Michael Filhol
email: michael.filhol@limsi.fr

John Mcdonald

Rosalee Wolfe
email: wolfe@cs.depaul.edu

Synthesizing Sign Language by

Keywords: Sign language, computational linguistics, computer animation, avatar

published or not. The documents may come

Introduction

Being a comparatively young field of research, Sign Language (SL) linguistics is still one of lively debate more than a set of tested and established scientific theories. Linguistic features and properties, for which a wide theoretical consensus exists, are not numerous. Most propositions to formalize SLs remain in question, including the most basic premises of language studies such as the nature or even existence of lexical units and a phonological level. [START_REF] Cuxac | La langue des signes française, les voies de l'iconicité[END_REF] [2] [START_REF] Stokoe | Semantic phonology[END_REF] Computer generation is therefore a goal located between two potentially unfortunate choices. One would be to choose a set of description formalisms arbitrarily. The other would be to keep playing back recorded (e.g. with motion capture) or hand-crafted (drawn) data with no model of it at all, which would never allow or foster content editing tasks. This paper presents our first exploration of a way forward between those two problematic paths. It combines a language description model set on making as few assumptions as possible about language structure, and an animation system built on the principle that the language description should drive the animation while making as few assumptions and limitations on how language processes can combine to affect avatar motion.

2

The AZee Language Description

The initial goal of the AZee framework was to enable formal description, without assuming any of the still debated hypotheses, especially when they would lay the foundations of the entire system. For example, all formal approaches to NLP assume not only a syntactic level of description but also a clear-cut and finite list of categories (e.g. N, V, Adj…), and a fixed mapping from a set of known lexical units to those categories. Whereas, the nature, relevance or even existence of such categories and a syntactic level are not agreed on when considering Sign Languages.

To enable SL description and synthesis without unwisely categorizing language objects, the AZee model proposed to fall back on three very weak linguistic assumptions before formalizing its visible features [START_REF] Filhol | Juxtaposition as a form feature; syntax captured and explained rather than assumed and modelled[END_REF], which are explained in turn below.

Language productions create observable forms carrying intended meaning. Observable forms are visible states and movements of any of the language's articulators, and synchronisation features between them. For example, A, B and C below are all visible form descriptors.

A: "eyelids closed" B: "left fingertip in contact with left-hand corner of mouth" C: "orient palm downwards" When two or more articulations are involved on a shared time line, synchronisation constraints between them such as "B starts before A ends" or "C is fully contained in A" are also form descriptors.

AZee calls a function any meaning that could be associated with such forms, whether intended (when the forms are produced) or interpreted (when the forms are perceived). For example, D, E and F are functions in this sense. D: "house" (as a concept carried by this word in English) E: "pejorative judgement on [person/object]"

F: "[date & time] form the context of [event]"

Systematic links between the two are what specifies the language. The idea behind this assumption is that language is a system, i.e. is governed by rules, shared between the users of the language, and that an experimental approach can identify. So we define the modelling problem as one of finding systematic associations between forms and functions. In other words, AZee proposes to capture every:  invariant combination of forms observed for the occurrences of an identified function, which yields a production rule that can be animated by SL synthesis software;  function consistently interpreted from the occurrences of a given form, which yields an interpretation rule that can support recognition software.

With our objective of synthesis, the former rule type is the one we will be referring to in this paper.

Languages allow for compositional structures. Compositionality in language is an essential premise of all language studies, admittedly already assumed in both sections above since they refer to "combinations" of forms and "pieces" of the meaning. This is why AZee allows for rule parametrisation, whereby a structure is defined as a process applied to its arguments. Comparably to common formal grammars, it also allows recursion.

These assumptions have two noteworthy corollaries:

 Multi-linearity: allowing to describe any sort of form combination, enrolling multiple articulators and defining potentially complex synchronisation patterns calls for the ability to partition and layer articulation tracks on top of one another, like a music score arranges multiple instrument productions on a single time line.  No functional partitioning of the articulator set: any form feature, articulation or synchronisation constraint can be associated with a function if systematically observed, and a single articulator can take part in many different functions.

These are novel properties for a formal model describing SL, which we have argued were an advantage over those generally used to this day [START_REF] Filhol | Non-manual features: the right to indifference[END_REF]. For example, the widespread standard HamNoSys [START_REF] Prillwitz | HamNoSys version 2.0, Hamburg notation system for sign languages[END_REF] relies primarily on manual activity as the vector for the lexical function, and builds SL utterances as a strings of juxtaposed lexical units, which makes it mostly linear. The AZee framework proposes a methodology to identify, and a language to formalise, production rules turning interpretable functions into the forms to articulate, and parametrise them to account for rule composition. Every rule is a <H, P, f(P)> triplet:

 H is the rule header, usually named after the function it carries, e.g. "house";  P is a list of parameters on which both the interpretation and the form might depend, e.g. the location of the house;  f(P) is a description of the form to produce if the rule is applied and given its necessary parameters, including all necessary and sufficient articulations and syn-chronisations, e.g. finger extension, duration of contact, etc. (by analogy with production rules in formal grammars, we call this part the right-hand side of the rule, RHS henceforth).

For example in LSF, expression of non-subjectivity in a measure, estimate or judgement X (in the sense that the speaker reckons X is generally not disputed), requires that lips be moved forward over the time used to sign X, see [START_REF] Cuxac | La langue des signes française, les voies de l'iconicité[END_REF]. This yields a rule:

 header: "non-subjectivity";  parameters: X = the estimated value or judgement;  RHS: see box diagram in Fig. 1, and the lip form to produce in Fig. 2. Rules being combinable, any rule's production can be used in place of a parameter for another rule. For example, consider the LSF rule "good", named after the English word for its meaning, with no parameters and whose RHS results in the fixed manual form in Fig. 3. One can use the product of this rule as parameter X for "non-subjectivity", nesting the former function inside the latter in an AZee expression:

(E1) non-subjectivity(good())

This composes the meaning of "generally deemed good", and directly produces the resulting form (Fig. 4) by combining the RHS of the invoked rules, which results in a multi-track time-line specification of the full arrangement. In the case of (E1), this means layering the lip pout of Fig. 2 over the manual form of Fig. 3. We call the resulting arrangement a sign score.

Fig. 4. Resulting score for (E1)

An AZee expression is generally represented by a tree structure, called a functional tree, whose nodes are the contained rules' headers used in the expression, and in which node A has child nodes Bi, if Bi are the arguments of A in the expression. The tree for our simple expression (E1) is:

We give another example, of a more complex tree below, for expression (E2):

(E2) info-about(open-list-non-mutex(cinema(), restaurant()), non-subjectivity(good()))

where:

 "info-about" has parameters A and B, carries function "B is the information given about A" [START_REF] Filhol | Juxtaposition as a form feature; syntax captured and explained rather than assumed and modelled[END_REF] and has the RHS of Fig. 5;  "open-list-non-mutex" has any number of parameters Ei, carries the function of itemising every Ei to form a non-exhaustive list of non-mutually exclusive items, and whose RHS is the succession in time of a Fig. 6 for each Ei;  "cinema" and "restaurant" have an optional point parameter each for location, carry the respective concepts they are named after in English, and have the manual RHSs illustrated in pictures Fig. 7 and Fig. 8. Layering all the RHS forms contained in the rules invoked in the expression, (E2) results in the score in Fig. 9, while composing the meaning: "cinema[s], restaurant[s], etc. are generally deemed good". The hope from there is that there exists a set of rules able to cover all language productions by combination, and that it remains small enough to be acceptable as a grammar model.

The Paula Animation System

Animating signed languages with an avatar is an ongoing challenge with difficulties arising both from the nature of human motion and the requirements of the input linguistic model. The flexibility and multi-track nature of the AZee description requires a commensurate level of flexibility in an animation system. An animation system such as [START_REF] Elliott | Linguistic modelling and language-processing technologies for Avatar-based sign language presentation[END_REF], which is based on the HamNoSys and SiGML representations, scripts animation in a sequential manner and would present many challenges in connecting to AZee. The Paula sign synthesis system [START_REF] Wolfe | An Avatar to Depict Sign Language: Building from Reusable Hand Animation[END_REF] provides a multi-track animation system that resembles AZee's representation. This section explores the nature of computer animation as it relates to sign synthesis, and provides an overview of the Paula system. Synthesizing sign directly from linguistic descriptions that specify key poses is well known to create robotic motion, but does have the advantage of being able to convey any utterance that can be encoded. Several other options for human character animation are also available, including traditional keyframe, procedural and motion-capture based animation. Each of these has distinct advantages and challenges when applied to the animation of signed languages.

Traditional hand-crafted key frame techniques rely on an animator for the composition of the movements of the avatar. The movements are controlled by setting key postures and interpolation data [START_REF] Parent | Computer animation: algorithms and techniques[END_REF]. Depending on the skill of the artist, the resulting animations can be extremely natural and appealing, with the advantage that the sparse nature of the keyframe data can be relatively easy to edit, modify and connect together to build animations from individual segments [START_REF] Mcdonald | An automated technique for real-time production of lifelike animations of American Sign Language[END_REF]. The main disadvantage of handcrafted keyframe animation is the high cost of animator time and the fact that new animations cannot be produced at will based on linguistic descriptions such as AZee.

Motion capture can produce some of the most realistic animation on an avatar since it is recording all of the subtleties of human motion. It can also be an invaluable tool for studying the motion of signers. Mocap recordings have provided insight into many aspects of sign including prosody and ambient non-linguistic motion [START_REF] Wilbur | A new technique for analyzing narrative prosodic effects in sign languages using motion capture technology[END_REF] [START_REF] Mcdonald | A new tool to facilitate prosodic analysis of motion capture data and a datadriven technique for the improvement of avatar motion[END_REF].

Mocap does come with challenges that make it difficult to use in a flexible sign synthesis system. First, motion capture can be nearly as costly to record and clean as hand-crafted animation is to produce [START_REF] Dent | What you need to know about 3D motion capture[END_REF]. It relies heavily on clean-up artists to both remove ambient noise in the system [START_REF] Mcdonald | A new tool to facilitate prosodic analysis of motion capture data and a datadriven technique for the improvement of avatar motion[END_REF] and also to fill in segments that are missing due to occluded sensors [START_REF] Gleicher | Animation from observation: Motion capture and motion editing[END_REF]. Second, the simple fact that mocap records all of the small details of human motion makes it challenging to stitch mocap recordings together in a natural way.

