Shrita Hassamal

Anne Abeillé

Stefan Müller

F Henri

D Godard

A Kihm

B Crysmann

O Bonami

G Fon

J-P Koenig

S Kriegel

Degree adverbs in Mauritian

Introduction 1

From a semantic point of view, degree words (very, a lot) can be analysed as modifying a value on a scale, which can be an intensity scale for properties (1a), or a quantity scale for objects (1b) or events (1c) [START_REF] Kennedy | Scale Structure, degree modification, and the semantics of gradable predicates[END_REF]. Some degree words are specialised for a certain type of scale (e.g. an intensity scale for extremely) and some are underspecified (e.g. a lot is compatible with all scales) (1b, c, d).

(1) a. John is very / extremely intelligent. b. A lot of books. c. John goes to the movies a lot. d. John likes the book a lot.

From a syntactic point of view, degree words usually differ with respect to their category or function: in English, very, very much are adverbs, the former modifying adjectives and adverbs (2a), the latter modifying verbs (2b); many, much are determiners (specifying count and mass nouns)(2c).

(2) a. John works very hard / drives very quickly. b. John likes the book very much. c. He does not have many books / much luck.

Comparative words, more and less, which compare two values on one (or more) degree scales, seem more polymorphic and combine with all the major syntactic categories:

(3) a. John is more intelligent/ laughs more often. b. John goes more to the movies/ likes the book more. c. He has more luck / more books.

Mauritian, a French-based Creole spoken in Mauritius, with no official status, is an understudied language (Baker 1970, Henri 2010, Allessaib 2012…) It has a very recent standardised spelling [START_REF] Hookoomsing | Grafi larmoni[END_REF]), one dictionary [START_REF] Carpooran | Diksioner Morisien, Koleksion Text Kreol[END_REF]) and very few written corpora: some literary works of Dev Virahsawmy (www.boukiebanane.orange.mu) and the journal of a political party (www.lalitmauritius.org). We conducted several surveys with informants, in situ and on line; the first author is a native speaker.

Interestingly, Mauritian degree words exhibit an extreme syntactic polymorphism: the same form tro 'too' may combine with all major categories: nouns (4a), verbs (4b), adjectives and adverbs (4c).

(4) a. Tro liv / tro per. 'Too many books, too much fear' b. Paul tro travay.

'Paul works too much' c. Tro long / tro vit 'Too long / too fast'

When there is more than one form (mari / boukou 'very, a lot'), they seem to be distinguished on semantic more than syntactic criteria, mari requiring an intensity scale (5a,b) and boukou a quantity scale (5a,c).

(5) a. Mari per / boukou liv. very fear / a-lot books ' a lot of fear / a lot of books.' b. Mo mari kontan liv.

I very like book 'I like the book(s) a lot' c. Mo al sinema boukou.

'I go to the movies a lot' We will focus on the syntax and semantics of four degree adverbs in Mauritian: mari ('very'), boukou ('a lot'), pli/plis ('more') and mwin/mwins ('less'), considering their category (adverb? determiner?) and their syntactic function (specifier? adjunct? complement?), as well as their combination with a scalar predicate: do they select it on syntactic or semantic criteria?

2

High degree adverbs in Mauritian 2.1

High degree adverbs in French

French also has some polymorphic degree adverbs, like trop ('too, too much, too many'), which may combine all major categories (7).

(7) a.

Trop grand / trop vite. 'too big / too fast' b.

Paul travaille trop. 'Paul works too much' c.

Trop peur/ trop de livres.

'too much fear/ too many books' [START_REF] Abeillé | The Syntactic Flexibility of French Degree Adverbs[END_REF] propose that French degree adverbs have a double life: as complements when they follow the verb, and as adjuncts when they precede it or combine with other categories.

French also has a high degree adverb pair: beaucoup 'a lot' / très 'very'. [START_REF] Abeillé | Adverbs and quantification[END_REF]) analyse beaucoup as an adverb combining with any verb and with non-verbal categories that have a quantity scale. Très2 on the other hand, combines only with non-verbal categories that have an intensity scale (table 1) Interestingly, Mauritian verbs have a conjugation with a short form (SF) and a long form (LF)4 , the choice of form being determined by syntactic and discourse factors: the verb is in SF when followed by a non-clausal complement and not focalised (12b); the LF is used otherwise (12a,c) [START_REF] Henri | Verb Form alternations in Mauritian[END_REF] and [START_REF] Henri | A constraint-based approach to verbal constructions in Mauritian: morphological, syntactic and discourse-based aspects[END_REF] In Mauritian, both boukou and mari are adverbs with a complementary distribution. Syntactically, boukou follows the verb and triggers the verbal short form, as a complement, while mari precedes it as an adjunct. With nonverbal categories, both are adjuncts. Semantically, boukou modifies predicates with a quantity scale while mari modifies predicates with an intensity scale.

The inequality comparative adverbs in Mauritian

Mauritian has two forms: pli and plis for the superior comparative adverb ('more') and mwin and mwins for the inferior comparative adverb ('less').

3.1

Pli / plis ('more') in Mauritian The two forms pli / plis6 , coming from French plus ('more')7 , are in complementary distribution (table 3), plis being the default form (used in isolation).

The distribution between the two forms is not phonologically determined since both forms can be followed by a vowel or a consonant (24). The inequality comparatives pli, plis, mwin and mwins are all adverbs. In the non-verbal domain, they are adjuncts, preceding the category they modify. On the semantic level, pli and mwin only combine with predicates that have an intensity scale and plis and mwins with predicates that have a quantity scale.

In the verbal domain, only plis and mwins are possible, and they have a double life : they can precede the verb, as adjuncts, but may also follow it, as complements, triggering the short verbal form.

4

An HPSG analysis of Mautitian degree adverbs

In HPSG, degree adverbs have been analysed as specifiers (Pollard and Sag 1994), adjuncts or complements [START_REF] Abeillé | The Syntactic Flexibility of French Degree Adverbs[END_REF] or as functors (Van Eynde 2007). Semantically, they have been analysed as modifiers [START_REF] Abeillé | The Syntactic Flexibility of French Degree Adverbs[END_REF] or quantifiers [START_REF] Kay | Cleaning up the big mess[END_REF]. We follow [START_REF] Kennedy | Scale Structure, degree modification, and the semantics of gradable predicates[END_REF] in analysing them as modifiers of scalar predicates and not as quantifiers.

Degree adverbs as polymorphic adjuncts

Lexical entries for degree adverbs In Mauritian, tro ('too') can combine with any scalar predicate whereas boukou ('a lot') and mari ('very') specify a subtype of scale. We rely on the following hierarchy for types of scales, ignoring the possible distinction between open and closed scales (intensity scales are always open).

scale quant-scale intens-scale event-q-scale object-q-scale ale, absan liv lafarinn contan, long, per Syntactically, they may underspecify the category they combine with [START_REF] Abeillé | The Syntactic Flexibility of French Degree Adverbs[END_REF] We have seen that a postverbal degree adverb is enough to trigger the short verbal form, and we thus analyse it as a complement. As shown by Henri 2010, manner adverbs have the same property: Degree and manner adverbs can thus optionally be added to the COMPS list of verbs [START_REF] Abeillé | The Syntactic Flexibility of French Degree Adverbs[END_REF], except pli ('more'), mwin ('less'), which cannot modify verbs, and mari, which is [LEX +]. 12 Lexical rule for Mauritian verbs

[ 𝑣𝑒𝑟𝑏 𝑆𝑌𝑁 [ 𝐻𝐸𝐴𝐷 [0] 𝑉𝐴𝐿 | 𝐶𝑂𝑀𝑃𝑆 𝐿 ] 𝑆𝐸𝑀 [1] ]=> [ 𝑆𝑌𝑁| 𝑉𝐴𝐿 |𝐶𝑂𝑀𝑃𝑆 𝐿 + 〈[ 𝑑𝑒𝑔 -𝑚𝑎𝑛 -𝑎𝑑𝑣 𝑀𝑂𝐷 [ 𝑆𝑌𝑁 | 𝐻𝐸𝐴𝐷 [0] 𝑆𝐸𝑀 [1] ] 𝐿𝐸𝑋 - ]〉
] 11 Henri 2010 analyses clausal complements as extraposed.

12 Degree adverbs can also head a verbless copular clause [START_REF] Henri | The syntax of copular constructions in Mauritian[END_REF] : 3000 roupis, li boukou ('3000 roupies, it is a lot'), Sa liv la plis ('This book is more'). A lexical construction can be used to provide them with a SUBJ feature [START_REF] Müller | On Predication[END_REF].

As complements, they follow the verb and trigger the short verbal form: HEAD < COMPS In preverbal position, we consider them as [LEX+] (Arnold & Sadler 1994): they can be premodified (boukou plis 'a lot more') but cannot head a full adverbial phrase (*plis ki pou twa 'more than yours') (28). They adjoin to the lexical verb, except pli ('more'), mwin ('less'), which cannot modify verbs, and boukou ('a lot'), which is [LEX -] :

Adjunct [LEX +] < Head [ 𝑣𝑒𝑟𝑏 𝐿𝐸𝑋 + ]
Other adverbs adjoin to VP (follow the verb and its complements) or S (are sentence initial or sentence final), like vit ('quickly'), yer ('yesterday').

