Hui Li

Dr Jérôme Ollivier

Prof. Dr Sophie M Guillaume
email: sophie.guillaume@univ-rennes1.fr

Jean-François Carpentier
email: jean-francois.carpentier@univ-rennes1.fr

J.-F Carpentier

Tacticity Control of Cyclic Poly(3-Thiobutyrate) Prepared by Ring-Opening Polymerization of Racemic β-Thiobutyrolactone

We report here on the ring-opening polymerization (ROP) of racemic β-thiobutyrolactone (rac-TBL), as the first chemical synthesis of poly(3-thiobutyrolactone) (P3TB), the thioester analogue of the ubiquitous poly(3-hydroxybutyrate) (P3HB). The ROP reactions proceed very fast (TOF > 12,000 h -1 at r.t.) in the presence of various metal-based catalysts. Remarkably, catalyst systems based on nonchiral yttrium complexes stabilized by tetradentate amino alkoxy-or diamino-bis(phenolate) ligands {ONXO R1,R2 } 2-(X = O, N) provide access to cyclic P3TB with either high isoselectivity (Pm up to 0.90) or high syndiotactic bias (Pr up to 0.70). The stereoselectivity can be controlled by manipulation of the substituents on the ligand platform and adequate choice of the reaction solvent and temperature as well. The cyclic polymer topology is evidenced by MALDI-ToF MS, NMR and TGA. Highly isotactic cyclic P3TB is a semi-crystalline material as revealed by DSC.

Introduction

Poly(hydroxyalkanoate)s (PHAs) are a valuable class of polyesters, representing a possible alternative to more and more controversial petrochemical-based plastics. [1] They feature different side-chain substituents along the backbone, of which both the chemical nature along with the overall architecture, especially the stereoregularity of the polymer chain, dictate / modulate the physical, thermo-mechanical, and biological properties of the resulting polymer materials. PHAs find a variety of packaging, agricultural, biomedical, and pharmaceutical applications, in particular thanks to their biocompatibility and degradability. Among the large variety of PHAs, poly(3hydroxybutyrate) (P3HB) stands out. When prepared by fermentation of naturally occurring biomass-derived sugars (e.g., sucrose, glucose) with microorganisms, it is a purely (R)-isotactic, highly crystalline thermoplastic polyester that resembles to some extent to isotactic polypropylene. [1a,e] Alternatively, P3HB can be prepared chemically by the ring-opening polymerization (ROP) of racemic β-butyrolactone or diolide, where the relief of ring-strain is the driving force for polymerization (Scheme 1). Key advantages of ROP include fine control of molar mass, dispersity, end-groups and, when relevant, of stereoselectivity. [START_REF] Dechy-Cabaret | ROP of cyclic esters mediated by metal catalysts[END_REF] Yet, rather few catalysts/initiators enable the ROP of these β-lactones and diolides, and even less address stereoselectivity. [2e] Scheme 1. Chemical preparation of simple and functional poly(3hydroxyalkanoate)s (PHAs) as syndiotactic or isotactic (as 1:1 mixture of enantiomeric chains) polyesters by ROP of racemic β-lactones or diolides with yttrium-based catalysts. [4,9] Our group reported that yttrium catalysts stabilized by nonchiral tetradentate amino alkoxy-or diamino-bis(phenolate) ligands {ONXO R1,R2 } 2-(X = O, N) [3] provide access to highly syndiotactic P3HB with a probability for racemic enchainment (Pr) up to 0.96, provided that bulky aryl-containing R 1 substituents are installed at the ortho positions of the phenolate platform. [4] Such stereoselective yttrium-based ROP catalysis has been extended to racemic functional or higher β-lactones, [5,6] in particular to βmalolactonates (4-alkoxycarbonyl-β-propiolactones, MLA R ; R = allyl, benzyl), [7] which are of high interest because of the readily tunable/transformable pendant carboxylate moieties. Surprisingly enough, for this latter class of β-lactones, only yttrium catalysts with an uncrowded ortho-substituted {ONXO R1,R2 } 2-ligand, e.g. R 1 = Me or Cl, allowed for highly syndiotactic polymerizations (Pr > 0.95). Catalysts with sterically crowded R 1 alkyl/aryl orthosubstituents on the {ONXO R1,R2 } 2-ancillary afforded only modest syndioselectivity. Such stereoselective ROP by yttrium complexes was extended to rac-4-alkoxymethylene-βpropiolactones (rac-BPL OR s). [8] Interestingly, the simple modification of the ortho R 1 substituents on the {ONXO R1,R2 } 2- platform of the yttrium catalyst induced a complete switch from syndioselective to isoselective polymerization of this specific class of rac-BPL OR s: an yttrium complex bearing an o,pdihalogeno (F, Cl, Br)-substituted ligand generated highly isotactic polyesters (probability for meso enchainment, Pm = 1 -Pr up to 0.95), while complexes with bulky tert-butyl and cumyl-substituted R 1 ligands gave syndio-enriched PBPL OR s (Pr = 0.90). DFT computations performed for the first elementary steps of the ROP of rac-BPL OR suggested that the origin of the isoselectivity could be traced back to attractive C-H … X non-covalent interactions between hydrogen from the alkoxymethylene group in the last ring-opened monomer unit in the growing chain and the ortho-halogeno substituents of the ligand platform. [5,8] Chen's group recently developed the highly effective synthesis of P3HB by the stereoselective ROP of the bio-sourced racemic eight-membered diolide (rac-8DL Me ), the cyclic dimer of 3-hydroxybutyric acid. By using racemic yttrium catalysts supported by C2-symmetric cyclohexyldiamino-based salen-type ligands, perfectly isotactic P3HB was recovered (Scheme 1). [9] The methodology was successfully extended to the preparation of higher isotactic homoPHAs as well as crystalline isotactic PHA copolymers made of rac-8DL Me and rac-8DL R (R = Et, nBu), that exhibit polyolefin-like thermal and mechanical properties. [10] It is also well-known that the introduction of sulfur atoms into the polymer backbone can impart polymers with special properties, significantly differentiated from those of regular hydrocarbon-based polymer analogues. [11] Hence, polythioesters (PTEs), that is polyester counterparts with thioester linkages within the backbone, are of great interest due to their biocompatibility and degradability properties for biomedical applications, their high refractive index for optical applications, and their sorbent properties for wastewater treatment. [12] PTEs remain, however, less explored than their corresponding polyesters. Poly(3-thiobutyrate) (P3TB) ( a ) is the first biohomopolymer with thioester linkages in the backbone that was isolated from a recombinant strain of E. coli cultured with thioalkanoic acids (Scheme 2). [13] This fermentation approach generates P3TB with a high molar mass (Mn = ca. 175 kg mol -1 ) and moderately broad dispersity (ÐM = 1.9); yet, the composition, microstructure (presumed to be highly isotactic, b vide infra) and thermo-mechanical performance of biosynthetic P3TB are difficult to adjust. On the other hand, there are a limited number of reports on the ROP conversion of thiolactones, and even less of βthiolactones, to high molar mass PTEs. Notable examples of suitable monomers for chemical synthesis of PTEs via ROP include amino-functional β-thiolactones, thiocarboxyanhydrides, dithiolactones, δ-and ε-thiolactones (See Scheme S1 in the ESI). [14,15,16] Early examples traced back to 1960s report the ROP of δ-thiovalerolactone and ε-thiocaprolactone with simple inorganic bases, [14] and of (S)-α-p-toluenesulfonamido-βpropiothiolactone, initiated in the presence of water or benzyl thioalcohol, or even thermally without any initiator, to produce the corresponding optically active P3TEs. [15] In 2016, Suzuki et al. reported the ROP of a cysteine-derived β-thiolactone mediated by N-Boc-L-cysteine methyl ester. [16c] In these latter two examples, the produced P3TEs have relatively low molar mass and broad dispersity, which is at least in part attributable to the extensive transthioesterification side-reactions occurring during the polymerization. Noteworthy, thioester bonds are more reactive than their oxoester analogs and some PTEs can readily revert back to cyclic thioester monomers. [16g] To alleviate transthioesterification and tune more favorably the a This material is sometimes also referred to as poly(3-mercapto-butyrate) (P3MB) but, for the sake of consistency, the thio nomenclature is used throughout this manuscript. b The stereoregularity of P3TB prepared by fermentation is not explicitly documented in the literature; see reference [13]. Yet, it can be presumed to be thermodynamics closer to equilibrium, Lu et al. recently modified the β-thiolactone structure by introducing a geminal dimethyl group on the four-membered ring (N R -PenTL); interestingly, its ROP organocatalyzed by DBU returned poly(N R -PenTL) with high molar mass but also improved the propensity of degradation back to the original monomer. [16g] Hence, almost all of the examples of the ROP of thiolactones reported so far used organocatalysts (Scheme S1), c and returned atactic PTEs, unless starting from enantiomerically pure monomers.

Scheme 2. a) Fermentation approach towards P3TB; [13] b) ROP of racemic 3thiobutyrolactone (rac-TBL) mediated by discrete catalysts (Scheme 3) towards stereoregular cyclic poly(3-thiobutyrolactone) (P3TB) reported in this study.

We herein report on the chemical synthesis of P3TB via ROP of rac-thiobutyrolactone (rac-TBL) promoted by metal-based catalysts (Scheme 2). The reactions are shown to proceed with very high rates under mild conditions and return PTEs with unprecedented microstructure and topology. The stereoselective ROP of rac-TBL is best achieved by discrete yttrium complexes, where the substituents installed on the phenolate rings induce the switch from iso-enriched to syndio-biased P3TBs. To our knowledge, this is the first highly stereoselective ROP of a racemic chiral β-thiolactone reported, and a rare case where fine tuning of a single family of discrete ROP catalysts allows accessing both isotactic and syndiotactic polymers from a single monomer. [5,8,17] The resulting P3TBs were characterized by MALDI-ToF MS analysis, revealing, also quite unexpectedly, the formation of purely cyclic polymers.

Results and Discussion

The results of the preliminary screening of the ROP of rac-TBL (see ESI for the preparation of the monomer) with a variety of wellestablished metal-based catalysts / initiators (Scheme 3) [START_REF] Dechy-Cabaret | ROP of cyclic esters mediated by metal catalysts[END_REF] are summarized in Table 1. As anticipated, a simple highly nucleophilic initiator such as NaOMe enables the ROP of rac-TBL at room temperature, but returns ill-defined P3TB (vide infra). Surprisingly, the zinc β-diketiminate complex (BDI)ZnO i Pr (2a), which is quite active for the polymerization of β-lactones and lactide, and the alternating copolymerization of carbon dioxide and epoxides, [18]

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Angewandte Chemie International Edition temperature. Similarly, no activity was observed under the same conditions with Zn[N(SiMe3)2]2 (2b) (alone or with 2 equiv. of BnOH as co-initiator), an established initiator for the isoselective ROP of cyclic O-carboxyanhydrides. [19] The chiral aluminum complex (R)-(SalBinam)AlO i Pr (1), another effective catalyst for the stereoselective ROP of rac(meso)-lactide via enantiomorphicsite control, [20] allowed full conversion of 25 equiv. of rac-TBL at room temperature. Yet, the recovered P3TB featured a Mn value as determined by SEC much higher than the theoretical one, suggesting a poor initiating efficiency and/or undesirable sidereactions. The lanthanum trisamide 3, alone or in combination with an external co-initiator, proved also a quite active system at room temperature. However, inspection of the microstructure of the polymers formed from NaOMe, 1 or 3 by 13 C NMR (vide infra), revealed the formation of atactic P3TBs (see the ESI, Figures S33, S54-S55).

