processes observed continuously in time over the interval . He obtained the property under the assumption of differentiability, according to the parameter , of the characteristics of . With the same asymptotic, Luschgy [4] obtained the local asymptotic mixed normality (LAMN) property on models associated with semimartingales. As a notion, LAMN property is more general than the LAN because it allows the Fisher information quantity to be random. With the asymptotic , the estimation methods do not seem to be feasible in practice, for this reason, several recent works focused on discretized schemes, i.e., observations of the process along the discrete scheme

In practice, the most interesting case of the discretization path turns out to be relatively difficult. The classical case of a Brownian motion in (1) has been widely treated [5]. Clément and Gloter [6] studied the LAN for the model in (1), in the case where is a Lévy process attracted by a symmetric stable process with index . Aït-Sahalia and Jacod [7], Masuda [8], and Kawai and Masuda [9, 10] studied LAN property for the model in (1) in case of constant coefficients, i.e.,

Our investigation goes to same direction of Aït-Sahalia and Jacod [11], who studied the LAN property and the problem of estimation of the parameter involved in the model of a log-asset price , solution of (3) with being a standard symmetric stable process with index . Section 4 completes their situation in case where is a general stable process, eventually mixed. The last direction was initiated Rammeh [12] with observations according to random schemes for the scale model:where is a real unknown real parameter, and is a symmetrical standard -stable process. Rammeh showed that the LAN property always occurs, and his main arguments strongly rely to the linearity in to the fact that stable processes have the temporal scaling property and to the asymptotic behavior of the stable densities. Theorem 2 generalizes Rammeh’s results in the context of deterministic discrete scheme .

Because of the intricacy of the case (1), we first focus on the following model, which contains (5) and intercepts (1): we assume that for all , under , is a Lévy process, null at , such that its Lévy exponent is given by the so-called Lévy-Khintchine formula:where , and is a positive measure on which integrates .

For sake of clarity, we take is the open interval . As in the precited literature, we will assume the following.(i)The existence of densities , such that is regular enough,(ii)The convergence, as , of some integrals depending on .

Theorem 1 and Corollary 1 provide conditions ensuring the LAN property for the model (6), when the process is observed along the discrete scheme (3). Denoting , the logarithmic derivative of relative to , the asymptotic Fisher information quantity at each should satisfy

It is difficult to find Lévy processes fulfilling (7), and the reasons are numerous, for instance, the existence of the densities , the fact that they are not explicit in general, and their degeneracy as . For these reasons, Corollary 1 focuses on the linear dependance (4) of the characteristics relative to . In this case, we may assume, without loss of generality, that contains a reference value, 1 for example, and the value 0 is excluded in order to avoid trivialities. In this case, we only need to assume some regularities of the function and conditions of the kind (7) for . Let and be the logarithmic derivatives of . The asymptotic Fisher information quantity should then satisfy

The case of the discretization with constant path is quite obvious since the scale model (5) becomes a regular i.i.d. one, that is, to say is finite and nonnull. If , the situation is more intricate because degenerates when . It turns out that even the linear model (5) is falsely simple to handle. Intuitively, one looks at special Lévy processes attracted by stable processes on the sense of (10). The price to pay is to exhibit refined controls on the probability density function of . In a second step, we restrict our attention to the scale model (5). For simplicity’s sake, it is easier in this case to express the probabilities in the form (5) rather than considering them as solutions of martingale problems associated with the family of characteristics because of the intricacy inherent in the truncation functions [13]. Generic examples of Lévy processes are stable processes. They characterized Lévy exponent as follows. Let and

A stable process, with parameters , is a Lévy process , such that the corresponding Lévy exponent is given by

The parameter is the stability coefficient, is the skewness coefficient, is the scale coefficient, and is the drift parameter. The corresponding triplet of characteristics is given by