Another challenge with motion capture from a synthesis perspective is that the sheer detail and data density can often record more than we actually want when trying to layer linguistic forms. A motion capture clip will have all of the linguistic relevant forms (the ones meant to be described by the AZee framework) the signer is expressing baked in with all other observable forms, and swapping out only part of the effects on a set of joints for a different form will be akin to attempting to remove the sugar from a baked cake to try to replace it with honey. In contrast, a skilled animator with knowledge of sign language structure can produce animations which are very clean in their expression of the linguistic parameters of the utterance. This facilitates layering linguistic processes in synthesized utterances.

Finally, procedural techniques in animation model the movements of the body mathematically and are useful for synthesis in several situations including:  Providing time saving utilities for keyframe animation [START_REF] Wolfe | Inferring biomechanical kinematics from linguistic data: A case study for role shift[END_REF]. Such tools can shorten the time that animators need to spend in fine tuning details of the animation and reduce the cost of this form of animation.  Modelling the motion for linguistic forms that affect specific parts of the body [START_REF] Mcdonald | Analysis for synthesis: Investigating corpora for supporting the automatic generation of role shift[END_REF]. Such processes can be provided with a set of parameters to tune the motion to the linguistic form.

In addition, procedural techniques can be used to produce non-linguistic forms that nevertheless increase legibility of sign synthesis by adding carefully tuned procedural noise to an avatar's joints to simulate the ambient motion that all human signers exhibit even when still or holding a pose [START_REF] Mcdonald | A new tool to facilitate prosodic analysis of motion capture data and a datadriven technique for the improvement of avatar motion[END_REF].

In short, different animation techniques are useful or appropriate for different situations, and the Paula animation system is designed to leverage multiple animation techniques and capitalize on each technique's strengths. It strives to use coarser hand-animated blocks to increase the naturalness of the motion while leveraging procedural blocks to both increase the naturalness of the motion and layer additional linguistic functions over the base movement. The Paula animation system is thus a multi-level hybrid system that strives to strike a balance between flexibility and naturalness in the avatar's motion through two components.

The first component is a set of hand-crafted animations built in a sign transcriber that resembles a traditional animation package, but with controls tied to the linguistic components of sign language, see Fig. 10. This component allows the animation of basic lexical signs and other animation components that rely on artist-driven key-frame techniques [START_REF] Wolfe | An Avatar to Depict Sign Language: Building from Reusable Hand Animation[END_REF].

Fig. 10. Paula's animation interface

The artist's work is augmented by procedural techniques such as a spine assist computation that augments the forward and inverse kinematics controls on the model's arms to move the torso in a natural way, thus reducing work for the artist [START_REF] Mcdonald | An automated technique for real-time production of lifelike animations of American Sign Language[END_REF]. Another important aspect of these controls and procedural techniques is that they help assure that animations encode as many linguistic parameters as possible and are as sparse, i.e. contain as few "key postures", as possible to enable easier editing and combining of animations.

The second component of the Paula system composes complete signed utterances. The system structures signed language animation as a collection of parallel tracks. Each track contains a sequence of animation blocks which can be timed and synchronized arbitrarily, much like a sound recording.

Unlike traditional animation systems, each track can affect entire sections of the avatar's anatomy, and multiple tracks can affect the same anatomy without disruption or conflict. For example, in the example in Fig. 11, the system animates several overlapping processes present in (E2). The base lexical items previously animated are combined with a collection effects that influence the face, torso and head. For more details on the techniques used by this system see [START_REF] Wolfe | An Avatar to Depict Sign Language: Building from Reusable Hand Animation[END_REF].

Appropriateness of the Two Systems for Connection

As we have explained, Paula implements a time line of parallel tracks, on which timed animation blocks are placed so that performing the tracks simultaneously produces a full Sign Language utterance. The animation blocks on the time line have so far mainly been filled with:

 hand-crafted animations, more or less computer-assisted at the time of creation but ultimately fixed to be played back;  procedural specifications, i.e. parametrised descriptions directly producing articulations, e.g. extent and duration of mouth aperture or position of a hand in space.

In the Paula system, virtually any contents can fill an animation block, as long as it specifies an articulation. On the AZee side, by construction, a node in a functional tree can only exist if a production rule was identified, i.e. together with a parametrised description of a form to produce (the rule's RHS). This RHS mixes invariant descriptors (fixed articulations like the lip pout in "non-subjectivity") with parameter dependencies like X in the same rule, usually containing more function calls. In any case, it always results in a score of synchronised parallel blocks, containing the layered fixed articulations contained in the RHS terms of the input expression.

In short, AZee provides with a mechanism that turns any functional tree into a resulting score that is similar to the internal animation format of the Paula animation system. We propose to take advantage of this by mapping every fixed form descriptor to a Paula animation block. This way, any resulting score from AZee would be trivially translated to a Paula time line. For example, one could map:

 the lip pout descriptor in the RHS of "non-subjectivity" over to a Paula mouth shape animation block [OO] performing the lip movement, fitting on the lip articulator track;  and the fixed form in Fig. 3 of "good" to, say, a hand-crafted animation block

[GOOD] on the manual track.

Expression (E1) generates a Paula timeline with [OO] and [GOOD] directly in place.

And similarly for (E2) if all blocks of the resulting score in Fig. 9 is mapped to a Paula block.

This proposition allows to consider full synthesis of Sign Language animations straight from semantically-loaded linguistic expressions. We emphasise that it is possible mostly thanks to the fact that the two corollaries of AZee's axioms were assumed in the design of Paula from the start. That is, both systems use parallel tracks and free block synchronisation, with no need to project unit composition a 1D time axis or split the tracks early in the process. However, linguistically meaningful (i.e. systematically interpretable) compositions in SL are meant to be accounted for in AZee, even within an RHS. For example, the form for "restaurant" in LSF involves an iconic weak hand shape reference to a flat surface, in this case of a waiter's notepad (see Fig. 8). This flat palm and finger arrangement being interpretable in other contexts the same way, good AZee practice would be to incorporate a function call "flat surface" in the RHS of "restaurant".

This basically captures the morphemic function of the hand shape, and is a token of AZee's flexibility as it can capture any sort of linguistic function in any sort of context. This has the power of accounting for as much compositionality of the language as possible. But in our circumstance of generating realistic animations, we understand that this proposition leads us to generate and combine many low-level animation blocks, which we know leads to robotic gestures, and is generally unacceptable for final rendering [START_REF] Kipp | Assessing the deaf user perspective on sign language avatars[END_REF]. Thus, the connection between the two systems is guided by the following principle:

Guiding principle: The coarser the basic animation blocks, the more natural the final animation.

To generate coarser elements and honour this principle, we have to avoid processing the deeply nested AZee parts, and recognise subtrees for which we could have Paula blocks available, whether as pre-recorded natural-looking animations or as procedures defined on the Paula side.

For example, function "resto" applied with no arguments, represented by the simple one-node subtree, always generates the same form, namely that in the picture in Fig. 8 and enclosed in the thick box (*) in Fig. 9. If we have that sign recorded, we can shortcut the AZee production rule and avoid reading and developing its RHS (incl. its flat weak hand description). We may play back the recorded form directly from the recognised subtree, which allows for a natural looking play back of the whole rule's RHS. Generalising this, an alternative solution is to match Paula animation blocks from full AZee expressions instead of just the basic AZee form descriptors. Applied to our first example (E1), this causes the system to search for a Paula block ready for play-back and meaning "generally deemed good".

Clearly, pushing the principle to the extreme rapidly goes out of scale. There is no formal limit against wishing for direct Paula matches for higher-level subtrees, or indeed for entire inputs like the full expression (E2), which would require an infinite number of recorded forms. So, we need some counter-balance.

We thus propose to prefer coarse building blocks when they are available, but allow the system to fall-back on finer pieces otherwise. Therefore, we propose to create mappings to Paula animation blocks from both of the following sets:  a subset of all possible AZee expressions;  the complete set of basic AZee form descriptors.

From any AZee input, we only then need to read the functional tree from the root down, try matching the full subtree against the inventory of available Paula blocks at every step (giving preference to the taller tree matches when they exist), and developing the RHS if no match is found (which takes the process one level down inside the score building expression).

Validation and Moving Forward

The expression (E2) is a first example that validates the direction of this mapping and makes a first test of the two systems as potential components of a complete synthesis solution. Initially, the two systems were each built as separate research efforts, yet:

1. The design of the Paula animation system has always had as a goal the eventual connection to a structured linguistic system such as AZee. To this point, however, the linguistic features expressed by the avatar have been incorporated in an adhoc manner, driven partially by the physical capabilities of the avatar. The result has been a system that displays increasingly fluid, legible sign, but which needs a more complete linguistic structure to drive its animation. 2. AZee also has been designed with the intention of connecting to an animation system, but to this point has had to itself perform all the work of translating its forms into articulatory specifications for connection to an avatar that would simply interpolate those articulator specifications.

So, while both recognized the need and intention to connect to a system much like the other, they have each pursued their own goal and separate evaluations. However, each have now reached a level of maturity that the bridge between them described in the last section is feasible. Expression (E2) applied the AZee to Paula mapping for a set of test utterances in French Sign language (LSF). In each case, AZee built the required LSF structure and produced a high level score that coordinated linguistic forms as shown in Fig. 9. This score was translated in to a preliminary version of the XML encoding of the utterance. An excerpt of this XML encoding is displayed in Fig. 12. Notice that in the XML output, AZee explicitly produces a series of hooks for the Paula system to use to shortcut if it has the capability.

