Dorian Sonet

Dr Mattéo Cayla

Raphaël Méreau

Estelle Morvan

Dr Aline Lacoudre

Dr Nicolas Vanthuyne

Muriel Albalat

Prof Dario M Bassani

Antoine Scalabre

Dr Emilie Pouget

Dr Brigitte Bibal
email: brigitte.bibal@u-bordeaux.fr

Dr A Scalabre

Chiral Anthranyl Trifluoromethyl Alcohols: Structures

Keywords:

come

Introduction

-Trifluoromethyl alcohols represent a well-known class of compounds exploited for their physical organic properties owing to their high polarity and ionizing power, mild acidity (pK a similar to phenols), strong hydrogen bond donor/poor acceptor groups and low chemical reactivity. [1,2] Compared to their hexafluorinated homologues such as hexafluoroisopropanol (HFIP), -trifluoromethyl alcohols offer the advantage of less extreme properties, such as a moderate steric hindrance that facilitates their incorporation into organic architectures (Figure 1). [3,4] Concerning their applicability, -trifluoromethyl alcohols have amphiphilic properties that are exploited by the pharmaceutical industry to enhance the drug availability or activity (befloxatone, LX-301). [5] The H-bond donor properties of RCH(OH)CF 3 molecules were initially exploited within chiral solvating agents, such as Pirkle's alcohol, for the determination of the enantiomeric excess (Figure 1B). [6] 2,2,2-Trifluoroethanol (TFE, Figure 1A) is widely employed as a solvent or an additive in organic synthesis to optimize different reactions: (photo)oxidations, nucleophilic substitutions and additions, cycloadditions, electrophilic and dearomative reactions. [1][2][3] On the other hand, the nucleophilic reactivity of - trifluoromethyl alcohols is far less documented. For TFE, some tandem nucleophilic addition reactions [7] on aromatic or aliphatic carbocations were reported to have poor stereoselectivity whereas nucleophilic substitutions [8] on carbohydrates [9] via an oxocarbenium intermediate were stereoselective (Figure 1C).

Beyond the successful example of TFE, few trifluoromethyl [6,[10][11][12][13] or hexafluoromethyl [14] alcohol groups were implemented on more elaborated skeletons, possibly due to selfaggregation issues that may limit their benefit. [15] In addition, the control of the non-symmetric induction (i. e., the diastereoisomers are obtained in a non statistical ratio, without any asymmetric induction) and the nucleophilicity of thetrifluoromethyl alcohol group have received little attention.

To further explore the physical properties and the reactivity of chiral -trifluoromethyl alcohols, we envisioned to implement this functional group on 9,10-diphenylanthracenes (DPA) by preparing compounds 1 (Figure 1D) which present two hindered groups on adjacent carbons C1' and C2'. Belonging to the class of chiral atropisomers, [START_REF] Barrett | chiral atropisomers with two stereogenic center/axis at least[END_REF] compounds 1 exist as syn and anti isomers due to the locked rotation along the C 1'-Ar ÀC 9-anthryl bond. In this environment, the rotation of the vicinal C 2'-Ar ÀC HðOHÞCF 3 bonds may be either restricted (rotamers) or orientated by electronic factors (favored conformers). [6d] Firstly, the structural properties of architectures 1 were investigated by NMR, X-ray diffraction and molecular modeling. Secondly, the oxidative dearomatization of anthracenes 1 into spiro-benzofuranes 4 was achieved through the synergistic oxidation of anthracene [17] and the nucleophilic addition of CH(OH)CF 3 moieties. This pathway for the dearomatization of anthracenes is novel compared to the classical [4 + 2] cycloadditions, metal-catalyzed or organolithium additions. [18] In addition, this access to chiral fluorinated spiro-heterocycles is a new route, complementary to the rare examples of fluorospirocyclizations [START_REF] Wilkinson | For reviews about fluorocyclizations[END_REF]20] and post-fluorination of spirocycles. [21,[START_REF] Wang | A photo-induced dearomative oxidation of gem-difluoroindoles was described without stereocontrol[END_REF] Thirdly, the chiroptical properties of enantiopure compounds 1 and 4 was investigated and compared. The variation of chiroptical properties by an oxidative stimulus can be attractive and less demanding in synthetic efforts compared to the classical approaches, that is, the synthesis of compounds series or the preparation of more sophisticated molecules. [23,24]

Results and Discussion

Synthesis of fluorinated alcohols

Syn and anti atropisomers 1 were obtained by using a two-step pathway from 9,10-bis(2-formylphenyl)-anthracene [25] syn-and anti-2, respectively. The retention of relative configuration along the synthesis was expected due to the high rotational barrier for ortho,ortho'-disubstituted 9,10-diphenyl anthracenes. [26][START_REF] Gawronski | For atropisomeric molecular triads based on smaller aromatic compounds than anthracene[END_REF] The functionalization of each dialdehyde syn-2 or anti-2 was independently conducted in the presence of the nucleophilic Prakash-Ruppert's reagent (2.3 equiv.) to create two asymmetric carbons (Scheme 1). The resulting bis-( -trifluorometh-yl)trimethylsilyl ethers syn-3 (88 % yield) and anti-3 (90 % yield) were isolated in high yields, as a mixture of stereoisomers that were separated at the following step. Finally the deprotection of each rotamer 3 with TBAF independently afforded bis-(trifluoromethyl)alcohols syn-1 (89 % yield) and anti-1 (93 % yield) respectively. The purification of these compounds 1 on column chromatography on silica gel allowed the separation of the meso R,S diastereoisomer from the racemic mixture of R,R/ S,S compounds.

Interestingly, the stereoisomeric ratios for products 1 depend on the relative (syn or anti) configuration of the starting material 2 (Scheme 1). Anti-1 was obtained as a quasi statistical ratio of stereoisomers, that is, 47 % of the racemic R,R/S,S compound ((AE)-anti-1) and 46 % of the (R,S)-meso-anti-1 instead syn-1 was isolated as 69 % of racemic R,R/S,S mixture (named (AE)-syn-1) and only 20 % of the R,S meso compound. This nonsymmetric induction revealed in the last step obviously occurred during the second step when the aldehyde syn-2 undergoes a double nucleophilic addition of the trifluoromethyl carbanion. Indeed, the addition of the first equivalent of the Prakash's reagent results in the anchorage of a bulky (trifluoromethyl)trimethylsilyl ether group close to the remaining aldehyde function which probably accounts for this nonsymmetric induction observed on the syn-products 1 and not on anti-1.

Racemic and meso stereoisomers of both syn-and anti-1 were fully characterized. As expected from their structural differences, the syn and anti relative configurations that respectively place the ÀCH(OH)CF 3 groups in close proximity or on opposite half-spaces with respect to the anthracene core, have an impact on the physical properties of compounds. For instance, all stereoisomers of syn-1 are fully soluble in chlorinated solvents whereas those of anti-1 are only partially soluble. The melting point of each diastereoisomer 1 was different: 218 °C for (AE)-syn-1, 227 °C for meso-syn-1, 345 °C for (AE)-anti-1 and 349 °C for meso-anti-1. These large differences between syn-and anti-1 could be attributed to the nature of Hbonded networks in solution and in the solid state that will be disclosed below.

DFT calculations at the M06-2X/6-311G (d,p) level of theory concerning the rotation of the C sp 2ÀC sp 2 bond (Figure 1, blue arrow) indicated that the syn/anti relative configurations are very stable with a high rotational barrier of 41.3 kcal mol À1 for syn-1 and 43.2 kcal.mol À1 for anti-1 (see the Supporting Information). This theorical result corroborates the thermal stability we observed when heating neat compounds 1 at 110 °C for 12 h without any structural modification.

Finally, racemic mixtures of syn-and anti-1 were separated by chiral HPLC allowing the obtention of the four chiral stereoisomers with excellent enantiomeric excess (ee) in the range of 97.5-99.5 % and very good yields: (R,R)-syn-1 in 31 % yield, (S,S)-syn-1 in 28 % yield, (R,R)-anti-1 and (S,S).anti-1 both in 20 % yield.

H-bonded atropisomers in the solid state

A few crystallographic structures of chiral ( -trifluoromethyl)alcohols are known. [28] Pioneering work of Berkessel pointed out that the racemic mixture of ( -trifluoromethyl)alcohols containing a ketone group crystallized as heterochiral dimers whereas the S enantiomer preferentially forms as zigzag ribbons. Herein we present the first example of bis-(trifluoromethyl)alcohols whose structures in the solid state are controlled by both relative and absolute configurations.

The syn/anti relative and the absolute configurations of stereoisomers 1 were unambiguously determined by X-ray diffraction on single crystals of (AE)-syn-1, meso-syn-1, (AE)-anti-1 and meso-anti-1.

Firstly, the single crystals were grown in non-competitive Hbonding solvents, by the slow diffusion of pentane into a solution of 1 in dichloromethane. Interestingly, the trifluoromethyl alcohols self-assemble through strong intermolecular H-bonds between the alcohol groups (distance HÀO•••H inferior to 2.01 Å). In addition, the nature of the H-bonding assemblies depends on the relative configuration: meso and racemic syn-1 form dimers (Figure 2a) whereas meso-anti-1 is organized as oligomers in the solid state (Figure 2b).

It is noteworthy that the crystallization of racemic syn-1 led to the co-crystallization of (R,R)-and (S,S)-syn-1 as self-sorted homochiral dimers: ((R,R)-syn-1) 2 and ((S,S)-syn-1) 2 (Figure 2a). This homochiral self-sorting observed by crystallization was also confirmed by molecular modeling for H-bonded dimer syn-1 (see the Supporting Information).

Secondly, due to their low solubility in apolar solvents, single crystals of anti-1 were also grown in dichloromethane with two drops of a H-bond-accepting solvent. Then, the crystallographic structures of (AE)-anti-1.3(DMSO) and meso-anti-1.1(CH 3 COCH 3 ) were obtained without any self-assembly of anthracene moieties. In these cases, isolated molecules of 1 presented strong H-bonds between the alcohol groups and 2d).

DMSO (distance HÀO•••O=S is 1.77-1.81 Å) or acetone (distance HÀO•••O=C is 1.94 Å) molecules, respectively (Figure
All crystallographic structures of atropisomers 1 have a common structural feature for the ÀCH(OH)CF 3 groups whatever the H-bonding network: each asymmetric carbon has a preferred arrangement in which the C sp 3ÀH bond is directed towards the anthracene core and thus presents less steric hindrance to the proximal bulky OH and CF 3 substituents, that seat above the 1,8 and 4,5 positions of anthracene (Figure 2d). This particular arrangement of ÀCH(OH)CF 3 groups of 1 in solution is discussed below.

Hydrogen-bonded atropisomers in solution

To investigate the structure and the H-bonding behavior of compounds syn-and anti-1 in solution, 1D and 2D NMR experiments were performed on the different atropisomers.

Diffusion-ordered spectroscopy (DOSY) experiments were conducted on atropisomers syn-and (AE)-anti-1 (1 mM) in CD 2 Cl 2 and indicated similar diffusion coefficients (D) of 1.00-1.03 × 10 À9 m 2 s À1 (Table 1). [29] Using the Stokes-Einstein equation, the calculated hydrodynamic volumes V H were found to be in the range of 550-600 Å 3 , which correspond to the volume of single molecules based on X-ray data (V X-ray = 530-600 Å 3 ).

Concerning meso-anti-1 which was only soluble in [D 6 ]DMSO, the calculated hydrodynamic V H in this solvent was slightly larger (830 Å 3 ) but was still in a good agreement with the volume occupied by a single molecule in the solid state compared to a H-bonded dimer (ca. 3800 Å 3 ).