As yttrium catalysts have shown excellent performances in a variety of (stereoselective) ROPs of cyclic esters, [3][4][5][6][7][8][9][10] we next explored two different families of such achiral complexes. The amido yttrium-salan complex 4, which produces -thanks to chainend stereocontrol -highly alternating poly(3-hydroxyalkanoate) copolymers via ROP of equimolar mixtures of opposite enantiomers of different chiral β-lactones, [21] features very high activity toward rac-TBL in toluene at ambient temperature: with or without a co-initiator (such as i PrOH), full conversion of 25-100 equiv. of the monomer proceeded within less than 1 min. d However, the resulting polymers were again atactic, as assessed by 13 C NMR spectroscopy (see the ESI, Figures S33,S57). Scheme 3. Chemical structure of the catalysts/initiators used for the ROP of rac-TBL in this study. g Pm is the probability of meso enchainment between TBL units as determined by 13 C{ 1 H} NMR analysis of the isolated P3TBs.

d At this stage of our studies, because of the very fast reactions at room temperature with La and Y systems, detailed kinetics were not established.

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Angewandte Chemie International Edition More promising results were obtained with the discrete yttrium amido complex 5a supported by a diamino-bis(phenolate) ligand bearing sterically bulky tBu substituents, that is a prototypical example among the family developed by our group since the early 2000s. [3] The ROP of rac-TBL in toluene promoted by 5a, with or without a protic co-initiator, proceeded also very rapidly at ambient temperature. Although no detailed kinetics could be determined under these conditions, d the turnover frequencies (TOF) exceed 3,000 h -1 ; these values are at least comparable, possibly higher than those obtained in the ROP of the parent β-lactone rac-BL under the same conditions. [3,4] 13 C NMR spectroscopy revealed that all P3TBs formed under these conditions are syndiotactic-enriched (Pm = 0.31-0.37), providing further opportunities in terms of stereocontrol improvement upon ligand tuning and optimization of the reaction conditions (vide infra). On the other hand, the experimental molar mass values determined by SEC, Mn,SEC, did not vary proportionally with the values calculated from the monomer-to-initiator ratio; also the dispersity values were somewhat high (ĐM, Mw/Mn = 1.42-1.66), and definitively higher than those typically observed (ĐM = 1.05-1.3) for P3HB obtained from ROP of rac-BL under similar conditions [4] (vide infra).

The 1 H and 13 C{ 1 H} NMR spectra of the P3TBs isolated from catalysts 1-5a (Table 1) clearly displayed the characteristic signals corresponding to TBL repeat units (see the ESI, Figures S9-S10 and S17-S18). However, whether 5a was used alone e or in the presence of a co-initiator such as i PrOH, i PrSH, or BnSH, no chain-end groups were detected in the NMR spectra of the polymers. Similar observations were made in the NMR spectra of P3TBs prepared with the aluminum complex 1, the trisamido lanthanum precursor 3/BnOH (1:3) system and the yttrium-salan complex 4. Only for the P3TB prepared from NaOMe (entry 1) was the methoxycarbonyl end-group unambiguously observed by 1 H and 13 C NMR spectroscopy (see the ESI, Figures S17-S18). For this latter polymer, the molar mass estimated by NMR (Mn,NMR = ca. 3900 g mol -1 ) was higher than the theoretical one (Mn,theo = ca. 2400 g mol -1 ), indicating a poor initiation efficiency and/or unselective formation of linear and cyclic macromolecules under these conditions.

To get a better insight, the topology of the isolated polymers was further investigated by matrix-assisted laser desorption / ionization time-of-flight mass spectrometry (MALDI-ToF MS). In line with the abovementioned NMR data, analysis of P3TB samples prepared from NaOMe evidenced the formation of mixtures of linear MeO-[C4H6OS]n-H and cyclic [C4H6OS]n macromolecules; these experiments were hardly reproducible as little changes in the reactions conditions returned variable amounts of linear and cyclic structures, as assessed by MS (see the ESI; Figures S19-S21). Conversely, P3TBs resulting from the ROP of rac-TBL promoted by discrete complexes 1, 4 or 5a revealed a single population of cyclic macromolecules, f i.e. without chain-end groups. For the latter complex, regardless of the use of a co-initiator ( i PrOH, i PrSH, or BnSH, 1 equiv. vs. Y) or not, and no matter the ROP solvent (toluene or THF), the observed m/z values match the data calculated for [C4H6OS]n macromolecules (Figure 1 and Figures S11-S16). Another indirect support of the formation of cyclic (vs. linear) polymers arises from thermal studies (vide infra). Cyclic polymers are a unique class of macromolecules with specific properties, e.g., higher density, higher glass transition temperature and melting point, lower intrinsic viscosity, increased crystallization rate, and higher refractive index than their linear counterparts. [22] Cyclic P3HB has been prepared by ROP of BL with lipase, [23] organic spirocyclic initiators, [24] or amidooxazolinate zinc complexes; [25] note that in the latter case, in the presence of a mono-or bifunctional co-initiator (e.g., EtOH, 1,4cyclohexanediol), the same zinc catalysts promote the formation of linear P3HBs. Other studies have noted that the co-initiator in metal-catalyzed ROP may have a significant effect on the topology of the resulting polymer. g Hence, Chen et al. reported the ROP of γ-butyrolactone using La[N(SiMe3)2]3; in the absence of a co-initiator, controlled cyclic topology was then achieved, but upon addition of 1, 2 or 3 equiv. of BnOH or i PrOH vs. La, the resulting polymer structure shifted from a predominantly cyclic one to a mixture of cyclic and linear structures, to a mostly linear structure, respectively. [26] The authors proposed that the formation of the cyclic polyester results from back-biting, and that this pathway is favored within chains initiated by -N(SiMe3)2, a better leaving group.

We assume that, similarly to this state of the art, [24,25] backbiting process occurs during the ROP of rac-TBL promoted by the present metal-catalysts. However, in our ROPs, the co-initiator / initiating group does not play such a decisive role in determining the topology of the resultant polymers. Indeed, catalyst systems herein generated in situupon addition of 1 equiv. of i PrOH, respectively, towards the formation of cyclic P3TB (Table 1, entries 8-15; ESI, Figures S11-S16). In fact, the intrinsically labile thioester bonds make transthioesterification a facile process (vide supra), [14e,h,27] much easier than transesterification, [START_REF] Dechy-Cabaret | ROP of cyclic esters mediated by metal catalysts[END_REF] and make back-biting possible at any electrophilic site within the P3TB chain.

A simple representation of the typical coordination-insertion putative mechanism at play in the ROP of cyclic (thio)esters for initiation and propagation steps, as well as dynamic processes that lead to the formation of cyclic P3TB and ring-expansion, are depicted in Scheme 4. Obviously, to account for the formation of high molar mass P3TB, this model implies a propagation rate significantly faster than the cyclization rate. [22b] Note that an increase in the monomer-to-catalyst ratio leads to the formation of cyclic P3TBs with a corresponding increase of their molecular weights (as illustrated with 150 up to 400 equiv. vs. 5c/ i PrOH; see entries 21-23, Table 2, below). Another indication of easy transthioesterification is provided by the narrowing of dispersity values of P3TBs upon running the ROP of rac-TBL at low temperatures (typically ĐM = Mw/Mn = 1.3-1.7 at r.t. vs. 1.15-1.35 at T ≤ -70 °C; see below, Table 2). Similar observations have been made in the polymerization of lactones with highly reactive metal-alkoxide catalyst systems. [2,18b] The tacticity in the various isolated P3TBs can be appreciated by 13 C NMR spectroscopy, especially from the carbonyl and methyl regions; at 100-125 MHz, the methine and methylene regions do not display resolved resonances (Figure 2). The carbonyl region of regular P3TBs 13 C{ 1 H} NMR spectra (δ ca. 195.8 ppm) essentially features two broad, yet well-resolved in the presence of BnOH favored the formation of a linear polymer; see: reference [19].

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Angewandte Chemie International Edition resonances, h, [28] the most downfield of which was assigned to a meso (m, isotactic) dyad and the most upfield one to the racemo (r, syndiotactic) dyad. These assignments were based on those made for the carbonyl region of P3HB. [4a,29] The methyl region of regular 13 C{ 1 H} NMR spectra (δ ca. 20.3-20.5 ppm) is better resolved and shows four resonances corresponding to triads; they actually form two (essentially non-overlapping) groups of two (partly overlapping) resonances, which are assigned, from highto-low field, to the four triads mm, rm, mr and rr, respectively. A similar triads distribution was also observed in the methine region (δ ca. 68.4-68.8 ppm) of the parent PMLA R s. [7a] Careful deconvolution of these four resonances allows extracting Pm (and Pr = 1 -Pm) values in good agreement with those determined from the carbonyl resonances, hence validating their assignment (see the ESI). Definitive confirmation of these assignments was gained from the 13 C NMR spectrum of a pure isotactic P3TB (Figure 2, top), independently prepared from the ROP of enantiopure (S)-TBL (see the ESI). 2, entry 48); c) complex 5g (50:1:0, -80 °C, THF, Pm = 0.89, Table 2, entry 34); d) system 5d/ i PrOH (100:1:1, 18 °C, toluene, Pm = 0.52, Table 2, entry 26); e) system 5a/ i PrOH (50:1:1, 18 °C, toluene, Pm = 0.34, Table 1, entry 12); f) system 5c/ i PrOH (100:1:1, -50 °C, toluene, Pm = 0.30, Table 2, entry 24).

Following the preliminary study of the ROP of rac-TBL promoted by catalyst 5a (Table 1), the impact of the catalyst structure on the tacticity of P3TBs was further investigated with a set of ligands a-l well established for the stereoselective ROP of lactide and functional β-lactones (Scheme 3). Representative results are summarized in Table 2 and complementary data are provided in Tables S1 andS2. The different yttrium catalyst systems based on 5a-l are quite active for the ROP of rac-TBL (TOF in the range 2,400~12,000 h -1 at r.t.), d thus allowing near quantitative conversion of 25-400 equiv. of rac-TBL within 6 min at ambient temperature. As mentioned above, the use of a coinitiator such as i PrOH, i PrSH or BnSH basically does not affect h In some of the regular 13 C{ 1 H} NMR spectra, four resonances can be identified in the carbonyl region; yet, they cannot be deconvoluted confidently. On the other hand, 13 C projections of 2D band-selective HMBC and HSQC experiments allows a better resolution of this carbonyl region, hence featuring the corresponding four triads mm, rm, mr and rr. Deconvolution and analysis the stereoselectivity of the polymerizations; only the nature of the ligand's substituents, of course of the reaction temperature, and, in some of cases, of the solvent, drive the tacticity of the produced P3TB.

As expected from previous works on the ROP of rac-BL and related functional β-lactones, [4,5,7,8] catalyst systems based on complexes 5a and 5c with bulky R 1 substituents ( t Bu or cumyl groups) installed on the ortho-positions of the phenolate rings generated syndiotactic-biased P3TBs (Pm ≈ 0.32-0.38) from the ROP of rac-TBL performed at ambient temperature in toluene. Decreasing the temperature down to -70 °C did not improve the syndioselectivity. Surprisingly enough, installation of a very bulky R 1 trityl substituent such as in 5l reversed the trend, as the resulting P3TB produced in toluene at ambient temperature featured an isotactic bias (Pm = 0.58, entry 49); the isoselectivity of this latter system improved significantly upon lowering the temperature down to -80 °C (Pm = 0.83, entry 50). On the other hand, significant differences were observed depending on the nature of the X donor group in the ''capping'' moiety of the {ONXO R1,R2 } 2-ligand (X = CH2OMe, CH2NMe2, CH2NEt2, 2-pyridyl). This pendant group X in the side arm of the ligand may affect the catalyst's electronics and/or sterics, and thus the catalytic activity and stereoselectivity in ROP reactions. [3,6,30] In the ROP of rac-TBL, changing the side arm donor from β-NMe2 in 5a or 5c to β-OMe as in 5b or 5d, respectively, reduced drastically the catalyst syndioselectivity and returned atactic P3TBs (Pm = 0.38, 0.32 vs. 0.48, 0.52, entries 17, 20 vs. 19 and 26, respectively). The β-NEt2 system 5e performed similarly to the β-NMe2 system 5b, in contrast to the pyridyl one (5f) which also produced atactic P3TB under these same conditions (entries 27, 19, 28).