The XML encoding was then used to populate the tracks of the Paula sentence generator, as seen in Fig. 11. It is up to Paula to translate and interpret the timing hints in such a way as to ensure smooth transitions between animation blocks on a given track and to preserve an overall flow in the animation. The head sweeps, mouth shapes and blinks are layered as procedures on top of the hand-crafted animations that provide a shortcut for the three hand-animated blocks CINEMA, RESTAURANT and GOOD. The resulting animation can be seen at the following URL: http://asl.cs.depaul.edu/Paula-Azee/Paula-Azee-Example.mp4

The mapping between the two systems provides a first validation of each system's approach from a technical standpoint, gives evidence of feasibility, and lays the groundwork for further development of both systems. <block_score> <block id="gl1"> <paulablock name="CINEMA"/> <sync/> </block> <block id="pn1"> <paulablock name="AZEE-PIDGEON"/> <sync> <param name="dur" value=".8"/> <param name="end" value="gl1:-1:.25"/> </sync> </block> Fig. 12. AZee XML output of the utterance

Conclusion

The results presented in this paper demonstrate the feasibility of the mapping between AZee and Paula and also an initial validation of the proposition in section 4. The power of this mapping, and thus the method for animating structured Sign Language linguistics lies in the fact that higher levels of animation can be used to animate entire sections of the AZee tree thus leveraging hand-animated or captured motion. In addition, the structured nature of the linguistic model allows us to exploit the layered nature of the AZee score to structure the corresponding animation. These lessons are important for any system that wishes to synthesize Sign Language animation based on a structured linguistic description. It is, however, important to note that the details of some of the elements presented here, such as the XML format in Fig. 12 are preliminary and will need further study to expand into a full system. Moving forward, we plan to expand the mapping between these two systems to more fully encompass the structures encoded by AZee and the range of expression available in Paula.

Fig. 1 .Fig. 2 .Fig. 3 .

 123 Fig. 1. RHS box diagram for "non-subjectivity"

Fig. 5 .Fig. 6 .Fig. 7 .Fig. 8 .

 5678 Fig. 5. RHS of "info-about"

Fig. 9 .

 9 Fig. 9. Resulting score for (E2)

Fig. 11 .

 11 Fig. 11. Paula's Sentence Generation Component


Michael Filhol
email: michael.filhol@limsi.fr

John Mcdonald
email: 3jmcdonald@cdm.depaul.edu

Extending the AZee-Paula Shortcuts to Enable Natural Proform Synthesis

Keywords: Sign Language Linguistics, Proforms, Classifiers, Synthesis, Avatars

Proform structures such as classifier predicates have traditionally challenged Sign Language (SL) synthesis systems, particularly in respect to the production of smooth natural motion. To address this issue a synthesizer must necessarily leverage a structured linguistic model for such constructs to specify the linguistic constraints, and also an animation system that is able to provide natural avatar motion within the confines of those constraints. The proposed system bridges two existing technologies, taking advantage of the ability of AZee to encode both the form and functional linguistic aspects of the proform movements and on the Paula avatar system to provide convincing human motion. The system extends a previous principle that more natural motion arises from leveraging knowledge of larger structures in the linguistic description.

Introduction

Producing natural synthesis of sign languages using an avatar is a goal with far-reaching applications for Deafhearing communication including improving sign language education tools, anonymizing online communication in sign language, and enabling translation for situations where hiring a certified interpreter is impossible. To support all of these applications, a sign synthesizer must be able to express all aspects of sign language including the full range of body signals used to communicate. Sign languages use a range of linguistic processes to communicate, the most basic of which being those listed in sign dictionaries. These are gestural units that have a standardized and stable meaning-form association. In addition, signers use a range of grammatical processes for rich, natural communication. All of these processes are communicated through signals involving the arms, eyes, face, torso and neck of the signer. Natural sign synthesis remains a challenge partially because of the fact that these structures can overlap and interact on the body [START_REF] Weast | American sign language tone and intonation: A phonetic analysis of eyebrow properties. Formational units in sign languages[END_REF] and animating such structures requires leveraging both sign language linguistics and knowledge of human motion [START_REF] Braffort | Aplus: A 3d corpus of french sign language[END_REF]. Most current sign synthesis systems are able to animate a stream of lexical signs [START_REF] Wolfe | Avatar to depict sign language: Building from reusable hand animation[END_REF][START_REF] Elliott | Linguistic modelling and languageprocessing technologies for avatar-based sign language presentation[END_REF][START_REF] Lombardo | An avatar-based interface for the italian sign language[END_REF]. However, more freehand constructs in sign languages such as classifier placement and movement or size and shape specifiers remain a challenge because of the inherent variability in form expressed through the signer's body. These structures use configurations on the signer's body known as proforms, wherein a part of the signer's body stands in for an object and is often iconic of its shape. For example, in American Sign Language (ASL) and French Sign Language (LSF), the index finger oriented vertically will represent a standing person, whereas a "C" handshape will represent a cylindrical object such as a glass [START_REF] Liddell | Grammar, gesture, and meaning in American Sign Language[END_REF]. Such proforms can be used to express the placement or movement of objects or to describe their size and shape. Proform structures have traditionally been a challenge for synthesis systems, both from a linguistic and from an animation standpoint, because they are highly productive. The placement and movement of body articulators cannot easily be captured by pre-set configurations, and so cannot be pre-animated or recorded. Thus systems relying on motion capture [START_REF] Gibet | The signcom system for data-driven animation of interactive virtual signers: methodology and evaluation[END_REF] or reusable hand animation [START_REF] Wolfe | Avatar to depict sign language: Building from reusable hand animation[END_REF] must fall back on more primitive synthesis techniques. In addition, while many proforms are predefined for a given sign language, variations in the shape or configuration of an object can be expressed through a near infinite range of hand or body configurations, especially when one considers size and shape specifiers [START_REF] Liddell | Grammar, gesture, and meaning in American Sign Language[END_REF]. Prior efforts have relied either on synthesis from phonetic components [START_REF] López-Colino | The synthesis of lse classifiers: From representation to evaluation[END_REF] or on predefined templates that encapsulate a limited set of standard proforms [START_REF] Huenerfauth | Spatial representation of classifier predicates for machine translation into american sign language[END_REF]. In both cases the results were far from natural involving awkward body configurations and robotic motion. The present work addresses three weaknesses that contribute to the robotic nature of the resulting animations.

1. Body gestures are never fully specified by the linguistics since there are a range of body configurations that can satisfy given linguistic constraints [START_REF] Filhol | Sign description: how geometry and graphing serve linguistic issues[END_REF]. Synthesis systems have had to fill in the missing body constraints for example by overspecifying the motion linguistically.

2. Synthesized motion has been limited to the avatar's arms, whereas natural human arm motion is always accompanied by supporting torso and clavicle motion. (McDonald et al., 2016a).

3. Motion specifications were limited to key positions for the handshape and didn't specify dynamic differences such as acceleration or speed profiles that can profoundly affect perception.

To fully specify proform movement and placement, and to produce natural motion, the synthesizer must take into consideration both:

• the linguistically defined constraints, which abstract human motion into meaningful gestural units;

• the range of human motion that accompanies such gestural units, but which are not encoded linguistically.

This paper extends the work in [START_REF] Filhol | Synthesizing sign language by connecting linguistically structured descriptions to a multi-track animation system[END_REF] which sought to bridge between a structured linguistic model of sign and a multitrack sign animation system. The system described here achieves natural linguistically driven proform motion through two key features. First, by separating the task into separate linguistic and animation components, it allows the linguistic component to encode the necessary information for the proform while allowing it to remain underspecified at a geometric level. This gives the avatar the needed freedom to move naturally within linguistic constraints. Second, it builds on the prior model's principle that natural motion is best achieved with large linguistic structures rather than from very basic phonetic specifications.

The next section explores the perspectives of each system on proforms and what each offers for building a combined synthesizer.

Perspectives of the two systems on proforms 2.1. AZee for descriptions

The Sign Language description model AZee [START_REF] Filhol | Nonmanual features: the right to indifference[END_REF] has several advantages that are relevant to the use of proforms in SL. The major one is its fully embedded geometric system that allows to build and describe points, vectors and paths in the signing space. From the origin of its predecessor Zebedee, geometric specification of body locations and skeletal orientations as points and vectors in an affine 3D space1 have been an essential feature of the descriptions (Filhol and Braffort, 2006a). More than an alternative style of body posture description, such geometric approach accounts for at least three notable features of SL, which are difficult to capture with other, e.g. parametric, models. First, it does not rely on a discrete set of points for locations in space, or directions for orientations. It allows to define geometric objects in a continuous space. In other words given two points, it is always possible to take the midpoint of the two. When making free productive uses of space, e.g. placing proforms to indicate relative positions, it is therefore possible to account for any relative placement, such as a date "in the middle of" the two boundaries of a delimited period on a time axis. Second, dependencies between elements of the descriptions are made relevant. This is useful for depicting structures such as the proform placements or movements we are addressing because positions are often relative to (dependant on) each other. For example, in the predicate "rabbit near and on the right of tree", the target location point for the rabbit proform is relative to that of the tree. AZee allows to express the rabbit's position as a geometric translation of the point with the appropriate distance, instead of projecting to a grid defining everything relative to the chest. Third, not only hands can be specified target locations but any articulator of the body. We have argued the necessity of this in several papers, but it becomes all the more relevant in dealing with depicting structures. Placing two-hand classifiers (e.g. round plates) and placing a full-arm classifier (e.g. tree) pose a problem if we want to consider hand placement alone as changing the classifier inside any expression will require a change of location as well. In AZee, a plate and a tree can be placed at the same location using the same point, even though hands actually end up in completely different locations.

Essentially, AZee is a language to write rules mapping invariant and parameterized forms to identified semantic functions, regardless of the level of linguistic description. Classifier/proform placements are indeed units that are difficult to locate in those terms, as they are arguably both lexical and grammatical constructions, or neither (Johnston categorizes those separately as "partly lexical signs"). AZee bypasses this problem, as any function-to-form link is tackled using the same description model. Like any other rule, a classifier generates a set of articulatory constraints for a consistently interpreted meaning. For example, the upright index finger shape denotes a standing person, possibly with a wrist orientation depicting the direction in which the person is facing. Because the meaning conveyed with this finger arrangement is consistent, an AZee rule "proform-standing-person" can be defined to specify the appropriate articulatory constraints: index up, possibly facing along a parameterized direction, other fingers closed. As we stated with the AZee approach, only the set of necessary and sufficient constraints are to be specified. In this instance we therefore exclude palm orientation from "proform-standing-person", because only fingers matter. In contrast, a rule for "tree" would have to include all bones from the tip of the fingers down to the elbow, since not extending and spreading the fingers would result in breaking the meaning. The AZee approach also encourages and facilitates factoring similar forms into new rules when the interpreted meanings share a common factor across multiple productions. For example, take the placements of entities in the signing space of the kind Liddell glosses with -BE-AT ↓ suffixes. All are produced with a small settling movement towards the surface on which the object is placed (often downwards), and an eye gaze towards the target location. This is true regardless of what the object is, and regardless of what articulator set is conveying the object. In AZee, one would therefore factor this common form, parameterizing the proform prf and the location point loc to define a rule with semantic function:

"placement of prf in space at location loc"

producing the form:

"small straight movement of prf down to point loc + look at loc + synchronize eye gaze with a negative time offset".