In contrast to their behavior in the solid state, DOSY experiments clearly reveal that (AE)-syn, meso-syn-and (AE)-anti-1 at a concentration of 1 mM are not aggregated in CD 2 Cl 2 , an apolar aprotic solvent. This result is in line with the description of chiral (poly)fluorinated alcohols in which H-bonding behavior differs between solution and the solid state. [14a] Variable temperature 1 H NMR spectroscopy of racemic mixtures of syn-and anti-1 (2 mM, C 2 D 2 Cl 4 ) between 20 and 105 °C evidenced no changes in spectra, except complexation/ decomplexation of residual water molecules (see the Supporting Information). Along with the lack of any concentration dependence (0.2-10 mM, 3 ), this strongly suggests that 1 is present as discrete molecules in chlorinated solvents.

Solvation effects for racemic syn-and anti-1 were noticeable in the 1 H NMR spectra (see the Supporting Information). In particular, a marked anisochrony is seen for the aromatic protons when employing [D 8 ]toluene or chlorinated solvents (Figure 3). In C 2 D 2 Cl 4 , the anthracene protons of (AE)-syn-1 appeared as four distinct signals (H 1 /H 5 , H 4 /H 8 , H 3 /H 7 and H 2 /H 6 ) between 7.6 and 7.4 ppm, meanwhile those of less hindered dialdehyde syn-2 were characteristics of 9,10-substituted an-thracenes with two sets (dd) at 7.5 pm (central protons H 1 , 4 , 5 , 8 ) and at 7.4 ppm (peripheral protons H 2 , 3 , 6 , 7 ).

1D or 2D 1 H, [START_REF] Wilkinson | For reviews about fluorocyclizations[END_REF] F heteronuclear Overhauser effect spectroscopy (HOESY) [30] experiments conducted in CD 2 Cl 2 confirmed the spatial proximity of the CF 3 groups to two specific anthracene protons depending on each stereoisomer. In the case of (AE)-syn-1, the fluorine atoms were close to the H 1 and H 5 protons, without any proximity to the symmetric H 4 and H 8 protons (Figure 3). The 1 H, [START_REF] Wilkinson | For reviews about fluorocyclizations[END_REF] F HOESY analysis conducted on meso-syn-1, (AE)-anti-1 and meso-anti-1 confirmed a similar specific anisotropy of the anthranyl aromatic protons. However, these analyses conducted at 20 °C cannot rule out neither a restricted rotation of the C sp 3ÀC sp 2 bond nor a preferred conformation [6d,31] for the fluorinated group.

DFT calculations were carried out on 1 to consider the rotation of the C sp 3ÀC sp 2 bond. As the rotational barriers for the C sp 3ÀC sp 2 bond were found to be moderate for syn-1 (9.4 kcal.mol À1 ) and anti-1 (9.0 kcal.mol À1 ), these can be interpretated as free rotors. Among the optimized conformers for each stereoisomer, the more stable one consistently present a C sp 3ÀH bond pointing towards the anthracene group (Figure 2d) and is the major one by far, according to the Boltzmann populations: 83 % for (S,S) syn-1, 94 % for (R,S) syn-1 and 99 % for (S,S)-anti-1.

Notably, the latter syn structures do not favor any intramolecular H-bonds. The molecular modeling indicates a preferred orientation of each ÀCH(OH)CF 3 groups within atropisomers 1, with the C sp 3ÀH bond pointing inwards, probably due to electronic factors.

According to NMR studies, syn-and anti-atropisomers 1 behave as isolated molecules in CD 2 Cl 2 , without self-associations. NMR and DFT calculations strongly suggest that the ÀCH(OH)CF 3 asymmetric carbons adopt a preferred conformation (CÀH bond directed inwards, Figure 2d), that consequently positions the CF 3 groups in a precise quarter space depending on the relative and absolute configurations of each compound 1.

Controlled tandem reactivity

The dearomatization reactions of nonactivated arenes are of particular interest as these transformations provide complex and value-added compounds from cheap starting materials. [18,32] Concerning anthracene and 9,10-disubstituted anthracenes, the classical routes to their dearomatization (Scheme 2A and C) are either cycloadditions [33] or tandem reactions such as the nucleophilic dearomatization reaction [34,35] that consists of a nucleophilic addition on the arene followed by an electrophilic quench.

Concerning hindered 9,10-diphenylanthracenes, their dearomatization is mainly reported using a regioselective [4 + 2] cycloaddition with singlet oxygen on the 9,10 positions (Scheme 2C). To the best of our knowledge, an alternative oxidative dearomatization of 9,10-diphenylanthracenes has not been reported. We envisioned that the two-electrons oxidation of the anthracene core in 1 can provide two electrophilic sites Table 1. DOSY experiments and comparison between measured diffusion constants (D), [a] the calculated hydrodynamic volumes (V H ) and estimated volumes (V XRay ) from X-ray structures for atropisomers 1.

D [× 10 À9 m 2 s À1 ] V H [Å 3 ] V X-ray [Å 3
] single/dimer [b] (AE)-syn-1 on the C-9 and C-10 positions, which may then intramolecularly react with the fluorinated alcohol groups, possibly leading to dearomatized bis-spirocyclic compounds 4 whose stereochemistry could be controlled by a "syn/anti atropisomeric lock" (Scheme 2D, Table 2). Control experiments conducted using light activation revealed that anthranyl syn-and anti-1 were chemically stable under an irradiation at 365 nm in an inert atmosphere (Table 2, entry 1). The direct [START_REF] Gupta | For the photosensitization of dioxygen by DPA[END_REF] or photosensitized addition of singlet oxygen on 1 (syn or anti) in dichloromethane at 0 °C led to the formation of a mixture of unidentified products (Table 2, entries 2 and 3), instead of the formation of the expected 9,10or 1,4-endoperoxide. This uncontrolled reactivity probably results from the large steric hindrance on the anthranyl moiety in both syn-and anti-1 which leads to the uncontrolled reactivity of 1 O 2 , possibly through an intermediate peroxy radical that might then evolve into various oxidation products.

The thermal activation of atropisomers 1 was evaluated under two conditions (Table 2,entry 3). Under neat conditions at 300 °C, syn-1 is isomerized into anti-1, as already observed for less hindered ortho-substituted DPA. [26b] Interestingly, during these experiments, the compounds partially sublimated. In contrast, when isomers 1 were heated in refluxing C 2 H 2 Cl 4 (147 °C) in a pressure vial, under argon or air, two new products were cleanly formed: the 9,10-bis-spiro-isobenzofurananthracene 4 and a mono-oxidized product 5. Whatever the relative syn or anti configuration of the starting material 1, the products 4 and 5 were isomerized and isolated as a mixture of syn/anti isomers. For instance, (AE)-anti-1 was transformed into compound (AE)-4 in 35 % yield (syn/anti ratio = 54 : 46) and (AE)-5 in 41 % yield (syn/anti ratio = 51 : 49). This is thus the first report of a tandem oxidative dearomatization/spirocyclization of substituted anthracenes.

To understand the appearance of ketone 4, a control experiment was conducted using PhCH(OH)CF 3 in refluxing tetrachloroethane and the slow oxidation into PhCOCF 3 was observed (see the Supporting Information). This oxidation of an aromatic -trifluoromethyl alcohol thus occurs in the absence Table 2. Screening for reactivity of bis-trifluoromethyl-alcohols 1 under chemical, light, thermal and oxidative activations. [a] Conditions Reactant Products (yield) [b] 1 Ar, 365 nm syn-or anti-1 0 % conv. 21 O 2 O 2 , 365 nm syn-or anti-1 syn-or anti-1 n. d. [c] n. d. [c]

(R,R) syn-1 meso-syn-1 (AE)-anti-1 (R,R) anti-1 meso-anti-1
(R,R) syn-4 (99 %) meso-syn-4 (100 %) (AE)-anti-4 (100 %) (R,R) anti-4 (99 %) meso-anti-4 (100 %)

[a] all reactions were conducted with compound 1 in dichloromethane at room temperature, under an argon atmosphere (unless specified) and monitored by thin layer chromatography (TLC) until completion. [b] enantiomeric excess for products 4 are in the range of 99.3-99.8 %. [c] not determined (n. d.) as several unidentified compounds were formed. [d] the reaction was conducted in a pressure tube.

of the anthracene moiety and in the presence of residual oxygen.

To control the oxidative dearomatization and avoid thermal isomerization, we investigated the reactivity of 1 in the presence of organic oxidants capable of single-electron transfer (SET) at room temperature (Table 2, entry 4). Bobbitt's oxoammonium salt, [37] a TEMPO derivative triggered the expected tandem reaction transforming (AE)-anti-1 exclusively into the bis-spiro-isobenzofuranyl anthracene (AE)-4 in 100 % yield. To perform the reaction in 24 h, the Bobbitt's reagent was used in excess (10 equiv.) and in the presence of 2,6-ludidine (9 equiv.). Notably, the oxoammonium salt presented an unexpected chemoselectivity towards anthracene, as the reported oxidation of the fluorinated alcohol groups into the corresponding ketone was not observed. [38] Even more efficient, the reaction was completed in 1 h using "Magic Blue" (p-BrC 6 H 4 ) 3 NSbCl 6 ), [39] in an exact stoichiometric amount (2 equiv.) and led to bis-spiro-9,9',10,10'-anthracenes 4 in 100 % yield, with a retention of relative (syn/anti) and absolute stereochemistry (Table 2, entry 4). Importantly, the successful use of these two SET organic oxidants (oxoammonium salt and Magic Blue) represents a new extension of their known ability to functionalize aromatic and polyaromatic compounds such as porphyrinoids.

The proposed mechanism (Scheme 3) for this oxidative dearomatization is a double tandem reaction which consists of a single electron transfer (SET) followed by a nucleophilic addition. The first SET step between the oxidant and the anthracene core affords the corresponding anthranyl radical cation A that is trapped by an intramolecular nucleophilic addition of one trifluoromethyl-alcohol on position 9, to form 9,9'-spiro anthranyl radical B, which is probably too hindered to dimerize. The second SET step allows the formation of the stable cation C, which again was subjected to the same intramolecular nucleophilic addition of the second alcohol on position 10, offering the bis-spiro dearomatized product 4.

This proposed redox mechanism is fully consistent with the redox potentials of DPA derivatives, [39] Magic Blue [40] and TEMPO derivatives [41] under similar conditions. Concerning the stereochemistry of products 4, the atropisomerism in 1 maintains the nucleophilic alcohol on its half-space defined by the polyaromatic core, thus ensuring the preservation of relative syn or anti configuration. The absolute configurations are unchanged, as the reactivity does not involve the two benzylic carbons, in agreement with the proposed mechanism.

The bis-spiro-isobenzofuran anthracenes 4 have shown a remarkable chemical stability. Attempts to open the rings of the spiro-lactone groups under the classical conditions [18b] for the rearomatization of 9,10-bis-diols-9',10'-dihydroanthracenes with heating (Zn/AcOH, SnCl 2 /TFA or KI/AcOH) had no impact on 4. Indeed, the rearomatization of 9,10-dialkyl-9,10-dimethoxy anthracene derivatives was only described under harsh conditions (alkali metal or TiCl 3 /LiAlH 4 , 2 : 1) from robust precursors. [42] The double tandem oxidation/spirocyclization of atropisomers 1 provides an attractive alternative oxidative pathway to dearomatize anthracene derivatives, complementary to the well-known cycloadditions and purely ionic tandem reactions. In addition, this access to chiral fluorinated spiro-heterocycles 4 is a complementary route to the rare examples of fluorospirocyclizations [START_REF] Wilkinson | For reviews about fluorocyclizations[END_REF]20] and post-fluorination of spirocycles. [21,[START_REF] Wang | A photo-induced dearomative oxidation of gem-difluoroindoles was described without stereocontrol[END_REF]

Chiroptical properties

The access to tunable chiroptical properties is highly desirable to broaden the scope of applications, especially in the fields of detection and materials. [43] Classically, this is achieved by synthetizing a series of different structures. [23c,24,44] Recently, several switchable compounds were designed and offered insitu tunable and reversible chiroptical properties thanks to an external stimulus (light, pH, redox or molecular inputs). A third approach consists in designing tunable compounds, whose chiroptical properties can be modified by chemical postmodification. This strategy allows to consider the exploitation of two different chiroptic signatures, from a single compound. As such, the chiroptical properties of enantiopure compounds 1 and the corresponding dearomatized molecules 4 were evaluated to verify the impact of the stereospecific oxidative dearomatization reaction upon anthracene derivatives.