Also unexpected and unprecedented to our knowledge, yttrium complexes bearing non-bulky substituents, namely the o,p-dimethyl-substituted systems 5g-5j gave isotactic-enriched P3TBs. At ambient temperature in toluene or THF, these catalysts are highly active for the ROP of rac-TBL, however with only a mild preference for isoselectivity (Pm = 0.54-0.58). Upon lowering the polymerization temperature down to -70 or -80 °C, isotacticity raised in toluene (Pm = 0.73, entry 30), and improved even more significantly in THF (Pm up to 0.89; entry 34). Also, the X capping moiety affected the performance in this series of o,p-dimethylsubstituted catalysts and the best isotacticity was again observed with the NMe2-capped complex 5g. The o,p-dichloro-substituted complex 5k had the same behavior as its dimethyl-substituted analogue 5g (Pm up to 0.90 in THF at -80 °C; entry 48); this suggests that, in the ROP of rac-TBL, methyl and chloro substituents apparently behave similarly to a non-bulky group (and without electronic contribution for Cl [8] ); this is reminiscent of the ROP of rac-β-malolactonates, [7] which yet offered syndiotactic PMLA R s in contrast to the present isotactic P3TBs. The high activity of the o,p-dichloro-substituted system 5k, even at low temperature (TOF > 3,000 h -1 at r.t. and 88 h -1 at -50 °C), is also noteworthy, as it contrasts strikingly with the ROP of rac-BL where only alkyl/aryl-substituted complexes are highly active, while {ONOO Cl2 }Y catalysts proved poorly active (TOF = 12 h -1 at r.t.). [4c] This further testifies of the high intrinsic reactivity of TBL and its singular behavior as compared to parent β-lactones.

of these four resonances returned Pm/Pr values fully consistent with those determined from the two m and r dyads of regular 13 C{ 1 H} NMR spectra and also from the triads in the methyl region. See the ESI, Figures S37-S40 and Tables S4-S5); see reference [28].

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Angewandte Chemie International Edition f Pm is the probability of meso enchainment between TBL units as determined by 13 C{ 1 H} NMR analysis of the isolated P3TBs.

The aforementioned positive influence of THF on the isoselective ROP of rac-TBL mediated by 5g and 5k prompted us to explore in a more systematic manner the influence of the solvent. A variety of solvents (n-hexane, 1.4-dioxane, toluene, 2-MeTHF, THF, dichloromethane, acetonitrile) was investigated using both the isoselective catalyst 5k (Table S1) and the syndioselective catalyst 5c (Table S2). Yet, at ambient temperature, no significant difference and further possible improvements were noted. Only THF seemed to have a differentiated behavior, making non-stereoselective the catalyst 5c that is otherwise syndioselective in other solvents. We speculate this may be due to THF coordination (in contrast to the more sterically crowded 2-MeTHF) onto the active site of the propagating species.

Analysis of the triad stereosequence distribution in both syndio-and iso-enriched P3TBs evidenced that it agrees well with Bernoullian statistics for a chain-end stereocontrol. [31] In fact, the Bernoulli model triad test, B = [(mm) × (rr)]/[(rm) × (mr)], returned values very close to 1, usually in the range 0.8-1.35 (see the ESI; Tables S4-S6), which falls within an acceptable range due to errors in deconvoluting the four respective triad peaks. [31] This indicates that, in agreement with the achiral nature of these yttrium catalysts, the stereocontrol in this rac-TBL ROP process is a consequence of either (1) interactions between the last two TBL units in the P3TB chain (penultimate control) or (2) interactions between the last TBL unit in the chain and the incoming monomer (ultimate control).

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Angewandte Chemie International Edition Figure 3. (a, left) TGA thermograms of P3TBs with different topologies: an essentially linear P3TB produced from NaOMe (blue line, Mn,SEC = 3.9 kg mol -1 Pm = 0.50, Table 1, entry 1) and a cyclic P3TB produced from complex 5a (red line, Mn,SEC = 4.9 kg mol -1 , Pm = 0.37, Table 1,entry 10 Preliminary studies were conducted to determine the thermal behavior of P3TBs by thermal gravimetric analysis (TGA) and differential scanning calorimetry (DSC). Typical TGA curves of P3TB samples are shown in Figures 3 andS70-S73. A cyclic atactic P3TB produced by complex 5a features a degradation temperature (Td, defined as the temperature for 5% weight loss) of ca. 198 °C, that is 12 °C higher than the corresponding value for a linear (as assessed by MS; see Figure S19) atactic P3TB produced with NaOMe system with a comparable molar mass (Figure S17). Also, the degradation extent at higher temperature is significantly more pronounced on the linear than on the cyclic polymer (e.g., at 300 °C, 83.5% vs. 66% weight loss, respectively). This higher thermal stability of cyclic P3TB can be accounted for by the extra energy needed to cleave cyclic polymers as compared to linear ones, as reported recently for other polyesters. [22a,c,26] A noticeable enhancement of the degradation temperature of cyclic P3TBs from Td = 195 to 230 °C was observed when the monomer-to-initiator ratio increased from 50:1 to 150:1, thus presumably with a P3TB of higher molar mass (Figure 3). On the other hand, the stereochemistry of cyclic P3TBs with similar molar mass, with Pm varying in the range 0.35-0.90, had a little influence on the thermal stability (Figure S73). This is in line with the weak tacticity effect noted on the thermal stability of syndio-and isotactic (linear) polystyrenes, [32] but contrasts with previous studies on hetero-, syndio-and isotactic (linear) PLAs where Td (5%) values spanned over a rather broad range (258-323 °C). [33] The DSC curve of a cyclic isotactic P3TB sample (Mn = 25.0 kg mol -1 , Pm = 0.90) features a glass-transition temperature (Tg) of 4 °C (Figure 4); in addition, a single exothermic transition with a crystallization temperature (Tc) of 95 °C and a single endothermic transition with a melting-transition temperature (Tm) of 101 °C are observed. The latter Tm value is very close to that determined for pure isotactic P3TB (Tm = 102 °C) prepared by ROP of enantiopure (S)-TBL and also to that reported for P3TB prepared with E coli (Tm = 100 °C), [13] indirectly corroborating that the latter material is indeed also highly (purely) isotactic. The semi-crystalline isotactic P3TB produced by ROP of rac-TBL has a notably lower Tm than isotactic P3HB (Tm ca. 170 °C) [9a] , highlighting that the introduction of the sulfur atom into the backbone of P3HB greatly impacts chain folding and packing into the crystal lattice. [26,34]

Conclusion

In summary, we have reported herein an efficient chemical synthesis of P3TB via ring-opening polymerization; the reactions proceed fast and with complete conversion of the TBL monomer 10.1002/anie.202202386

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Angewandte Chemie International Edition into the corresponding polythioester. With the most effective yttrium-based catalysts, cyclic polymers are formed, most likely as a result of easy transthioesterification. Interestingly, using a single family of achiral catalysts, upon simple tuning of the substituents on the bis(phenolato) platform, the stereoselectivity of the ROP of rac-TBL can be switched from highly isoselective to syndioselective. To our knowledge, this is quite a rare feature in ROP catalysis and observed previously only for rac-lactide with aluminum complexes supported by tetradentate aminophenoxide ligands [17] and for rac-BPL CH2OR (R = Me, allyl, benzyl) monomers with the same yttrium complexes as those used in this study. [5,8] However, the mechanisms at work for the stereocontrol are obviously not the same within the two classes of monomers: for TBL, a simple "uncrowded" dimethyl-substituted catalyst affords highly isotactic P3TB, while for BPL CH2OR only a catalyst with a dichloro-substituted ligand platform enabled reaching isotactic polymers, thanks to attractive Cl … H non-covalent interactions in the alkoxymethylene group in the last ring-opened monomer unit of the growing polymer chain. [5,8] Also noteworthy and quite distinctive from previous works is the possibility to access isotactic P3TB with the bulky trityl-ortho-substituted catalyst, which evidences how subtle the stereocontrol is. At this stage of our studies, we cannot elaborate more confidently on the origin of this stereocontrol; neither can we state on the exact nature of isotactic P3TB: it may be a racemic mixture of two isotactic homopolymers, a stereoblock copolymer, or a tapered blocked copolymer. Kinetic studies as well as preparation of model compounds are underway to decipher mechanistic considerations. We anticipate that this stereoselective ROP of the simplest, prototypical four-membered racemic-β-thiolactone may open up opportunities for the development of new valuable polymer materials.

Entry for the Table of Contents

Isotactic or syndiotactic cyclic poly(3-thiobutyrolactone) (P3TB) was obtained from ring-opening polymerization of racemic thiobutyrolactone (rac-TBL) promoted by achiral yttrium catalysts. The tacticity of cyclic polythioesters can be readily switched upon tuning the bulkiness of the substituents on the catalyst ligand.

Institute and/or researcher Twitter usernames: @HuiLI39831976 @chimie_ISCR @ENSCR @UnivRennes1 @gfpolymeres @INC_CNRS 10.1002/anie.202202386

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a

  Polymerization performed at ambient temperature (15-20 °C), [TBL]0 = 0.5 M. b (co)Initiator. c Reaction times were not optimized. d Conversion of rac-TBL as determined by 1 H NMR analysis of the crude polymerization mixture. e Molar mass calculated according to Mn,theo = [TBL]0/[Cat.]0 × Conv.TBL × M TBL. f Numberaverage molar mass (Mn,SEC) and dispersity (ĐM) determined by SEC analysis in THF at 30 °C vs. polystyrene standards (uncorrected values).

Figure 1 .

 1 Figure 1. MALDI-ToF mass spectrum of a cyclic P3TB produced by Y{ONNO tBu2 }{N(SiHMe2)2}(THF) (5a) in toluene in the absence of co-initiator (Table 1, entry 10), with a plot of experimental m/z values vs. theoretical number of TBL repeat units.

i

  PrSH or BnSH onto 4 or 5a (see the Experimental section)revealed as the real yttrium-alkoxy or -thioalkoxy initiating species,g The ROP of O-carboxyanhydride with La[N(SiMe3)2]3 as catalyst in the absence of co-initiator gave a cyclic polymer, while the use of Zn[N(SiMe3)2]2

Figure 2 .

 2 Figure 2. Carbonyl, methine, methylene and methyl regions of the 13 C{ 1 H} NMR spectra (125 MHz, CDCl3, 25 °C) of : a) a highly isotactic cyclic P3TB (Pm > 0.95, Mn = 3.2 kg mol -1 , ĐM = 1.52) prepared from the ROP of enantiopure (S)-TBL (with the 5a/ i PrOH (1:1) system); and isotactic-, syndiotactic-enriched cyclic P3TBs prepared from the ROP of rac-TBL mediated by: b) complex 5k ([TBL]0/[5]/[init.] = 50:1:0,-80 °C, THF, Pm = 0.90, Table2, entry 48); c) complex 5g (50:1:0, -80 °C, THF, Pm = 0.89, Table2, entry 34); d) system 5d/ i PrOH (100:1:1, 18 °C, toluene, Pm = 0.52, Table2, entry 26); e) system 5a/ i PrOH (50:1:1, 18 °C, toluene, Pm = 0.34, Table1, entry 12); f) system 5c/ i PrOH (100:1:1, -50 °C, toluene, Pm = 0.30, Table2, entry 24).

  Figure 3.(a, left) TGA thermograms of P3TBs with different topologies: an essentially linear P3TB produced from NaOMe (blue line, Mn,SEC = 3.9 kg mol -1 Pm = 0.50, Table1, entry 1) and a cyclic P3TB produced from complex 5a (red line, Mn,SEC = 4.9 kg mol -1 , Pm = 0.37, Table1, entry 10). (b, right) TGA thermograms of cyclic P3TBs produced from complex 5a or 5c in toluene at room temperature as a function of the monomer-to-initiator ratio: Td = 195 °C ([TBL]0/[5a] = 50:1, Mn,SEC = 3.8 kg mol -1 , Pm = 0.34; Table1, entry 12), 205 °C ([TBL]0/[5a] = 80:1, Mn,SEC = 7.8 kg mol -1 , Đ = 1.65, Pm = 0.34), 230 °C ([TBL]0/[5c]/[ i PrOH] = 150:1:1, Mn,SEC = 25.0 kg mol -1 , Pm = 0.35; Table 2, entry 21). Heating rate = 10 °C min -1 in both cases.