Proform prf and point loc become empty placeholders, and necessary arguments of the created rule. More factoring can take place, this time with a rule already found and reported elsewhere: the "category" rule, whose semantic function is to give a category in which to interpret a second argument item. It allows to juxtapose "town" and "Berlin", or "profession" and "bakery" to specify a sort of hypernym for the second item, likely but not necessarily ambiguous on its own. Classifier constructions in LSF often involve juxtapositions of a dictionary sign and a placement like the one described above, applied to a proform. The overall form then, including the juxtaposition, a slight head tilt and a specific inter-sign transition timing, is the same as that specified by the "category" rule. Plus, the meaning is to us all but similar: the first item gives a class of which the second is an instance. Therefore, from the three rules below:

• category

• proform-vehicle

• place-prf and a point loc, one can build the complex expression:

(E1) category(car(), place-prf(proform-vehicle, loc))

producing the sequence traditionally glossed as:

CAR VEHICLE-BE-AT ↓loc and taking care of the precise timing and adding the gaze towards loc.

Paula for Natural Animations of Sign

The Paula sign synthesis system compliments AZee's linguistic proposition in supporting natural animations of SL.

As described in previous publications, it is a hybrid animation system that supports layering motion from a variety of sources including procedural, keyframe, etc. [START_REF] Mcdonald | An improved framework for layering linguistic processes in sign language generation: Why there should never be a brows tier[END_REF]. From the synthesizer's perspective, proform movements can be modeled as a collection of keyframe data. In this respect, Paula offers a range of features that allow it to produce more natural animation from such data, and allows leveraging animator/sign-expert skills to a larger degree than prior systems. The following features are key to producing natural animations of proform movement from the linguistic specifications:

1. Key postures can be set by using either forward (FK) or inverse-kinematics (IK) systems tuned for sign linguistics (McDonald et al., ). The IK system allows any point defined relative to the hand or arm to be placed at a chosen target in space, or at a chosen site on the avatar's body, and allows full exploration of the redundant degrees of freedom in the IK chain.

2. Keyframes can be scheduled completely asynchronously on different articulatory chains [START_REF] Mcdonald | An improved framework for layering linguistic processes in sign language generation: Why there should never be a brows tier[END_REF].

3. Interpolations between key postures are accomplished using nonlinear rotation controllers that create transitions following the natural arcs of human motion. These interpolators also allow independent control of speed and trajectory along paths.

4. Procedural techniques automatically allow the torso and shoulders of the system to naturally support and accompany arm movements (McDonald et al., 2016a), and also add sub-linguistic ambient motion [START_REF] Mcdonald | A new tool to facilitate prosodic analysis of motion capture data and a data-driven technique for the improvement of avatar motion[END_REF].

These features all contribute to allow the synthesizer to produce smooth, natural movement from the linguistic features described above.

The first of these features allows any articulator on or near the arm to be used for targeting, and facilitate the natural positioning of two-handed proforms such as in the roundplate classifier example from the last section. The fourth feature in this list extends the avatar's arm motion through the trunk all the way to the hips, directly addressing the second cause of robotic motion cited in the introduction. Finally and Paula's nonlinear rotation controllers provide for independent velocity control on articulatory chains, needed to animate the linguistic categories of dynamics, thus addressing the third cause of robotic motion.

To address the underspecified nature of proforms, Paula also offers components that help leverage animator and sign-expert skills as far, and as deep, as possible in the synthesis process. The first component most often used in sign animation is Paula's Sign Transcriber, which scripts for prerecordable segments [START_REF] Wolfe | Avatar to depict sign language: Building from reusable hand animation[END_REF]. In spite of the fact that proforms are highly variable, Paula's Sign Transcriber allows animators to provide significant body posture and movement hints, which the proform generation system can then use to produce more natural animations than would be possible from the linguistic data alone. What remains is to build a coherent bridge between these two systems so that the necessary constraints are communicated while giving the animation system the freedom it needs.

3. Extending "the coarser the better"

Last proposal (and clarification)

The present work builds on the AZee-Paula bridge proposed in [START_REF] Filhol | Synthesizing sign language by connecting linguistically structured descriptions to a multi-track animation system[END_REF], which was based on the principle that working with larger blocks of animation or procedural motion will generate more natural sign synthesis compared to animating from individual constraints and joint settings. The previous bridge mapped from AZee expressions that could be recognized when reading block descriptions output by the AZee parser. This allowed the animation system to shortcut the application of a block if a prerecorded or procedural animation is directly available, rather than reconstructing the entire block's animation from low-level primitives nested in the generated form description. This principle, which the authors called "the coarser the better", relies on AZee's organization of scores as a hierarchical nesting of blocks, and asserts that the Paula animation system is able to produce more natural animation with larger parent blocks compared to combinations of many small child blocks. The system thus considers parent blocks in the hierarchy first, and falls back on developing child blocks only if no match is found.

The system accomplished this with a library of preanimated segments and procedural techniques that correspond to blocks named after their AZee source expression.

If the system recognized the expression in its entirety, the resulting animation or procedure would be invoked on the required portions of the avatar. In the absence of a match for an animation block, the system recursively falls back on animating child blocks until individual articulatory constraints would be required, sacrificing naturalness in the resulting synthesis. The multilinear nature of Paula's skeleton integrated these animation blocks into a seamless whole even when overlapping blocks simultaneously used a given set of articulations.

Limitation for Synthesizing Proforms

What the prior work did not insist on (and indeed possibly confused by basing its narrative mostly on functional AZee trees) is that all short-cuts described and exemplified were taken based on the nesting of the score blocks, therefore only dealt with the specified forms to animate. The contents of blocks in the shown XML structure contain a source AZee expression which is functional, but it was used as a mere string label for a stored form with which to shortcut. This means that nothing on the functional (semantic) side of the AZee system was considered during matching. This becomes a significant issue when considering proform synthesis.

For instance, consider applying this matching scheme to the proform expression below:

(E2) place-prf(proform-standing-person, translate(@ST, along(medium, fwd)))

The previous system would first run the linguistic AZee interpreter to generate an XML specification of the resulting score of nested blocks [START_REF] Filhol | Synthesizing sign language by connecting linguistically structured descriptions to a multi-track animation system[END_REF], illustrated in Figure 1. It would then search for a procedural or prerecorded animation short-cut whose name perfectly matched the source expression (E2), labelling the outer-most level of block nesting. It is unlikely that a perfect match, including the exact expression for the second argument, would be available as a prerecorded animation due to the infinite number of possible values for this argument. The system would then fall back on the block's child constituents (the inner blocks), which we see already consist of low-level constraints. Synthesizing from those would already be sacrificing naturalness. In this case such fallback is therefore a leap from too high a level to shot-cut, to one that is too low.

Yet the placement of a standing person does have a consistent natural dynamics, regardless of its location. There is a consistent handshape, with the upright index finger being slightly over-extended, and a consistent arcing motion, resulting from elbow and shoulder rotations which the description would specify as "downward" though it never comes out as such. If the animator has provided examples of this kind of motion, or if the system has a procedural specification for the right dynamics based on corps study, the animation system should be able exploit them, and producing in a more natural animation than one from a sparse "straight movement down" description. The "coarser the better" principle should somehow apply here despite the fact that the top-level expression cannot fully be matched.

To do this, the system needs a way to look into the AZee expression and enable some form of short-cutting there. In other words, in addition to matching blocks in the form tree as covered by our previous bridge, we need to expand our short-cutting scheme to look into the information on the functional side of the input as well, i.e. the AZee expression itself.

For example, to apply a generic proform placement procedure when animating (E2), we need to recognize part of its contents, with a template like the one below:

place-prf(proform-standing-person, X)

where X can be anything, provided it is an expression that evaluates to a point where to place the proform, and can be retrieved for actual use with the matched animation procedure.

Proposal for new system and results

To extend the "coarser the better + fall-back" principle, the new model proposes to allow an intermediate check for matching such templates via the following extended fallback mechanism:

1. match as label for form shortcut;

2. match with template for functional shortcut; 3. recursively process child blocks if no match (recursion terminates when reaching a block consisting of lowlevel constraints only).

The matching in step 2 is similar to the characterization of classifier motions as abstract templates [START_REF] Liddell | Grammar, gesture, and meaning in American Sign Language[END_REF], with a set of parameters provided by the linguistic system. It is then up to the animation system to read and interpret those parameters. For example, consider processing the expression below, where "midssp" is a pre-defined name representing a point in the centre of the signing space.

place-prf(proform-vehicle, midssp)

Assuming that Paula cannot find a full match for the complete expression, the animation system will perform the following steps. Notice that we are using the word template here both for the AZee expression matching and also for the animation data specified by the artist for the specific proform proform-vehicle.

1. look for a template procedure for place-prf ;

2. look for a procedure or animated version of the "proform-vehicle";

3. Evaluate the midsp expression to obtain the point for placement;

4. Animate the avatar using the template data.

In evaluating the proform-vehicle argument, the system will look for an artist specified (or motion-capture if available) template to use for the animation. Paula will then be able to leverage a range of information from that template to use in the formation of the key-frames for the action. Among these will be the handshape, the torso and clavicle parameters, the arm and elbow height as well as the articulator point on or near the arm, used for targeting the proform.