See [14]. In model (6), a candidate for the unknown parameter could be any the parameters , or . Since stable processes enjoy the scaling property, with ,then a candidate for the unknown parameter in the model (5) is clearly the parameter . For more account on Lévy processes, the reader is referred to [13] or [15] and for stable distributions, we suggest [16] and also [17]. Section 4 provides nontrivial examples of LAN models associated with Lévy processes attracted by stable ones. That means that there exist measurable functions and a nondegenerate distribution , such that

In [18], Far focused on the LAMN property for the model (5) discretized along the scheme , when the process is of the form , the sum of a standard Brownian motion and an independent compound Poisson process. She obtained LAMN property under the condition that the Lévy measure of has no diffuse singular part and that if is absolutely continuous, then the model has the LAN property. Our development in Section 5 constitutes a complement to Corollary 1 for the scale model (5) and also to Far’s work [18] and illustrates how to build a LAN scale model from another LAN scale model.

2. Definition of the LAN Property

In Section 3, we provide some theoretical results on models associated with observations, at times , of the process , and to illustrate by some examples. To this end, we consider the sequence of i.i.d. random variables and the family of -fields:

Denoting and , we introduce the sequence of filtered statistical models:

For any fixed , we denoteand we introduce the statistical experiments localized around :where the last statistical experiment is a Gaussian Shift. By a Gaussian shift, we mean, that for all is the unique probability on equivalent to on each and that its associated likelihood process is the geometric Brownian motion defined bywhere is a Wiener process, and then, under , the process is again a Wiener process. The quantity is called the asymptotic Fisher information quantity; it is a positive constant related to the sequence of statistical experiments in (17) and has to be determined. The asymptotic Fisher information quantity is crucial in any estimation procedure. Indeed, under the LAN property, is the lower bound of the variance of any estimator of . More precisely, HAJEK’s asymptotic convolution theorem says that if satisfiesthen the distribution is the convolution product , where and is a probability measure on . See [19] for more.

Local asymptotic normality of the sequence of models in (17), in a value , is actually equivalent to the weak functional convergence in time of the sequence of statistical experiments to the Gaussian shift in (17). This fact is explained as follows: let and be the likelihood processes defined, for all and at each time , bywith the convention , if . According to [5], the likelihood process of the statistical experiment is represented by

The notion of weak functional convergence in time was introduced by Lecam [2] and developed by Strasser [19] and Jacod [20]. It is expressed as follows: for every finite subset of , and every , we havein the sense of the weak convergence for the Skorohod topology.

3. When Does LAN Property Hold for Lévy Models?

Our aim is to give sufficient conditions on the p.d.f. of under , ensuring the LAN property for the sequence of filtered statistical models .

3.1. LAN Property for the Model (6)

In this section, we will consider the model (6). If , integrates , and (respectively, ), then the support of the distribution of is

In all other cases, the distribution of has a support equal to . There are many situations in which for all has a probability p.d.f. which is infinitely differentiable in . For instance, the latter holds iffor any and for some and some , see [21]. Later on, we may assume the following:

We denote and we define, on the interior of , the following functions:

When the number appears, it is always understood that is big enough so that and are in . For all , and , we denote

For statisticians, is a familiar quantity and corresponds to the Fisher information quantity at stage . The quantity is less intuitive; it is a localized quantity around the true value and corresponds to the rest of Taylor approximations at the order 1 of Hellinger integrals of the model.

We are now able to state our first result, that is, the LAN property for the model (6).

Theorem 1. Assume (H0) and the following conditions:

Then, the sequence of sequence of filtered statistical models (15), corresponding to (6), has the LAN property at with the speed and the asymptotic Fisher information quantity .

Remark 1. (i)Cauchy–Schwarz inequality gives , and both conditions (H1) and (H2) are implied by(ii)Under different conditions and a different proof, Masuda obtained ([8], Theorem 2.12) the same conclusion as in Theorem 1.