Each pair of enantiomers syn-1 and anti-1 (Figure 4A andB) displayed symmetric circular dichroism (CD) spectra, with opposite Cotton effects. The syn/anti relative configuration has no impact upon the CD spectra (see the Supporting Information). Remarkably, the chiroptical response of the anthracene core in the region of 340-400 nm is observed, even if moderate. Despite several bonds separating the anthracene core from the chiral carbon, the 3D architecture of compounds 1 allowed a proximity between both moieties as confirmed by solid state structures. The dimensionless anisotropic ratio g CD = / was calculated, with being the differential molar extinction coefficient of left and right circularly polarized light. The obtained g CD factors are around 7.5 × 10 À4 on the aromatic and Scheme 3. Proposed mechanism for the dearomative spirocyclization of anthranyl fluorinated alcohols syn-and anti-1 (shown: (R,R)-syn-1), with retention of the absolute and relative configurations. This reaction was controlled in the presence of the organic SET oxidants Bobbitt's reagent and Magic Blue in excellent yields (99-100 %).

2.3 × 10 À4 on the anthracene moieties (Figure 4A and the Supporting Information). For comparison, Pirkle's alcohol (2,2,2trifluoro-1-(9-anthryl)ethanol) presented a CD spectrum with a very low intensity without any obvious contribution from the anthranyl substituent (Figure 4B).

Circularly polarized luminescence (CPL) was detected from enantiomers of 1 upon excitation at 370 nm, which is specific to the anthracene core (Figure 4C). The luminescence dissymmetric factors g lum of syn-and anti-1 are about 5-6 × 10 À4 ( ex = 260 or 370 nm), which are similar to those obtained for small organic discrete polyaromatic compounds, that is, 10 À5 -10 À3 (see the Supporting Information). [45] As seen with CD spectra, these remarkable CPL values arise from the spatial proximity between the emitter (anthracene) and the chiral center (CH-(OH)CF 3 group) generated by this controlled atropisomeric architecture.

In comparison, the oxidation products 4 presented weak or absent chiroptical properties. As for Pirkle's alcohol, the CD contribution of the aromatics is very low in the 240-270 nm region and the one corresponding to the anthracene obviously disappears (Figure 4B). For CPL, no g lum was detected for the enantiomers of syn-4 either with an excitation on the aromatics or the anthracene. An obvious quasi extinction of the chiroptical properties (Figure 4B andC) was obtained by a stereospecific oxidative post-modification of 1.

Conclusion

Novel chiral syn-and anti-9,10-diphenylanthryl -trifluoromethyl alcohols have been synthesized. The vicinal arrangement of the groups allows the preferred orientation of the ÀCH(OH)CF 3 function in the different quadrants close to the anthracene core depending on the both relative and absolute configurations of each isomer. This structural feature was then exploited to vary the physical properties and reactivity. In the solid state, the compounds self-assemble into H-bonded syn homochiral dimers and anti ribbons, whereas in chlorinated solvents, they behave as discrete molecules. Taking advantage of the preorganization of anthracene and alcohols groups and their synergic reactivity, the dearomative oxidation of compounds 1 was stereospecifically achieved to 9,10-bis-spiro-isobenzofurananthracene 4 by using organic SET oxidants. The chiroptical properties (CD, CPL) of these -trifluoromethyl alcohols 1 and the corresponding products 4 were compared and showed an important modulation in intensity. Less demanding in synthesis efforts, this variation of chiroptics by an oxidant stimulus is an attractive way of tuning properties that could be implemented in the future.

Experimental Section

Deposition Numbers 2159931 (meso-syn-1), 2159932 (meso-anti-1), 2159933 ((AE)-syn-1), 2159949 (meso-anti-1.(CH 3 COCH 3 ), 2159954 ((AE)-anti-1. (DMSO) 3 ), 2159984 (meso-anti-4), 2159985 (meso-anti-3), 2159986 ((AE)-syn-4), 2159987 ((AE)-syn-3), 2160002 ((AE)-anti-5). contain the supplementary crystallographic data for this paper. These data are provided free of charge by the joint Cambridge Crystallographic Data Centre and Fachinformationszentrum Karlsruhe Access Structures service.

Figure 1 .

 1 Figure 1. -Trifluoromethyl alcohols A) as solvents in organic synthesis, B) as chiral solvating agents in analytical chemistry, C) as nucleophiles in a few cases, and D) within hindered chiral architectures 1 to explore their preorganization and unusual oxidative dearomatization into isobenzofurans 4. The chiroptical properties of 1 and 4 were also examined.

Figure 2 .

 2 Figure 2. Solid-state structures of syn-and anti-1 determined by X-ray diffraction showing H-bonded associations. a) Meso-syn-1 is organized as a dimer with off-set anthracene moieties, whereas (AE)-syn-1 forms homochiral dimers with anthranyl moieties shifted by 60°. b) Meso-anti-1 is arranged as H-bonded ribbons. c) In the presence of H-bond-donor solvents, the crystal structures are individual compounds (shown racemic and meso-anti-1). d) A systematic positioning of the ÀCH(OH)CF 3 group with the C sp 2ÀH pointing towards the anthracene core is observed in all crystals (only anti-1 is represented).

Figure 3 .

 3 Figure 3. Left: structure of (S,S)-syn-1 with the 1 H, 19 F NOE correlations; right: aromatic region of the 1D (blue) and 1D selective 1 H, 19 F HOESY (red) NMR spectra of (AE)-syn-1 (400 MHz, CD 2 Cl 2 , 2 mM).

Scheme 2 .

 2 Scheme 2. Routes for the dearomatization of anthracenes. a) Nucleophilic dearomatization followed by an electrophilic quench, b) acid/nucleophile tandem additions catalyzed by molybdenum, c) [4 + 2] cycloadditions with dienophiles, and d) the new access through a tandem oxidation/spirocyclization reaction when 9,10-diarylanthracene possesses -trifluoromethyl alcohol substituents in ortho' positions.

Figure 4 .

 4 Figure 4. A) Chiroptical properties of enantiopure anthracene derivatives before (compounds 1) and after (compounds 4) oxidation; B) UV-Vis. and CD spectra for syn-1 and syn-4, compared to Pirkle's alcohol. C) Photoluminescence and CPL spectra of enantiopure syn-1 at an excitation wavelength of 370 nm. All spectra were recorded at 17 M in dichloromethane at 20 °C, under argon.

Scheme 1. The two-step synthesis of chiral ortho,ortho'-( -trifluoromethyl)alcohols implemented on syn-and anti-9,10-diphenylanthracenes 1. A nonsymmetric induction is observed for syn-1.


Lauric Cécillon
email: cecillon@cetiom.fr

Nathalie Cassagne

Sonia Czarnes

Raphaël Gros

Jean-Jacques Brun

Variable selection in

Keywords: NIRS, PLSR, Wavelength selection, Soil biogenic structures, Soil macrofauna, Microbiological activity

come

Near infrared reflectance spectroscopy (NIRS) is a rapid and non-destructive analytical technique involving diffuse reflectance measurement in the near infrared (NIR) region (1000-2500 nm). NIR spectra depend on the number and type of chemical bonds in the analysed material [START_REF] Foley | Ecological applications of near-infrared spectroscopy -a tool for rapid, cost-effective prediction of the composition of plant and animal tissues and aspects of animal performance[END_REF]. NIRS is now widely used to predict soil carbon and nitrogen content (e.g. [START_REF] Cécillon | Near-infrared reflectance spectroscopy (NIRS): a practical tool for the assessment of soil carbon and nitrogen budget[END_REF], usually using partial least squares regression (PLSR) applied over the whole spectrum which is not an optimal solution according to various chemometric studies (e.g. [START_REF] Roger | Using genetic algorithms to select wavelengths in nearinfrared spectra : application to sugar content prediction in cherries[END_REF]. Recent developments of NIRS in Soil Ecology have shown its efficiency to predict soil biological properties including microbial biomass, potential nitrification, soil enzyme activities [START_REF] Reeves | Near infrared reflectance spectroscopy for the determination of biological activity in agricultural soils[END_REF][START_REF] Cohen | Visible-near infrared spectroscopy for rapid, nondestructive assessment of wetland soil quality[END_REF][START_REF] Terhoeven-Urselmans | Usefulness of nearinfrared spectroscopy to determine biological and chemical soil properties: Importance of sample pre-treatment[END_REF] and to discriminate soil biogenic structures (e.g. earthworm casts) from surrounding soil [START_REF] Hedde | Specific functional signature in soil macro-invertebrates biostructures[END_REF][START_REF] Velasquez | This ped is my ped: Visual separation and near infrared spectra allow determination of the origins of soil macroaggregates[END_REF]. Hitherto, the predictive capacity of NIRS for biological attributes of soil biostructures has not been assessed. In addition, the prediction of soil microbiological activity like soil enzyme activities with NIRS is not well-established and needs more research.

The objectives of this study were to assess (i) the potential of NIRS to predict various biological attributes of soil and earthworm casts; (ii) the efficiency of selecting the most important NIR wavelengths (process variables which really have an effect on the response) for improving PLSR models.

A sample set was collected in the Maures mountains (Var, France) during spring 2007 in 25 plots depicting a heterogeneous mosaic of Mediterranean forests. This mosaic is representative of a large range of Mediterranean forest ecosystems generated by various wildfire frequencies. The sample set comprises as much samples of earthworm casts (sampled at the soil surface) as topsoil samples (0-5 cm) which could contain some below-ground casts. In each plot, one composite sample was made for topsoil and another one for casts, resulting in a total of 49 samples (Table 1).

Casts were mainly produced by Nicodrilus nocturnus, a deep-burrowing species found in these forest soils classified as Cambisols (IUSS Working Group WRB, 2006). Organic carbon, total nitrogen content and pH were determined from conventional analyses. Biological analyses included microbial carbon (C mic ), two extracellular enzymes (FDA hydrolase, cellulase), potential nitrification and potential denitrification. Microbial-to-organic carbon ratio (C mic :C org ) was computed from obtained data. The soil analyses performed and their basic statistics are summarised in Table 1. Dry-sieved (2 mm) samples were ground (0.25 mm) to obtain homogeneous powders for NIRS analysis. Diffuse reflectance measurements (1000-2500 nm) were carried out using a Fourier-transform NIR spectrophotometer Antaris II (Thermo electron).

Each spectrum comprised 32 averaged scans of the sample. Data were collected at a 0.5 nm resolution resulting in 6224 absorbance values per spectrum.

Inference from spectra was done using PLSR which tackles the autocorrelation of spectral data [START_REF] Tenenhaus | La régression PLS[END_REF] with all available samples (casts and soils), since comparing topsoils and biogenic structures was beyond the scope of this paper. Each PLSR model was tuned in order to minimize the root-mean-square error of cross-validation (RMSECV), to maximize the Q² value (cross-validated R², which gives the predicting ability of the model) and the ratio of performance-todeviation (RPD) obtained from a full-model leave-one-out cross-validation (X-Val). When using small data sets (40-120 samples) in quantitative multivariate modelling, X-Val gives the best estimate of the predictive performance of an obtained model [START_REF] Martens | Validation and verification of regression in small data sets[END_REF].

Tuning options included removing outliers detected from a principal component analysis (PCA) on NIR data, multiplicative scatter correction (MSC), computing derivatives of spectra and selecting the most important NIR wavelengths using the variable importance on the projection (VIP) method [START_REF] Tenenhaus | La régression PLS[END_REF]. The VIP method computes a score for each wavelength. VIP scores obtained by the PLSR correspond to an importance measure of each explanatory variable (i.e. wavelength).