Conclusion

In Mauritian, an understudied French-based creole, degree words like tro ('too') should be analysed as polymorphic adverbs, combining with all the major categories. From a syntactic point of view, they are complements after the verb and adjuncts when they precede it. With other categories, they are always adjuncts. Mauritian also has a pair of high degree adverbs (boukou -mari) in complementary distribution: while boukou follows the verb as a complement, mari precedes it as an adjunct. From a semantic point of view, boukou selects predicates with a quantity scale, and mari predicates with an intensity scale. Comparatives also come in pairs (pli, plis, 'more'; mwin, mwins, 'less') in a complementary distribution: plis, mwins select verbs with any scale and predicates with a quantity scale while pli, mwin combine with non verbal predicates with an intensity scale. Like tro, plis, mwins are complements after the verb and adjuncts when they precede it. We have provided new arguments for the adverb as complement analysis proposed in HPSG by Bouma et al. 2001, Kim and[START_REF] Kim | Negation without Head Movement[END_REF], and the polymorphic analysis proposed for French adverbs by Abeillé & Godard 2003, using HPSG underspecification.

  pli intelizan / pli gran. 'Paul is more intelligent / more tall.' b. Paul plis admir li. / plis travay. 'Paul admires her/him more / works more.'

  manz bien. Paul eat-SF well 'Paul eats well' b. Paul roul vit. Paul drive-SF fast 'Paul drives fast'

  worked / Paul worked yesterday'

Table 1 :

 1

	with verbs			with noun		with adjectives	with adverbs
	intensity	quantity	intensit	quantity	intensity	quantity	intensity	comp
	scale	scale		y scale	scale	scale	scale	scale	arativ
									es
	Il l'aime	Il sort	trop	trop de	trop	trop absent	trop vite	-
	trop 'he	trop		peur	livres	grand	'too absent'	'too fast'
	likes it	'he		'too	'too	'too big'
	too	goes		much	many
	much'	out too	fear'	books'
		much'
	Il	Il sort	-	beauco	-	beaucoup	-	beauc
	l'admire	beauco		up de		absent		oup
	beaucou	up'he			livres		'absent a		mieux
	p	goes			'a lot of		lot'		'a lot
	'he	out	a		books'				better
	admires	lot'							'
	it a lot'
	-	-		très	-	très	-	très vite
				faim		grand		'very fast'
				'much		'very tall'
				hunger'

distribution of trop ('too'), beaucoup ('a lot') and très ('very') in French

2.1 Boukou ('a lot') in Mauritian

In Mauritian, boukou ('a lot') combines with nouns and verbs, but only if they have a quantity scale (table

2

). It precedes nouns and follows verbs.

Table 2 :

 2 tro ('too'), boukou ('a lot') and mari ('very') in Mauritian Mauritian has developed another form mari 5 ('very'), which also combines with nouns and verbs, but selects predicates with an intensity scale, and always precedes them. Boukou and mari are thus in complementary distribution (table 2).

	with verbs			with nouns		with adjectives	with adverbs
	intensity	quantity		intensity	quantity	intensity	quantity	intensity	compa
	scale		scale		scale	scale	scale	scale	scale	ratives
	Mo	tro	Mo	al	tro per	tro liv	tro long	tro absan	tro vit
	admir li	sinema
			tro
	%	Mo	Mo	al	*boukou	boukou	*boukou	absan	*boukou	bouko
	admir li	sinema		per	liv	long	boukou	vit	u pli
	boukou		boukou 'I		'a lot of		'absent a		vit
	'I admire	go to the		books'		lot'		'a lot
	him a lot'	movies a						faster'
			lot'
	Mo mari	*Mo mari	mari per	* mari liv	mari long	*mari	mari vit	*mari
	admir li 'I	al sinema	'very		'very	absan	'very	pli vit
	admire				scared'		long'		fast'
	him very
	much'

POSS1SG

shoes PROG shine-LF/ shine-SF very Certain verbs can be associated with both types of scales. For example plore ('to cry') can have a quantity scale (object scale: quantity of tears or eventscale: frequency) or an intensity scale. The two degree adverbs thus trigger a different meaning, and a different verb form: the SF appears with boukou (20b), LF with mari (20a). Some adjectives, like malad (sick), may have both scales: mari triggers an intensive reading (22a) whereas boukou a count-event or frequency interpretation (22b).

Table 3 :

 3 Distribution of pli and plis 'more' in Mauritian With nouns With nouns, pli only combines with predicative gradable nouns (25a) and plis is used with count and mass nouns (25b,c)

	with verbs		with nouns		with adjectives	with
	intensity	quantity scale intensity	quantity	intensity	quantity	adverbs
	scale		scale	scale	scale	scale
	*Mo	*Mo al sinema	pli per	*pli liv	pli gran	*pli absan	pli vit
	admir li	pli	'more		'more		'more
	pli		scared'		tall'		fast'
	Mo	Paul al sinema	*plis per	plis liv	*plis	plis absan	* plis
	admir li	plis 'I go to the		'more	gran	'more	vit
	plis 'I	movies more'		books'		absent'
	admire
	her
	more'

b. Paul al sinema plis/*pli (ki Rita).

'Paul goes to the movies more than Rita.'

Plis can either precede (27a) or follow (27b) the verb. When it follows, it triggers the short verbal form and thus behaves as a complement (27b). It can also appear after a complement (27c). a. Paul pli malad ki Rita. 'Paul is sicker than Rita.' b. Paul malad plis ki Rita. 'Paul is sick more often than Rita.' Adverbs, on the other hand, only combine with pli.

Table 4 :

 4 distribution of mwin and mwins ('less') in Mauritian With nouns With nouns, mwin only combines with predicative gradable nouns (35a) and mwins is preferred with count and mass nouns (35b). Only mwins is recorded in Carpooran (2011)'s dictionary, and is coded as an adjective affecting quantity. 9 French has one form for the inferior comparative moins ('less') which is pronounced /mwɛ/, and /mwɛz/ in liaison contexts. A non standard realization /mwɛs/ may appear in the South West of France, in conditions to be determined, but it is unlikely that it could have influenced Mauritian creole.

	with verbs			with nouns		with adjectives	with
	intensity	quantity		intensity	quantity	intensity	quantity	adver
	scale	scale		scale	scale	scale	scale	bs
	*Mo	*Mo	al	mwin per	*mwin liv	mwin gran	*mwin	mwin
	admir li	sinema		'less		'less tall'	absan	vit
	mwin	mwin		scared'				'less
								fast'
	Mo	Mo	al	*mwins	mwins liv	*mwins	mwins	*mwi
	admir li	sinema		per	'less	gran	absan	ns vit
	mwins 'I	mwins 'I		books'		'less
	admire	go to the				absent'
	her less'	movies
		less'

8

Degree adverbs in the non verbal domain In

  10 .Comparative adverbsThe comparative adverbs have two arguments: the degree of the predicate they modify and the degree of their ki ('than') marked dependent (usingKay and Sag's 2012's EXTRA feature). They also exhibit a double selectionboth syntactic and semantic, where pli ('more') and mwin ('less') combine with non-verbal categories associated with an intensity scale, whereas plis and mwins combine with verbs, and non-verbal categories associated with a quantity scale. Mauritian, count and mass nouns can be used as bare NPs and are associated with a quantity scale. With nouns and other non verbal categories, we analyse degree adverbs as optional adjuncts, and not specifiers, as they do not have the same distribution as possessive or demonstrative determiners.Mari precedes the verb, boukou follows it, and plis and mwins either precede or follow it. We use the feature LEX (Arnold and Sadler 1994) as for the English adjectives to distinguish mari from the others. Most degree adverbs are unspecified for LEX, while mari is [LEX +] and boukou [LEX -].Degree adverbs as complements of verbsThe short verbal form implies the presence of a (non clausal) complement.