Genon-Catalot and Jacod [5] exhibited discretized models according to random sampling schemes associated with a diffusion process driven by Brownian motions (with coefficients dependent on and by an homogeneous way on ) and proved the LAMN property under conditions similar to (H0), that is, differentiability to the third order relative to and integrability of the densities of the processes. Their proofs have a general vocation in the sense that they only use the Markovian property of the processes and are based on a method of approximation of the log-likelihood. Because of the intricate form 16 of the likelihood processes, we show the weak functional convergence of to via the convergence of the Hellinger processes, and with a tool, one can find in [20].

Proof of Theorem 1. Fix . The Hellinger process of order between and , relative to , is deterministic and has the formAccording to ([20], Theorem 5.3), it is enough to show that the Hellinger processes between and , relative to , satisfy the following: there exists , such that for every , , and , and the convergence in lawholds. We will use this method because in our framework, the processes are also deterministic and have the following quite simple form one can find in [5]: with being the integer part of , we have(1) For , , take , and observe thatAccording to (32) and (33), it is enough to show that , and , and we have(2) Assume (H0), (H1), and (H2) for a fixed . Applying Taylor expansion at the first order of for and big enough, we get for , the representation of on :where for all , the functionsare defined on the interior of . Also, observe the following relations:Because of (38) and (39), one hasUsing (36), finally write(3) For all and large enough, so that , writeThen, use (39) and obtainAccording to (41), we havewhere(4a) We will now prove the following convergence that will imply (35): for all ,(4b). To prove (46), we use both (26) and (37), and for all and , we have the representation and the control.Let , and let . Using (26), (37), and Taylor expansion at the first order of , we obtain the galloping control, valid for all :Taking in (48), applying Cauchy–Schwarz inequality and assuming (H1) and (H2), we obtainWe then can writeand clearly (50) implies (46).
(4c). To prove (47), we use same the same arguments as in (4b), the Taylor expansion at the first order of , and the representationBy (48) and Cauchy–Schwarz inequality, we have the following control, valid for all :The latter impliesUsing (48), we also haveand reproducing the method, we used to get (54), and we obtainAccording to (38) and (54), we also haveFurthermore, (38), (48), and Cauchy–Schwarz inequality implyFinally, according to (45), (54), (56)–(58), we obtain the controland we conclude with the fact that assumptions (H1) and (H2) imply (47).

3.2. LAN Property for the Scale Model (5)

As a consequence of Theorem 1, we obtain a result for the scale model (5). The main argument is that the r.v. has a p.d.f under if and only if has the p.d.f. under . From now on, the functions , and denote, respectively, , the p.d.f. of under , the first and the second derivatives of . For and , we have the expression

Therefore, if we want (H0) to be satisfied, we need to impose . Then, for all , we havewhere the functions and are given by

Making a change of variables, we see that the quantities and , defined in (27), satisfy

Using Cauchy–Schwarz inequality and again a change of variables, we obtain the following: for all , the quantities , defined in (28), satisfy

Corollary 1. For the scale model (5), the sequence of statistical models has the LAN property with speed at each if the following conditions are satisfied:In this case, the asymptotic Fisher information quantity is .

Proof. It is immediate that (C0) implies (H0). Representation (63) shows that (C1) implies (H1) and (64) shows that (C2) implies (H2).

Remark 2. Since is the p.d.f. of , then we should be aware that the discretization path intervene in the assumptions (C1) and (C2). In Theorem 2, we will see that in some favorable cases, the effect of the path has a quite negligible effect. In general, the LAN property depends strongly on the asymptotic of .

4. Examples of LAN Property for Lévy Processes Attracted by Stable Processes

In this section, we provide some examples of Lévy processes satisfying the conditions of Corollary 1. This corollary provides the LAN property for the scale model (5), , under regularity and integrability conditions on the p.d.f. of . Unfortunately, these p.d.f., if they exist, are not explicit in general; for this reason, we focus on processes which belong to the domain of attraction of stable processes.