In the case of the place-prf action above, the system would then set up two key-frames with associated velocity controls to "settle" the proform at point "midssp". This settle action is an example of one of Johnson's ballistic transitions that passes through the first position with a smooth speed and then eases to rest in the position of the second key. In setting up the keyframes and animating the segment, Paula will leverage the features described in section 2.2.. In particular, it will:

• read the handshape, initial elbow configuration, preferred comfortable height and other data for the proform from the artist template;

• use the IK system to set up the two keyframes for the motion (the first one with a target point that is above the final point by a small amount), adjusting the data from the artist template as needed;

• use the spine assist and livening procedures to move the avatar's torso in concert.

From these keyframes the nonlinear motion controllers will move the avatar's arms along natural arcs, which will be straight enough here to provide a perceptual "straightdown" motion. Note that in the first item above, the system reads a name for the handshape and triggers a procedure without looking into the articulatory constraints that compose it on the child level. The second item similarly applies a generic two-keyframe layout to implement a recognized AZee pattern. We emphasize that by doing so, the system is performing a shortcut on an element of the functional expression.

If at some point in this process Paula fails to find a match, for example if Paula does not have a template for the proform action, or the profrom specification deviates from one of the "known" forms, the system then falls back to the expression's child blocks which will give a set of primitive constraints for the movement. Again, this would necessarily sacrifice quality but provides robustness for the system. This part of the system has not yet been integrated.

In addition to the place-prf template, the system currently also supports a move-prf template to provide movement of a proform along a path. Other proform templates will be added as the system matures. Figures 2 and3 show two The slight blur in the torso and face in 2 is indicative of the subtle spine movement that automatically supports the motion. Larger arm motions across the body as in Figure 3 are naturally accompanied by a larger torso motion, just as they would be for a live signer. This avatar torso movement is an improvement over the results of prior efforts in which the torso was stationary, and contributes to greater naturalness in the resulting animation.

Figures 4 and 5 demonstrate the generated placement movement for a selection of proforms and placement sites.

The system can use any site in front of the body for placement and movement, both for one-handed and two-handed proforms. These figures further illustrate the responsiveness in the torso algorithm through the degree of blur in the head caused by the superposition of the two frames. The torso reacts more to the two-handed motion in figures 5a and 5b, causing a larger blur relative to the motion shown in figures 4a and 4b.

An animated version that demonstrates placement and movement of a selection of proforms is available online at http://asl.cs.depaul.edu/proforms/ proformPlacementAndMovement.mp4. This animation also provides examples of placement in several locations to show the flexibility of the system. In all of these cases, Paula uses its procedural models of human motion along with data from the artist template to provide natural poses and transitions.

Conclusion and future work

This paper represents a first attempt at achieving more natural proform placement and movement using the structured linguistic model AZee to drive the hybrid animation system Paula. By extending the existing XML matching scheme for shortcutting to allow templating on the functional AZee expression, the system provides a flexible way for the animation system to leverage animator specified data for known proforms to improve postures and movement, while specifying a robust fallback algorithm, that still requires implementation. This sacrifices quality when the proform actions or the proforms themselves are not recognized.

Moving forward, the bridge currently supports a selection of movement procedures, but more work needs to be done both linguistically and geometrically to identify other proform structures that can be shortcut. In particular, the present work has focused on placement and straight movement of isolated one and two-handed proforms. More generally, proforms in sign languages exhibit a wide range of motion styles including bounce and wavy motions that indicate styles of movement, very general spatial movements to trace the size and shapes of objects, and relative proform placement and movement for complex scene descriptions. The current models have been built to support many of these structures, but extended study is needed for such complex motions to both refine the linguistic descriptions and naturally coordinate motion between interacting proforms. Finally, in the future, we hope to set up a web interface providing rendered videos from input AZee expressions. This way, users will be able to connect and sandbox with our system.

Figure 1 :

 1 Figure 1: Score for expression (E2).

Figure 2 :

 2 Figure 2: A standing person moving to center

Figure 5 :

 5 Figure 5: A plate placed on the right of the signing space (a); a plate placed on the left (b)

In geometry, an affine space is a vector space with no chosen metric or origin. In our case everything is defined relative to the body, including directions and distances. This allows implementation with any avatar. No body geometry is assumed by AZee.

Acknowledgements

The authors would like to thank Université Paris Sud for their support of Dr. John McDonald through their "recrutement de professeur invité" grant program, Dr. Annelies Braffort for her valuable input and suggestions concerning proforms and Dr. Rosalee Wolfe for her valuable input on avatar posture.


Maxime Compastié
email: .maxime.compastie@orange.com

Rémi Badonnel
email: remi.badonnel@loria.fr

Olivier Festor
email: olivier.festor@loria.fr

Ruan He
email: ruan.he@orange.com

Mohamed Kassi-Lahlou
email: mohamed.kassilahlou@orange.com

A Software-Defined Security Strategy for Supporting Autonomic Security Enforcement in Distributed Cloud

Keywords: computer security, network and service management, cloud computing, autonomic computing, policy based management

We propose in this paper a software-defined security framework, for supporting the enforcement of security policies in distributed cloud environments. These ones require security mechanisms able to cope with their multi-tenancy and multi-cloud properties. This framework relies on the autonomic paradigm to dynamically configure and adjust these mechanisms to distributed cloud constraints, and exploit the software-defined logic to express and propagate security policies to the considered cloud resources. The proposed framework is evaluated through a set of validation scenarios corresponding to a realistic use cases including cloud resource allocation/deallocation, cloud resource state change, and dynamic access control.

I. INTRODUCTION

Cloud computing defines a new architectural model for building ellaborated infrastructures and applications based on multiple computing resources, such as virtual machines, network devices, software components, provided as a service that can be easily deployed through the Internet. According to the NIST Institute [START_REF] Mell | The NIST Definition of Cloud Computing[END_REF], this model is mainly characterized by the following features: on-demand self-service, broad network access, resource pooling, rapid elasticity, and measured service. It supports an as a service scheme that permits a transparent access to resources and the outsourcing of part of the management to the cloud provider. This separation enables optimizing the resource allocation and usage, but may also introduce management complexity due to its distributed nature. In particular, the cloud infrastructure and its applications may typically be divided into isolated sets of resources called tenants, corresponding to different ownerships and requirements, defining the multi-tenancy property. Another property comes to the facts that the resources may be distributed among several infrastructures, as each of them may be specialized in a dedicated processing. Distributed cloud can be defined by the conjuction of the multi-tenancy and multi-cloud properties.

In this context, security management has become a major challenge. The dynamics of cloud infrastructures induced by their on-demand self-service, rapid elasticity and distribution has outrun traditional security management, while the ubiquity and high availability of cloud resources make them attractive targets for attackers [START_REF]Top Threats to Cloud Computing v1. 0[END_REF]. The paper introduces a software-defined security strategy for supporting autonomic security enforcement in distributed cloud. It investigates how autonomics and programmability mechanisms could support such a security management. Autonomic computing permits to address the scalability issues generated by large and distributed cloud infrastructure resources, by delegating part of the management tasks to the environment itself. In addition, network programmability has already showed its benefit for softwaredefined networking by separating the network infrastructure into two separate planes, i.e. data-plane and control plane, and contributing to its dynamic configuration and adaptation.

Similarly, there is an important need for supporting software-defined security in distributed cloud. We propose in that context a dedicated framework to specify security policies at this scale, and permit their autonomic enforcement in a multi-tenant multi-cloud environment. Security mechanisms should be dynamically aligned and adjusted based on changes that may occur in the distributed cloud. We detail the corresponding architecture and confront it to a set of validation scenarios corresponding to a realistic use-case.

The following of this paper is organized as follow: an overview of existing works and their limits is given in Section II. The proposed architecture and its components are described in Section III. They are evaluated through a set of validation scenarios in Section IV. Finally, Section V concludes the paper and points out future research efforts.

II. RELATED WORK

The security of cloud infrastructures has already been largely explored in the literature. In particular [START_REF] Waller | Policy Based Management for Security in Cloud Computing[END_REF] highlights several challenges related to policy-based security management, such as the specification of a cloud security policy, the assurance of the security decisions, as well as the the certification of security components in that context. In the same manner, the TCloud framework [START_REF] Bessani | The TClouds Platform: From the Concept to the Implementation of Benchmark Scenarios[END_REF] proposes to enforce a security policy with a hardened cloud stack. This one provides infrastructure-level and platform-level security components, that might be compatible with multi-cloud environment, with a hardened build of OpenStack environment. However, these solutions do not specifically address self-configuration mechanisms, nor the management issues generated by multi-cloud and multi-tenancy properties. The author of [START_REF] Atinuke | A Policy-Based Management Framework for Cloud Computing Security[END_REF] proposes also a cloud management framework able to deal with multi-tenancy, but this one is limited to access control policies and cannot support other security mechanisms. The proposed architecture is independent from the available security mechanisms and adresses their self-configuration in a distributed cloud.

In the area of programmability, software-defined networking (SDN) permits to separate the control plane making decisions about where the traffic should be sent from the data plane forwarding of packets. This paradigm enables a dynamic and adaptive policy enforcement. It may also serve as a support for chaining security functions. For instance, the Flowtags framework described in [START_REF] Kaveh Fayazbakhsh | Flowtags: Enforcing Network-Wide Policies in the Presence of Dynamic Middlebox Actions[END_REF] enables the integration of middleboxes whose composition is supported by SDN controller. [START_REF] Koorevaar | Dynamic Enforcement of Security Policies in Multi-Tenant Cloud Networks[END_REF] propose a framework for enforceing a network security policy through a set of middleboxes. But, this solution only considers middleboxs for instanciating security mechanisms. IETF is also working on SDN-based security services using interface to network security functions [START_REF] Jeong | Software-Defined Networking Based Security Services using Interface to Network Security Functions[END_REF]. Such approaches the advantages of SDN with respect to security policy enforcement. We provide a software-defined framework, dedicated to security management and not limited to the networks enforcement field.

The autonomic computing paradigm defines a lifecycle featuring self-management and details the related architecture [START_REF] Jeffrey | The Vision of Autonomic Computing[END_REF]. It includes self-configuration, self-optimization, self-protection and self-healing activities. Although it does not bring a formal distributed cloud support, it may introduce the negotiation among independent components. This approach may deal with exhaustive enforcement issues, as autonomic components can continuously enforce the security policy and adapt to the changes in their action perimeters. Even if the two previous paradygms do not directly deal with distributed cloud issues, they provide important building blocks for supporting security policy enforcement and defining a security management architecture in that context and in our framework.