	Adverbial adjuncts precede nouns and non verbal categories (pli vit 'more
	quickly', tro gran 'too big', mari long 'very long', boukou liv 'a lot of
	mari books'):					pli
	[ 𝑆𝐸𝑀 [ 𝑆𝑌𝑁 [ ℎ𝑖𝑔ℎ -𝑑𝑒𝑔 -𝑟𝑒𝑙 𝐻𝐸𝐴𝐷 [ 𝑎𝑑𝑣 𝑀𝑂𝐷 𝑆𝐸𝑀 [ 𝐿𝐸𝑋 + 𝐴𝑅𝐺 𝑑 ] Adjunct < Head [HEAD non-verbal] 𝑆𝐶𝐴𝐿𝐸 𝑖𝑛𝑡𝑒𝑛𝑠 -𝑠 𝑠𝑢𝑝𝑒𝑟 -𝑟𝑒𝑙 ] [ 𝐸𝑋𝑇𝑅𝐴 𝐿 + 〈[ 𝑉𝐴𝐿𝑈𝐸 𝑑 ]] ] ] 𝑆𝑌𝑁 𝐻𝐸𝐴𝐷 [ 𝑎𝑑𝑣 𝑀𝑂𝐷 [ 𝑆𝐸𝑀 [𝑉𝐴𝐿𝑈𝐸 𝑑2] 𝑆𝑌𝑁[𝑀𝐴𝑅𝐾𝐼𝑁𝐺 𝑘𝑖] 𝑆𝑌𝑁 [ 𝐻𝐸𝐴𝐷 𝑛𝑜𝑛 -𝑣𝑒𝑟𝑏𝑎𝑙 ]〉 𝐸𝑋𝑇𝑅𝐴 𝐿 𝑆𝐸𝑀 [ 𝑆𝐶𝐴𝐿𝐸 𝑖𝑛𝑡𝑒𝑛𝑠 -𝑠 𝑉𝐴𝐿𝑈𝐸 𝑑1 ] ] ] 4.3 Degree adverbs in the verbal domain	] ]
								[	𝑆𝐸𝑀 [ 𝐴𝑅𝐺2 𝑑2 𝐴𝑅𝐺1 𝑑1	]	]
	plis
				𝑎𝑑𝑣
	𝑆𝑌𝑁	𝐻𝐸𝐴𝐷	[ 𝑀𝑂𝐷 [	𝑆𝑌𝑁 [ 𝐸𝑋𝑇𝑅𝐴 𝐿 𝐻𝐸𝐴𝐷 𝑣𝑒𝑟𝑏 𝑆𝐸𝑀 [ 𝑆𝐶𝐴𝐿𝐸 𝑠 𝑉𝐴𝐿𝑈𝐸 𝑑1	] ]	𝑆𝑌𝑁 [𝐸𝑋𝑇𝑅𝐴 𝐿] 𝑆𝐸𝑀 [ ] 𝑉 [ 𝑆𝐶𝐴𝐿𝐸 𝑞𝑢𝑎𝑛𝑡 -𝑠 𝑉𝐴𝐿𝑈𝐸 𝑑1	] ] ]
			[ 𝐸𝑋𝑇𝑅𝐴 𝐿 + 〈[	𝑆𝑌𝑁 [𝑀𝐴𝑅𝐾𝐼𝑁𝐺 𝑘𝑖] 𝑆𝐸𝑀 [𝑉𝐴𝐿𝑈𝐸 𝑑2]	]〉	]
			𝑠𝑢𝑝𝑒𝑟 -𝑟𝑒𝑙
	[	𝑆𝐸𝑀 [ 𝐴𝑅𝐺2 𝑑2 𝐴𝑅𝐺1 𝑑1	]			]
	4.2
		tro						boukou
	[ 𝑆𝑌𝑁 [𝐻𝐸𝐴𝐷 [ (43) a. Mo'nn aste lafarinn pou fer gato 𝑎𝑑𝑣 𝑀𝑂𝐷 SEM [ 𝑆𝐶𝐴𝐿𝐸 𝑠 ] ] 𝑆𝑌𝑁 [ 𝐻𝐸𝐴𝐷 [ 𝑎𝑑𝑣 𝑀𝑂𝐷 𝑆𝐸𝑀 [ 𝑉𝐴𝐿𝑈𝐸 𝑑 𝑆𝐸𝑀 [ 𝑒𝑥𝑐𝑒𝑠𝑠 -𝑣𝑎𝑙 -𝑟𝑒𝑙 𝐴𝑅𝐺 𝑑 ] ] [ 𝐿𝐸𝑋 -𝑆𝐸𝑀 [ ℎ𝑖𝑔ℎ -𝑑𝑒𝑔 -𝑟𝑒𝑙 𝐴𝑅𝐺 𝑑 'I bought flour to make a cake/ cakes' ] b. boukou dilo 'a lot of water'	𝑆𝐶𝐴𝐿𝐸 𝑞𝑢𝑎𝑛𝑡 -𝑠 𝑉𝐴𝐿𝑈𝐸 𝑑	]	]	]	]
			c. mwins rob 'less dresses'
			d. plis lafarinn 'more flour'

11

Lexical constraint on Mauritian verbs

[START_REF] Henri | A constraint-based approach to verbal constructions in Mauritian: morphological, syntactic and discourse-based aspects[END_REF]

:

verb [HEAD [VFORM short]] => [VAL [COMPS non-empty-list]]

  SF with POSS.3sg brother a-lot/ a-lot with POSS.3sg brother 'Paul speaks with his brother a lot / a lot with his brother.' Degree adverbs as adjuncts They are adjuncts when they precede the verb. Like two manner adverbs (bien 'well', mal 'badly'), and a few other adverbs (nek 'only', fek 'just')), they occur between the TAM marker (pe, inn, pou…) and the lexical verb (Henri & Kihm forthcoming).

	As oblique complements, they follow the direct complements (45a) and may
	permute with other oblique complements (45b)
	(45)	a. Paul zwenn so frer boukou.
		Paul meet-SF POSS.3sg a-lot
		'Paul meets his brother a lot.'
		b. Paul koz [ar so frer] boukou / boukou [ar so frer].
	Paul speak-(46) a. Paul inn	mal / mari	travay.
		Paul PERF badly/ very	work
		'Paul has worked badly / a lot'
		b. Paul pe	nek	plore.
		Paul PROG only cry-SF
		'Paul is only crying'

Très ('very') also combines with participles -a mixed category with adjectival propertieswith an intensity scale: Ça m'a très étonné ('This has a-lot surprised me').

We gloss the TAM markers as PAST (past), PERF (perfective), PROG (progressive), FUT (future). SF stands for short verbal form and LF long verbal form.

[START_REF] Baker | Kreol: A description of Mauritian Creole[END_REF] shows that 70% of the verbal lexicon in Mauritian have two forms, the others being syncretic, like travay (work), ena (have), vo (be worth), ekrir (write).

Mari belongs to an informal register, and comes from the French noun mari ('husband'), which also gave an adjective ('superior') in Mauritian. Mauritian has not retained French très. Extra, with the same distribution as mari, is also an innovation (French extra, from Latin, is a prefix or an adjective).

[START_REF] Carpooran | Diksioner Morisien, Koleksion Text Kreol[END_REF]`s dictionary codes plis as a conjunction (because of the additive use: 2 plis 2 = 4) and pli as an adverb which may affect quantity or quality.

French has one form of the superior comparative plus ('more') and some synthetic comparatives: meilleur ('better'), pire ('worse'). Spoken French has three forms for plus: /ply/, /plys/ and /plyz/ before vowel for the liaison form. A precise study of the distribution of ply/plys in French still has to be done. French also has a negative homonym plus ('no longer') which in Mauritian has become nepli.

In SBCG, Kay and Sag 2012 analysed them as functors (Van Eyde 2007), with a SEL feature instead of a MOD feature, triggering a MARKING value deg. As far as we can see, nothing would be changed if we adopt this analysis, as long as they may also be complements in the verbal domain.

This work is part of the program Investissements d'Avenir, overseen by the French National Research Agency, ANR-10-LABX-0083, (Labex EFL).


N Krell
email: nathalie.krell@univ-rennes1.fr

Statistical estimation of jump rates for a specific class of Piecewise Deterministic Markov Processes

Keywords: Piecewise Deterministic Markov processes, Nonparametric estimation, jump rate estimation, ergodicity of Markov chains. Mathematical Subject Classification: 62M05, 62G05, 62G20, 60J25

We consider the class of Piecewise Deterministic Markov Processes (PDMP), whose state space is R * + , that possess an increasing deterministic motion and that shrink deterministically when they jump. Well known examples for this class of processes are Transmission Control Protocol (TCP) window size process and the processes modeling the size of a "marked" Escherichia coli cell. Having observed the PDMP until its nth jump, we construct a nonparametric estimator of the jump rate λ. Our main result is that for D a compact subset of R * + , if λ is in the Hölder space H s (D), the squared-loss error of the estimator is asymptotically close to the rate of n -s/(2s+1) . Simulations illustrate the behavior of our estimator.

Introduction

The Piecewise deterministic Markov processes were first introduced in the literature by Davis ([18] and [START_REF] Davis | Markov models and optimization[END_REF]), they form a family of càdlàg Markov processes involving a deterministic motion punctuated by random jumps. We refer to the paper [START_REF] Azaïs | Piecewise deterministic Markov process (pdmps)[END_REF] and its references for an overview of PDMPs. Let us detail the special case of PDMPs that will be considered in this paper. The motion of the PDMP (X(t)) t≥0 depends on three local characteristics, namely the jump rate λ, the flow φ and a deterministic increasing function f which governs the location of the process at the jump time (in the general case it depends on a Markov kernel Q). The process starts from x and follows the flow φ(x, t) until the first jump time T 1 which occurs spontaneously in a Poisson-like fashion with rate λ(φ(x, t)). The location of the process at new bacterium of equal size. In a general case, the instantaneous growth rate (constant for a given bacteria) would depend on the bacteria itself. Consequently, the size of the bacterium is no longer a Markov process. However, on the other hand, the size of the process and the instantaneous growth rate, together form a Markov process. The present paper contains the case of a "marked" bacterium, in its simple case, where the instantaneous growth rate is the same for all bacteria.