We recall that a process is stable, if it is a Lévy process with Lévy exponent given in (10). In the case and , the process is a subordinator, i.e., a positive increasing the Lévy process, and the distribution of , has a support equal to . In all cases, , has a p.d.f. which is infinitely differentiable and is explicit only for the particular values . These values correspond, respectively, to the first passage times of the Brownian motion, the Cauchy process, and the Brownian Motion. Otherwise, is be expressed only as the inverse Fourier transform of . As announced, the introduction, we focus in this section on the case where the Lévy processes , in the scale model (5), is attracted by a stable process, i.e., satisfying (13). Bertoin and Doney [22] showed that if (10) holds, then the process is attracted by a stable one, and then, the following holds:(i)There exists , , and a slowly varying function (i.e., a function satisfying ), such that and (ii), where is the p.d.f. of some stable r.v. ;(iii)The p.d.f. of , is infinitely differentiable and satisfies(iv)The convergence in (13) can be entirely expressed with the behavior of the tail of the Lévy measure of or by the existence of a Brownian component.

Observe that the p.d.f. of is represented by

As in (8), let us denote by and the logarithmic derivative of and , respectively. After the change of variable , the asymptotic Fisher information quantity in (63) takes the form

The convergence (66) does not ensure that is finite and nonnull; it only ensures that if it happens, then

The stronger controlis sufficient to prove (69), but it is hard to obtain. In [14, 17], we provided controls for the p.d.f. in the space variable and uniformly in small or big time , the processeswhere the coefficients vary in a set , the processes are independent, and is independent of the subordinator . We emphasize that in (71), satisfies (13) and does not enjoy the scaling property (12) anymore.(i)If and the stability coefficients are pairwise different, we have an asymptotic scaling property; if and and (respectively, ), then The last property will be useful in Subsection 4.2 when taking the path (respectively, ).(ii)If , we assume that the subordinator is itself attracted by a stable subordinator: for some and some (deterministic) speed ,

The function is necessarily a regularly varying function of order at 0 (respectively, ): and is slowly varying, i.e., . Then, there exist and , such that

4.1. The Scale Models Associated with Stable Processes Have the LAN Property

The following result is a first application of Corollary 1.

Theorem 2. For the scale model (5), assume is a stable process, such that is null if and . Recall is defined in (12). Let the path satisfy and one of the following conditions:(i)(ii) and , for some (iii) and , for some Then, the sequence of sequence of filtered statistical scale models (15) have the LAN property with speed at each . The asymptotic Fisher information quantity is given bywhere is the p.d.f. of .

Remark 3. If , then support of the law of is (respectively, ) if (respectively, ); otherwise, the support is whole . The assumption ensures that the support of the law of satisfies the condition (C0) of Corollary 1. We are aware that the cases are not very realistic by the statistical point of view. The most interesting cases are as follows:(a), then , and the assumptions of Theorem 2 are satisfied. This is an essentially trivial result because we treat then a regular i.i.d. model.(b), then , and the assumptions of Theorem 2 are satisfied if and only if one of the following holds:(i), and then, ;(ii), and then, ;(iii), and then, .

Proof. of Theorem 2. (1a) Recall that denotes the infinitely differentiable p.d.f of . Due to the scaling property (12), we have the following representations: for all and ,where(1b) In [14, 17], we provided several properties of the p.d.f. . For instance, there exist positive constants that depend explicitly on , such thatwhereand means . Furthermore, with the convention , the functionsare continuous on , given by (77), and there exist positive numbers , depending explicitly on and and , such thatFrom the last equivalences, we see that for any nonnegative integer , we have the implicationThus, since , then(2) We need to verify the assumptions of Corollary 1, i.e., to check (C1):and also (C2); there exists , such that for , one has(3) The scaling property (76) and the corresponding change of variables give the following representation of the quantity in (84):Using (82) and the fact that , we obtainDeveloping the last expression, integrating by parts and using the fact that and both tend to 0 as goes to each endpoint of the support (77), we recover (84).
(4) Again, by the change of variables corresponding to (76) and by the representation (85), one hasThanks to (83) and to the fact that , and it is clear that if , thenis integrable. Let . To prove the convergence (85), it is enough to have(5) Now, we distinguish between the values of .(a)If and if , then (91) is true. By assumptions, we have , for large . If , then we have (91) as soon as , which is equivalent to . Thus, we only have to choose and to get (90) to obtain (91) for .(b)If , then (91) is always true.(c)If and if , then (91) is true. As in (5a), we have for large . If then we have (91) as soon as , which is equivalent to . Thus, we only need to choose to obtain (90) and (91) for .