Since the average of squared VIP scores equals 1 [START_REF] Chong | Performance of some variable selection methods when multicollinearity is present[END_REF], only influential wavelengths with a VIP score greater than 1 were kept in the model. A new PLSR was then performed with selected wavelengths. Statistical treatments were conducted using R software version 2.6.1 (R Development Core Team, 2007) with ade4 package for PCA [START_REF] Chessel | The ade4 package -I : One-table methods[END_REF], signal package for computing derivatives with Savitzky-Golay filters [START_REF] Fearn | Savitzky-Golay filters[END_REF] and pls package for PLSR [START_REF] Mevik | The pls package: principal component and partial least squares regression in R[END_REF] with the VIP algorithm of [START_REF] Chong | Performance of some variable selection methods when multicollinearity is present[END_REF].

PCA revealed no spectral outliers (data not shown). Some concentration outliers were removed for FDA hydrolase, potential denitrification and C mic :C org (Table 1). For all properties, second derivative of spectra without MSC was chosen as spectral pretreatment. Three PLSR models and their X-Val statistics for each of the six biological properties are presented (Table 2).

They correspond to the same spectral processing with zero, one or two successive steps of variable selection with the VIP method.

Predictive performance of PLSR models using the whole spectrum was very poor with RPD close to 1 for most properties (Table 2, Figure 1). One step of variable selection generated reliable models for all properties with RPD above 1.9 (Table 2). Best models were computed with a second step of variable selection (Figure 2); C mic and potential denitrification obtaining "reasonable" [START_REF] Williams | What does the raw material have to say?[END_REF] X-Val statistics with Q² above 0.9 and RPD above 3 (Table 2). These results are in accordance or even better than previously published studies with Q² usually ranging from 0.6 to 0.8 for soil biological properties (e.g. [START_REF] Reeves | Near infrared reflectance spectroscopy for the determination of biological activity in agricultural soils[END_REF][START_REF] Rinnan | Application of near infrared reflectance (NIR) and fluorescence spectroscopy to analysis of microbiological and chemical properties of arctic soil[END_REF]. They can be considered as positive for such a heterogeneous sampling design, as confirmed by the strong standard deviation of reference data (Table 1).

Implementing the VIP method resulted in the selection of intervals rather than individual wavelengths with an important contribution from the spectral region 1750-2500 nm (Figure 3). For all biological properties, selected intervals contained wavelengths assigned to organic matter (e.g. 1725 nm, 1930 nm) by [START_REF] Ben-Dor | The reflectance spectra of organic matter in the visible near-infrared and short wave infrared region (400-2500 nm) during a controlled decomposition process[END_REF] but also most signature wavelengths of microbial biomass identified by [START_REF] Vaidyanathan | Fundamental investigations on the nearinfrared spectra of microbial biomass as applicable to bioprocess monitoring[END_REF], except for cellulase (Table 3). Thus, our models for biological properties could reflect not only the changes in macronutrient availability (amount and quality) which control soil biological activity (Palmborg and Nordberg, 1996;[START_REF] Rinnan | Application of near infrared reflectance (NIR) and fluorescence spectroscopy to analysis of microbiological and chemical properties of arctic soil[END_REF] but also information directly related to microbial biomass even if mean C mic represents only 1.5 mg g -1 soil (Table 1).

Improvement of models with variable selection is obvious and should be generalized when using PLSR. Despite the increasing use of NIRS in Soil Science, this strategy has been applied only once: Palmborg and Nordberg (1993) selected 15 wavelengths within NIR spectra of soil samples according to their regression coefficients. But selecting intervals is more robust than working with few selected variables [START_REF] Höskuldsson | Variable and subset selection in PLS regression[END_REF]. Our approach of wavelength selection in NIR spectra with the VIP method provided PLSR models with a reliable prediction performance for microbiological properties of soil and earthworm casts. Next step will be the validation of these models on independent samples. One possible application could be mapping soil conditions with fair predictive precision. Abbreviations: Abbreviations: * : similar results were obtained for all properties

  h = number of latent variables used to perform PLSR Whole NIR = PLSR performed over the whole NIR spectrum VIP 1 = PLSR performed after one step of variable selection with the VIP method VIP 2 = PLSR performed after two steps of variable selection with the VIP method Q² = coefficient of determination after a full-model leave-one-out cross-validation RMSECV = root mean squared error of cross-validation RPD = ratio of performance-to-deviation (calculated as RPD = SD / RMSECV)

Y

  Figure 1: Scatter plots of predicted vs. actual values for biological properties using the whole NIR region. The dashed lines indicate 1:1.

Figure 2 :

 2 Figure 2: Scatter plots of predicted vs. actual values for biological properties after two steps of variable selection with the VIP method (VIP 2). The dashed lines indicate 1:1.

Figure 3 :

 3 Figure 3: Relative frequency (%) of selected wavelengths within NIR intervals after one (VIP 1) or two (VIP 2) steps of variable selection for microbial carbon * (C mic ).

Table 2 :

 2 Prediction of microbiological properties with NIRS : X-Val results

	Property	Variable selection	Selected wavelengths (%)	h Q² RMSECV RPD
	Potential denitrification Whole NIR	100	3 0.38	0.16	1.3
		VIP 1	34	4 0.87	0.07	2.8
		VIP 2	16	4 0.91	0.06	3.4
	Potential nitrification	Whole NIR	100	1 0.43	0.20	1.3
		VIP 1	26	3 0.70	0.14	1.9
		VIP 2	13	3 0.89	0.09	3.1
	Microbial carbon (C mic ) Whole NIR	100	2 0.55	0.43	1.5
		VIP 1	28	3 0.83	0.26	2.5
		VIP 2	13	4 0.90	0.20	3.3
	C mic :C org ratio	Whole NIR	100	2 0.14	0.0064	1.0
		VIP 1	35	3 0.73	0.0031	1.9
		VIP 2	16	4 0.84	0.0024	2.5
	Cellulase	Whole NIR	100	3 0.18	0.0083	0.9
		VIP 1	33	4 0.72	0.0041	1.9
		VIP 2	12	6 0.81	0.0034	2.3
	FDA hydrolase (Fdase) Whole NIR	100	3 0.03	0.00029	1.1
		VIP 1	34	3 0.78	0.00014	2.2
		VIP 2	15	3 0.84	0.00012	2.6

Table 3 :

 3 Selection of signature wavelengths of microbial biomass after two steps of the VIP

	method (VIP 2)
			Main absorbance peaks of biomass * (nm)
	Property	1580 1690 1730 1760 2060 2270 2310 2350
	Potential denitrification	N	Y	Y	N	Y	Y	Y	Y
	Potential nitrification	N	N	Y	Y	Y	Y	Y	N
	Microbial carbon (C mic )	N	N	Y	N	Y	Y	Y	Y
	C mic :C org ratio	N	Y	Y	N	Y	Y	Y	Y
	Cellulase	Y	N	N	N	N	N	N	N
	FDA Hydrolase (Fdase)	Y	N	Y	N	Y	Y	Y	Y

Acknowledgements

This study was part of the Forest Focus IRISE project (Coordinator: M. Vennetier, http://irise.mediasfrance.org/). Authors are grateful to J. Grenet, S. Figarol, F. Ruaudel, N. Guillaumaud, Y. Paillet, B. Couchaud for their technical support and to participants of Grenoble workshop "NIRS in Soil Science" (http://spirsolgrenoble2007.free.fr/) for their valuable comments. This work was supported by the French Agency for Environment and Energy Management (ADEME) and Cemagref.

Cellulase (U 1 g -1 dw) 49 0.015 0.007 [START_REF] Deng | Cellulase activity of soils[END_REF] FDA hydrolase (U 2 g -1 dw) 48 (1) 0.0012 0.0003 [START_REF] Adam | Development of a sensitive and rapid method for the measurement of total microbial activity using fluorescein diacetate (fda) in a range of soils[END_REF] Abbreviations:

n = number of samples used to calculate summary statistics and to perform PLSR SD = standard deviation; FDA = fluorescein di-acetate; dw = dry weight equivalent;

U 1 = µmol of glucose released per minute; U 2 = µmol of fluorescein released per minute * : the number of concentration outliers removed is in parentheses. In one of the 25 plots, no earthworm casts could be collected above-ground, hence a total of 49 samples ** : only 36 measurements were available for potential nitrification


The survival probability of a weakly subcritical multitype branching process in iid random environment

Keywords: multitype branching process, survival probability, random environment, product of random matrices, weakly subcritical case. AMS classification: Primary 60J80; secondary 60F17, 60K37

We study the asymptotic behavior of the probability of non extinction of a weakly subcritical multitype branching process in iid random environment. Under suitable assumptions, the survival probability is of order of ρ n n -3/2 for some ρ ∈ (0, 1) to specify.

1 Introduction and main results

Historical context

Branching processes in random environment (BPRE) is an important subject of the theory of branching processes. This model was introduced at the beginning of the 1960's; it takes into account the random fluctuations of the reproduction laws over time. Various properties of this and more general models of BPRE have been analyzed these last decades, through a large number of varied articles. There exists a relatively complete description of the basic properties of many models of BPRE either under the annealed approach or the quenched one. In particular, the behavior of the single type BPRE is mainly determined be the properties of the so-called "associated random walk" constructed by the logarithms of the expected population sizes of particles of the respective generations; this random walk divides in a natural way the set of all single type BPRE into the classes of supercritical, critical and subcritical processes (see [START_REF] Dyakonova | The survival probability for a class of multitype subcritical branching processes in random environment[END_REF] for more detail). Notice that a precise estimate for single type branching process in finite state space markovian environment does exist [START_REF] Grama | The survival probability of critical and subcritical branching processes in finite state space Markovian environment[END_REF]; the fact that the underlying Markov chain is finite is essential in this study, this hypothesis is not relevant in the case of product of random matrices.

Analogue statements known for the single type case do exist for the multitype BPRE with finitely types p ≥ 2. The role of the associated random walk is played in this case by the logarithms of the norms of products of the mean random matrices associated to the environment. Recent useful results on fluctuations of ordinary random walks on the real line have been extended in terms of p × p random matrices. In particular, the description of the asymptotic behavior of the survival probability for the critical multitype BPRE under general conditions is done in [START_REF] Page | The survival probability of a critical multitype branching process in iid random environment[END_REF] and [START_REF] Dyakonova | Multitype branching processes in random environment: probability of survival for the critical case[END_REF] where it is proved, under natural and quite general assumptions, that the probability of survival up to time n is of order n 1/2 for a multitype branching process evolving in iid random environment. The case of supercritical multitype branching processes is studied in [START_REF] Grama | Convergence in L p for a supercritical multitype branching process in a random environment[END_REF], where a Kesten-Stigum type theorem is established. At last, as in the single type case, subcritical BPRE are divided in three categories: strongly, intermediately and weakly subcritical. The first two subcases are the subject of recent work: in the strongly subcritical subcase, the annealed survival probability at time n is equivalent to ρ n [START_REF] Vatutin | Multitype subcritical branching processes in a random environment[END_REF] while in the intermediately subcritical subcase it is of order ρ n n -1/2 [START_REF] Dyakonova | The survival probability for a class of multitype subcritical branching processes in random environment[END_REF], where ρ ∈ (0, 1) is a constant defined in terms of the Lyapunov exponent for products of the mean value matrices of the laws of reproduction of particles.

In the present paper, we establish a rough asymptotic estimate of the annealed survival probability at time n in the weakly subcritical case. This case was resistant since there was no local limit theorem for the norm of products of random matrices conditioned to remain greater than 1 until time n. Such a local limit theorem is obtained in [START_REF] Peigné | A conditioned local limit theorem for non negative random matrices[END_REF], its proof can be adapted to the present context to show that the probability of survival at time n is of order ρ n n -3/2 where ρ ∈ (0, 1).