  P3TBs with lower isotacticity (Pm < 0.73), as well as atactic and syndiotactic (Pr up to 0.70) ones, do not feature semi-crystallinity, as revealed by DSC. On the other hand, the stereoregularity of cyclic P3TB has a negligible influence on their Tg values (see the ESI, Figure S74-S77).

Figure 4 .

 4 Figure 4. DSC curve of a highly isotactic cyclic P3TB produced from complex 5k in toluene at room temperature (Pm = 0.90, Table 2, entry 48) (2 nd heating cycle; heating rate = 10 °C min -1 , cooling rate = 10 °C min -1 ).

  turned out inactive towards rac-TBL at room

	highly isotactic, as compared to other PHAs prepared by microorganisms,
	such as P3HB; see reference [1].
	c Chen et al. mentioned briefly the ROP of the bridged bicyclic thiolactone
	monomer BTL (see Scheme S1) mediated by the La[N(SiMe3)2]3/BnOH
	system, which provided a modest conversion and no stereocontrol; see
	reference [14d].

Table 1 .

 1 Selected results for the preliminary screening of the ROP of rac-TBL. a

	Entry	Cat.	Init. b	[TBL]0/[Cat.]	Solvent	Reaction	TBL Conv. d	Mn,theo e	Mn,SEC f	ĐM f	Pm g
				/[Init.]		time c (min)	(%)	(kg mol -1 )	(kg mol -1 )
	1	-	NaOMe	23:0:1	C6D6	60	>99	2.4	3.9	1.61	0.51
	2	1	-	25:1:0	Toluene	72	>99	2.6	14.5	1.46	0.48
	3	2a	i PrOH	50:1:1	Toluene	120	0	-	-	-	-
	4	2b	-	50:1:0	Toluene	180	0	-	-	-	-
	5	2b	BnOH	50:1:2	Toluene	180	0	-	-	-	-
	6	3	-	25:1:0	C6D6	1	>99	2.6	5.2	1.53	0.50
	7	3	BnSH	100:1:3	Toluene	6	>99	3.4	1.7	1.17	0.48
	8	4	-	25:1:0	Toluene	1	>99	2.6	6.0	2.15	0.54
	9	4	i PrOH	100:1:1	Toluene	1	>99	10.3	46.5	1.45	0.52
	10	5a	-	25:1:0	Toluene	5	>99	2.6	4.9	1.47	0.37
	11	5a	-	50:1:0	Toluene	1	>99	5.1	3.7	1.48	0.32
	12	5a	i PrOH	50:1:1	Toluene	1	>99	5.1	3.8	1.42	0.34
	13	5a	i PrSH	50:1:1	Toluene	1	>99	5.1	3.9	1.66	0.32
	14	5a	BnSH	50:1:1	Toluene	1	>99	5.1	3.5	1.65	0.32
	15	5a	BnSH	100:1:1	Toluene	5	>99	10.3	6.1	1.59	0.31

Table 2 .

 2 Stereoselective ROP of rac-TBL mediated by yttrium complexes 5a-k. Init. = co-initiator = i PrOH. b The reaction times were not necessarily optimized. c Conversion of rac-TBL as determined by 1 H NMR analysis of the crude polymerization mixture (refer to the ESI). d Molar mass calculated according to Mn,theo = [TBL]0/[Cat.]0 × Conv.TBL × MTBL. e Number-average molar mass (Mn,SEC) and dispersity (ĐM) as determined by SEC analysis in THF at 30 °C vs. polystyrene standards (uncorrected values).

	Entry	Cat.	[TBL]0/[Cat.]	Reaction	Solvent	[TBL]0	Reaction	TBL	Mn,theo	Mn,SEC	ĐM e	Pm f
			/[Init] a	temp. (°C)		(mol/L)	time (min) b	Conv.(%) c	(kg/mol) d	(kg/mol) d
	16	5a	50:1:1	18	Toluene	0.5	1	>99	5.1	3.8	1.42	0.34
	17	5a	100:1:1	18	Toluene	0.65	6	>99	10.2	18.7	1.43	0.38
	18	5a	100:1:1	-70	Toluene	0.5	16	>99	10.2	46.5	1.14	0.42
	19	5b	100:1:1	18	Toluene	1.0	1	>99	10.2	7.6	1.39	0.48
	20	5c	100:1:0	18	Toluene	1.0	1	>99	10.2	42.7	1.68	0.32
	21	5c	150:1:1	18	Toluene	1.0	2	>99	15.4	25.0	1.58	0.35
	22	5c	200:1:1	18	Toluene	1.0	1	>99	20.4	30.2	1.59	0.35
	23	5c	400:1:1	18	Toluene	1.0	2	>99	40.8	59.2	1.68	0.34
	24	5c	100:1:1	-50	Toluene	0.5	15	98	10.0	31.9	1.50	0.30
	25	5c	100:1:1	-70	Toluene	0.5	30	50	5.1	21.9	1.37	0.32
	26	5d	100:1:1	18	Toluene	0.65	5	>99	10.2	7.2	1.42	0.52
	27	5e	50:1:1	18	Toluene	0.5	1	>99	5.1	21.4	1.31	0.41
	28	5f	50:1:1	18	Toluene	0.5	2	>99	5.1	13.3	1.67.	0.52
	29	5g	75:1:1	18	Toluene	0.5	2	>99	7.6	6.8	1.42	0.58
	30	5g	100:1:1	-70	Toluene	0.5	2	69	7.2	24.3	1.22	0.73
	31	5g	50:1:0	18	THF	0.5	1	>99	5.1	4.9	1.80	0.53
	32	5g	50:1:0	-50	THF	0.5	40	>99	5.1	43.4	1.62	0.63
	33	5g	100:1:0	-70	THF	0.5	11	50	5.1	23.0	1.28	0.87
	34	5g	50:1:0	-80	THF	0.5	45	83	4.2	21.4	1.22	0.89
	35	5h	100:1:1	18	Toluene	1.0	5	>99	10.2	2.5	1.24	0.54
	36	5h	100:1:1	18	Toluene	1.0	5	>99	10.2	32.5	1.59	0.57
	37	5h	50:1:1	18	THF	0.5	1	>99	5.1	3.2	1.28	0.55
	38	5h	50:1:1	-20	THF	0.5	60	>99	5.1	4.2	1.48	0.65
	39	5h	50:1:1	-50	THF	0.5	450	26	1.3	n.d.	n.d.	0.73
	40	5i	100:1:1	18	Toluene	1.0	1	41	4.1	1.9	1.21	0.54
	41	5i	50:1:0	18	THF	0.5	1	>99	5.1	4.8	1.61	0.58
	42	5i	50:1:0	-80	THF	0.5	95	>99	5.1	5.9	1.79	0.69
	43	5j	50:1:0	18	THF	0.5	1	>99	5.1	4.5	1.42	0.58
	44	5j	50:1:0	-50	THF	0.5	60	>99	5.1	5.9	1.65	0.75
	45	5k	50:1:0	18	Toluene	0.5	1	>99	5.1	4.2	1.48	0.55
	46	5k	50:1:0	18	THF	0.5	1	>99	5.1	4.8	1.53	0.57
	47	5k	50:1:0	-50	THF	0.5	34	>99	5.1	5.2	1.37	0.72
	48	5k	50:1:0	-80	THF	0.5	45	76	3.9	25.0	1.28	0.90
	49	5l	50:1:1	18	Toluene	1.0	1	88	9.0	18.2	1.83	0.58
	50	5l	50:1:1	-80	Toluene	0.5	30	90	4.6	5.1	1.21	0.83
	51	5l	50:1:0	-80	THF	0.5	40	29	1.5	6.3	1.33	0.63

a

e Note that the -C(=O)N(SiHMe2)2 terminal group is usually not observed in polymers (PLA, P3HB…) prepared from ROP with a M-N(SiHMe2)2 initiator, as this functional group is sensitive to hydrolytic workup and readily converts into -C(=O)NH2 (See: H. Ma, J. Okuda, Macromolecules 2005, 38, 2665-2673).f No MALDI-ToF mass spectra could be obtained for P3TB prepared with the lanthanum trisamide 3; the corresponding ESI mass spectra were complicated, showing several populations of macromolecules, but no cyclic population was observed.10.1002/anie.202202386Accepted ManuscriptAngewandte Chemie International Edition

Acknowledgements

We are grateful to the financial support provided by the China Scholarship Council (CSC Ph.D. grant No. 201804910783 to H.L.). We also thank Philippe Jehan and Elsa Caytan for extensive MS and NMR analyses, respectively. Part of this work has been performed using NMR spectrometers belonging to the CRMPO (UAR ScanMAT, CNRS-Université de Rennes 1) and PRISM (UAR BIOSIT, CNRS-INSERM-Université de Rennes 1) core facilities.

Conflict of interest

The authors declare no conflict of interest.

Keywords: β-thiolactone • ring-opening polymerization • yttrium • polythioester • stereoselective catalysis • cyclic polymer


Remy Garnier

Raphaël Langhendries

Joseph Rynkiewicz

HOLD-OUT ESTIMATES OF PREDICTION MODELS FOR MARKOV PROCESSES

Keywords: Statistical Learning Theory, Concentration Inequalities, Model Selection, Markov processes (1) CDiscount

We consider the selection of prediction models for Markovian time series. For this purpose, we study the theoretical properties of the hold-out method. In the econometrics literature, the hold-out method is called out-of-sample and is the main method to select a suitable time series model. This method consists of estimating models on a learning set and picking up the model with minimal empirical error on a validation set of future observations. Hold-out estimates are well studied in the independent case, but, as far as we know, this is not the case when the validation set is not independent of the learning set. In this paper, assuming uniform ergodicity of the Markov chain, we state generalization bounds and oracle inequalities for such method; in particular, we show that the out-of-sample selection method is adaptative to noise condition.