4.2. The Scale Models Associated with the Sum of Independent Stables Processes Have the LAN Property

This subsection gives a second example which also generalizes the previous one and achieves the situation (13). Define if and and assume the following restrictions on the skewness parameters:

Let and be the processes defined by

The processes satisfy the asymptotic scaling property (72). Denote by the p.d.f of ,and by , the one of . Then, satisfies

In [14, 17], we showed that the following functions and given byare well defined, and the above convergence holds uniformly in and still hold for the successive derivatives. As one can guess, we are going to exploit identity (95) and state the following result.

Theorem 3. Let be a sequence satisfying one of the following conditions:(i)(ii), and there exists , such that ;(iii), and there exists , such that .For the scale model (5) with , assume . Then, the sequence of filtered statistical scale models (15) have the LAN property with speed at each value . With given in (95), the asymptotic Fisher information quantity has the following expression:

Remark 4. (i)Let us briefly explain the nature of the assumption in Theorem 3. In [17], conditions of type allowed us to show that is distributed as an -stable variable mixed on the skewness and scale parameters by other processes. More precisely, for all , we have these identities in distribution: for all in the interior of the domain (given in ) and for all , there exist a r.v. and , such that The processes and are Where is a sum of independent standard stable subordinators, and the nonnegative numbers depend only on . In the case where is a constant for all , then , is distributed as a normalized sum of independent stable subordinators, and the converge in the following distribution holds. Furthermore, notice that the assumption is satisfied in the symmetrical cases(ii)In general, when are r.v.’s lying in the set of admissible parameters and is the -field generated by them, we gave in [17] a structure of the -conditional Lévy process to . See also [13] for the notion of the -conditional Lévy process. If is deterministic, is simply distributed as , and with our construction, we allow the same identity even if is random and correlated with . We also considered in [17] the densities of some families of mixed stable variables and gave several examples when these densities and their derivatives behave like the proper stable densities and this uniformly in .(iii)It is also possible to state a version of Theorem 3 with stable processes with drifts. It is enough to strengthen the conditions on the asymptotic as done in Theorem 2. We consider that Theorem 3 is far from being exhaustive. It is produced in the aim of illustrating the difficulty of this case. If one wants to reduce the assumption on the coefficients, then additional controls on the densities are needed.

Before tackling the proof of Theorem 3, we need the following result borrowed from [17].

Theorem 4 (See [17]). Controls of the densities of some mixed stable variables . Let be a pure jump subordinator characterized byAssume , where , and is a slowly varying function. For , define(a)If is slowly varying at infinity and or if is slowly varying at zero, then(b)Moreover, assume that there exist and , such that Let and denote one of these families: , and , with slowly varying at , and , with slowly varying at 0. Let and be any family of r.v.’s, such that , for some , (, if ). Then, the p.d.f. of the mixed stable variables are infinitely differentiable and satisfy the following: for all and all , we have For all , we have(c)If is a family of r.v.’s, such that for some and all , then the controls (106) and (17) remain true for the p.d.f.’s obtained by replacing by , where is given by (103).

Remark 5. If is a pure jump -stable subordinator, then , and the scaling property gives . Furthermore, if , is deterministic, then , for some . The estimates (106) and (17) are an immediate consequence of the behavior of the stable densities given in the Proof of Theorem 2.