This article has two main features that may be useful for future studies. First, it can be used as a tool to verify the PDMP jump rate proposed in the existing literature, for example λ(x) = x in the TCP case. Secondly, it can suggest an estimator for some special cases, such as for Escherichia coli bacteria, where the jump rate is not known.

In [START_REF] Azaïs | Nonparametric estimation of the conditional distribution of the inter-jumping times for piecewise-deterministic Markov processes[END_REF] Azaïs et al. give an estimator of the conditional distribution of the inter-jump times for a PDMP, which is uniformly consistent when only one observation of the process within a long time is available. They deal with PDMPs which jump when they hit the boundary (this case is not considered in our paper). Their method relies on a generalization of Aalen's multiplicative intensity model [START_REF] Aalen | Statistical inference for a family of counting processes[END_REF][START_REF] Aalen | Weak convergence of stochastic integrals related to counting processes[END_REF][START_REF] Aalen | Nonparametric inference for a family of counting processes[END_REF]. But they only prove the uniform consistency of their estimator. They also have to assume that the process (X(t)) t≥0 evolves in a bounded space. Here we do not make this assumption. As a consequence the tools of their paper and of the present one are different. To the best of my knowledge, [START_REF] Azaïs | Nonparametric estimation of the conditional distribution of the inter-jumping times for piecewise-deterministic Markov processes[END_REF] is the only work investigating the nonparametric estimation of the conditional distribution of the inter-arrival times for PDMPs. This paper relies on [START_REF] Azaïs | Nonparametric estimation of the jump rate for non-homogeneous marked renewal processes[END_REF] in which the authors focus on the non parametric estimation of the jump rate and the cumulative rate for a class of non homogeneous marked renewal processes. The case where the post-jump locations of the PDMP do not depend on inter-arrival times was considered in [START_REF] Azaïs | Nonparametric estimation of the jump rate for non-homogeneous marked renewal processes[END_REF].

We refer to [START_REF] Azaïs | Nonparametric estimation of the conditional distribution of the inter-jumping times for piecewise-deterministic Markov processes[END_REF] for an overview of the statistical methods related to this kind of process, as well as to [START_REF] Aalen | Weak convergence of stochastic integrals related to counting processes[END_REF][START_REF] Aalen | Nonparametric inference for a family of counting processes[END_REF][START_REF] Aalen | Statistical inference for a family of counting processes[END_REF] for statistical inference related to the multiplicative intensity model. The book of Andersen et al. [START_REF] Andersen | Statistical models based on counting processes[END_REF] gives a comprehensive account of estimation for jump rates which depend both on time and spatial variable.

As far as I know, the only other paper dealing with general PDMP is the work of Azaïs [START_REF] Azaïs | A recursive nonparametric estimator for the transition kernel of a piecewise-deterministic Markov process[END_REF], where the author focuses on a non parametric recursive estimator for the transition kernel of the PDMP.

Other works dealing with specific cases of PDMP can been seen as ruin probability, for example, as found in the references of [START_REF] Asmussen | Ruin probabilities[END_REF]. In addition, the PDMP modeling the quantity of a given food contaminant in the body has been studied in [START_REF] Bouguet | Quantitative speeds of convergence for exposure to food contaminants[END_REF][START_REF] Bertail | A storage model with random release rate for modeling exposure to food contaminants[END_REF][START_REF] Bertail | Statistical analysis of a dynamic model for dietary contaminant exposure[END_REF], assuming that the inter-intake times are i.i.d.. In this paper we do not make this assumption.

The paper of Azaïs and Genadot [START_REF] Azaïs | Semi-parametric inference for the absorption features of a growth-fragmentation model[END_REF] consider a growth-fragmentation model where λ is constant and the Markov kernel Q is absolutely continuous with respect to the Lebesgue measure. The case considered is totally different from the present one.

The paper is organized as follows. In section 2 we introduce the class of PDMPs, that will be studied, and we give an explicit construction of the PDMP. Section 3 concerns the statistical estimation of the jump rate. We first define the observation scheme and the class of functions for the 3 parameters of the PDMP concerned, and the assumptions used (in Subsection 3.2). In subsection 3.3, some ergodicity results are stated uniformly over the class of functions previously defined. We explicitly construct an estimator λ n of λ. In subsection 3.5 an upper bound for the squared-error loss is given in the main Theorem 1. In 3.6, we illustrate our result with simulations of a TCP process which could not been seen as a "marked" bacteria process, amongst others.

Finally, section 4 is reserved for the proofs. In subsection 4.1 we prove the ergodicity result of Subsection 3.3. Subsection 4.2 presents the intermediate results needed in 4.3 to prove the major results given in Subsection 3.5.

PDMP

In general a PDMP is defined by its local characteristics, namely, the jump rate λ, the flow φ and the transition measure Q according to which the location of the process is chosen at the jump time. In this article, we consider a specific class of PDMP which includes the control of congestion TCP/IP used in communication networks (V. Dumas and al [START_REF] Dumas | A Markovian analysis of additiveincrease multiplicative-decrease algorithms[END_REF], V. Guillemin and al. [START_REF] Guillemin | AIMD algorithms and exponential functionals[END_REF]), for which the transition measure Q is a Dirac mass function, which means that when the process jumps, the size after the jump is a deterministic function of its size before. More precisely, Assumption 1.

• The flow φ : R + × R + → R + is a one-parameter group of homeomorphisms: φ is C 1 , φ(., t) is an homeomorphism for each t ∈ R + , satisfying the semigroup property: φ(., t + s) = φ(φ(., s), t) and φ x (.) := φ(x, .) is an C 1 -diffeormorphism.

• The jump rate λ : R + → R + is assumed to be a measurable function satisfying

∀x ∈ R + , ∃ ε > 0 such that ǫ 0 λ(φ(x, s))ds < ∞. • f : R + → R + is an increasing C 1 -diffeomorphism and Q(u, {y}) = 1l {f (u)=y} .
Given these three characteristics, it can be shown ( [START_REF] Davis | Markov models and optimization[END_REF], pages 62-66), that there exists a filtered probability space (Ω, F , {F t }, {P x }) such that the motion of the process {X(t)} starting from a point x ∈ R + may be constructed as follows. Consider a random variable T 1 such that

P x (T 1 > t) = e -Λ(x,t) , (1)
where for x ∈ R + and t

∈ R + Λ(x, t) = t 0 λ(φ(x, s))ds.
If T 1 is equal to infinity, then the process X follows the flow, i.e. for t ∈ R + , X(t) = φ(x, t).

Otherwise let

Z 1 = f (φ(x, T 1 )). (2)
The trajectory of {X(t)} starting at x, for t ∈ [0, T 1 ], is given by

X(t) = φ(x, t) for t < T 1 , Z 1 for t = T 1 .
Inductively starting from X(T n ) = Z n , we now select the next inter-jump time T n+1 -T n and post-jump location X(T n+1 ) = Z n+1 in a similar way. This construction properly defines a strong Markov process {X(t)} with jump times {T k } k∈N (where T 0 = 0). A very natural Markov chain is linked to {X(t)}, namely the jump chain (Z n ) n∈N .

{X(t)} is a Markov process with infinitesimal generator G:

Gh(y) = φ ′ x (y)h ′ (y) + λ(y) h(f (y)) -h(y) (3)
for h : R + → R a bounded measurable functional. Thanks to (1), we get that

P(T 1 ∈ dt|Z 0 = x) = λ(φ x (t))e -t 0 λ(φx(s))ds dt.
Using (2), the monotonicity of f • φ x and a simple change of variables, we get the transition probability of the Markov chain (Z n ) n∈N :

P(Z 1 ∈ dy|Z 0 = x) = λ(f -1 (y))e -y f (x) λ(f -1 (s))gx(s)ds g x (y)1l {y≥f (x)} dy, (4)
where

g x (y) = (f • φ x ) ′ (f • φ x ) -1 (y) -1
and φ x (.) := φ(x, .).

3 Statistical estimation of the jump rate

The observation scheme

Statistical inference is based on the observation scheme:

(X(t), t ≤ T n )
and asymptotics are considered when the number of jumps of the process, n, goes to infinity. Actually the simpler observation scheme:

(X(T i ), 1 ≤ i ≤ n) = (Z i , 1 ≤ i ≤ n) is sufficient.