Introduction

Many models in time series involve the one-step prediction of the next value knowing past values, and hold-out or out-of-sample (OOS) method is probably the most commonly used model selection method in practice. This method consists in splitting the sample of size n + m in two parts: a training set of length n and a validation set of size m. The training set is used to derive a nite collection of candidate prediction functions and we chose the function with the best performance on the validation set. It turns out to look like picking a prediction function from a nite collection; however, we must be careful with the dependence between the learning set and the validation set for the Markov case. For time series, splitting the data into a training subset and a validation subset of future observations is an option implemented in most statistical or machine learning software. This standard evaluation procedure works very well in practice (see, for example, Tashman [START_REF] Tashman | Out-of-sample tests of forecasting accuracy: an analysis and review[END_REF] for OOS to assess the model's accuracy). In the machine learning community, Cerqueira et al. [START_REF] Cerqueira | Evaluating time series forecasting models: an empirical study on performance estimation methods[END_REF] compare the performances between OOS and other methods empirically; they found that OOS produces the most accurate estimates for real time series. The authors think that the main reason for the performance of OOS method is the preservation of the temporal order of the observations. For the independent and identical distributed (i.i.d.) case, the hold-out theoretical properties are well known, and, for example, in the classication case, it adapts to the noise conditions (see Blanchard and Massart [START_REF] Blanchard | Discussion: Local rademacher complexities and oracle inequalities in risk minimization[END_REF]). However, as far as we know, there are few theoretical results for the OOS method for dependent data. Some studies assess the asymptotical performance of related methods like cross-validation (see Arlot and Celisse [START_REF] Arlot | A survey of cross-validation procedures for model selection[END_REF]) in the context of regression (Burman and Nolan [START_REF] Burman | Data dependent estimation of prediction functions[END_REF]). But, for model selection, in the case of dependent observations, cross-validation is known to be severely aected by dependence (see Chu and Marron [START_REF] Chu | Comparison of two bandwidth selectors with dependent errors[END_REF]). Other authors investigate methods of model selection for dependent data, which focus more on penalization (see Alquier and Wintenberger [START_REF] Alquier | Model selection for weakly dependent time series forecasting[END_REF]) or on complexity measure such that Rademacher complexity (Mohri and Kuznetsov [19]) or stability bounds (Mohri and Rostamizedeh [20]). Empirical risk minimization has also been studied in the framework of uniformly ergodic Markov chains (see Bin et al. [START_REF] Bin | Learning from uniformly ergodic markov chains[END_REF]). To be exhaustive, we can also cite some studies on the asymptotic convergence rate for estimating nite Markov chain transition matrices (see Falahatgar et al. [START_REF] Falahatgar | Learning markov distribution: Does estimation trump compression[END_REF], Hao et al. [START_REF] Hao | On learning markov chains[END_REF], and [START_REF] Han | Optimal prediction of markov chains with and without spectral gap[END_REF]). Note that such results are dicult to apply to massive models like Deep networks (Zhang et al. [START_REF] Zhang | Understanding deep learning (still) requires rethinking generalization[END_REF]). Indeed, these authors show, through extensive systematic experiments, that these traditional approaches fail to explain why large neural networks generalize well in practice. Eventually, OOS is still the standard method for model selection in time series, especially for Deep learning models. If the data are drawn from a process indexed in time order, the validation set is no longer independent from the learning set, and the classical i.i.d. theory of hold-out does not hold anymore. This paper aims to provide generalization bounds and oracle inequalities for the selected prediction model by the OOS method in a Markovian framework. Our paper is organized as follows: in the next section, we will present the observations, models, notations, and concentration inequalities for uniformly ergodic Markov chain. In the third section, we give rst exponential inequalities, generalization bounds, and oracles inequalities for the model selected with OOS method. In this section, we use only the boundedness property of the loss function, and under additional assumptions, we improve these bounds in the fourth section. Finally, in the fth section, we rene these bounds under noise conditions and show our main result: the OOS method is still adaptative to noise conditions for uniformly ergodic Markov chain. We postpone long proofs in the Appendix.

2. The model 2.1. Assumptions and denitions for the observed process. We consider (Y t ) t∈Z , an Y-valued, k-order Markov chain, where Y is a Polish state space. We assume that (Y t ) t∈Z is time homogenous, stationary, and uniformly ergodic. Let p be xed a integer with p ≥ k, then the markovization (X t ) t∈Z of (Y t ) t∈Z with: (1)

X t := (Y t , • • • , Y t-p ) T ,
will be a time homogenous, stationary, uniformly ergodic Markov chain of order 1. We are going to state some denitions from the theory of general state space Markov chains, based on Gareth and Rosenthal [START_REF] Gareth | General state space markov chains and mcmc algorithms[END_REF].

Denition 1. Let us denote K its transition kernel. K(x, .) is the distribution of X n+1 conditioned on X n = x. Denote by Q its stationary law, i.e. the probability distribution such that

x∈X Q(dx)K(x, dz) = Q(dz).
We dene the total variational distance of two distributions P and Q dened on the same state space (X , A) as (2)

d T V (P, Q) := sup A∈A |P (A) -Q(A)|.
Since (Y t ) t∈Z is uniformly ergodic, (X t ) t∈Z will be uniformly ergodic, and constants C > 0 and 0 < ρ < 1 exist such that:

(3)

sup x∈X d T V (K n (x, .), Q) ≤ Cρ n .
2.2. Estimated functions and loss functions. We want to estimate a one-step prediction model:

g * (Y t-1 , • • • , Y t-p ),
which minimizes the expectation of a real bounded loss function L. According to equation (1), the observations are a realization of

(X 1 , • • • , X n , X n+1 , • • • , X n+m ). the variables (X 1 , • • • , X n ) constitute
the learning set, and the variables (X n+1 , • • • , X n+m ) the validation set. Let us introduce some denitions: Denition 2. We will introduce the notion of prediction and loss functions.

• A measurable function g from Y p into Y will be called a prediction function.

• Let L be a real, positive, measurable, bounded function dened on Y 2 . Without loss of generality, we can always rescale the function L such that |L(y, y )| ≤ 1. L will be called a loss function.

• With a slight abuse of notation, we will denote: (4)

L(g(X t )) := L (g(Y t-1 , • • • , Y t-p ), Y t ) .
• Let X be a random vector with the stationary law Q of the Markov chain (X t ) t∈Z . The expected loss of a measurable function g will be:

(5)

L(g) = E Q (L(g(X))) .
• Let F be the set of measurable functions from Y p into Y. The best prediction function, g * , is the function that minimizes the loss function applied to the one-step prediction under the stationary law:

(6)

g * = arg min g∈F E Q L (g(X t )) .
For example, if Y is nite, and L is the misclassication loss function: L(y, y ) = 1 y =y , g * will the best prediction of the following state

Y t knowing Y t-1 , • • • , Y t-p .
To estimate g * , we seek prediction function ĝ among a set of possible functions G ⊂ F by minimizing an empirical loss function on the learning set

(X 1 , • • • , X n ):
Denition 3. Let L be a loss function.

• The empirical estimation of g * , among a set of possible functions G, will be dened as:

ĝ (X 1 , • • • , X n ) = arg min g∈G n t=1 L (g(X t )) . (7)
• The estimated function ĝ depends on X 1 , • • • , X n and is a random function. If we observe the realization

x 1 , • • • , x n of X 1 , • • • , X n , we will observe a realization ĝ(x 1 , • • • , x n ) of ĝ(X 1 , • • • , X n ).
For convenience, since our results are valid for any realization ĝ(

x 1 , • • • , x n ) of ĝ(X 1 , • • • , X n ),
we will denote (8)

ĝn 1 := ĝ(x 1 , • • • , x n ).
The set of possible function G depends on user choice. In pratice, we have to chose between several sets {G k } 1≤k≤N , where N is a nite integer. Each set G k denes a empirical minimizer:

(9)

(ĝ n 1 ) k = arg min g∈G k n t=1 L (g(x t )) .
Our goal is to chose the best prediction function among ((ĝ n 1 ) k ) 1≤k≤N . Note that the learning loss L need not be equal to L; it can also be a proxy function easier to optimize. If the set of possible functions G k is large, the function (ĝ n 1 ) k may have poor performances on future observations of the process (X t ) t∈Z , even if the empirical learning loss is small. We say that (ĝ n 1 ) k overts the learning set (X 1 , • • • , X n ). Hence, it is wise to assess the performance of the estimated model on an out-of-sample set X n+1 , • • • , X n+m : Denition 4. Let g be any measurable function from Y p into Y. The empirical, out-of-sample loss of g will be: [START_REF] Chu | Comparison of two bandwidth selectors with dependent errors[END_REF]

Lm (g) = 1 m n+m k=n+1 L(g(X k )).
Note that, by the law of large number, the empirical loss Lm (g) converges, almost surely, towards the theoretical loss L(g):

L(g) a.s.

= lim m→∞ Lm (g).

Remark 1. For an integer b ≥ 0, let us write L b (ĝ n 1 ), the expected loss of any minimizer ĝn 1 for a vector X n+1+b of future observations conditionally to the realization

(X 1 = x 1 , • • • , X n = x n ): (11) L b (ĝ n 1 ) = E (L(ĝ n 1 (X n+b+1 )) |X 1 = x 1 , • • • , X n = x n ) .
A direct application of the uniform ergodicity of the Markov chain shows that, for a nite b, we can approximate the expectation of the loss L b by the theoretical loss L. Hence, with the notations of denition 1, for any realization

x 1 , • • • , x n of X 1 , • • • , X n : ( 12
)
|L b (ĝ n 1 ) -L(ĝ n 1 )| ≤ Cρ b .

Exponential inequalities.

To study the link between the empirical loss [START_REF] Chu | Comparison of two bandwidth selectors with dependent errors[END_REF] and the theoretical loss (5), we need uniform inequalities between the empirical mean and the expected mean (as in Lugosi [START_REF] In Györ | Principles of nonparametric learning, chapter Pattern classication and learning theory[END_REF]). This section aims to give such inequalities. First, let us introduce the notion of mixing time, which allows us to evaluate the speed of the convergence of a Markov Chain to its stationary distribution.

Denition 5. Let (X t ) t∈Z be a time homogeneous, uniformly ergodic, Markov chain. Let the total variation distance be dened by equation ( 2). The mixing time t mix is dened by:

d(t) := sup x∈X K t (x, •) -Q T V , t mix (ε) = min{t : d(t) ≤ ε}, and t mix = t mix ( 1 4
).

The fact that t mix is nite is equivalent to the uniform ergodicity of the chain (see Roberts and Rosenthal [START_REF] Roberts | General state space markov chains and mcmc algorithms[END_REF]). If we introduce an integer gap b between the learning and validation sample as a technical tool, a straightforward adaptation of Corollary 2.10 and equation (3.27) of Paulin [START_REF] Paulin | Concentration inequalities for markov chains by marton couplings and spectral methods[END_REF] yields the following proposition:

Proposition 1. Let C ≥ 1 and 0 ≤ ρ < 1 be the positive constants in equation (3). For any realization

x 1 , • • • , x n of X 1 , • • • , X n , real number 0 ≤ ε ≤ 1, integers 0 ≤ b < m,
with the notations of denitions 4, 3, and 5:

(13)

P ± 1 m -b n+m k=n+1+b L(ĝ n 1 (X k )) -L(ĝ n 1 ) > ε ≤ exp -2 (m -b)ε 2 9t mix + Cρ b .
Moreover, according to equation (3.30) of Paulin [START_REF] Paulin | Concentration inequalities for markov chains by marton couplings and spectral methods[END_REF], we have

Cρ b ≤ 2 exp -b ln(2) tmix , so (14)
P ± 1 m -b n+m k=n+1+b L(ĝ n 1 (X k )) -L(ĝ n 1 ) > ε ≤ exp -2 (m -b)ε 2 9t mix + 2 exp - b ln(2) t mix .
Now, we introduce the notion of pseudo spectral gap briey; see Paulin [START_REF] Paulin | Concentration inequalities for markov chains by marton couplings and spectral methods[END_REF] for a detailed presentation. Denition 6. For a Markov chain with transition kernel K(x, dz) and stationary distribution Q, we dene the spectrum of the chain as

S 2 := λ ∈ C\0 : (λI -K) -1
does not exist as a bounded linear operator on L 2 (Q) . We also dene the time reversal of K as the Markov kernel

K * (x, dz) := K(z, dx) Q(dx) Q(dz).
Then, the linear operator K * is the adjoint of the linear operator K on L 2 (Q). We dene a new quantity, called the pseudo spectral gap of K, as

γ ps := max k≥1 γ (K * ) k K k /k , where γ (K * ) k K k denotes the spectral gap of the self-adjoint operator (K * ) k K k .
Now, a straightforward adaptation of Theorem 3.4 of Paulin [START_REF] Paulin | Concentration inequalities for markov chains by marton couplings and spectral methods[END_REF] gives: Proposition 2. Let (X t ) t∈N be a stationary Markov chain with spectral gap γ ps , and a real number

0 ≤ ε ≤ 1. Let f ∈ L 2 (Q) with, for every x, |f (x) -E Q (f )| ≤ B. Let V f = V ar Q (f ),
and

S = n i=1 f (X i ), then: (15)
P (± (S -E Q (S)) > nε) ≤ exp - n 2 ε 2 γ ps 8(n + 1/γ ps )V f + 20nεB
.