Notations and assumptions

We fix an integer p ≥ 2 and denote R p (resp., N p ) the set of p-dimensional column vectors with real (resp., non negative integers) coordinates; for any column vector x = (x i ) 1≤i≤p ∈ R p , we denote x the row vector x := (x 1 , . . . , x p ). Let 1 (resp., 0) be the column vector of R p whose all coordinates equal 1 (resp., 0). We fix a basis {e i ,

1 ≤ i ≤ p} in R p and denote | • | the corresponding L 1 norm.
The multitype Galton-Watson process is a temporally homogeneous vector Markov process (Z n ) n≥0 whose states are column vectors in N p . For any 1 ≤ i ≤ p, the i-th component Z n (i) of Z n may be interpreted as the number of objects of type i in the n-th generation.

A multivariate probability generating function (f (i) ) 1≤i≤p , denote by f , is a function from (R + ) p to R + defined by

f (i) (s) = α∈N p p (i) (α)s α , for any s = (s i ) 1≤i≤p ∈ (R + ) p , where (i) α = (α i ) i ∈ N p and s α = s α 1 1 . . . s αp p ; (ii) p (i) (α) = p (i) (α 1 , . . . , α p )
is the probability that an object of type i has α 1 children of type 1, . . . , α p children of type p.

From now on, the multivariate generating function f is a random variable defined on a probability space (Ω, F, P); let f= (f n ) n≥0 be a sequence of iid random copies of f , called the (random) environment. The Galton-Watson process with p types of particles in the random environment f describes the evolution of a particle population Z n = (Z n (i)) 1≤i≤p for n ≥ 0.

For any s = (s

i ) 1≤i≤p , 0 ≤ s i ≤ 1, E s Zn |Z 0 , . . . , Z n-1 , f 0 , . . . , f n-1 = f n-1 (s) Z n-1 which yields E s Zn |Z 0 = ẽi , f 0 , . . . , f n-1 = f (i) 0 (f 1 (. . . f n-1 (s) . . .)). 2
The probability of non extinction q (i)

n at generation n given the environment f when the ancestor is of type i is

q (i) n := P(|Z n | > 0 | f (i) 0 , f 1 , . . . , f n-1 ) = 1 -f (i) 0 (f 1 (. . . f n-1 (0) . . .)) = ẽi (1 -f 0 (f 1 (. . . f n-1 (0) . . .))), so that E[q (i) n ] = P(Z n = 0|Z 0 = ẽi ) = E[ẽ i (1 -f 0 (f 1 (. . . f n-1 (0) . . .)))].
The asymptotic behavior of the quantity above is controlled by the mean of the offspring distributions. We assume that the offspring distributions have finite first and second moments; the generating function f = (f (i) ) 1≤i≤p , is thus C 2 -functions on [0, 1] p and we introduce:

(i) the mean matrix M = (M (i, j)) i,j = ∂f (i) ∂s j (1)
i,j

;

(ii) the Hessian matrices

B (i) = ∂ 2 f (i) ∂s k ∂s ℓ (1) k,ℓ
, for i = 1, . . . , p.

We denote by (M n ) n≥0 (resp. (B

(i) n ) n≥0 , i = 1, . . . , p
) the sequence of iid random mean matrices (resp. Hessian matrices) associated with the sequence (f n ) n≥0 . These matrices belong to the semigroup S of p × p matrices with positive entries; we endow S with the L 1 -norm.

We assume that the distribution µ of M satisfies the following assumptions. P1 Moment assumption: E(|M |) < +∞. P2 Irreducibility assumption: The support of µ acts strongly irreducibly on the semigroup of matrices with non negative entries, i.e. there exists no affine subspaces A of R p such that A ∩ (R + ) p is non empty, bounded and invariant under the action of all elements of the support of µ.

P3 There exists B > 1 such that for µ-almost all M and any 1 ≤ i, j, k, ℓ ≤ p,

1 B M (k, ℓ) ≤ M (i, j) ≤ B M (k, ℓ). (1.1)
From now on, we denote by S B the subset of S of p × p-matrices M satisfying condition (1.1).

P4 Subcriticality: The upper Lyapunov exponent

γ µ := lim n→+∞ E[ln |M 0 • • • M n-1 |] n is negative. P5 There exists ε > 0 such that µ{M ∈ S : ∀x ∈ (R + ) p suchthat |x| = 1, ln |xM | ≥ ε} > 0 and µ{M ∈ S : ∀x ∈ (R + ) p suchthat |x| = 1, ln |M x| ≥ ε} > 0.
As it is usual in studying local probabilities, one has to distinguish between "lattice" and "non lattice" distributions. The "non lattice" assumption ensures that the process (ln |xM 0,n-1 |) n≥1 does not live in the translation of a proper subgroup of R; in the contrary case, when µ is lattice, a phenomenon of cyclic classes appears which involves some complications which are not interesting in our context. In section 2, we give a precise definition of this notion in the context of products of random matrices. P6 Non lattice assumption: The measure µ is non lattice.

We also introduce the following hypotheses concerning the environment f . P7 There exist ε 0 , K 0 > 0 such that for any i, j ∈ {1, . . . , p} and any n ≥ 0, (a)

P(Z n+1 (i) ≥ 2 | Z n = ẽj ) ≥ ε 0 , (b) P(|Z n+1 | = 0 | Z n = ẽj ) ≥ ε 0 , (c) E[|Z n+1 | 2 | Z n = ẽj ] ≤ K 0 .
Let us first explain some consequences of hypotheses P1 and P4 and the way three different subcases appear in the subcritical case. Using the standard subadditivity argument, one infers that, under P1, for any θ ∈ [0, 1] the limit

λ(θ) := lim n→+∞ E |M n-1 • • • M 0 | θ 1/n
< +∞ is well defined. The function Λ is the analogue of the moment generating function for the associated random walk in the case of branching processes in random environment with single type of particles. The function

θ → E[|M | θ ] is at least twice continuously differentiable on [0, 1[. We set: for θ ∈ [0, 1[, Λ(θ) := ln λ(θ).
This function is also twice continuously differentiable and strictly convex on [0, 1[ (see [START_REF] Buraczewski | On multidimensional Mandelbrot's cascades[END_REF], Proposition 3.1). There are three cases to consider. 1. Λ ′ (1) < 0 (hence Λ(1) < 0, by convexity). This case corresponds to the strongly subcritical case for single type branching processes, it is studied in [START_REF] Vatutin | Multitype subcritical branching processes in a random environment[END_REF] when p ≥ 2. Under suitable conditions, it holds

E[q (i) n ] = P(|Z n | > 0 | Z 0 = ẽi ) ∼ c i λ(1) n , i = 1, . . . , p
with c i > 0 and λ(1) ∈]0, 1[. 2. Λ ′ (1) = 0 (hence Λ(1) < 0). It corresponds to the intermediately subcritical case for single type branching processes, it is studied in [START_REF] Dyakonova | The survival probability for a class of multitype subcritical branching processes in random environment[END_REF] when p ≥ 2. Under suitable conditions, it holds the following rough estimate

E[q (i) n ] = P(|Z n | > 0 | Z 0 = ẽi ) ≍ c i λ(1) n √ n , i = 1, . . . , p with c i > 0 and λ(1) ∈]0, 1[. 3. Λ ′ (1)
> 0. This is the weakly subcritical case when p = 1 and this is the aim of the present article in the case when p ≥ 2, we only obtain a rough estimate.

We denote by θ ⋆ the unique value of

θ in ]0, 1[ such that Λ ′ (θ ⋆ ) = 0. We set ρ ⋆ = λ(θ ⋆ ); it is obvious that ρ ⋆ ∈]0, 1[.
Notation. Let c > 0 and φ, ψ be two functions of some variable u; we shall write φ

Main statement

Now, we may state the main result of this article which concerns a rough estimate of the probability of extinction in the weakly subcritical case.

Theorem 1.1 Assume that conditions P1-P7. Assume that Λ ′ (1) > 0 and let θ ⋆ be the unique

value of θ ∈]0, 1[ such that Λ ′ (θ ⋆ ) = 0.
Then, there exists positive constants c and C such that for

1 ≤ i ≤ p, c ρ n ⋆ n 3/2 ≤ P(|Z n | > 0 | Z 0 = ẽi ) ≤ C ρ n ⋆ n 3/2 with ρ ⋆ = λ(θ ⋆ ) ∈]0, 1[.

On products of positive random matrices

The random variables M n and

B (i)
n are iid with values in the semigroup S of p × p matrices with positive coefficients. We set, for any 0

≤ k ≤ n, M k,n = M k • • • M n and M n,k = M n • • • M k .
In order to control the asymptotic behavior of the matrices M n,0 (resp. M 0,n ), we study their action on the cone of column (resp. row) vectors with positive entries.

Let C (resp. C) be the set of column vector (resp. row vectors) in R p with positive entries. For any x ∈ C, we denote by x the corresponding row vector. We also set X := {x ∈ C, |x| = 1} and X = {x | x ∈ X}. We consider the following (left and right) actions of S:

• the linear action of S on C (resp. C) defined by (M, x) → M x (resp. (M, x) → xM ) for any M ∈ S and x ∈ C,

• the projective action of S on X (resp. X) defined by (M, x)

→ M • x := M x |M x| resp. (M, x) → x • M = xM |xM | for any M ∈ S and x ∈ X. For any M ∈ S, set v(M ) := min 1≤j≤d p i=1 M (i, j) . Then, for any x ∈ C, 0 < v(M ) |x| ≤ |M x| ≤ |M | |x|.
Similarly, noticing t M the transpose of the matrix M , it holds

0 < v( t M ) |x| ≤ |xM | ≤ |M | |x|. Consequently, hypothesis P5 holds when µ{M | v(M ) > 1} > 0 and µ{M | v( t M ) > 1} > 0.
In the context of branching processes, we focus on the right action of S on X. Therefore, we naturally endow X with a distance d which is a variant of the Hilbert metric: this distance is bounded on X and any element M in S acts on the metric space ( X, d) as a contraction. More precisely: for any x = (x i ) 1≤i≤p , ỹ = (y i ) 1≤i≤p ∈ X, we write m(x, ỹ) = min

x i y i i = 1, . . . , p such that y i > 0 and we set d(x, ỹ) := ϕ m(x, ỹ)m(ỹ, x) , where ϕ(s)

:= 1 -s 1 + s for 0 ≤ s ≤ 1. For M ∈ S, we set c(M ) := sup{d(x • M, ỹ • M ) | x, ỹ ∈ X}.
By [START_REF] Hennion | Limit theorems for products of positive random matrices[END_REF], the function d is a distance on X which satisfies the following properties.

Properties 2.1 1. sup{d(x, ỹ) | x, ỹ ∈ X} = 1.
2. For any M ∈ S,

c(M ) = max i,j,k,l∈{1,...,p} |M (i, j)M (k, l) -M (i, l)M (k, j)| M (i, j)M (k, l) + M (i, l)M (k, j) .
In particular, there exists

c B ∈ [0, 1) such that c(M ) ≤ c B < 1 for any M ∈ S B . 3. d(x • M, ỹ • M ) ≤ c(M )d(x, ỹ) ≤ c(M ) for any x, ỹ ∈ X and M ∈ S B . 4. c(M M ′ ) ≤ c(M )c(M ′ ) for any M, M ′ ∈ S B .
The following lemma plays a crucial role in controlling the behavior of the norm of the product of random matrices M 0 , M 1 , . . . (see Lemma 2.2 in [START_REF] Brofferio | On the affine recursion on M d + in the critical case[END_REF]). We denote by T B the closed semigroup generated by S B . Lemma 2.2 Under hypothesis P3, for any M ∈ T B and 1 ≤ i, j, k, ℓ ≤ p, M (i, j)

B 2 ≍ M (k, ℓ).
In particular, there exists δ > 1 such that for any M, N ∈ T B and x, y ∈ X,

1. |M x| δ ≍ |M | and |ỹM | δ ≍ |M |, 2. | yM x| δ ≍ |M |, 3. |M ||N | δ |M N | ≤ |M ||N |.
On the product space X × S, we define the function ρ by setting ρ(x, M ) := log |xM | for (x, M ) ∈ X × S. This function satisfies the following cocycle property:

ρ(x, M N ) = ρ(x, M ) + ρ(x • M, N )
for any M, N ∈ S and x ∈ X.