Class of functions

We want to bound from above the squared-loss error of our estimator over compact intervals D of R * + . We need to specify the local smoothness properties of λ over D, together with general properties that ensure that the empirical measurements of the PDMP converge toward the invariant probability with an appropriate speed of convergence. So we have to impose technical assumptions on λ in particular near the origin and infinity.

m : [0, ∞) → (0, ∞) and M : [0, ∞) → (0, ∞) such that for all x ∈ [0, ∞) : M(x) ≥ m(x) > 0, we introduce the class F (c, m, M) of triples of continuous functions λ : [0, ∞) → [0, ∞), f : R + → R + and φ : R + × R + → R + such that f (r) 0 M(s)λ(f -1 (s))ds ≤ L, (5)
∞ f (r) m(s)λ(f -1 (s))ds = ∞, (6) r f (r)
M(s)λ(f -1 (s))ds ≥ l, (7) λ
(x) ≥ a(f (x)) b m(f (x)) ∀x ≥ r, (8)
∀x > 0, 0 < f (x) ≤ κx, (9)
∀y > 0, ∀x ≥ 0, m(y) ≤ g x (y) ≤ M(y), (10)
where

g x (y) = (f • φ x ) ′ (f • φ x ) -1 (y) -1 and φ x (•) = φ(x, •).
We notice that f (0) = 0. Typically an interesting case would be f (x) = κx with κ ∈ (0, 1) and then [START_REF] Azaïs | Semi-parametric inference for the absorption features of a growth-fragmentation model[END_REF] would simply be

M -1 (y) ≤ κφ ′ x (φ -1
x (y/κ)) ≤ m -1 (y). This seems quite reasonable because, in view of the definition of the infinitesimal generator of the PDMP defined in (3), we would like φ ′ x (•) not to be identically zero. Also the cases where φ x (t) = xe αt or φ x (t) = x + t satisfy [START_REF] Azaïs | Semi-parametric inference for the absorption features of a growth-fragmentation model[END_REF]

over D. Define δ(c, f ) := 1 1 -κ b+1 exp -(1 -κ b+1 ) am b+1 (f (r)) b+1
. The last assumption that we will need is:

Assumption 2. ∃b > 0 : (κ b+1 -1) am b + 1 (f (r)) b+1 < log(1 -κ b+1 ), (11)
so that we have δ(c, f ) < 1 Fix a vector of positive constants c = (r, κ, l, L, a), a constant b and a function f .

Geometric ergodicity of the discrete model

Let x ∈ R + . Introduce the transition kernel

P λ (x, dx ′ ) = P Z n ∈ dx ′ Z n-1 = x
of the size of the process at the nth jump time, given the size of the process at the (n -1)th jump time. From (4), we infer that P(Z

1 ∈ dy|Z 0 = x) is equal to λ(f -1 (y))e -y f (x) λ(f -1 (s))gx(s)ds g x (y)1l {y≥f (x)} dy.
Thus we obtain an explicit formula for

P λ (x, dy) = P λ (x, y)dy with P λ (x, y) = λ(f -1 (y))e -y f (x) λ(f -1 (s))gx(s)ds g x (y)1l {y≥f (x)} . (12)
Denote the left action of positive measures µ(dx) on R + for the transition kernel P λ by

µP λ (dy) = R + µ(dx)P λ (x, dy)
and the right action of a function ψ on R for the transition P λ by

P λ ψ(x) = R + ψ(y)P λ (x, dy)
We now give the geometric ergodic theorem that we will need for the statistical part. We need an uniformity on the class of functions F (c, m, M) defined in subsection 3.2. We introduce the Lyapunov function

V(x) = exp a b + 1 (f (x)) b+1 for x ∈ R + . ( 13
)
The function V controls the rate of the geometric ergodicity of the chain with transition P λ and appears in the proof of Proposition 1.

Proposition 1. Under Assumption 1, for every λ such that (λ, f, φ) ∈ F (c, m, M) there exists a unique invariant probability measure of the form ν λ (dx) = ν λ (x)dx on R + . Moreover there exist 0 < γ < 1, a constant R and a function V :

R + → [1, ∞) such that sup λ∈F (c,m,M ) sup |ψ|≤V P k λ ψ(x) - R + ψ(z)ν λ (z)dz ≤ RV(x)γ k ( 14
)
for every x ∈ R + , k ≥ 0, where the supremum is taken over all functions ψ : R

+ → R satisfying |ψ(x)| ≤ V(x) for all x ∈ R + . The function V is ν λ -integrable for every λ such that (λ, f, φ) ∈ F (c, m, M).
For all y ∈ R + we have the relation:

λ(y)E ν λ (g Z 0 (f (y))1l {f (Z 0 )≤f (y)} 1l {Z 1 ≥f (y)} ) = ν λ (f (y)). (15)

Construction of a nonparametric estimator

By formula [START_REF] Bouguet | Quantitative speeds of convergence for exposure to food contaminants[END_REF],

λ(y) = ν λ (f (y)) E ν λ (g Z 0 (f (y))1l {f (Z 0 )≤f (y)} 1l {Z 1 ≥f (y)} ) ,
provided the denominator is positive. This representation suggests an estimation procedure, replacing the marginal density ν λ (f (y)) and the expectation in the denominator by their empirical counterparts. To that end, pick a kernel function

K : R → [0, ∞), R K(y)dy = 1,
and set K h (y) = h -1 K h -1 y for y ∈ R and h > 0. Our estimator is defined by

λ n (y) = n -1 n k=1 K hn (Z k -f (y)) n -1 n k=1 g Z k-1 (f (y))1l {Z k ≥f (y), f (y)≥f (Z k-1 )} ∨ ̟ n . ( 16
)
where ̟ n > 0 is a threshold that ensures that the estimator is well defined in all cases and x ∨ y = max{x, y}. Thus ( λ n (y), y ∈ D) is specified by the choice of the kernel K, the bandwidth h n > 0 and the threshold ̟ n > 0.

Assumption 3. The function K has compact support, and for some integer n 0 ≥ 1, we have

R x k K(x)dx = 1 {k=0} for 0 ≤ k ≤ n 0 .

Rate of convergence

We are now ready to state our main result. For s > 0, with s = ⌊s⌋ + {s}, 0 < {s} ≤ 1 and ⌊s⌋ an integer, introduce the Hölder space H s (D) of functions f : D → R possessing a derivative of order ⌊s⌋ that satisfies

|f ⌊s⌋ (y) -f ⌊s⌋ (x)| ≤ c(f )|x -y| {s} . ( 17
)
The minimal constant c(f ) such that (17) holds defines a semi-norm |f | H s (D) . We equip the space H s (D) with the norm

f H s (D) = f L ∞ (D) + |f | H s (D)
and the associated Hölder balls

H s (D, M 1 ) = {λ : λ H s (D) ≤ M 1 }, M 1 > 0.
Theorem 1. Work under Assumption 1 and Assumption 2. Specify λ n with a kernel K satisfying Assumption 3 for some n 0 > 0 and

h n = c 0 n -1/(2s+1) , ̟ n such that lim n→∞ ̟ n = 0
For every M 1 > 0 and M 2 > 0, there exist c 0 = c 0 (c, M 1 , M 2 ) and d(c) ≥ 0 such that for every 0 < s < n 0 and every compact interval

D ⊂ (d(c), ∞) such that inf D ≥ f (r), we have sup λ E µ λ n -λ 2 L 2 (D) 1/2 ̟ -2 n n -s/(2s+1) ,
where the supremum is taken over

(λ, f, φ) ∈ F (c, m, M), λ ∈ H s (D, M 1 ), g x ∈ H s (D), f L ∞ (D) ≤ M 2 , and f -1 ∈ H s (D)
and E µ [•] denotes expectation with respect to any initial distribution µ(dx) for (Z 0 ) on R + such that V(x) 2 µ(dx) < ∞.

We observe that we recover the result for the marked bacteria of [START_REF] Doumic | Statistical estimation of a growth-fragmentation model observed on a genealogical tree[END_REF]. In this case φ(x, t) = xe κ 0 t with κ 0 ∈ R + * and f (x) = x/2, so that g x (y) = 1 κ 0 y . We find the same estimator but the speed of convergence is a little bit better, as ̟ n need not be log(n); rather we only require that lim n→∞ ̟ -1 n = 0.

Numerical implementation

The goal of this subsection is to illustrate the asymptotic behaviour of our estimator via numerical experiments. More precisely we first investigate numerical simulations for the TCP.

The TCP window-size process appears as the scaling limit of the transmission rate of a server uploading packets on the Internet according to the algorithm used in the TCP (Transmission Control Protocol) in order to avoid congestion (see [START_REF] Dumas | A Markovian analysis of additiveincrease multiplicative-decrease algorithms[END_REF] for details on this scaling limit). This PDMP takes values in R + and the jump rate λ is the identity function. The function f which represents the proportion of the size kept after the jump is f (x) = x/2. The flow is φ(x, t) = x + t.

As a consequence the size of the process after the n-th jump Z n , conditional on Z n-1 , has the same law as (1/4)Z 2 n-1 + e n /2, where (e n ) n≥0 is a family of i.i.d. random variables with exponential distribution of parameter 1. The variable e n is also independent of (Z i ) i≤n-1 . As a consequence it is easy to generate the (Z n ) n≥0 recursively. A trajectory of such a PDMP is given in Figure 1. These processes satisfy the assumptions required for our Theorem, with

κ = 1/2, m(•) = 1, M(•) = 1.1, g • (•) = 1/2, r = 1, a = 1, b = 1/2, ̟ n = (log(n)) -1 , h n = n -1/3 and K(x) = (2π) 1/2 exp(-x 2 /2).
With the Gaussian Kernel, for which n 0 = 1 for Assumption 2, we expect a rate of convergence of order n 1/3 at best. We display our numerical results as specified above in Figures 1, 2 and3.

Figure 2 displays the reconstruction of λ for different simulated samples, for n = 1000, n = 10000 and n = 100000. As expected, the estimation is better for larger n. The estimator performs worse fo small x as these sizes are rarely reached by the TCP process.

In figure 3, we plot the empirical mean error of our estimation procedure on a log-log scale. The numerical results agree with the theory.