From this proposition, we deduce a lemma, proven in the Appendix, that will be used in the last section:

Lemma 1. With the same assumptions than the previous proposition 2, for any 0 < δ < 1:

P ± (E Q (S) -S) ≤ 8 (γ ps + 1) γ 2 ps nV f log 1 δ + 20 γ ps B log 1 δ ≥ P ± (E Q (S) -S) ≤ 8 γ ps (n + 1/γ ps )V f log 1 δ + 20 γ ps B log 1 δ ≥ 1 -δ (16)
Finally, proposition 2 and equation (3.27) of Paulin [START_REF] Paulin | Concentration inequalities for markov chains by marton couplings and spectral methods[END_REF] yield the following proposition: Proposition 3. Let (X t ) t∈Z be a stationary Markov chain with pseudo spectral gap γ ps . For any function g, let us denote V g = V ar(L(g(X t )) the variance of the loss function computed with the stationary law. Then, for any realization

x 1 , • • • , x n of X 1 , • • • , X n , real number 0 ≤ ε ≤ 1, integers 0 ≤ b < m,
with the notations of denitions 4 and 3, since Cρ b ≤ 2 exp -b ln(2) tmix :

P ± 1 m -b n+m k=n+b+1 L(ĝ n 1 (X k )) -L(ĝ n 1 ) > ε ≤ exp - (m -b) 2 ε 2 γ ps 8((m -b) + 1 γps )V ĝn 1 + 20(m -b)ε + 2 exp - b ln(2) t mix . (17)

First bounds with Hoeffding-type inequality

In this section, we will use only the boundedness property of the loss functions. We obtain bounds valid for all models, but they can be loose under particular noise conditions.

x 1 , • • • , x n of X 1 , • • • , X n , integers b and m, with 0 ≤ b < m: (18) P ± Lm (ĝ n 1 ) -L(ĝ n 1 ) > ε + b m ≤ exp -2 (m -b)ε 2 9t mix + 2 exp - b ln(2) t mix .
Proof. We begin to prove [START_REF] Massart | Ecole d'été de probabilité de Saint-Flour XXXIII[END_REF] with the sign +. We have, on the event Lm (ĝ n

1 ) -L(ĝ n 1 ) ≥ b m : 0 ≤ 1 m n+m k=n+1 L(ĝ n 1 (X k )) -L(ĝ n 1 ) - b m ≤ 1 m n+m k=n+b+1 L(ĝ n 1 (X k )) - (m -b) m × L(ĝ n 1 ) = m -b m 1 m -b n+m k=n+b+1 L(ĝ n 1 (X k )) -L(ĝ n 1 ) ≤ 1 m -b n+m k=n+b+1 L(ĝ n 1 (X k )) -L(ĝ n 1 ).
So, by proposition 1:

P 1 m n+m k=n+1 L(ĝ n 1 (X k )) -L(ĝ n 1 ) > ε + b m ≤ P 1 m -b n+m k=n+b+1 L(ĝ n 1 (X k )) -L(ĝ n 1 ) > ε ≤ exp -2 (m -b)ε 2 9t mix + 2 exp - b ln(2) t mix . Now, on the event Lm (ĝ n 1 ) -L(ĝ n 1 ) < b m , we have P 1 m n+m k=n+1 L(ĝ n 1 (X k )) -L(ĝ n 1 ) > ε + b m = 0.
So,

P Lm (ĝ n 1 ) -L(ĝ n 1 ) > ε + b m ≤ exp -2 (m -b)ε 2 9t mix + 2 exp - b ln(2) t mix .
This shows equation ( 18) with the sign +.

For the sign -, remark that, on the event

L(ĝ n 1 ) -Lm (ĝ n 1 ) ≥ b m : 0 ≤ L(ĝ n 1 ) - 1 m n+m k=n+1 L(ĝ n 1 (X k )) - b m ≤ m -b m L(ĝ n 1 ) - 1 m n+m k=n+b+1 L(ĝ n 1 (X k )) = m -b m L(ĝ n 1 ) - 1 m -b n+m k=n+b+1 L(ĝ n 1 (X k )) ≤ L(ĝ n 1 ) - 1 m -b n+m k=n+b+1 L(ĝ n 1 (X k )),
and by the same argument as previously, we get equation [START_REF] Massart | Ecole d'été de probabilité de Saint-Flour XXXIII[END_REF] for the sign -Using this proposition, we can state an exponential bound for the theoretical loss function. The proof is based on a suitable choice of b and is given in the Appendix.

Theorem 3.1. With the notations of proposition 1, for any realization

x 1 , • • • , x n of X 1 , • • • , X n , and
0 ≤ ε ≤ 1: (19)
P ±( Lm (ĝ n 1 ) -L(ĝ n 1 )) > ε ≤ 2 exp ln(2) t mix + 1 exp - mε 2 ln(2) (1 + 9 ln(2))t mix .

A rst oracle inequality. Let ((ĝ n

1 ) k ) k=1,••• ,N denote a nite collection of prediction functions obtained by processing a realization of training sample of length n. In an ideal world, a benevolent oracle would tell us which index k minimizes the theoretical loss: [START_REF] Mohri | Stability bounds for stationnary φ-mixing and β-mixing processes[END_REF]

k = arg min k∈{1,••• ,N } L((ĝ n 1 ) k ).
However, all we can do is to chose the index k that minimizes the empirical validation loss:

k = arg min k∈{1,••• ,N } Lm ((ĝ n 1 ) k ). (21)
An oracle inequality between the optimal and the empirical choices k and k may be written:

(22) E L((ĝ n 1 ) k) -L(g * ) ≤ C L((ĝ n 1 ) k) -L(g * ) + γ(n) n .
Where, C is a factor at least as large as 1, γ(n) is a slowly growing function, and L(g * ) is the best expected loss. The term inf k (L ((ĝ n

1 ) k ) -L(g * )) = L((ĝ n 1 ) k) -L(g * )
is called the bias term, and the term γ(n) n , the variance term. The concept of oracle inequality was advocated in Donoho and Johnstone [START_REF] Donoho | Ideal spatial adaptation by wavelet shrinkage[END_REF] and is now widely used (see Candes [START_REF] Candes | Modern statistical estimation via oracle inequalities[END_REF]). To establish the oracle inequality, we will begin by inequalities between empirical and theoretical losses for k and k. The theorem 3.1 and union bound give the following theorem: Theorem 3.2. For any realization

x 1 , • • • , x n of X 1 , • • • , X n , and 0 ≤ ε ≤ 1: P L((ĝ n 1 ) k) -Lm ((ĝ n 1 ) k) > ε ≤ N 2 exp ln(2) t mix + 1 exp - mε 2 ln(2) (1 + 9 ln(2))t mix ,
and

P Lm (ĝ n 1 ) k -L (ĝ n 1 ) k > ε ≤ N 2 exp ln(2) t mix + 1 exp - mε 2 ln(2) (1 + 9 ln(2))t mix .
We deduce then, an upper bound of expectations between the empirical et theoretical losses:

E L((ĝ n 1 ) k) -Lm ((ĝ n 1 ) k) ≤ ln eN 2 exp ln(2) tmix + 1 (1 + 9 ln (2)) t mix ln (2) m ,
and

E Lm (ĝ n 1 ) k -L (ĝ n 1 ) k ) ≤ ln eN 2 exp ln(2) tmix + 1 (1 + 9 ln (2)) t mix ln (2) m .
Proof. We can write:

E L((ĝ n 1 ) k) -Lm ((ĝ n 1 ) k) ≤ E max L((ĝ n 1 ) k) -Lm ((ĝ n 1 ) k) , 0 ≤ E max L((ĝ n 1 ) k) -Lm ((ĝ n 1 ) k) , 0 2 = 1 0 P L((ĝ n 1 ) k) -Lm ((ĝ n 1 ) k) > √ t dt ≤ ln eN 2 exp ln(2) tmix + 1 (1 + 9 ln (2)) t mix ln (2) m .
The proof of the second inequality is symmetric. Now, remark that,

L (ĝ n 1 ) k -L (ĝ n 1 ) k = L((ĝ n 1 ) k) -Lm ((ĝ n 1 ) k) + Lm ((ĝ n 1 ) k) -Lm (ĝ n 1 ) k + Lm (ĝ n 1 ) k -L (ĝ n 1 ) k ,
with, by denition, Lm ((ĝ n

1 ) k) -Lm ((ĝ n 1 ) k) ≤ 0.
Hence, we have

L (ĝ n 1 ) k -L (ĝ n 1 ) k ≤ sup k∈{1,••• ,N } (L ((ĝ n 1 ) k ) -Lm ((ĝ n 1 ) k )) + sup k∈{1,••• ,N } ( Lm ((ĝ n 1 ) k ) -L ((ĝ n 1 ) k )),
and we get the following oracle inequality:

Theorem 3.3. For any realization

(x 1 , • • • , x n ) of (X 1 , • • • , X n ),
we have:

E L((ĝ n 1 ) k) -L (ĝ n 1 ) k ≤ 2 ln eN 2 exp ln(2) tmix + 1 (1 + 9 ln (2)) t mix ln (2) m .
Or, if we denote by g * the best prediction function:

E L((ĝ n 1 ) k) -L(g * ) ≤ L (ĝ n 1 ) k -L(g * ) + 2 ln eN 2 exp ln(2) tmix + 1 (1 + 9 ln (2)) t mix ln (2) m .
Note that this bound rate is of the same order as in the independent case when m goes toward innite.

All the bounds in this section depend on the unknown constant t mix , however, this constant can be estimated from the data (see Wolfer and Kontorovich [START_REF] Wolfer | Estimating the mixing time of ergodic markov chains[END_REF]).

Fast rates with Bernstein-type inequality

We can remove the square root in the bound of theorem 3.3 by increasing the empirical hold-out error estimate by a small constant factor and using Berstein-type inequality (like in Bartlett et al. [START_REF] Bartlett | Model selection and error estimation[END_REF]). When the theoretical loss is small, the inequalities obtained may be better than the previous inequalities.

4.1. Exponential bound for L(ĝ n 1 ). We begin to establish exponential bounds for slightly modied empirical losses. As in section 3.1, we will write a generalization bound by considering the last m -b validation data. Hence, an application of the proposition 3 yields the following proposition, proven in the Appendix: Proposition 5. With the notations and assumptions of proposition 3, let 0 < a < 1 be a xed constant.

For, any realization

(x 1 , • • • , x n ) of (X 1 , • • • , X n ),
P L(ĝ n 1 ) - 1 1 -a Lm (ĝ n 1 ) > ε + b m ≤ exp   - (m -b)γ ps a(1 -a)ε 8 1 + 1 γps + 20   + 2 exp - b ln(2) t mix . (24)
Using this proposition, with a suitable choice of b, we can get exponential bounds for P 1 1+a Lm (ĝ n 1 ) -L(ĝ n 1 ) > ε and P L(ĝ n 1 ) -1 1-a Lm (ĝ n 1 ) > ε . These bounds are stated in the following theorem. The proof may be found in the Appendix. Theorem 4.1. With the notations and assumptions of proposition 3, for any realization

x 1 , • • • , x n of X 1 , • • • , X n , 0 < a < 1, and 0 ≤ ε ≤ 1: (25) P 1 1 + a Lm (ĝ n 1 ) -L(ĝ n 1 ) > ε ≤ 1 + 2 exp ln(2) t mix exp   - a(1 + a)mε 4t mix 8 1 + 1 γps + 20   ,

and

(26

) P L(ĝ n 1 ) - 1 1 -a Lm (ĝ n 1 ) > ε ≤ 1 + 2 exp ln(2) t mix exp   - a(1 -a)mε 4t mix 8 1 + 1 γps + 20   .
4.2. Bound for the generalization error. In the same framework as section 3.2, we consider

((ĝ n 1 ) k ) k=1,••• ,N
, a nite collection of prediction functions obtained by processing a realization of training sample of length n. We recall that the function with index k minimizes the empirical validation loss [START_REF] Paulin | Concentration inequalities for markov chains by marton couplings and spectral methods[END_REF], and the function with index k minimizes the theoretical loss [START_REF] Mohri | Stability bounds for stationnary φ-mixing and β-mixing processes[END_REF]. Now, the theorem 4.1 and union bound give the following result: Theorem 4.2. With the notations and assumptions of proposition 3, for any realization

x 1 , • • • , x n of X 1 , • • • , X n , 0 < a < 1 and 0 ≤ ε ≤ 1: P 1 1 + a Lm (ĝ n 1 ) k -L (ĝ n 1 ) k > ε ≤ N 1 + 2 exp ln(2) t mix exp   - a(1 + a)mε 4t mix 8 1 + 1 γps + 20   ,
and