We achieve this paragraph with the definition of a "lattice" distribution µ (see hypothesis P6) in the context of products of random matrices. It may be stated as follows: the measure µ is lattice if there exist t > 0, ǫ ∈ [0, 2π[ and a function ψ :

X → R such that ∀g ∈ T µ , ∀x ∈ X, exp {itρ(g, x) -iǫ + i(ψ(g • x) -ψ(x))} = 1, (2.1)
where T µ is the closed sub-semigroup generated by the support of µ.

Exponential change of measures

Under hypothesis P1, for any θ ∈ [0, 1], we introduce the operator P θ defined by: for any bounded Borel function ϕ : X → C and x ∈ X,

P θ ϕ(x) := E[|xM | θ ϕ(x • M )].
For any n ≥ 1, it holds

P n θ ϕ(x) := E[|xM 0,n-1 | θ ϕ(x • M 0,n-1 )
]. The operators P θ are positive, they act continuously on the space C( X) of C-valued continuous functions on X endowed with the norm of uniform convergence | • | ∞ . Under condition P3, their spectral radius on C( X) equals λ(θ).

We denote by B θ the space of θ-Hölder continuous functions ϕ :

( X, d) → C such that m θ (ϕ) := sup x, ỹ∈ X x =ỹ |ϕ(x)-ϕ(ỹ)| d(x,ỹ) θ < +∞. Endowed with the norm | • | θ := | • | ∞ + m θ (•), the space (B θ , | • | θ ) is a C-Banach space. Furthermore, for any ϕ ∈ B θ and n ≥ 1, m θ (P n θ ϕ) ≤ E[c(M 0,n-1 ) θ |M 0,n-1 | θ ] m θ (ϕ) + 2 θ E[|M 0,n-1 | θ ] |ϕ| ∞ ≤ r n E[|M 0,n-1 | θ ] m θ (ϕ) + 2 θ E[|M 0,n-1 | θ ] |ϕ| ∞ with r = (c B ) θ where c B is introduced in Properties 2.1. By hypothesis P3, it holds c B ∈ [0, 1[, hence r < 1.
In other words, the operator P θ satisfies the Doeblin-Fortet inequality on B θ : for any ϕ ∈ B θ and n ≥ 1,

|P n θ ϕ| θ ≤ r n E[|M 0,n-1 | θ ]|ϕ| θ + C × E[|M 0,n-1 | θ ]|ϕ| ∞
for some constants C > 0 and r ∈]0, 1[. By [START_REF] Hennion | Sur un théorème spectral et son application aux noyaux Lipchitziens[END_REF], the operator P θ is quasicompact on B θ , with spectral radius λ(θ). By [START_REF] Guivarc | Spectral gap properties for linear random walks and Pareto's asymptotics for affine stochastic recursions[END_REF] and [START_REF] Buraczewski | On multidimensional Mandelbrot's cascades[END_REF], the real positive number λ(θ) is the unique eigenvalue of P θ with modulus λ(θ), it is simple and there exist a unique strictly positive function v θ ∈ B θ and a unique probability measure ν θ on X such that ν θ (v θ ) = 1 satisfying

ν θ P θ = λ(θ)ν θ and P θ v θ = λ(θ)v θ .
For any θ ∈ [0, 1], we thus introduce the new transition probability kernel Pθ on X defined by: for any bounded Borel function ψ : X → C and any x ∈ X,

Pθ ψ(x) := 1 λ(θ)v θ (x) E[|xM | θ v θ (x • M )ψ(x • M )].
By the above, this Markovian operator is quasicompact on X with 1 as a simple and isolated eigenvalue and without any other eigenvalue of modulus 1. The powers of Pθ are given by: for any n ≥ 1,

P n θ ψ(x) := 1 λ(θ) n v θ (x) E[|xM 0,n-1 | θ v θ (x • M 0,n-1 )ψ(x • M 0,n-1 )].
Let (X θ n ) n≥0 be the Markov chain on X with transition operator Pθ . For a ∈ R + fixed, any x ∈ X and any n ≥ 1, we set

S 0 = a, S n = S n (x, a) = a + ln |xM 0,n-1 |.
We associate to Pθ the Markov walk (X θ n , S n ) n≥0 on X × R with transition operator Pθ defined by: for any bounded Borel function Ψ : X × R → C and any (x, a) ∈ X × R,

Pθ Ψ(x, a) := 1 λ(θ)v θ (x) E[|xM | θ v θ (x • M )Ψ(x • M, a + ln |xM |)].
In order to study the behavior of the (X θ n , S n ) n≥0 , we introduce in a natural way the family of "Fourier operators" ( Pθ,t ) t∈R defined by: for any bounded Borel function ψ : X → C, for any θ ∈ [0, 1] and t ∈ R,

Pθ,t ψ(x) := 1 λ(θ)v θ (x) E[|xM | θ+it v θ (x • M )ψ(x • M )]. = 1 λ(θ)v θ (x) E[e (θ+it) ln |xM | v θ (x • M )ψ(x • M )]
In the sequel, it is thus natural to consider, for any x in X and a ∈ R, the probability measures P θ x,a (with associated expectation E θ x,a ) on (Ω, F) whose restriction to the σ-algebras σ(M 0 , . . . , M k-1 ), k ≥ 1 is given by: for any positive Borel function Φ :

S k → C, E θ x,a (Φ(M 0 , . . . , M k-1 )) := 1 λ k (θ)v θ (x) E[Φ(M 0 , . . . , M k-1 )e θ(a+ln |xM 0,k-1 )| v θ (x • M 0,k-1 )] (2.2)
and with the same conditional probability measure P M 0 =m 0 ,...,M k-1 =m k-1 as P, for any m 0 , . . ., m k-1 in S. Now, we fix θ = θ ⋆ . The matrices M n are iid with respect to the measure P but this property is no longer relevant with respect to P θ⋆ x,a . Neverthelesss, due to the quasicompactness of the operators Pθ⋆ established above, the fluctuations of the process (xM 0,n-1 ) with respect to P θ⋆ x,a may be controlled as in [START_REF] Pham | Conditioned limit theorems for products of positive random matrices[END_REF]; the Markov chain (M n , x•M 0,n-1 ) n≥1 driven by the Markov operator P θ⋆ falls within the scope of the article [START_REF] Grama | Limit theorems for Markov walks conditioned to stay positive under a spectral gap assumption[END_REF] and leads to the following statement.

We denote by τ x,a := min{n ≥ 1 | S n (x, a) ≤ 0} the first moment when the sequence (S n (x, a)) n≥0 enters the set ]-∞, 0]. Modifying in a natural way the arguments used in [START_REF] Pham | Conditioned limit theorems for products of positive random matrices[END_REF] one can conclude that under the conditions P1-P6 the function h θ⋆ : X × [0, +∞[→ [0, +∞[ specified by the equality

h θ⋆ (x, a) = lim n→+∞ E θ⋆ (S n (x, a), τ x,a > n)
satisfies the property E θ⋆ (h θ⋆ (x • M 0 , S 1 (x, a)); τ x,a > 1) = h θ⋆ (x, a). This function h θ⋆ satisfies the following properties (see Theorems 2.2 and 2.3 in [START_REF] Grama | Limit theorems for Markov walks conditioned to stay positive under a spectral gap assumption[END_REF]).

Proposition 2.3 There exist constants C > 1 and A > 0 such that for any x ∈ X and a > 0,

P θ⋆ (τ x,a > n) ∼ h θ⋆ (x, a) √ 2πn as n → +∞ with P θ⋆ (τ x,a > n) ≤ C h θ⋆ (x, a) √ n and C -1 max(1, a -A) ≤ h θ⋆ (x, a) ≤ C(1 + a).
In the sequel, we also need a control of the quantities

P θ⋆ (S n (x, a) ∈ [b, b + ℓ], τ x,a > n) for any interval [b, b + ℓ] included in R + .
The following statement, in the spirit of Stone's theorem for classical random walks, is stated in [START_REF] Grama | Conditioned local limit theorems for random walks defined on finite Markov chains[END_REF] for Markov walks over a finite state space and is meaningful when the parameter b is of order √ n. As announced in [START_REF] Grama | Conditioned local limit theorems for random walks defined on finite Markov chains[END_REF], this statement is valid in fact in a more general setting, namely when the underlying Markov chain is irreducible and its transition operator satisfies a spectral gap property. When µ is non lattice the spectral radius in B θ⋆ of the "Fourier operators" Pθ⋆,t , t ∈ R * , is strictly less than 1; this is crucial in the proof of the following statement.

Proposition 2.4 Under hypotheses P1-P6, for any fixed (x, a) ∈ X × R, ℓ > 0 and uniformly in b ∈ R + ,

lim n→+∞ nP θ⋆ (τ x,a > n, S n (x, a) ∈ [b, b + ℓ]) - 2 ℓ h θ⋆ (x, a) √ 2πσ 2 ϕ + b σ √ n = 0, where ϕ + (t) = te -t 2 /2 1 R + (t) is the Rayleigh density.
This is now of interest to introduce the following notations. In the sequel, we set ∆ := ln δ where δ is the constant which appears in Lemma 2.2; the proof is detailed in [START_REF] Peigné | A conditioned local limit theorem for non negative random matrices[END_REF]. Propositions 2.3 and 2.4 yield to following statement.

Corollary 2.5 Assume hypotheses P1-P6 hold. Then, there exists a positive constant C > 0 such that for any x ∈ X, a, b ≥ 0 and any ℓ > 0,

P θ⋆ (τ x,a > n, S n (x, a) ∈ [b, b + ℓ]) ≤ C n 3/2 h θ⋆ (x, a) hθ⋆ (x, b) ℓ. (2.3)
Furthermore, there exist constants c > 0 and ℓ 0 such that for ℓ > ℓ 0 and b ≥ B,

lim inf n→+∞ n 3/2 P θ⋆ x,a (τ x,a > n, S n (x, a) ∈ [b, b + ℓ]) ≥ c h θ⋆ (x, a) hθ⋆ (x, b) ℓ. (2.4)
Proof of corollary 2.5. When studying fluctuations of random walks (S n ) n≥1 with iid increments Y k on R p , p ≥ 1, we often reverse time as follows. For any 1 ≤ k ≤ n, the random variables S n -S k = Y k+1 + . . . + Y n and S n-k = Y 1 + . . . + Y n-k have the same distribution. In the case of products of random matrices, even when the M i are iid (which is not the case under P θ⋆ ), the cocycle property

S n (x) = S n (x, 0) = ln |xM 0,n-1 | = S k (x) + S n-k ( Xk ), with Xk = x • M 0,k-1 ,
decomposes the sum S n (x) into two terms which are not independent. Hence, the same argument as in the iid case cannot be applied directly. The fact that the matrices M 0 , M 1 , . . . belong to S B helps here: for any x, y ∈ X, we can compare the distribution of S n (x) -S k (x) to the one of ln |M n-k-1,0 y| (notice here that the non commutativity of the product of matrices forces us to consider in this last quantity the right linear action of the matrices M 0,n-k-1 ). It is the strategy that we apply to obtain the following result (see lemma 2.3 in [START_REF] Peigné | A conditioned local limit theorem for non negative random matrices[END_REF]) which is central in the sequel.