With respect to security policies, the OASIS consortium introduces two standardized languages: XACML (eXtensible Access Control Markup Language) for representing and exchanging security policies [START_REF] Godik | EXtensible Access Control Markup Language (XACML) version 1[END_REF] and SAML (Security Assertion Markup Language) for specifying security statements [START_REF] Maler | Assertions and Protocols for the OASIS Security Assertion Markup Language (SAML)[END_REF]. However, they do not handle any modifications of cloud policies nor its evolution propagation to enforcers. This approach remains relevant as the XACML defines modular components for security enforcement. Besides, an architecture and usecases featuring XACML and SAML in distributed environment have been detailled in [START_REF] Golbeck | Trust on the World Wide Web: a Survey[END_REF]. The latter validates the usability of XACML in distributed systems, underlining some limitations such as the need for a high granularity of sub-policies and the difficulty of maintaining an encoded security policy. The languages and formats introduced by SCAP protocol constitutes also an interesting support, as they cover many complementary specifications, such as vulnerability descriptions and scorings, that are exploitable for automating security in distributed cloud [START_REF] Waltermire | The Technical Specification for the Security Content Automation Protocol (SCAP): SCAP version 1.2[END_REF]. Those standards are usable in our architecture.

III. SOFTWARE-DEFINED SECURITY STRATEGY FOR DISTRIBUTED CLOUD ENVIRONMENTS

Our approach introduces a software-defined security (SD-Sec) strategy for supporting distributed cloud, and address- ing its multi-tenant and multi-cloud properties. It relies on software-defined scheme to provide a global security policy specification interface and exploit autonomic mechanisms within distributed cloud infrastructures to enable cloud resources to be dynamically and exhaustively protected according to this policy. More precisely, first consists in a global security policy (GSP) which formaly defines at a business level the security objectives of cloud resources and is then translated into several tenant-level security policies (TLSP), providing security statements that must be verified by specified resources at the tenant level within the distributed cloud.

These security statements are then enforced on cloud resources, i.e. virtualized infrastructures and software products. They aim at altering the behavior of these components and protecting them based on countermeasures available with distributed cloud. This application can be active if its application requires negotiation with a decisional instance. The enforcement should be performed dynamically, more precisely in an adaptive (it adapts to any change in the enforced resource state or in the infrastructure), automatic (no operator interventions are needed for it), and self-configured manner (policy decisions for it are automatically made according to several criteria including the security requirements).

The different components parts of this SDSec architecture for distributed cloud are as follows:

• The security orchestrator hosts a GSP specified by the system administrator, exposes TLSP, and receives enforcement feedbacks from PDP to adapt them. It is reactive as exposed TLSP are modulated by such feedbacks and the cloud orchestrator interactions. • The policy decision point parses its local TLSP to address security statements to the security enforcement stack and responds its security decisional requests. Its decisions can be influenced by third party components providing external security attributes. • The security discovery protocol enables the PDP to identify the security orchestrator and to fetch its corresponding TLSP. It works on the top of a middleware joining PDP and security orchestrator.

• The policy enforcement point is a component executing the security statements. It is dedicated to the enforcement on one type of cloud resources. It executes PDP security statements and exposes hooks for security decisions. • The enforcement discovery protocol allows the PDP to identify and locate the PEPs in its security perimeter and identify their capabilities. It works between the PDP and the corresponding PEPs. • The statement protocol sends the security statement from the PDP to the PEP, and returns the feedback. It is asynchrone, and is based on existing standards for security statements encoding. • The decision requesting protocol enables the PEP to ask the PDP a security decision for a reactive enforcement in a synchronous manner.

Consequently, this architecture brings several benefits with respect to cloud security: a) Distributed cloud protection enablement: this architecture is compliant with the distributed cloud constraint as it matches the multi-tenancy and multi-cloud properties. This compatibility is promoted by the GSP which is localy translated into a TLSP and adapted to a local resource enforcement. Moreover, the multi-cloud-aware discovery and communication protocols assist the multi-cloud security enforcement. b) Software-defined management: this architecture implements the software-defined paradigm to configure the security mechanisms. This approach is inspired from the softwaredefined networking (SDN) but applied to the security field. Actually, in SDN, the control-plane corresponds to the SDN controllers and the data-plane matches the administrated network components. In our architecture, the control-plane is composed by the security orchestrator and all instanciated PDPs while all the instanciated PEPs and enforced cloud resources constitute the data-plane. The proposed architecture covers several areas of security policy enforcement. Also, this paradigm allows autonomic reaction to security state changes. c) Self-configuration: this architecture relies on the autonomic computing concepts which propose a lifecycle for handling a dynamic and local management. Besides, this management is highly adapted to specified components. This property is mainly brought by the PEPs, as they configure and monitor the components they apply the security policy on. Moreover, they are specifically conceived for the components they are in charge of. Another interesting aspect of this self-configuration is the lower coupling with respect to the orchestration. Instead of the traditional orchestration model addressing a request and expecting a feedback, the security orchestrator adopts a passive approach by exposing security requirements, and refines the security requirement for different PDPs in different contexts.

IV. ARCHITECTURE VALIDATION

In order to evaluate the proposed approach, we confronted to several scenarios, based on the following use-case of a distributed cloud: a Cloud Service Provider (CSP) offers a Platform-as-a-Service (PaaS) solution to its customers, providing several infrastructures in different geographic areas. To protect its information system from the client security negligences (or deliberate malevolences), the CSP enforces a security policy (GSP) on its own infrastructures, and gather with each resource of its customers.

In this context, the multi-tenancy aspect of that use-case is promoted by the exploitation of the same infrastructure by several independent customers, and the multi-cloud aspect is promoted by the need for each infrastructure of the CSP to collaborate about the customers preferences, customers resource intercommunication and billing.

We will focus on the case of the tenant customer #2 which regroups two virtual machines for hosting two web applications: one for a European version of his application and another one for the American version.

A. Resource allocation scenario

The CSP's client wants to set up a back-up server in his tenant for his web application. It must use SQL and FTP servers in a dedicated VM stored in the European infrastructure. The client requests the cloud orchestrator to create a VM instance from an image provided by the CSP and to deploy SQL and FTP servers. Each instanciated resource embeds a dedicated PEP. This deployment is notified to the security orchestrator.

Using the security discovery protocol, the PDP fetches the newly available TLSP from the security orchestrator. Then, using the enforcement discovery protocol, it identifies the new PEPs in its enforcement perimeter. As FTP and SQL services are newly deployed in the tenant, the security orchestrator assumes that the TLSP of the tenant's PDP is no more adapted and modifies the corresponding TLSP of this PDP. Those are again fetched by the security discovery protocol. Employing the security statement protocol, the PDP sends the new security statements to the PEP to enforce the TLSP, and fetches the feedback. Finally, the PDP transmits a positive enforcement feedback to the security orchestrator.

B. Resource evolution scenario

The VM hosting the European instance of the web application is targeted by a distributed denial of service (DDoS) attack: its memory and CPU consumption indicators increase and exceed the alert threshold of the PEP, which was specified by the PDP. The PEP uses the security statement protocol to alert the PDP of the failing enforcement on the web server resources consumption. The PDP uses the security statement protocol to order the infrastructure PEP to increase temporarily the VM resource, and informs the security orchestrator of the non-enforcement of the GSP.

Then, the security orchestrator asks the network security enforcement to block attackers' IP addresses, and order the tenant's PDP to switch the affected VM into a fail-safe mode. Once the network security enforcement has blocked the DDoS attack, the security orchestrator reverts back the TLSP exposed to the tenant customer #2 in order to restore the VM.

C. Access request scenario

The GSP imposes the credentials used for inter-cloud resource interactions to have an expiration delay. The verification of the validity is committed by the PDP using a tierce module. As the client has set-up back-up processes between the backup server hosted in the European infrastructure and the production server located in the american infrastructure, the related resources have to meet these requirements: when the production server connects to the back-up server, the connection is blocked and the related PEPs make decision requests to the PDP, providing hashes of the used credentials.

As the TLSP imposes the verification of the lifetime of the credentials as well, the PDP uses its third party module to check it. As this module has no precedent records of the hashes, it concludes that the transmitted credentials are newly created ones and are allowed to be used. The PDP responses to both security decision requests are positive. Thus, they are sent back to the two PEPs, allowing the two incoming connections.

D. Resource removal scenario

The client wants to update the VM supporting the American web application by proceeding to a fresh installation. To meet this objective, the client wants to completely remove it and reconfigure a new VM. He uses the cloud orchestrator to remove this VM, which is notified to the security orchestrator.

The security orchestrator updates its GSP to take into account the removal of the cloud resource and checks its consequences on the enforcement: the TLSP is updated. The PDP of tenant customer #2 fetches the new TLSP, and stores it. The security orchestrator starts deallocating resources to the American VM and the PEP addresses a security decisional request to its PDP for allowing the removal. According to its TLSP, the PDP grants the request. The PEP lets the cloud orchestrator to complete the resource removal.

V. CONCLUSIONS This paper proposes a software-defined security strategy for distributed cloud environments. It relies on the programmability of software-defined security, and exploits the autonomic paradigm for addressing the constraints induced by multi-tenancy and multi-cloud properties. The underlying architecture is composed of a security orchestrator, policy decision points (PDP) and policy enforcement point (PEP) interacting according to a dedicated set of protocols. Based on the specification of a security policy, it supports the dynamic configuration of security mechanisms to adjust to contextual changes, based on available resources and countermeasures. The proposed approach has been evaluated through a set of validation scenarios corresponding to a realistic use case, including cloud resource allocation/deallocation, cloud resource change, and dynamic access control.

The proposed architecture has raised several challenges with respect to the design of the considered components, and the specification of security policies in a multi-cloud and multitenant context. The PEPs will apply model-driven scheme to facilitate the inter-operability of heterogeneous enforcements. In the longer term, the security policy specification of distributed cloud, and the dedicated access mode will be investigated to complement the security orchestration.

Figure 1 .

 1 Figure 1. Proposed architecture for SDSec in distributed cloud

Figure 2 .