P L((ĝ n 1 ) k) - 1 1 -a Lm ((ĝ n 1 ) k) > ε ≤ N 1 + 2 exp ln(2) t mix exp   - a(1 -a)mε 4t mix 8 1 + 1 γps + 20   .
So we get an upper bound of the expectations of these expressions:

E 1 1 + a Lm (ĝ n 1 ) k -L (ĝ n 1 ) k ≤ 4t mix 8 1 + 1 γps + 20 ln eN 2 exp ln(2) tmix + 1 a(1 + a)m ,
and

E L((ĝ n 1 ) k) - 1 1 -a Lm ((ĝ n 1 ) k) ≤ 4t mix 8 1 + 1 γps + 20 ln eN 2 exp ln(2) tmix + 1 a(1 -a)m .
Proof. We can write:

E 1 1 + a Lm ((ĝ n 1 ) k)) -L((ĝ n 1 ) k)) ≤ E max 1 1 + a Lm ((ĝ n 1 ) k)) -L((ĝ n 1 ) k)) , 0 = 1 0 P 1 1 + a Lm ((ĝ n 1 ) k)) -L((ĝ n 1 ) k)) > t dt ≤ 4t mix 8 1 + 1 γps + 20 ln eN 2 exp ln(2) tmix + 1 a(1 + a)m .
The proof of the second inequality is symmetric. Now, remark that:

L((ĝ n 1 ) k) -L((ĝ n 1 ) k)) = L((ĝ n 1 ) k) - 1 1 -a Lm ((ĝ n 1 ) k) + 1 1 -a Lm ((ĝ n 1 ) k) - 1 1 -a Lm ((ĝ n 1 ) k)) + 1 1 + a Lm ((ĝ n 1 ) k)) -L((ĝ n 1 ) k) + 2 a 1 -a 2 Lm ((ĝ n 1 ) k ,
with, by denition,

1 1-a Lm ((ĝ n 1 ) k) -1 1-a Lm ((ĝ n 1 ) k)) ≤ 0. Hence, we have L((ĝ n 1 ) k) -L((ĝ n 1 ) k)) ≤ sup k∈{1,••• ,N } L ((ĝ n 1 ) k ) - 1 1 -a Lm ((ĝ n 1 ) k ) + sup k∈{1,••• ,N } 1 1 + a Lm ((ĝ n 1 ) k ) -L ((ĝ n 1 ) k ) + 2 a 1 -a 2 Lm ((ĝ n 1 ) k ,
and we get the following inequalities for the empirical choice of the index k:

Theorem 4.3. With the notations and assumptions of proposition 3, for any realization

(x 1 , • • • , x n ) of (X 1 , • • • , X n ),
and 0 < a < 1, we have:

E L((ĝ n 1 ) k) -L((ĝ n 1 ) k)) ≤ 4t mix 8 1 + 1 γps + 20 ln eN 2 exp ln(2) tmix + 1 a(1 -a)m + 4t mix 8 1 + 1 γps + 20 ln eN 2 exp ln(2) tmix + 1 a(1 + a)m + 2a 1 -a 2 L((ĝ n 1 ) k).
Or, if we denote by g * the best prediction function:

E L((ĝ n 1 ) k) -L(g * ) ≤ 1 + 2a 1 -a 2 L((ĝ n 1 ) k) -L(g * ) + 4t mix 8 1 + 1 γps + 20 ln eN 2 exp ln(2) tmix + 1 a(1 -a)m + 4t mix 8 1 + 1 γps + 20 ln eN 2 exp ln(2) tmix + 1 a(1 + a)m + 2a 1 -a 2 L(g * ).
These bounds are not exactly like an oracle inequality [START_REF] Roberts | General state space markov chains and mcmc algorithms[END_REF]; however, they are better than the bounds of the previous section if the theoretical loss L (ĝ n 1 ) k is small enough. An extreme and rare case will be when L (ĝ n 1 ) k = 0, and, in such case, the bounds of the previous theorem are of order O 1 m . Note that, all the bounds in this section depend on the unknown constants t mix and γ ps , but they can be estimated from the data (see Wolfer and Kontorovich [START_REF] Wolfer | Estimating the mixing time of ergodic markov chains[END_REF]).

Fast rate under noise conditions

The previous condition L (ĝ n 1 ) k = 0 may be seen as a rough noise condition. We can try to rene our analysis because hold-out enjoys excellent theoretical properties under noise conditions for the i.i.d. case (see Blanchard and Massart [START_REF] Blanchard | Discussion: Local rademacher complexities and oracle inequalities in risk minimization[END_REF], Boucheron et al. [START_REF] Boucheron | Theory of classication: A survey of some recent advances[END_REF] or Massart [START_REF] Massart | Ecole d'été de probabilité de Saint-Flour XXXIII[END_REF]). Hence, we will investigate OOS properties in the Makov case under similar conditions for the noise. In this section, we assume that the state space Y is discrete, that L is the misclassication loss function: L(y, y ) = 1 y =y , and the functions g

(Y t-1 , • • • , Y t-p ) are the predictions of the following state Y t knowing Y t-1 , • • • , Y t-p .
Hence, we take advantage that for discrete observations, the prediction of the following state is a classication task. First, we give the assumption on the noise.

Assumption on the noise (H):

• A function ω(.) exists such that ω(x)/

√

x is non-increasing and, for any function g,

V ar (1 g =g * ) ≤ ω(L(g) -L(g * )),
where the expection are computed under the stationary law of the Markov chain

(X t ) t∈Z . • Let τ * m denote the smallest positive solution of ω(ε) = √ mε.
Again, in the same framework as sections 3.2, and 4.2, we consider ((ĝ n 1 ) k ) k=1,••• ,N , a nite collection of prediction functions obtained by processing a realization of training sample of length n. The function with index k minimizes the empirical validation loss [START_REF] Paulin | Concentration inequalities for markov chains by marton couplings and spectral methods[END_REF], and the function with index k minimizes the theoretical loss [START_REF] Mohri | Stability bounds for stationnary φ-mixing and β-mixing processes[END_REF]. We can then set the following proposition. We prove this proposition in the Appendix. Proposition 6. Let us assume the assumption on the noise (H), for any realization

x 1 , • • • , x n of X 1 , • • • , X n ,
integers b and m, with 0 ≤ b < m, real numbers 0 ≤ ε ≤ 1, and 0 < θ < 1:

P L (ĝ n 1 ) k -L(g * ) -(1 + θ) L (ĝ n 1 ) k -L(g * ) > ε + (1 + θ)2b m ≤ N exp - 1 1 + θ θγ ps (m -b) 16(1 + 1 γps )mτ * m + 80θ ε + 2 exp - b ln(2) t mix . (27)
5.1. Exponential bound under noise condition. We can now state an exponential bound under noise condition. The following theorem is proven in the Appendix.

Theorem 5.1. Let us assume the assumption on the noise (H), for any realization

x 1 , • • • , x n of X 1 , • • • , X n ,
and 0 < θ < 1:

P L (ĝ n 1 ) k -L(g * ) -(1 + θ) L (ĝ n 1 ) k -L(g * ) > ε ≤ N + 2 exp ln(2) t mix exp - 1 4t mix (1 + θ) θγ ps mε 16(1 + 1 γps )mτ * m + 80θ . (28)
Finally, taking expectation, we get the oracle inequality:

E L (ĝ n 1 ) k -L(g * ) ≤ (1 + θ) × L (ĝ n 1 ) k -L(g * ) + 4t mix (16(1 + 1 γps )mτ * m + 80θ) θγ ps m ln e 2 exp ln(2) t mix + N . (29)
Remark 2. Assume the Mammen-Tsybakov noise condition with exponent α hold (see Mammen and

Tsybakov [START_REF] Mammen | Smooth discrimination analysis[END_REF]), that is, we can choose w(r) = r h α/2 for some positive h. Then, τ * m = (mh α )

-1/(2-α) , and the corollary translates into

E L (ĝ n 1 ) k -L(g * ) ≤ (1 + θ) × L (ĝ n 1 ) k -L(g * )+ 4t mix (16(1 + 1 γps )h -α/(2-α) m 1-1/(2-α) + 80θ) θγ ps m ln e 2 exp ln (2)
t mix + N = (1 + θ) × L (ĝ n 1 ) k -L(g * )+   320t mix γ ps m + 4t mix 16(1 + 1 γps )h -α/(2-α) θγ ps m 1/(2-α)   ln e 2 exp ln(2) t mix + N   . (30)
Remark 3. If the state space Y is equal to {0, 1}, and if the conditional expectation function:

η(Y t-1 , • • • , Y t-p ) = E (Y t |Y t-1 , • • • , Y t-p ) is such that for all y t-1 , • • • , y t-p ∈ Y p , |2η(y t-1 , • • • , y t-p ) -1| > h, then the Mammen-Tsybakov noise condition holds with α = 1.
The rate of the oracle inequality will then be fast:

E L (ĝ n 1 ) k -L(g * ) ≤ (1 + θ) × L (ĝ n 1 ) k -L(g * )+   320t mix γ ps m + 4t mix 16(1 + 1 γps )h -α/(2-α) θγ ps m   ln e 2 exp ln(2) t mix + N   . Hence, if L (ĝ n 1 ) k -L(g * ) = 0, E L (ĝ n 1 ) k -L(g * ) ≤ O 1 m .
Remark 4. Let p ≥ 1 be a xed integer and consider the set of homogenous ergodic Markov chains of order p with Y = {0, 1}. Let us denote A the set of possible transition kernels:

A = {P (Y t = 1|y t-1 , • • • , y t-p ), (y t-1 , • • • , y t-p ) ∈ {0, 1} p } = [0, 1] 2 p .
Now, put the uniform measure on A, then the Lebesgue measure of the set of models such that the Mammen-Tsybakov noise condition does not hold with α = 1 will be null. Indeed,

P (Y t = 0|y t-1 , • • • , y t-p ) = P (Y t = 1|y t-1 , • • • , y t-p ) = 1
2 for some y t-1 , • • • , y t-p , means that a linear constraint for the coecients of the transition kernel exists and such set is of Lebesgue measure 0. Hence, for almost all models, an h exists such that, for all

y t-1 , • • • , y t-p ∈ Y p , |2η(y t-1 , • • • , y t-p ) -1| > h,
and we get the previous rate for the oracle inequality.

6. Appendix 6.1. Proof of lemma 1. We will prove the lemma for the sign +, the proof for the signis the same. By the proposition 2, we have for any 0 < δ < 1:

P E Q (S) -S > 8 γ ps (n + 1/γ ps )V f log 1 δ + 20 γ ps B log 1 δ ≤ exp   - γ ps 8 γps (n + 1/γ ps )V f log 1 δ + 20 γps B log 1 δ 2 8 (n + 1/γ ps ) V f + 20B 8 γps (n + 1/γ ps )V f log 1 δ + 20 γps B log 1 δ    ≤ exp   - 8(n + 1/γ ps )V f log 1 δ + 40 8 γps (n + 1/γ ps )V f log 1 δ B log 1 δ + 1 γps 20B log 1 δ 2 8 (n + 1/γ ps ) V f + 20B 8 γps (n + 1/γ ps )V f log 1 δ + 20 γps B log 1 δ   ≤ exp   -log 1 δ 8(n + 1/γ ps )V f + 40B 8 γps (n + 1/γ ps )V f log 1 δ + 1 γps (20B) 2 log 1 δ 8 (n + 1/γ ps ) V f + 20B 8 γps (n + 1/γ ps )V f log 1 δ + 20 γps B log 1 δ   ≤ δ.
Noting that n, completes the proof 6.2. Proof of theorem 3.1. We prove [START_REF] Mohri | Generalization bounds for non-stationary mixing processes[END_REF] with the sign +, the proof for the signis symmetric.