Lemma 2.6 For any x, y ∈ X, a, b ≥ 0 and ℓ > 0,

P(τ x,a > n, S n (x, a) ∈ [b, b + ℓ]) ≤ P(τ y, b+ℓ+B > n, Sn (y, b + ℓ + B) ∈ [a, a + ℓ + 2B]).
Similarly, for a ≥ ℓ > 2B > 0 and b ≥ ∆,

P(τ x,a > n, S n (x, a) ∈ [b, b + ℓ]) ≥ P(τ y, b-B > n, Sn (y, b -B) ∈ [a -ℓ, a -2B]).
This allows us to apply the same strategy as in [START_REF] Peigné | A conditioned local limit theorem for non negative random matrices[END_REF]

P θ⋆ (τ x,a > n, S n (x, a) ∈ [b, b + ℓ]) ≥ P θ⋆ (τ x,a > n, S m (x, a) ∈ [ √ n, √ 2n], S n (x, a) ∈ [b, b + ℓ]) ≥ k∈N √ n≤ k ≤ √ 2n-1 P θ⋆ τ x,a > n, k ≤ S m (x, a) ≤ k + 1, b ≤ S n (x, a) ≤ b + ℓ ≥ k∈N √ n≤ k ≤ √ 2n-1 X×[k,k+1] P θ⋆ (τ x,a > m, (x • M 0,m-1 , S m (x, a)) ∈ dx ′ da ′ ) P θ⋆ (τ x′ ,a ′ > n -m, b ≤ S n-m (x ′ , a ′ ) ≤ b + ℓ) ≥ k∈N √ n≤ k ≤ √ 2n-1 X×[k,k+1] P θ⋆ τ x,a > m, (x • M 0,m-1 , S m (x, a)) ∈ dx ′ da ′ ) P θ⋆ (τ x,b-∆ > n -m, a ′ -ℓ ≤ Sn-m (x, b -∆) ≤ a ′ -2∆) ≥ k∈N √ n≤ k≤ √ 2n-1 P θ⋆ τ x,a > m, k ≤ S m (x, a) ≤ k + 1 P θ⋆ τx,b-∆ > n -m, k + 1 -ℓ ≤ Sn-m (x, b -∆) ≤ k -2∆ (2.5)
By Proposition 2.4, there exists a constant C 0 > 0 such that when

√ n ≤ k ≤ √ 2n -1, lim inf n→+∞ nP θ⋆ τ x,a > m, k ≤ S m (x, a) ≤ k + 1 ≥ C 0 h θ⋆ (x, a)
and lim inf n→+∞

P θ⋆ τx,b-∆ > n -m, k + 1 -ℓ ≤ Sn-m (x, b -∆) ≤ k -2∆ ≥ C 0 (ℓ-2∆-1) hθ⋆ (x, b -∆) hθ⋆ (x,b)
.

Hence, inequality (2.5) yields, for n large enough,

n 2 P θ⋆ (τ x,a > n, S n (x, a) ∈ [b, b + ℓ]) ≥ C 2 0 2 ( √ 2n - √ n)(ℓ -2∆ -1) h θ⋆ (x, a) hθ⋆ (x, b),
which implies, for such n,

P θ⋆ (τ > n, S n ∈ [b, b + ℓ]) h θ⋆ (x, a) hθ⋆ (x, b) n 3/2 (ℓ -2∆ -1).
This achieves the proof of the lower bound (2.4) taking ℓ 0 := 4∆ + 2. ✷ As a direct consequence, we obtain the following "rough local limit theorem" for the process (S n (x, a)) n≥0 and its minimum under the probability P.

Corollary 2.7 Under hypotheses P1-P6, there exist positive constants c, C and ℓ 0 such that for any a ≥ ℓ ≥ ℓ 0 , b ≥ ∆ and any x ∈ X,

P(τ x,a > n, S n (x, a) ∈ [b, b + ℓ]) ≤ C ρ n ⋆ n 3/2 h θ⋆ (x, a) hθ⋆ (x, b) e θ⋆(a-b) ℓ (2.6)
and

lim inf n→+∞ n 3/2 ρ n ⋆ P(τ x,a > n, S n (x, a) ∈ [b, b + ℓ]) ≥ c h θ⋆ (x, a) hθ⋆ (x, b) e θ⋆(a-b-ℓ) ℓ. (2.7)
Similarly, Proof. By using definition of E θ⋆ (see (2.2)) and the fact that the function v θ⋆ is non negative and continuous on X, we may write a) ] with

P(τ x,a > n) ≤ C ρ n ⋆ n 3/2 e θ⋆a h θ⋆
P(τ x,a > n, S n (x, a) ∈ [b, b + ℓ]) ≍ ρ n ⋆ e θ⋆a E θ⋆ [τ x,a > n, e -θ⋆Sn(x,a) 1 [b,b+ℓ] (S n (x, a))] with e -θ⋆(b+ℓ) 1 [b,b+ℓ] (S n (x, a)) ≤ e -θ⋆Sn(x,a) 1 [b,b+ℓ] (S n (x, a)) ≤ e -θ⋆b 1 [b,b+ℓ] (S n (x, a)). Simi- larly P(τ x,a > n) ≍ ρ n ⋆ e θ⋆a E θ⋆ [τ x,a > n, e -θ⋆Sn(x,
E θ⋆ [τ x,a > n, e -θ⋆Sn(x,a) ] ≍ k≥0 e -θ⋆k P θ⋆ (τ x,a > n, S n (x, a) ∈ [k, k + 1[).
Inequalities (2.6), (2.7) and (2.8) follow, by applying Corollary 2.5.

✷

3 Proof of the main theorem

Expression of non extinction probability

For every realization of the environmental sequence f = (f 0 , f 1 , . . .) and 0 ≤ k ≤ n -1, we define

F k,n-1 = f k • . . . • f n-1 and F n,n-1 = Id. For 1 ≤ i ≤ p, we set F (i) k,n-1 = f (i) k • . . . • f n-1 . From the definition of the process (Z n ) n≥0 , it holds E[s Zn | Z 0 = ẽi , f 0 , . . . f n-1 ] = f (i) 0 • . . . • f n-1 (s) = F (i) 0,n-1 (s) so that E[q (i) n ] = P(|Z n | > 0 | Z 0 = ẽi ) = E[1 -F (i) 0,n-1 (0)].
For a generating function f with corresponding mean matrix M and a matrix a = (a(k, ℓ)) 1≤k,ℓ≤p with positive entries, we set, for s ∈

[0, 1] p , ψ f,a (s) := |a| |a(1 -f (s))| - |a| |aM (1 -s)| .
We fix i ∈ {1, . . . , p}. Let a (i) be the matrix with a (i) (i, i) = 1 and a (i) (k, ℓ) = 0 for all (k, ℓ) = (i, i). Using the definition of the functions ψ f,a , we write for n ≥ 1,

1 1 -F (i) 0,n-1 (s) = 1 |a (i) M 0,n-1 (1 -s)| + n-2 k=0 1 |a (i) M 0,k | ψ f k ,a (i) M 0,k-1 (F k+1,n-1 (s)),
where, as previously M 0,k = M 0 . . . M k for any k ≥ 0. Consequently,

E[q (i) n ] = E[1 -F (i) 0,n-1 (0)] = E         1 |a (i) M 0,n-1 | + n-2 k=0 1 |a (i) M 0,k | ψ f k ,a (i) M 0,k-1 (F k+1,n-1 (0)) η k,n-1     -1     = E         n-1 k=0 1 |a (i) M 0,k | ψ f k ,a (i) M 0,k-1 (F k+1,n-1 (0)) η k,n-1     -1    
where the random variable η k,n-1 is defined as above with the convention η 0,n-1 = 1. By Lemma 3 in [START_REF] Dyakonova | Multitype branching processes in random environment: probability of survival for the critical case[END_REF], it holds 0

≤ η k,n-1 ≤ Bp 2 p i=1 |B (i) k |/|M k | 2 .
In the sequel, we also need a control of the lower bound of the random variables η k,n-1 . This is why we introduce the restrictive assumption P7, which readily implies, by a straightforward computation, that the random variables are bounded from below and above by non negative constants (see Proposition 2.1 and Lemma 3.1 in [START_REF] Dolgopyat | Multitype branching processes with time-dependent branching rates[END_REF]). Therefore, under this additive assumption, it holds

E[q (i) n ] ≍ E 1 |M 0,0 | -1 + . . . + |M 0,n-1 | -1 . (3.1)
This is not an exact formula but it is sufficient in our context since we only have rough estimates in Corollary 2.7. We cannot improve our result as long as we do not have a precise estimate in the local theorem for the norms of products of random matrices conditioned to remain strictly greater than 1 until time n.

The change of measure P x,a

In the case when the Lyapunov exponent of the sequence (M n ) n≥0 equals 0, it is natural to introduce the following new probability measure P x,a on the canonical path space ((X × R) ⊗N , σ(X n , S n : n ≥ 0), θ) of the Markov chain (X n , S n ) n≥0 (4) , whose associated expectation E x,a is characterized by the following property: for any bounded and non negative Borel function ϕ on ( X × R) k+1 with compact support,

E x,a [ϕ(X 0 , S 0 , . . . , X k , S k )] := lim n→+∞ E x,a [ϕ(X 0 , . . . , S k ) | τ > n],
when the limit exists. The fact that the Lyapunov exponent of the sequence (M n ) n≥0 equals 0 implies that the probability P x,a (τ > n) decreases towards 0 as 1/ √ n (see [START_REF] Pham | Conditioned limit theorems for products of positive random matrices[END_REF]) and the expectation E x,a is given explicitely as follows

E x,a [ϕ(X 0 , S 0 , . . . , X k , S k )] = 1 h(x, a) E x,a [ϕ(X 0 , . . . , S k )h(X k , S k ), τ > k],
for some non negative function h (see [START_REF] Page | The survival probability of a critical multitype branching process in iid random environment[END_REF] for the details). Furthermore, if (Y k ) k≥0 is a sequence of bounded real-valued random variables, adapted to the filtration (F k ) k≥0 and which converges in L 1 ( P x,a ) to some random variable Y ∞ , then

lim n→+∞ E x,a Y n | τ > n = E x,a [Y ∞ ]. (3.2)
This last property yields to the speed of convergence to 0 of the extinction probability in a rather luminous way [START_REF] Page | The survival probability of a critical multitype branching process in iid random environment[END_REF], [START_REF] Page | Central limit theorem for a critical multitype branching process in random environment[END_REF].

In the present case, a similar construction does exist under the probability P θ⋆ ; in this case, the sequence (M n ) n≥0 is not iid but the choice of the parameter θ ⋆ is done in such a way the corresponding Lyapunov exponent Λ ′ (θ ⋆ ) equals 0. Therefore, we introduce the following family of probability measures P θ⋆

x,a (with corresponding expectation E θ⋆ x,a ) whose restriction to the σalgebras σ(X 0 , S 0 , . . . , X k , S k ), k ≥ 0 are defined as follows: for any bounded and non negative Borel function ϕ on (X × R) k+1 with compact support

E θ⋆ x,a [ϕ(X 0 , S 0 , . . . , X k , S k )] := 1 h θ⋆ (x, a) E θ⋆ x,a [ϕ(X 0 , . . . , S k )h θ⋆ (X k , S k ), τ > k] = E x,a ϕ(X 0 , . . . , S k )h θ⋆ (X k , S k )e θ⋆S k v θ⋆ (X k ), τ > k ρ k ⋆ h θ⋆ (x, a) v θ⋆ (x) (3.3)
The rough estimate of the quantity P x,a (τ > n) given by (2.8) immediately yields

E θ⋆ x,a [ϕ(X 0 , S 0 , . . . , X k , S k )] ≍ E x,a ϕ(X 0 , . . . , S k )h θ⋆ (X k , S k )e θ⋆S k , τ > k ρ k ⋆ h θ⋆ (x, a) . (3.4)
Let us conclude this paragraph noticing that the property (3.2) does not hold anymore, even in a weaker form, as the reader can see by following the demonstration of lemma 4.1 in [START_REF] Page | Central limit theorem for a critical multitype branching process in random environment[END_REF]. The strategy to get the speed of convergence towards 0 of the survival probability is thus different and uses a clever decomposition of the denominator of the right hand side in (3.1), inspired by [START_REF] Page | A local limit theorem on the semi-direct product of R * + and R d[END_REF] and [START_REF] Geiger | Limit theorems for subcritical branching processes in random environment[END_REF].