Proof

Proof of Proposition 1

We will follow the same idea as in [START_REF] Doumic | Statistical estimation of a growth-fragmentation model observed on a genealogical tree[END_REF]. We prove a minorisation condition, strong aperiodicity and a drift condition for the transition operator P λ in order to use Theorem 1.1 of [START_REF] Baxendale | Renewal theory and computable convergence rates for geometrically ergodic Markov chains[END_REF].

Minorisation condition. Let λ be such that (λ, f, φ) ∈ F (c, m, M) and set C = (0, r) where r is specified by c. Fix a measurable A ∈ F and x ∈ C; thanks to (10), we have We introduce the function ϕ λ ϕ λ (y) := λ(f -1 (y))e -y 0 λ(f -1 (s))M (s)ds m(y)

P λ (x, A) ≥ A λ(f -1 (y))e -y f (x) λ(f -1 (s))M (s)ds m(y)1l {y≥f (x)} dy.
∀y ∈ R + , (18)
and the measure µ λ µ λ (dy) := ϕ λ (y) c λ 1l {y>f (r)} dy,

where c λ = ∞ f (r) ϕ λ (u)du. Thus, we have

P λ (x, A) ≥ µ λ (A)c λ .
By using ( 6) and ( 5), we get that

c λ = ∞ f (r) ϕ λ (u)du = -m(y)e -y 0 λ(f -1 (s))M (s)ds M(y) ∞ f (r) ≥ m(f (r))e -L M(f (r)) := β > 0.
This shows that the following minorisation condition holds for every x ∈ C and A ∈ F uniformly in λ such that (λ, f, φ) ∈ F (c, m, M):

P λ (x, A) ≥ µ λ (A) β. (19)
Strong aperiodicity condition. We have

µ λ (C) β = c -1 λ β r f (r) ϕ λ (y)dy = c -1 λ β -m(y)e -y 0 λ(f -1 (s))M (s)ds M(y) r f (r) = β(1 - m(r)M(f (r)) M(r)m(f (r)) e -r f (r) λ(f -1 (s))M (s)ds , ) (20)
using the computation we just did for c λ . Now we use [START_REF] Azaïs | Piecewise deterministic Markov process (pdmps)[END_REF] to get that

µ λ (C) β ≥ β 1 - m(r)M(f (r)) M(r)m(f (r)) e -l := β > 0. ( 21
)
Drift condition. Let λ be such that (λ, f, φ) ∈ F (c, m, M) and recall that V : R + → [1, ∞), which is defined in [START_REF] Bertail | A storage model with random release rate for modeling exposure to food contaminants[END_REF], is continuously differentiable and satisfies

lim y→∞ V(y) exp -a b+1 y b+1 = 0. (22)
For x ≥ r, using [START_REF] Azaïs | Semi-parametric inference for the absorption features of a growth-fragmentation model[END_REF] and integration by parts with the boundary condition (6), we have,

P λ V(x) = ∞ f (x)
V(y)λ(f -1 (y))e -y f (x) λ(f -1 (s))gx(s)ds g x (y)dy

≤ ∞ f (x)
V ′ (y)e -y f (x) λ(f -1 (s))m(s)ds dy.

Thanks to (8), we get that

P λ V(x) ≤ ∞ f (x) V ′ (y)e -y f (x) as b ds dy ≤ e a f (x) b+1 b+1 ∞ f (x)
V ′ (y)e -a y b+1 b+1 dy.

Integrating again by parts and using [START_REF] Grigorescu | Reccurence and ergodicity for a continuous aimd model[END_REF], we obtain that

P λ V(x) ≤ e a f (x) b+1 b+1 ∞ f (x)
V(y)ay b e -a y b+1 b+1 dy.

Now use the change of variable z = a y b+1 b+1 and the definition of V(x). As ( 22) is satisfied, we get

P λ V(x) ≤ V(x) ∞ a f (x) b+1 b+1 e a b+1 f (( z(b+1) a ) 1/(b+1) ) b+1 -z dz.
By using (9), we obtain, for x ≥ r

P λ V(x) ≤ V(x) ∞ a f (x) b+1 b+1
e (κ b+1 -1)z dz.

Therefore,

P λ V(x) ≤ V(x)δ(c, f ), (23)
with δ(c, f ) = 1 1 -κ b+1 exp -(1 -κ b+1 ) a b+1 (f (r)) b+1 ,
and we have δ(c, f ) < 1 by Assumption 2. We next need to control ), that is on the small set C. For every x ∈ C, we have

P λ V outside x ∈ [r, ∞
P λ V(x) ≤ f (r) f (x) V(y)λ(f -1 (y))g x (y)dy + ∞ f (r) V(y)λ(f -1 (y))e -y f (x) λ(f -1 (s))gx(s)ds g x (y)dy ≤ M sup y∈[0,f (r)] V(y)L + δ(c, f )V(r) =: K < ∞, (24)
where we used ( 5), ( 10), ( 23) for x = r and the fact that (λ, f, φ) ∈ F (c, m, M). Combining ( 23) and ( 24), we conclude that

P λ V(x) ≤ δ(c, f )V(x)1l {x / ∈C} + K1l {x∈C} . ( 25
)
End of the proof of Proposition 1. By Theorem 1.1 in Baxendale [START_REF] Baxendale | Renewal theory and computable convergence rates for geometrically ergodic Markov chains[END_REF] the minorisation condition [START_REF] Davis | Markov models and optimization[END_REF] together with the strong aperiodicity condition [START_REF] Dumas | A Markovian analysis of additiveincrease multiplicative-decrease algorithms[END_REF] and the drift condition (25) imply inequality [START_REF] Bertail | Statistical analysis of a dynamic model for dietary contaminant exposure[END_REF], with R and γ that explicitly depend on δ(c, f ), β, β, V and K. By construction, this bound is uniform in λ such that (λ, f, φ) ∈ F (c, m, M). More specifically, we have γ < min{max{δ(c, f ), γ λ,V }, 1}, with γ λ,V the spectral radius of the operator P λ -1 ⊗ ν λ acting on the Banach space of functions ψ : R + → R such that

sup |ψ(x)| V(x) , x ∈ R + < ∞.
Therefore, under Assumption [START_REF] Bardet | Total variation estimates for the TCP process[END_REF] we have γ < 1.

It remains to prove equality [START_REF] Bouguet | Quantitative speeds of convergence for exposure to food contaminants[END_REF]. As P λ (x, dy) = P λ (x, y)dy and

ν λ P λ = ν λ ,
we have that ν λ (dy) = ν λ (y)dy and

ν λ (y) = R + ν λ (x)P λ (x, y)dx = R + ν λ (x)λ(f -1 (y))e -y f (x) λ(f -1 (s))gx(s)ds g x (y)1l {f (x)≤y} dx.
Thanks to [START_REF] Azaïs | A recursive nonparametric estimator for the transition kernel of a piecewise-deterministic Markov process[END_REF], we get that

e -y f (x) λ(f -1 (s))gx(s)ds = ∞ y λ(f -1 (s))g x (s)e -s f (x) λ(f -1 (s ′ ))gx(s ′ )ds ′ ds.
Therefore,

ν λ (y) =λ(f -1 (y)) R + ν λ (x)g x (y)1l {f (x)≤y} ∞ y λ(f -1 (s))g x (s)e -s f (x) λ(f -1 (s ′ ))gx(s ′ )ds ′ dsdx =λ(f -1 (y)) R + ν λ (x)g x (y)1l {f (x)≤y} ∞ y 1l {s≥y} P λ (x, s)dsdx =λ(f -1 (y))E ν λ (g Z 0 (y)1l {f (Z 0 )≤y} 1l {Z 1 ≥y} ).

Rate of convergence for the empirical measure

We now give a few results that we will need for the proof of Theorem 3.2 in the next Subsection. In fact, we decompose the square loss error into a sum of three terms that we will study in the following Propositions.

The notation means inequality up to a constant that not depend on n.

Lemma 1. For any c such that Assumptions 2 and 1 are satisfied, there exists a constant d(c) ≥ 0 such that for any compact interval D ⊂ (d(c), ∞), we have

inf λ: (λ,f,φ)∈F (c,m,M ) inf x∈D ϕ λ (x) -1 ν λ (x) > 0,
where ϕ λ (x) is defined in [START_REF] Davis | Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models[END_REF].