Example 1. The following are examples of processes satisfying the conditions of Theorem 4. Let , and . Let be pure jump subordinators whose Lévy measures are, respectively, equal toThe process is the sum of independent stable subordinators, and is the independent sum of a stable subordinator and a standard Poisson process. With and , notice thatwhere are slowly varying at 0, is slowly varying at , and

The following result is a consequence of Theorem 4.

Corollary 2. Assume . Let be the p.d.f of , . Let , and

Then, for every nonnegative integer , we have

Proof. By Remark 5, notice that the Lévy measure of subordinator in (99) has the required conditions. According to Theorem 4 (c), these conditions imply that the family behaves like and that behaves like with and .

Proof of Theorem 3. We will check the assumptions of Corollary 1, which consist in the convergence of the integralsThe latter is guaranteed by Corollary 2, since the functions satisfyfor , and . The rest is obtained by reproducing the Proof of Theorem 2.

Remark 6. Notice that the main argument for proving Theorem 3 is the behavior of the densities uniformly in time. Theorem 4 provides many other examples. For example, with the same proof as in Theorem 3, one could state a version with stable processes time changed by any independent “nice” subordinator. The time change process could be the sum of a stable subordinator and a Poisson process (Remark 5). Finally, it appears that more investigation concerning the behavior in small time of p.d.f.’s of Lévy processes attracted by stable processes is crucial for statistical purposes.

5. How to Build a LAN Model from Another LAN Model?

In this section, we investigate to which extent the choice of the asymptotic is crucial. Assume we start from a LAN model associated to the observations of a Lévy process along a discretization scheme . Can we affirm that the model associated to the observations of , where is another independent Lévy process, also enjoys the LAN property with the same discretization scheme? We need some preliminaries and two lemmas to answer the last question.

Consider two independent Lévy processes and defined on some probability space with values in the Skorohod space (when the processes and are seen as infinite-dimensional random variables). Assume that is a nondrifted compound Poisson process with Lévy measure , and consider the process . Recall that the increment process of , observed along a scheme , is defined in (14) by

For , suppose we observe , and let

The probability measure (respectively, and the scale models (respectively, ) correspond to (respectively, ) as in (15) and (16).

Recall that if are two probability measures on some sample space, then the total variation distance is the quantity

Lecam’s lemma [2] is as follows:

Lemma 1. For every probability measures , we have the inequality

We also have the following result.

Lemma 2. If , then .

Proof. Sinceand since has stationary and independent increments, we haveIt is now clear that goes to 0, uniformly in as .
Now, we are able to complete Far’s problem [18], which treats the case where is a Brownian motion and where the discretization path is , i.e., .

Theorem 5. Assume . If the scale model (5) associated to the process has the LAN property with speed in a point , then so is the scale model associated to the process .

Proof. of Theorem 5. (1) Fix , and is a finite subset of and . We shall prove that the weak functional convergence (22) of the likelihood processes of yields the one of the likelihood processes of . The expression of the likelihood processes is given by (21), and for more convenience, we denote them from now on byWe need to show the following convergence in laws:or equivalently, for every -Lipschitz function , bounded by a constant , we need to show that(2) For such functions , we will control the difference:In virtue of Lemma 2, we haveFor every , we haveWith representation (21), observe that and are a step process, time-dependent, up to . Then, denotingand using Markov in equality, we obtainApplying Lemma 1 with , we obtainand then, Lemma 2 givesThe latter, together with (126), allows to conclude that

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest.

Acknowledgments

The work was supported by the “Research Supporting Project number (RSP-2021/162), King Saud University, Riyadh, Saudi Arabia.” The author would like to thank the referees for their valuable comments and recommendations that improved the content of the article. The author is also grateful to Jean Jacod who introduced him to the topic of statistic of Lévy processes.

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Copyright © 2021 Wissem Jedidi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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