 2 Figure 2. Use-case for supporting the validation of our SDSec architecture


Essomanda Konzou

Angelo Efoévi Koudou

About the Stein equation for the generalized inverse Gaussian and Kummer distributions

Keywords: Probability and Statistics, 21, 303-316] Generalized inverse Gaussian distribution, Kummer distribution, Stein characterization

We propose a Stein characterization of the Kummer distribution on (0, ∞).

This result follows from our observation that the density of the Kummer distribution satisfies a certain differential equation, leading to a solution of the related Stein equation. A bound is derived for the solution, under a condition on the parameters. The derivation of this bound is carried out using the same framework as in Gaunt 2017 [A Stein characterisation of the generalized hyperbolic distribution. ESAIM:

Introduction

2K p ( √ ab)

x p-1 e -1 2 (ax+b/x) , x > 0, where K p is the modified Bessel function of the third kind.

For details on GIG and Kummer distributions see [START_REF] Koudou | Characterizations of GIG laws: a survey complemented with two new results Proba[END_REF][START_REF] Koudou | Independence properties of the Matsumoto-Yor type[END_REF][START_REF] Piliszek | Change of measure technique in characterizations of the gamma and Kummer distributions[END_REF] and references therein, where one can see for instance that these distributions are involved in some characterization problems related to the so-called Matsumoto-Yor property.

In this paper, these two distributions are considered in the context of Stein's method. This method introduced in [START_REF] Stein | A bound for the error in the normal approximation to the distribution of a sum of dependent random variables[END_REF] is a technique used to bound the error in the approximation of the distribution of a random variable of interest by another probability (for instance the normal) distribution. For an overview of Stein's method see [START_REF] Chen | Normal approximation by Stein's method[END_REF][START_REF] Ross | Fundamentals of Stein's method[END_REF]. The first steps of this method consist in finding an operator called Stein operator characterizing the targeted distribution, then solving the corresponding socalled Stein equation.

One finds in [START_REF] Stein | A bound for the error in the normal approximation to the distribution of a sum of dependent random variables[END_REF] a seminal instance of the method, where Stein showed that a random variable X has a standard normal distribution if and only if for all real-valued absolutely continuous function

f such that E |f ′ (Z)| < ∞ for Z ∼ N(0, 1), E [f ′ (X) -Xf (X)] = 0. The corresponding Stein equation is f ′ (x) -xf (x) = h(x) -Eh(Z)
where h is a bounded function and Z a random variable following the standard normal distribution. The operator f → T f defined by (T f )(x) = f ′ (x) -xf (x) is the corresponding Stein operator.

If a function f h is a solution of the previous equation, then for any random variable U we have

|E(f ′ h (U) -Uf h (U))| = |E(h(U)) -E(h(Z))|.

Thus, in order to bound |E(h(U)) -E(h(Z))

| given h, its enough to find a solution f h of the Stein equation and to bound the left-hand side of the previous equation. The problem of solving the Stein equation for other distributions than the standard normal distribution and bounding the solution and its derivatives has been widely studied in the literature (see [START_REF] Goldstein | Stein's method for the Beta distribution and the Pòlya-Eggenberger urn[END_REF] among many others).

The aim of this paper is to solve the Stein equation and derive a bound of the solution for the Kummer distribution (which is new) and for the generalized inverse Gaussian distribution (which has been done in [START_REF] Gaunt | A Stein characterisation of the generalized hyperbolic distribution[END_REF], but there was a little mistake in the bound of the solution).

The idea of this paper emerged by reading the remarkable work by [START_REF] Gaunt | A Stein characterisation of the generalized hyperbolic distribution[END_REF] about a Stein characterization of the generalized hyperbolic distribution of which the generalized inverse Gaussian distribution (GIG) is a limiting case. Among many other results, [START_REF] Gaunt | A Stein characterisation of the generalized hyperbolic distribution[END_REF] solved the GIG Stein equation and bounded the solution by using a general result obtained in [START_REF] Schoutens | Orthogonal Polynomials in Steins Method[END_REF] when the targeted distribution has a density g satisfying (s(x)g(x)) ′ = τ (x)g(x)

(1.1)

for some polynomial functions s and τ . Also a bound was obtained for the solution under the condition that the function τ be a decreasing linear function. But since this linearity condition does not hold in the GIG case, the bound given by [START_REF] Gaunt | A Stein characterisation of the generalized hyperbolic distribution[END_REF] has to be slightly corrected. This is done in Theorem 3.1 after recalling the general framework of Schoutens [START_REF] Schoutens | Orthogonal Polynomials in Steins Method[END_REF] and adapting it to the cases where τ is decreasing but not necessarily linear. Indeed, we realized that the procedure adopted in [START_REF] Schoutens | Orthogonal Polynomials in Steins Method[END_REF] still works, via a slight change, even if τ is not linear.

Observing that the Kummer density also satisfies (1.1), we can use the same methodology (Theorem 4.2) for this distribution. We have to put the restrictions p ≤ -1 for the GIG density and 1 -b -c ≤ 0 for the Kummer density in order for the corresponding function τ to be decreasing on (0, ∞).

In Section 1 we recall the general framework established by [START_REF] Schoutens | Orthogonal Polynomials in Steins Method[END_REF] for densities g satisfying (1.1) without the assumption of linearity of τ . We retrieve the Stein operator given in [START_REF] Schoutens | Orthogonal Polynomials in Steins Method[END_REF] by using the density approach initiated in [START_REF] Stein | Use of exchangeable pairs in the analysis of simulations[END_REF] and further developed in [START_REF] Ley | Stein's density approach and information inequalities[END_REF].

In Section 2 we show the application of this method to the GIG distribution as mentioned in [START_REF] Gaunt | A Stein characterisation of the generalized hyperbolic distribution[END_REF] by giving the right bound for the solution of the Stein equation. Section 3 is devoted to the Stein characterization and the Stein equation related to the Kummer distribution.

Stein characterization in the Schoutens framework

Theorem 1 in [START_REF] Schoutens | Orthogonal Polynomials in Steins Method[END_REF] addressed the problem of establishing a Stein characterization for probability distributions with density g satisfying (1.1) for some polynomial functions s et τ , and proved that a Stein operator in this case is f → sf ′ + τ f . We realized (see the following theorem) that the same Stein operator can be arrived at by using the density approach designed in [START_REF] Stein | Use of exchangeable pairs in the analysis of simulations[END_REF] and [START_REF] Ley | Stein's density approach and information inequalities[END_REF]. The support of the density may be any interval, but here we take this support to be (0, ∞) in the purpose of the application to the GIG and Kummer distributions.

Theorem 2.1 Consider a density g on (0, ∞) such that (1.1) holds for some polynomial functions s and τ . Then a positive random variable X has density g if and only if for any differentiable function f such that lim

x→0 s(x)g(x)f (x) = lim x→∞ s(x)g(x)f (x) = 0, E [s(X)f ′ (X) + τ (X)f (X)] = 0.
Proof: We use Corollary 2.1 of [START_REF] Ley | Stein's density approach and information inequalities[END_REF]. According to this corollary, a Stein operator related to the density function g is

T g f (x) = 1 g(x) (f g) ′ (x.)
Applyng this operator to sf , we have

T g (sf )(x) = 1 g(x) (sf g) ′ (x) = 1 g(x) (f ′ (x)s(x)g(x) + f (x)(sg) ′ (x))
which, by (1.1), reads

T g (sf )(x) = 1 g(x) (f ′ (x)s(x)g(x) + f (x)τ (x)g(x)) = f ′ (x)s(x) + f (x)τ (x).
Theorem 2.1 shows that the Stein equation related to any density g satisfying (1.1) enjoys the tractable form

s(x)f ′ (x) + τ (x)f (x) = h(x) -Eh(W ) (2.1)
where W is random variable with density g. Schoutens [START_REF] Schoutens | Orthogonal Polynomials in Steins Method[END_REF] found a solution to the Stein equation (2.1) and established a bound for the solution, under the condition that the function τ be a decreasing linear function (which is the case for the so-called Pearson and Ord classes of distributions considered in [START_REF] Schoutens | Orthogonal Polynomials in Steins Method[END_REF]).

The following result comes from Proposition 1 in [START_REF] Schoutens | Orthogonal Polynomials in Steins Method[END_REF]. We again take the support of the density function to be (0, ∞).

f h (x) = 1 s(x)g(x) x 0 g(t) [h(t) -Eh(W )] dt = -1 s(x)g(x) +∞ x g(t) [h(t) -Eh(W )] dt.
(2.2) Remark 2.1 The proof of this proposition is given in [START_REF] Schoutens | Orthogonal Polynomials in Steins Method[END_REF] just by calculating the derivative of the function f h defined by (2.2) and checking directly that f h satisfies (2.1). Our following proposition complements this result. Proposition 2.2 Under the notation and assumptions of Proposition 2.1,

• The solutions of the Stein equation (2.1) are of the form

f h (x) = 1 s(x)g(x) x 0 g(t) [h(t) -Eh(W )] dt + C s(x)g(x) = -1 s(x)g(x) +∞ x g(t) [h(t) -Eh(W )] dt + C s(x)g(x) (2.3)
where C is constant.

• Suppose lim x→0 s(x)g(x) = 0. For the solution to be bounded, it is necessary that C = 0 in (2.3).

Proof:

Multiplying both sides of (2.1) by g(x) we have

s(x)g(x)f ′ (x) + τ (x)g(x)f (x) = g(x)(h(x) -Eh(W ))
which, by (1.1), can be written

s(x)g(x)f ′ (x) + (sg) ′ (x)f (x) = g(x)(h(x) -Eh(W )), i.e. (sgf ) ′ (x) = g(x)(h(x) -Eh(W )).
As a consequence, there exists a constant C such that

s(x)g(x)f (x) = x 0 g(t) [h(t) -Eh(W )] dt + C (2.4)
which implies (2.3). Suppose f is bounded. Since lim x→0 s(x)g(x) = 0, letting x tend to 0 in (2.4) yields C = 0.