Proof of Theorem 1.1

We set here m

k := min(|M 0,0 |, . . . , |M 0,k-1 |) = min(| 1M 0,0 |, . . . , | 1M 0,k-1 |) for any k ≥ 1.
Hence (m n ≥ e -a ) = (τ 1,a > n) for any a ≥ 0 and (2.7) and (2.8) may be restated in particular as follows: for ℓ > ℓ 0 , lim inf

n→+∞ n 3/2 ρ n ⋆ P(τ x,a > n, S n (x, a) ∈ [b, b + ℓ]) ≥ ch θ⋆ (x, a) hθ⋆ (x, b) e θ⋆(a-b-ℓ) ℓ (3.5)
and

P(m n ≥ e -a ) ≤ C ρ n ⋆ n 3/2 e θ⋆a h θ⋆ ( 1, a), (3.6)
where c and C are two strictly positive constants.

Proof of the upper bound

It holds

P(|Z n | > 0 | Z 0 = ẽi ) = E[q (i) n ] ≤ P(m n ≥ 1) + E[q (i) n ; m n < 1] P(m n ≥ 1) + E[m n ; m n < 1] ≤ P(m n ≥ 1) + +∞ j=1 e -j+1 P(e -j ≤ m n < e -j+1 ) ≤ P(m n ≥ 1) + +∞ j=1 e -j+1 P(m n ≥ e -j ) ρ n ⋆ n 3/2 + ρ n ⋆ n 3/2 +∞ j=1
h θ⋆ ( 1, j) e (θ⋆-1)j by (3.6),

ρ n ⋆ n 3/2 since h θ⋆ ( 1, j) 1 + j and θ ⋆ < 1.
The upper bound is established.

Proof of the lower bound

We fix 0 < k < n/2 and decompose

n-1 ℓ=0 1 |M 0,ℓ | as A + B + C where A := k-1 ℓ=0 |M 0,ℓ | -1 , B := n-k-1 ℓ=k |M 0,ℓ | -1 and C := n-1 ℓ=n-k |M 0,ℓ | -1 .
It holds

P(|Z n | > 0 | Z 0 = ẽi ) = E[q (i) n ] ≥ E 1 1 + A + B + C ; m n ≥ 1 ≥ E 1 1 + A + C ; m n ≥ 1 -E [B; m n ≥ 1] .
First step: control of the term E

1 1 + A + C ; m n ≥ 1 It holds E 1 A + C ; m n ≥ 1 ≥ E[ 1 1 + A + n-1 ℓ=n-k |M 0,ℓ | -1 ; m n-k ≥ 1, |M n-k,n-1 | ≤ 1, . . . , |M n-1,n-1 | ≤ 1] since [m n-k ≥ 1, |M n-k,n-1 | ≤ 1, . . . , |M n-1,n-1 | ≤ 1] ⊂ [m n ≥ 1] ≥ E[ 1 2 + A + n-1 ℓ=n-k |M ℓ,n-1 | ; m n-k ≥ 1, |M n-k,n-1 | ≤ 1, . . . , |M n-1,n-1 | ≤ 1] since |M ℓ+1,n-1 | ≥ |M 0,ℓ | -1 , n -k ≤ ℓ ≤ n -2 and |M 0,n-1 | -1 ≤ 1 on the event [|M 0,n-1 | ≥ 1] ≥ E 1 2 + A + n-1 ℓ=n-k |M ℓ,n-1 | ; m n-k ≥ 1, 1 ≤ |M 0,n-k-1 | ≤ K, 1 K ≤ |M n-k,n-1 | ≤ 1, . . . , |M n-1,n-1 | ≤ 1 for any constant K > 1, ≥ E 1 2 + A + k ℓ=1 |M ′ ℓ,1 | ; m n-k ≥ 1, 1 ≤ |M 0,n-k-1 | ≤ K, 1 K ≤ |M ′ k,1 | ≤ 1, . . . , |M ′ 1,1 | ≤ 1 by setting (M n-k , . . . , M n-1 ) = (M ′ k , . . . , M ′ 1 ) and M ′ ℓ,1 = M ′ ℓ . . . M ′ 1 = E 1 2 + A + C ′ ; m n-k ≥ 1, 1 ≤ |M 0,n-k-1 | ≤ K, 1 K ≤ |M ′ k,1 | ≤ 1, M ′ k ≤ 1 with C ′ := k ℓ=1 |M ′ ℓ,1 | and M ′ k := max(|M ′ 1,1 |, . . . , |M ′ k,1 |) ≥ 1 2 E 1 1 + A × 1 1 + C ′ ; m n-k ≥ 1, 1 ≤ |M 0,n-k-1 | ≤ K, 1 K ≤ |M ′ k,1 | ≤ 1, M ′ k ≤ 1 ≥ 1 2 E 1 1 + A ; m n-k ≥ 1, 1 ≤ |M 0,n-k-1 | ≤ K × E 1 1 + C ′ ; 1 K ≤ |M ′ k,1 | ≤ 1, M ′ k ≤ 1
by independence of the random variables 1 1 + A 1 [m n-k ≥1,1≤|M 0,n-k-1 |≤K] and

1 1 + C ′ 1 [ 1 K |M ′ k,1 |≤1,M ′ k ≤1] .
Now, we assume ln K ≥ ℓ 0 , where ℓ 0 is defined in Lemma 2.6; it holds, on the one hand, lim inf

n→+∞ n 3/2 ρ n-k ⋆ E 1 1 + A ; m n-k ≥ 1, 1 ≤ |M 0,n-k-1 | ≤ K ≥ lim inf n→+∞ n 3/2 ρ n-k ⋆ E 1 1 + A ; m k ≥ 1, P xk ,a k (m n-2k • θ k ≥ 1, 1 ≤ |M 0,n-2k • θ k | ≤ K) with xk = 1 • M 0,k-1 and a k = ln |M 0,k-1 | ≥ ρ -k ⋆ E 1 1 + A ; m k ≥ 1, lim inf n→+∞ n 3/2 ρ n-2k ⋆ P xk ,a k (m n-2k • θ k ≥ 1, 1 ≤ |M 0,n-2k • θ k | ≤ K)
≥c h θ⋆ (x k ,a k ) hθ⋆ (x k ,0)e θ⋆ (a k -ln K) ln K >0 by (3.5)

≥ c ln K K θ⋆ ρ -k ⋆ E 1 1 + A |M 0,k-1 | θ⋆ ; m k ≥ 1, h θ⋆ ( 1 • M 0,k-1 , ln |M 0,k-1 |) ≍ E θ⋆ 1,0 [ 1 1+A ]
On the other hand, since the function hθ⋆ associated with the process (M ′ n,1 •x, b+ ln |M ′ n,1 x|) n≥1 satisfies hθ⋆ (x, b) ≍ 1 for b in a compact set, it holds x on R * + , it holds

ρ -k ⋆ E 1 1 + C ′ ; 1 K ≤ |M ′ k-1,1 | ≤ 1, M ′ k-1 ≤ 1 ≥ c (ln K) E θ⋆
E θ⋆ 1,0 1 1 + A ≥ 1 E θ⋆ 1,0 [1 + A] ≥ 1 E θ⋆ 1,0 1 + +∞ ℓ=0 | 1M 0,ℓ | -1
. and it suffices to check that E θ⋆

  c ψ (or simply φ ψ) when φ(u) ≤ cψ(u) for any value of u. The notation φ c ≍ ψ (or simply φ ≍ ψ)

  For any y ∈ X and b > 0, we set S0 (y, b) = b and Sn (y, b) := b -ln |M n-1,0 y| for n ≥ 1. We denote by τy,b := min{n ≥ 1 | Sn (y, b) ≤ 0} the first moment when the sequence ( Sn (y, b)) n≥0 enters the set ] -∞, 0]. Propositions 2.3 and 2.4 also hold for the process (M n-1,0 • y, Sn (y, b)) n≥1 and the stopping time τy,b , we denote by hθ⋆ the function analogous to h θ⋆ which appears in the corresponding statements for the process (M n-1,0 • y, Sn (y, b)) n≥1 .

(x, a) and lim inf n→+∞ n 3

 3 x,a > n) ≥ c e θ⋆a h θ⋆ (x, a).(2.8)

7 ) 1 ℓ=kE 1 ≤n-k- 1 ℓ=k 2 - 1 ≤n-k- 1 ℓ=k 2 -

 71112112 Second step: control of the term E [B;m n ≥ 1] E [B; m n ≥ 1] = n-k-|M 0,ℓ | -1 ; m n ≥ j≥0 j P |M 0,ℓ | -1 ∈ [2 j , 2 j+1 [; m n ≥ j≥0 j P(|M 0,ℓ | -1 ∈ [2 j , 2 j+1 [; m ℓ ≥ 1, |M 0,ℓ+1 | ≥ 1, . . . , |M 0,n-1 | ≥ 1) P(|M 0,ℓ | -1 ∈ [2 j , 2 j+1 [; m ℓ ≥ 1, |M ℓ+1,ℓ+1 | ≥ 2 -j , . . . , |M ℓ+1,n-1 | ≥ 2 -j ) P(|M 0,ℓ | -1 ∈ [2 j , 2 j+1 [; m ℓ ≥ 1)Last step: choosing k large enoughBy the above supn≥ n 3/2 ρ n ⋆ E [B; m n ≥ 1] -→ 0 as k → +∞.It remains to check that the factors ′ in the right hand side of (3.7) do not converge to 0 as k → +∞. By(3.4) and by convexity of the function x → 1

|( 1 + 0 1 1 +

 101 1M 0,ℓ | -1 < +∞. Indeed, for any ℓ ≥ 0,E θ⋆ 1,0 | 1M 0,ℓ | -1 E θ⋆ 1,0 |M 0,ℓ | -1 h θ⋆ (X ℓ+1 , S ℓ+1 ), τ > ℓ + 1 E θ⋆ |M 0,ℓ | -1 (1 + ln |M 0,ℓ |), m ℓ+1 ≥ 1 ≤ +∞ j=0(1 + j)e -j P θ⋆ (e j ≤ |M 0,ℓ | < e j+1 , m ℓ+1 ≥ 1) = P θ⋆ 1,0 (j≤S ℓ+1 <j+1,τ >ℓ+1) j) 2 e -j by inequality (2.3), which yields the expected result. The same argument works for the factor E θ⋆ 1,C ′ by considering the random variables Sℓ (1, 0) and τ1,0 instead of S ℓ ( 1, 0) and τ 1,0 .

  to prove Corollary 2.5. Let us propose a sketch of the argument. Proof. Inequality (2.3) is proved in [17] Corollary 3.7. The proof of the lower bound (2.4) is based on Proposition 2.4. We set m = ⌊n/2⌋ ; by the Markov property and Lemma 2.6,

Institut Denis Poisson UMR 7013, Université de Tours, Université d'Orléans, CNRS France. email: peigne@univ-tours.fr,

CERADE ESAIP, 18 rue du 8 mai 1945 -CS 80022 -49180 St-Barthélemy d'Anjou ; email: dpham@esaip.org

LAREMA UMR CNRS 6093, Angers, France.

θ denotes the shift operator on (X × R) ⊗N defined by θ (x k , s k ) k≥0 = (x k+1 , s k+1 ) k≥0 for any (x k , s k ) k≥0 in (X × R)⊗N