Proof. Recall that V(x) = exp a b+1 (f (x)) b+1 for every x ∈ [0, ∞). By Proposition 1 (and, more precisely, equation ( 14)) we have

sup λ: (λ,f,φ)∈F (c,m,M ) [0,∞) V(x)ν λ (x)dx < ∞, (26)
additionally from (25) in the proof of Proposition 1, we have that sup λ: (λ,f,φ)∈F (c,m,M ) P λ V(x) < ∞ for every x ∈ R + . As a consequence, for every x ∈ (0, ∞), we have

∞ f -1 (x) ν λ (y)dy ≤ exp - a b + 1 (f (x)) b+1 [0,∞) V(y)ν λ (y)dy,
and this bound is uniform in λ such that (λ, f, φ) ∈ F (c, m, M) by (26). Therefore, for every x ∈ (0, ∞), we have

sup λ:(λ,f,φ)∈F (c,m,M ) ∞ f -1 (x) ν λ (y)dy ≤ c(c, f ) exp - a b + 1 (f (x)) b+1 , (27)
for some c(c, f

) > 0. Let d(c, f ) > f -1 ( b + 1 a log(c(c, f ))) 1/(b+1) . (28)
By the definition of ν λ and using [START_REF] Azaïs | Semi-parametric inference for the absorption features of a growth-fragmentation model[END_REF], for every y ∈ (0, ∞), we now have

ν λ (y) = ∞ 0 ν λ (x)λ(f -1 (y))e -y f (x) λ(f -1 (s))gx(s)ds g x (y)1l {f (x)≤y} dx ≥ e -y 0 λ(f -1 (s))M (s)ds λ(f -1 (y))m(y) f -1 (y) 0 ν λ (x)dx ≥ e -y 0 -1 (s))M (s)ds λ(f -1 (y))m(y) 1 - ∞ f -1 (y) ν λ (x)dx ≥ e -y 0 λ(f -1 (s))M (s)ds λ(f -1 (y))m(y) 1 -c(c, f ) exp - a b + 1 (f (y)) b+1
where we used (27) for the last inequality. By (28), for y ≥ d(c, f ) we have

1 -c(c, f ) exp - a b + 1 (f (y)) b+1 > 0,
and the conclusion follows readily by the definition of ϕ λ .

For every y ∈ (0, ∞), define

D(y) = E ν λ g Z 0 (f (y))1l {Z 1 ≥f (y), f (y)≥f (Z 0 )} , (29)
D n (y) = n -1 n k=1 g Z k-1 (f (y))1l {Z k ≥f (y), f (y)≥f (Z k-1 )} , (30)
and

D n (y) ̟n = n -1 n k=1 g Z k-1 (f (y))1l {Z k ≥f (y), f (y)≥f (Z k-1 )} ̟ n . (31)
Proposition 2. Work under Assumptions 2 and 1. Let µ be a probability measure on

R + such that R + V(x) 2 µ(dx) < ∞. If 1 ≥ ̟ n → 0 as n → ∞, we have sup y∈D E µ D n (y) ̟n -D(y) 2 n -1 (32)
uniformly in λ such that Proof . By [START_REF] Bouguet | Quantitative speeds of convergence for exposure to food contaminants[END_REF] and the definition of ϕ B in ( 18), we readily have that

D(y) = ν λ (f (y)) λ(y) = ν λ (f (y)) ϕ λ (f (y)) e -f (y) 0 λ(f -1 (s))M (s)ds m(y). Since (λ, f, φ) ∈ F (c, m, M) and λ ∈ H s (f (D), M 1 ),
by applying (5), we obtain

f (y) 0 λ(f -1 (s))M(s)ds ≤ sup D 0 λ(f -1 (s))M(s)ds ≤ L + sup D f (r) λ(f -1 (s)) sup y∈D |M(y)|ds ≤ L + sup y∈D |M(y)|M 1 sup D < ∞,
where we used that inf D ≤ f (r). It follows that

inf y∈D exp - f (y) 0 λ(f -1 (s))M(s)ds ≥ exp -(L + sup y∈D |M(y)|M 1 sup D) > 0
and Lemma 2 follows by applying Lemma 1.

Proof of Proposition 2. Since D n (y) is bounded by M, we have

D n (y) ̟n -D(y) 2 D n (y) -D(y) 2 + 1 {Dn(y)<̟n} . (33)
Thus, by Lemma 2 we may choose n sufficiently large that By the Bienaymé-Tchebychev inequality, this quantity is less than a constant times

0 < ̟ n ≤ q = 1
E µ D n (y) -D(y) 2 .
Set G(x, z, y) = g x (f (y))1l {z≥f (y), f (y)≥f (x)} and note that G(•, •, •) is bounded on R + by sup y∈D |M(y)|. It follows that

D n (y) -D(y) = n -1 n k=1 G(Z k-1 , Z k , y) -E ν λ G(Z k-1 , Z k , y) .
Therefore,

E µ D n (y) -D(y) 2 = 1 n 2 k,k ′ ∈{1,..,n} E µ G(Z k-1 , Z k , y) -E ν λ G(Z k-1 , Z k , y) G(Z k ′ -1 , Z k ′ , y) -E ν λ G(Z k ′ -1 , Z k ′ , y) . ( 34
)
For |k -k ′ | ≥ 2, applying Markov's property, we get that

E µ G(Z k-1 , Z k , y) -E ν λ G(Z k-1 , Z k , y) G(Z k ′ -1 , Z k ′ , y) -E ν λ G(Z k ′ -1 , Z k ′ , y) |Z i ∀i ≤ k ∧ k ′ = P k∨k ′ -k∧k ′ λ (Z k∧k ′ , dz)P λ (z, dz ′ ) G(z, z ′ , y) -E ν λ G(Z 0 , Z 1 , y) G(Z k∧k ′ -1 , Z k∧k ′ , y) -E ν λ G(Z 0 , Z 1 , y) , with k ∧ k ′ = min{k, k ′ }. Applying Proposition 1 with h(z) = P λ (z, dz ′ )G(z, z ′ , y), we get E µ G(Z k-1 , Z k , y) -E ν λ G(Z k-1 , Z k , y) G(Z k ′ -1 , Z k ′ , y) -E ν λ G(Z k ′ -1 , Z k ′ , y) ≤ RE µ V(Z k∧k ′ ) G(Z k∧k ′ -1 , Z k∧k ′ , y) -E ν λ G(Z 0 , Z 1 , y) γ k∨k ′ -k∧k ′ E P k∧k ′ λ V(x)µ(dx) γ k∨k ′ -k∧k ′ ,
as the function G is bounded by sup y∈D |M(y)|.

For |k -k ′ | = 1, we suppose for example that k ′ = k -1. Applying the Markov property, we get that

E µ G(Z k-1 , Z k , y) -E ν λ G(Z k-1 , Z k , y) G(Z k ′ -1 , Z k ′ , y) -E ν λ G(Z k ′ -1 , Z k ′ , y) |Z i ∀i ≤ k -1 = P λ (Z k-1 , dz) G(Z k-1 , z, y) -E ν λ G(Z k ′ -1 , Z k ′ , y) G(Z k-2 , Z k-1 , y) -E ν λ G(Z 0 , Z 1 , y) .
Applying Proposition 1 again, we get

E µ G(Z k-1 , Z k , y) -E ν λ G(Z k-1 , Z k , y) G(Z k ′ -1 , Z k ′ , y) -E ν λ G(Z k ′ -1 , Z k ′ , y) ≤ RE µ V(Z k∧k ′ ) G(Z k-2 , Z k-1 , y) -E ν λ G(Z 0 , Z 1 , y) γ k∨k ′ -k∧k ′ E P k∧k ′ λ V(x)µ(dx) γ k∨k ′ -k∧k ′ ,
as the function G is bounded by sup y∈D |M(y)|.

For k = k ′ , E µ G(Z k-1 , Z k , y) -E ν λ G(Z k-1 , Z k , y) 2 |Z i ∀i ≤ k -1 γ k∨k ′ -k∧k ′ ,
as the function G is bounded by sup y∈D |M(y)|. Moreover as V satisfies (25), we get

sup λ: (λ,f,φ)∈F (c,m,M ) P k∧k ′ λ V(x) 1 + V(x) (35)
and, thus, for any k and k ′ ,

E µ G(Z k-1 , Z k , y) -E ν λ G(Z k-1 , Z k , y) G(Z k ′ -1 , Z k ′ , y) -E ν λ G(Z k ′ -1 , Z k ′ , y) R + 1 + V(x) µ(dx) γ k∨k ′ -k∧k ′ .
Since V is µ-integrable by assumption, thanks to (34), we have

E µ D n (y) -D(y) 2 1 n 2 k,k ′ ∈{1,2,..,n} γ k∨k ′ -k∧k ′ n -1 , (36)
uniformly in y ∈ D and λ such that (λ, f, φ) ∈ F (c, m, M).