Since 0 ≤ ε ≤ 1, if b = mε 2 1+9 ln(2) , then ε -b m > 0.
Using the proposition 4, we have:

P ±( Lm (ĝ n 1 ) -L(ĝ n 1 )) > ε ≤ exp -2 (m -b)(ε -b m ) 2 9t mix + 2 exp - b ln(2) t mix . Now, if b = mε 2 1+9 ln(2)
, then b -1 ≤ b ≤ b, and we get

P ±( Lm (ĝ n 1 ) -L(ĝ n 1 )) > ε ≤ exp -2 (m -b)(ε -b m ) 2 9t mix + 2 exp ln(2) t mix exp - b ln(2) t mix .
Moreover,

exp -2 (m -b)(ε -b m ) 2 9t mix + 2 exp ln(2) t mix exp - b ln(2) t mix = 2 exp ln(2) t mix + exp -m 2(1 -b m )(ε -b m ) 2 9t mix - b ln(2) mt mix exp - b ln(2) t mix = 2 exp ln(2) t mix + exp - m t mix 2(1 -b m )(ε -b m ) 2 9 - b ln(2) m exp - b ln(2) t mix . Now, 2(1 -b m )(ε -b m ) 2 9 - b ln(2) m = 2(1 - ε 2 1+9 ln(2) )(ε - ε 2 1+9 ln(2) ) 2 9 - ε 2 ln(2) 1 + 9 ln(2) ≥ ε 2 2(1 - 1 1+9 ln(2) )(1 - 1 1+9 ln(2) ) 2 9 - ln(2) 1 + 9 ln(2) = ε 2 2 9 1 - 1 1 + 9 ln(2) 3 - ln(2) 1 + 9 ln(2) > 0,
and nally 6.3. Proof of proposition 5. First, we will prove that for a stationary uniformly ergodic Markov chain (X t ) t∈Z and any function g ∈ G:

exp -2 (m -b)(ε -b m ) 2 9t mix + 2
(31)

P 1 1 + a 1 m m k=1 L(g(X k )) -E(L(g(X))) > ε ≤ exp   - mγ ps a(1 + a)ε 8 1 + 1 γps + 20   ,
where E(L(g(X))) is computed under the stationay law of (X t ) t∈Z .

Since 0 ≤ L(g(X)) ≤ 1, 0 ≤ L(g(X)) 2 ≤ L(g(X)) ≤ 1, and V (L(g(X))) ≤ E(L(g(X)))(1 -

E(L(g(X)))) ≤ E(L(g(X))).
Moreover,

P 1 1 + a 1 m m k=1 L(g(X k )) -E(L(g(X))) > ε = P 1 m m k=1 L(g(X k )) -E(L(g(X))) > aE(L(g(X))) + (1 + a)ε .
Let t = aE(L(g(X))) + (1 + a)ε, by the proposition 2, we have:

P 1 m m k=1 L(g(X k )) > t ≤ exp - m 2 t 2 γ ps 8(m + 1/γ ps )E(L(g(X))) + 20mt = exp - mt 2 γ ps 8(1 + 1 mγps )E(L(g(X))) + 20t
Now, and we deduce equation (31). Now, using equation (3.27) of Paulin [START_REF] Paulin | Concentration inequalities for markov chains by marton couplings and spectral methods[END_REF], we get

P 1 1 + a 1 m -b n+m k=n+b+1 L(ĝ n 1 (X k )) -L(ĝ n 1 ) > ε ≤ exp   - (m -b)γ ps a(1 + a) 8 1 + 1 γps + 20 ε   +2 exp - b ln(2) t mix .
Finally, by the same arguments as in the proof of proposition 4, we get equation [START_REF] Tashman | Out-of-sample tests of forecasting accuracy: an analysis and review[END_REF]. The proof of equation ( 24) is symmetric. . Since b > b -1, equation ( 23) yields

P 1 1 + a Lm (ĝ n 1 ) -L(ĝ n 1 ) > ε ≤ exp   - γ ps a(1 + a) 8 1 + 1 γps + 20 m - b m ε - b m   + 2 exp ln(2) t mix exp - b ln (2)
8 1 + 1 γps + 20     1 - a(1 + a) 4 ln(2) 8 1 + 1 γps + 20 ε     1 - a(1 + a) 4 ln(2) 8 1 + 1 γps + 20   - 1 4t mix γ ps       × exp   - a(1 + a)mε 4t mix 8 1 + 1 γps + 20   .
Now, since ε ≤ 1, and, by equation (3.9) of Paulin [START_REF] Paulin | Concentration inequalities for markov chains by marton couplings and spectral methods[END_REF],

γ ps ≥ 1 2tmix ⇔ 2 ≥ 1 2tmixγps , we get P 1 1 + a Lm (ĝ n 1 ) -L(ĝ n 1 ) > ε ≤   2 exp ln(2) t mix + exp   -γps a(1 + a)mε 8 1 + 1 γps + 20      1 - a(1 + a) 4 ln(2) 8 1 + 1 γps + 20   2 - 1 2          × exp   - a(1 + a)mε 4t mix 8 1 + 1 γps + 20   .
Finally, noting that a(1+a)

g k = min k∈{1,••• ,N } 1 m + b m+b t=1 L(g k (X t )) and g k = min k∈{1,••• ,N } L(g k ).
We will give bounds involving the m last variables: Lm (g k ) := 1 m m+b t=b+1 L(g k (X t )). Note that, for any function g k : (32)

1 m + b m+b t=1 L(g k (X t )) - 1 m m+b t=b+1 L(g k (X t )) ≤ 1 m + b m+b t=1 L(g k (X t )) - 1 m + b m+b t=b+1 L(g k (X t )) ≤ b m + b , and
1 m m+b t=b+1 L(g k (X t )) - 1 m + b m+b t=1 L(g k (X t )) ≤ m + b m(m + b) m+b t=b+1 L(g k (X t )) - m m(m + b) m+b t=1 L(g k (X t )) ≤ b m(m + b) m+b t=b+1 L(g k (X t )) ≤ b m + b , (33) so
(34) 1 m m+b t=b+1 L(g k(X t )) - 1 m m+b t=b+1 L(g k (X t )) ≤ 2b m + b .
By the lemma 1 and the union bound, with probability at least For 0 < θ < 1 we have:

1 -δ, for all k ∈ {1, • • • , N }, L (g k ) -L(g * ) ≤ Lm (g k ) -Lm (g * ) + 8(1 + 1 γps ) log N δ γ ps m × ω (L(g k ) -L(g * )) + 40 log N δ γ ps m , and
L(g * ) -L g k ≤ Lm (g * ) -Lm g k + 8(1 + 1 γps ) log N δ γ ps m × ω L g k -L(g * ) + 40 log N δ γ ps m . Since L g k -L(g * ) ≤ L (g k ) -L(g * ),

3. 1 .

 1 Exponential bound. Let us consider out-of-sample data of length m in the future of the last learning observation X n . We will write a generalization bound by taking into account the last mb validation data. By doing so, we omit to take into account the b rst observations, but these observations account for at most b m in the empirical validation error because the loss function L is bounded by 1. So, we get: Proposition 4. With the notations of proposition 1, for any realization

  integers b and m, with 0 ≤ b < m, we get:

γ

  ps a(1 + a)mε

  for any k ∈ {1, • • • , N }, by summing the two inequalities, we obtainL (g k ) -L g k ≤ Lm (g k ) -Lm g k + 2 8(1 + 1 γps ) log N δ γ ps m × ω L g k -L(g * ) + 80 log N δ γ ps m .As Lm g k -Lm g k ≤ 2b m+b , with probability larger than 1 -δ,L g k -L g k ≤ 2b m + b + 2 8(1 + 1 γps ) log N δ γ ps m × ω L g k -L(g * ) + 80 log N δ γ ps m .Let τ * m be dened as the statement of the theorem.If L g k -L(g * ) ≥ τ * m+b , then ω L g k -L(g * ) / √ m + b ≤ L g k -L(g * )τ *m+b , and we haveL g k -L g k ≤

θ 2 2 L 2 8 2 L 2 ≤ 1 +L

 22221 g k -L(g * )g k -L(g * ) ≤ L g k -L(g * ) θ, we get, with probability larger than 1 -δ:L g k -L(g * ) ≤ (1 + θ)× L g k -L(g * ) g k -L(g * )-(1 + θ) L g k -L(g * ) + 2b m + b ≤ (1 + θ) 16(1 + 1 γps )(m + b)τ * m+b + 80θ θγ ps m log N δ .Note that we have done the reasoning ifL g k -L(g * ) ≥ τ * m+b . However, if L g k -L(g * ) < τ * m+b , the bound (35) is obvious. Now, we deduce from equation (35) thatP L g k -L(g * ) -(1 + θ) L g k -L(g * ) + 2b m + b > ε ≤ N exp -1 1 + θ θγ ps mε 16(1 + 1 γps )(m + b)τ * m+b + 80θ. Now, considering the actual chain (X t ) t∈N , in the framework of section 3.2, we get for any realizationx 1 , • • • , x n of X 1 , • • • , X n , integers b and m, with 0 ≤ b < m: P L (ĝ n 1 ) k -L(g * ) -(1 + θ) L (ĝ n 1 ) k -L(g * ) > ε + (1 + θ)2b m ≤ N exp -1 1 + θ θγ ps (m -b)ε 16(1 +

  6.4. Proof of theorem 4.1. We prove[START_REF] Zhang | Understanding deep learning (still) requires rethinking generalization[END_REF], the proof for (26) is symmetric. Let us dene b =

	ma(1+a)ε
	4 ln(2) 8 1+ 1 γps +20

  Proof of proposition 6. Let us considere a nite collection of functions {g 1 , • • • , g N }. For any integers b and m, with 0 ≤ b < m, and a sample (X 1 , • • • , X m+b ) where the m last variables (X b+1 , • • • , X m+b ) follow the stationary law of the Markov chain (X t ) t∈Z . Let us dene:

								2
			8 1+ 1 γps +20	≤ 1 14 , we have 1 -	a(1+a) 4 ln(2) 8 1+ 1 γps +20	-1 2 ≥ 0, and
									
	P	1 1 + a	Lm (ĝ n 1 ) -L(ĝ n 1 ) > ε ≤ 2 exp	ln(2) t mix	+ 1 exp	 -	a(1 + a)mε 4t mix 8 1 + 1 γps + 20	
	6.5.

  .6. Proof of theorem 5.1. Applying the proposition 6 to 2(1 + θ)ε, we get . Following the same reasoning as in the proof of theorem 4.1, we getP L (ĝ n 1 ) k -L(g * ) -(1 + θ) L (ĝ n 1 ) k -L(g * ) > ε ≤ 

			P	L (ĝ n 1 ) k -L(g * ) 2(1 + θ)	-	L (ĝ n 1 ) k -L(g * ) 2	> ε +	b m	≤
			N exp -2	θγ ps (m -b) 16(1 + 1 γps )mτ * m + 80θ	ε + 2 exp -	b ln(2) t mix	.
	θmε γps )mτ * (16(1+ 1  m +80θ)2 ln(2)  2 exp Let us dene b = ln(2) t mix + N exp  -2θγ ps mε (16(1 + 1 γps )mτ * m + 80θ)	  1 -	θ γps )mτ * (16(1 + 1 m + 80θ)2 ln(2)	2	-	1 2	      ×
	exp -	θγ ps εm γps )mτ * (16(1 + 1 m + 80θ)2t mix	.
	Noting that	θ γps )mτ * 16(1+ 1 m +80θ ≤ 1 80 , we have
								2
				1 -	θ γps )mτ * (16(1 + 1 m + 80θ)2 ln(2)	-	1 2	≥ 0,
			2 exp	ln(2) t mix		+ N exp -	1 4t mix (1 + θ)	θγ ps εm 16(1 + 1 γps )mτ * m + 80θ	.
							1 γps )mτ * m + 80θ	+ 2 exp -	b ln(2) t mix	.

6and

P L (ĝ n 1 ) k -L(g * ) -(1 + θ) L (ĝ n 1 ) k -L(g * ) > ε ≤

This work was partly involved in Project EcoDep PSI-AAP2020-0000000013.