The second expression for f h follows from the fact that, since W has density g,

+∞ 0 g(t) [h(t) -Eh(W )] dt = 0.
The following proposition proves that the solution given by (2.2) is bounded indeed if h is bounded, and thus is the unique bounded solution to the Stein equation associated to the density g. A bound is provided. Proposition 2.3 Consider a density function g > 0 on (0, ∞) satisfying Equation (1.1), where s and τ are polynomial functions such that s > 0 on (0, ∞) and τ is decreasing and has a unique zero α on (0, ∞). Assume that lim where

M = max 1 s(α)g(α) α 0 g(t)dt; 1 s(α)g(α) +∞ α g(t)dt and ||f h || = sup x>0 |f h (x)| .
Remark 2.2 This result is a reformulation of Lemma 1 in [START_REF] Schoutens | Orthogonal Polynomials in Steins Method[END_REF] without the assumption that τ is linear. With this assumption, [START_REF] Schoutens | Orthogonal Polynomials in Steins Method[END_REF] established the same bound with α = E(X) (for a random variable X with density g), which is not true if τ is not linear. The proof given below follows the lines of that of [START_REF] Schoutens | Orthogonal Polynomials in Steins Method[END_REF] where we observed that the assumption of linearity of τ was used nowhere except to state that its only zero is α = E(X).

The proof of Proposition 2.3 uses the following lemma : Proof: Suppose x < α. Since τ is positive and decreasing on (0, α), we have

τ (t) τ (x)
≥ 1 for all t ≤ x. Therefore

x 0 g(t)dt ≤ x 0 g(t) τ (t) τ (x) dt = 1 τ (x) x 0 τ (t)g(t)dt = s(x)g(x) τ (x)
because of (1.1) and as lim t→0 s(t)g(t) = 0.

For x > α, since τ is negative and decreasing on (α, ∞), we have

τ (t) τ (x) ≥ 1 for all t ≥ x. As a consequence, +∞ x g(t)dt ≤ +∞ x g(t) τ (t) τ (x) dt = -s(x)g(x) τ (x) since lim t→∞ s(t)g(t) = 0.
Now, let us prove (2.5).

Proof:

For x < α, |f h (x)| = 1 s(x)g(x) x 0 g(t) [h(t) -Eh(W )] dt ≤ 1 s(x)g(x) x 0 g(t) |h(t) -Eh(W )| dt ≤ ||h(.) -Eh(W )|| 1 s(x)g(x) x 0 g(t)dt. Let l(x) = 1 s(x)g(x)
x 0 g(t)dt. Then l is differentiable on (0, ∞) and

l ′ (x) = -(s(x)g(x)) ′ (s(x)g(x)) 2 x 0 g(t)dt + 1 s(x) = -τ (x) s 2 (x)g(x) x 0 g(t)dt + 1 s(x)
Using Lemma 2.1, we conclude that l ′ (x) ≥ 0. Then l(x) ≤ l(α).

For x > α,

|f h (x)| = ||h(x) -Eh(W )|| 1 s(x)g(x) +∞ x g(t)dt. Let u(x) = 1 s(x)g(x) +∞ x g(t)dt. The function u is differentiable on (0, ∞) and u ′ (x) = -τ (x) s 2 (x)g(x) +∞ x k(t)dt - 1 s(x)
By Lemma 2.1, we conclude that u ′ (x) ≤ 0. Then u(x) ≤ u(α).

In the two next sections we apply the previous results to the GIG and Kummer distributions.

About the Stein equation of the generalized inverse Gaussian distribution

Recall that the density of the GIG distribution with parameters p ∈ R, a > 0, b > 0 is

g p,a,b (x) = (a/b) p/2 2K p ( √ ab) x p-1 e -1 2 (ax+b/x) , x > 0, (3.1)
where K p is the modified Bessel function of the third kind. Let

s(x) = x 2 and τ (x) = b 2 + (p + 1)x - a 2 x 2 . (3.2)
Then, as observed by [START_REF] Gaunt | A Stein characterisation of the generalized hyperbolic distribution[END_REF], the GIG density g p,a,b satisfies

(s(x)g p,a,b (x)) ′ = τ (x)g p,a,b (x).
This enables us to apply Theorem 2.1 to retrieve the following Stein characterization of the GIG distribution given in [START_REF] Koudou | Characterizations of GIG laws: a survey complemented with two new results Proba[END_REF] and [START_REF] Gaunt | A Stein characterisation of the generalized hyperbolic distribution[END_REF]:

Proposition 3.1 A random variable X
E X 2 f ′ (X) + b 2 + (p + 1)X - a 2 X 2 f (X) = 0.
The corresponding Stein equation is

x 2 f ′ (x) + b 2 + (p + 1)x - a 2 x 2 f (x) = h(x) -Eh(W ) (3.3)
where h is a bounded function and W a random variable following the GIG distribution with parameters p, a, b.

We apply Proposition 2.2 and Proposition 2. Secondly, observe that if p ≤ -1, then the function τ defined by (3.2) is decreasing on (0, ∞) and that its only zero on (0, ∞) is α = p + 1 + (p + 1) 2 + ab a . Thus, by using Proposition 2.2 and Proposition 2.3, we obtain the following theorem. Remark 3.1 This result was claimed by Gaunt (see [START_REF] Gaunt | A Stein characterisation of the generalized hyperbolic distribution[END_REF]) with α = E(X) by applying Proposition 1 of [START_REF] Schoutens | Orthogonal Polynomials in Steins Method[END_REF]. The only slight mistake is that τ is not a polynomial function of degree one as in [START_REF] Schoutens | Orthogonal Polynomials in Steins Method[END_REF].

About the Stein equation related to the Kummer distribution

Recall that for a > 0, b ∈ R, c > 0, the Kummer distribution K(a, b, c) has density

k a,b,c (x) = 1 Γ(a)ψ(a, a -b + 1; c) x a-1 (1 + x) -a-b e -cx , (x > 0)
where ψ is the confluent hypergeometric function of second kind. Let Remark 4.1 These results could be used in future work to provide rates of convergence in limit problems related to the GIG and Kummer distributions.

s(x) = x(1 + x) and τ (x) = (1 -b)x -cx(1 + x) + a. ( 4

For

  a > 0, b ∈ R, c > 0, the Kummer distribution with parameters a, b, c has densityk a,b,c (x) = 1 Γ(a)ψ(a, a -b + 1; c) x a-1 (1 + x) -a-b e -cx , (x > 0)where ψ is the confluent hypergeometric function of the second kind. The generalized inverse Gaussian (hereafter GIG) distribution with parameters p ∈ R, a > 0, b > 0 has density g p,a,b (x) = (a/b) p/2

Proposition 2 . 1

 21 Consider a density function g > 0 on (0, ∞) satisfying Equation (1.1), for some polynomial functions s and τ . Then a solution of the Stein equation (2.1) is

  )g(x) = 0. If h is a bounded continuous function, then ||f h || ≤ M ||h(.) -Eh(Z)|| (2.5)

Lemma 2 . 1

 21 Under the assumptions of Proposition 2.3, dt ≤ -s(x)g(x)τ (x) for x > α.

  follows the GIG distribution with density g p,a,b if and only if, for all real-valued and differentiable function f such that lim x→∞ g p,a,b (x)f (x) = lim x→0 g p,a,b (x)f (x) = 0, we have:

  3 to solve Equation(3.3) and bound the solution. Let us check that the assumptions of these propositions are true in the GIG case.Firstly, we note that, by (3.1),s(x)g p,a,b (x) = (a/b) p/2 2K p ( √ ab)x p+1 e -1 2 (ax+b/x) , x > 0, which shows that lim x→∞ s(x)g p,a,b (x) = lim x→0 s(x)g p,a,b (x) = 0.

Theorem 3 . 1

 31 The GIG Stein equation (3.3) has solutionf h (x) = 1 s(x)g p,a,b (x) x 0 g p,a,b (t) [h(t) -Eh(W )] dt = -1 s(x)g p,a,b (x) +∞ x g p,a,b (t) [h(t) -Eh(W )] dt (3.4)where W follows the GIG distribution with parameters p, a, b.If h is a bounded continuous function and p ≤ -1, then the function defined by (3.4) is the unique bounded solution of (3.3) and ||f h || ≤ M ||h(.) -Eh(W )|| where α = p + 1 + (p + 1) 2 + ab a , M = max 1 s(α)g p,a,b (α) α 0 g p,a,b (t)dt; 1 s(α)g p,a,b (α) +∞ α g p,a,b (t)dt .

Theorem 4 . 2

 42 x)k a,b,c (x)) ′ = τ (x)k a,b,c (x.) Then we can use Theorem 2.1 to obtain the following Stein characterization of the Kummer distribution: Theorem 4.1 A random variable X follows the K(p, a, b) distribution if and only if, for alldifferentiable function f , E [X(X + 1)(f ′ (X) + ((1 -b)X -cX(1 + X) + a) f (X)] = 0.The corresponding Stein equation isx(x + 1)f ′ (x) + [(1 -b)x -cx(1 + x) + a] f (x) = h(x) -Eh(W ) (4.2)where W has density k a,b,c . We haves(x)k a,b,c (x) = 1 Γ(a)ψ(a, a -b + 1; c) x a (1 + x) 1-a-b e -cx , x > 0and we see that lim x→∞ s(x)k a,b,c (x) = lim x→0 s(x)k a,b,c (x) = 0. Note also that if 1 -b -c ≤ 0, then the function τ defined by (4.1) is decreasing on (0, ∞) and its only zero on this interval is α = 1 -b -c + (1 -b -c) 2 + 4ac 2c . Then we use again Proposition 2.2 and Proposition 2.3 to obtain the following result: The Kummer Stein equation (4.2) has solutionf h (x) = 1 s(x)k a,b,c (x) x 0 k a,b,c (t) [h(t) -Eh(W )] dt = -1 s(x)k a,b,c (x) +∞ x k a,b,c (t) [h(t) -Eh(W )] dt(4.3) where W ∼ K(a, b, c). If h is a bounded continuous function and 1 -b -c ≤ 0, then f h defined by (4.3) is the unique bounded solution of (4.2) and ||f h || ≤ M ||h(.) -Eh(W )|| where α = 1 -b -c + (1 -b -c) 2 + 4ac 2c , M = max 1 s(α)k a,b,c (α) α 0 k a,b,c (t)dt; 1 s(α)k a,b,c (α) +∞ α k a,b,c (t)dt .