Proposition 3. Work under Assumptions 2, 3 and 1. Let µ be a probability measure on R + such that R + V(x) 2 µ(dx) < ∞. Then we have

sup y∈D E µ K hn ⋆ ν n (y) -K hn ⋆ ν λ (y) 2 (nh n ) -1 (37) uniformly in λ such that (λ, f, φ) ∈ F (c, m, M) and f L ∞ (D) ≤ M 2 with ν n (•) = 1 n k∈{1,...,n} 1l {Z k } (•) Proof. By definition, E µ K hn ⋆ ν n (y) -K hn ⋆ ν λ (y) 2 = (nh n ) -2 E µ k∈{1,...,n} K Z k -y hn -E ν λ K Z 0 -y hn 2 = (nh n ) -2 k,k ′ ∈{1,...,n} E µ K Z k -y hn K Z k ′ -y hn , with K( Z k -y hn ) = K Z k -y hn -E ν λ K Z 0 -y hn .
As in the proof of Proposition 2, thanks to the Markov property we obtain

E µ E µ ( K Z k -y hn K Z k ′ -y hn Z k∧k ′ = P k∨k ′ -k∧k ′ λ J(Z k∧k ′ ) -E ν λ (K Z 0 -y hn ) K( Z k∧k ′ -y hn ) -E ν λ (K Z 0 -y hn ) ,
with J(•) = K( •-y hn ). First, as K has bounded support, K V and so we can apply ( 14) from Proposition 1. We obtain

P k∨k ′ -k∧k ′ λ J(Z k∧k ′ ) -E ν λ (K Z 0 -y hn ) ≤ RV Z k∧k ′ γ k∨k ′ -k∧k ′ . (38)
Moreover, thanks to ( 15) and ( 10) and the fact that λ ∈ H s (f (D), M 1 ), we have that

sup x∈f (D) ν λ (x) ≤ M 1 M, (39)
so that

E ν λ (K Z 0 -y hn ) ≤ [0,∞) K x-y hn ν λ (x)dx h n . (40)
Putting together (38) and (40) we derive

E µ E µ ( K Z k -y hn K Z k ′ -y hn Z k∧k ′ E µ (RJ(Z k∧k ′ )V Z k∧k ′ )γ k∨k ′ -k∧k ′ + E µ (RV Z k∧k ′ )γ k∨k ′ -k∧k ′ h n .
On the one hand, by using the Markov property, the fact that

e -u f (Z k∧k ′ -1 ) λ(f -1 (s))g Z k∧k ′ -1 (s)ds ≤ 1, (10)
, that λ ∈ H s (f (D), M 1 ) and as f is increasing, we can bound V(u)1l {u∈D} by e a b+1 M b+1 2 , we get that

E µ J Z k∧k ′ V Z k∧k ′ ≤ E µ ∞ f (Z k∧k ′ -1 ) K u-y hn λ(f -1 (u))e -u f (Z k∧k ′ -1 ) λ(f -1 (s))g Z k∧k ′ -1 (s)ds g Z k∧k ′ -1 (u)V(u)du ≤ sup x∈D |M(x)| M 1 [0,∞) E µ K h -1 n (u -y) du h n
as K has compact support. On the other hand, as V 2 and V are µ integrable by assumption we get that

E µ E µ ( K Z k -y hn K Z k ′ -y hn Z k∧k ′ h n γ k∨k ′ -k∧k ′ .
Therefore,

E µ K hn ⋆ ν n (y) -K hn ⋆ ν λ (y) 2 (nh n ) -2 k,k ′ ∈{1,...,n} h n γ k∨k ′ -k∧k ′ (nh n ) -1 .

Proof of Theorem 3.2

Recall that

λ n (y) = n -1 n k=1 K hn (Z k -f (y)) n -1 n k=1 g Z k-1 (f (y))1l {Z k ≥f (y), f (y)≥f (Z k-1 )} ̟ n and λ(y) = ν λ (f (y)) E ν λ (g Z 0 (f (y))1l {f (Z 0 )≤f (y)} 1l {Z 1 ≥f (y)} )
.

We will use the decomposition λ n (y) -λ(y) = (I + II + III),

where for some continuous function ψ.

I = K hn ⋆ ν λ (f (y)) -ν λ (f (y)) D(y) , II = K hn ⋆ ν n (f (y)) -K hn ⋆ ν λ (f (y)) D n (y) ̟n , III = K hn ⋆ ν λ (f (y)) D n (
Proof of Lemma 3. We first recall that ν λ (y) = λ(f -1 (y)) E ν λ (x)e -y f (x) λ(f -1 (s))gx(s)ds g x (y)1l {f (x)≤y} dx.

Define Λ λ (x, y) = e -y f (x) λ(f -1 (s))gx(s)ds g x (y).

If λ ∈ F (c), then Λ λ (x, .) ∈ H s (D) for every y ∈ [0, ∞), and we have Λ λ (x, .) H s (D) ≤ ψ 1 ( λ H s (D) , g x H s (D) , f -1 H s (D) )

for some continuous function ψ 1 . The result is then a consequence of the representation ν λ (y) = λ(f -1 (y))

f -1 (y) 0 Λ λ (x, y)dx.

Returning to (41) we deduce from Lemma 3 that ν λ H s (f (D)) is bounded above by a constant that depends on D, g x H s (f (D)) , f The term V. We have

E µ [V ] ≤ ̟ -2 n |D| sup y∈D E µ K hn ⋆ ν n (f (y)) -K hn ⋆ ν λ (f (y)) 2 .
By (37) of Proposition 3 we derive that

E µ [V ] ̟ -2 n (nh n ) -1 (43)
uniformly in λ ∈ F (c)

The term VI. First, thanks to Lemma 2, we get that K hn (z -y)ν λ (z)dz ≤ sup

y∈D 1 ν λ (y) K L 1 ([0,∞)) , (44)
where D 1 = {y + z, y ∈ f (D), z ∈ supp(K hn )} ⊂ D, for some compact interval D since K has compact support by Assumption 3. Thanks to (39), we see that (44) holds uniformly in λ such that (λ, f, φ) ∈ F (c, m, M). We derive that

E µ V I ̟ -2 n sup y∈f (D) E µ D n (y) ̟n -D(y) 2 .
Applying (32) of Proposition 2, we conclude that

E µ V I ̟ -2 n n -1 , (45)
uniformly in λ such that (λ, f, φ) ∈ F (c, m, M).

End of the proof of Theorem 3.2. We put together the three estimates (42), ( 43) and (45) to obtain

E µ λ n -λ 2 L 2 (D) h 2s n + ̟ -2 n (nh n ) -1 + ̟ -2 n n -1
uniformly in λ ∈ H s (D, M 1 ) and (λ, f, φ) ∈ F (c, m, M). The choice h n ∼ n -1/(2s+1) yields the rate ̟ -2 n n -2s/(2s+1) .

Definition 1 .

 1 For b > 0, a vector of positive constants c = (r, κ, l, L, a), and two positive functions

Figure 1 :

 1 Figure 1: Evolution of the TCP process when 10 jumps occur.

Figure 2 :

 2 Figure 2: Reconstruction of λ for n = 1000, n = 10000 and n = 100000 in the TCP case.

Figure 3 :

 3 Figure 3: The empirical mean error of the estimation procedure v.s. the theoretical rate on a log-log scale in the TCP case.

Figure 4 :

 4 Figure 4: Reconstruction of λ for n = 100000 in the marked bacteria case.

2 .

 2 (λ, f, φ) ∈ F (c, m, M), and λ ∈ H s (f (D), M 1 ), We first need the following estimate Lemma Work under Assumptions 2 and 1. Let d(c, f ) be defined as in Lemma 1. For every compact interval D ⊂ (d(c, f ), ∞) such that inf D ≤ f (r), we have inf λ: (λ,f,φ)∈F (c,m,M ) and λ∈H s (f (D),M 1 ) inf y∈D D(y) > 0.

2 infλ:

 2 (λ,f,φ)∈F (c,m,M ) and λ∈H s (f (D),M 1 ) inf y∈D D(y). Since {D n (y) < ̟ n } ⊂ {D n (y) -D(y) < -q}, by integrating (33), we have that E µ D n (y) ̟n -D(y) 2 is less than a constant times E µ D n (y) -D(y) 2 + P µ |D n (y) -D(y)| ≥ q .

K 2 L 2 Lemma 3 .

 223 y) ̟n D(y) D(y) -D n (y) ̟n , and where D(y) and D n (y) ̟n are defined in (29) and (31) respectively. It follows thatλ n -λ 2 L 2 (D) = D λ n (y) -λ(y) 2 dy IV + V + V I,whereIV = D K hn ⋆ ν λ (f (y)) -ν λ (f (y)) hn ⋆ ν n (f (y)) -K hn ⋆ ν λ (f (y)) 2 D n (y) -2 ̟n dy V I = D D n (y) ̟ -D(y) 2 K hn ⋆ ν λ (f (y)) 2 D n (y) ̟n D(y) -2 dy.The term IV. We get rid of the term 1 D(y) 2 using Lemma 2. By Assumption 3 and classical kernel approximation, we have for every 0 < s ≤ n 0IV K hn ⋆ ν λ -ν λ (f (D)) |ν λ | 2 H s (f (D)) h 2s n .We work under Assumption 1. Let D ⊂ (0, ∞) be a compact interval, λ ∈ F (c) and (λ, f, φ) ∈ F (c, m, M) for some c satisfying Assumption 2, g x ∈ H s (D) and f -1 ∈ H s (D). Then we haveν λ H s (D) ≤ ψ D, λ H s (D) , g x H s (D) , f -1 H s (D)

- 1 H

 1 s (f (D)) and λ H s (f (D)) only. It follows that IV h 2s n (42) uniformly in λ ∈ H s (D, M 1 ).

infλ:

  (λ,f,φ)∈F (c,m,M ) and λ∈H s (f (D),M 1 ) inf y∈f (D) D n (y) ̟ D(y) ̟ n . Next, sup y∈f (D) |K hn ⋆ ν λ (y)| = sup y∈f (D) [0,∞)

Acknowledgements

The research of N. Krell is partly supported by the Agence Nationale de la Recherche PIECE 12-JS01-0006-01.