Mehdi Kabani
email: mehdi.kabani@cnrs.fr

Ronald Melki

The Yarrowia lipolytica orthologs of Sup35p assemble into thioflavin Tnegative amyloid fibrils

Keywords:

The translation terminator Sup35p assembles into self-replicating fibrillar aggregates that are responsible for the [PSI þ ] prion state. The Q/N-rich N-terminal domain together with the highly charged middle-domain (NM domain) drive the assembly of Sup35p into amyloid fibrils in vitro. NM domains are highly divergent among yeasts. The ability to convert to a prion form is however conserved among Sup35 orthologs. In particular, the Yarrowia lipolytica Sup35p stands out with an exceptionally high prion conversion rate. In the present work, we show that different Yarrowia lipolytica strains contain one of two Sup35p orthologs that differ by the number of repeats within their NM domain. The Y. lipolytica Sup35 proteins are able to assemble into amyloid fibrils. Contrary to S. cerevisiae Sup35p, fibrils made of fulllength or NM domains of Y. lipolytica Sup35 proteins did not bind Thioflavin-T, a well-known marker of amyloid aggregates.

Introduction

In yeast, the eukaryotic release factor Sup35p, required for translation termination [1e3], has prion properties responsible for the [PSI þ ] trait [4e6]. In [PSI þ ] cells, soluble Sup35p is dramatically depleted by its aggregation into self-replicating fibrillar protein assemblies. As a consequence, defective translation termination leads to increased nonsense suppression events [4e10].

The domains responsible for the translation termination and prion propagation functions within Sup35p are well defined [START_REF] Ter-Avanesyan | Deletion analysis of the SUP35 gene of the yeast Saccharomyces cerevisiae reveals two non-overlapping functional regions in the encoded protein[END_REF][START_REF] Kabani | Yeast prions assembly and propagation: contributions of the prion and non-prion moieties and the nature of assemblies[END_REF]. The N-terminal domain (spanning amino acid residues 1e123) is critical for prion propagation, driving the switch from the monomeric, functional [psi À ] state to the aggregated [PSI þ ] prion state. This domain is unusually rich in glutamine and asparagine residues, and contains several imperfect repeats of the aggregation-prone sequence PQGGYQQ-YN [START_REF] Tuite | Yeast prions and their prion-forming domain[END_REF]. The overexpression of the N-terminal domain in cells expressing full-length Sup35p triggers prion conversion [START_REF] Derkatch | Genesis and variability of [PSI] prion factors in Saccharomyces cerevisiae[END_REF], whereas replacing Sup35p by a variant containing only the C-terminal domain causes the irreversible loss of [PSI þ ] [15e17]. Various mutations within the N-terminal Q/N-rich region or the oligopeptide repeats in the prion domain have been previously shown to affect the propagation, strength and stability of [PSI þ ] [START_REF] Depace | A critical role for aminoterminal glutamine/asparagine repeats in the formation and propagation of a yeast prion[END_REF][START_REF] Shkundina | The role of the N-terminal oligopeptide repeats of the yeast Sup35 prion protein in propagation and transmission of prion variants[END_REF]. The highly charged middle domain (M, amino acids 124e253) is dispensable for prion propagation, yet it increases the stability of the protein, ensures steady [PSI þ ] propagation during cell division and is an interaction site for molecular chaperones [START_REF] Liu | Changes in the middle region of Sup35 profoundly alter the nature of epigenetic inheritance for the yeast prion [PSIþ[END_REF]. The translation termination function of Sup35p requires the highly conserved and compactly folded C-terminal GTPase domain (amino acids 254e685) [START_REF] Ter-Avanesyan | Deletion analysis of the SUP35 gene of the yeast Saccharomyces cerevisiae reveals two non-overlapping functional regions in the encoded protein[END_REF][START_REF] Salas-Marco | GTP hydrolysis by eRF3 facilitates stop codon decoding during eukaryotic translation termination, TL -24[END_REF]. We previously showed that point mutations within the C-terminal domain of Sup35p affect [PSI þ ] propagation [START_REF] Kabani | A mutation within the C-terminal domain of Sup35p that affects [PSIþ] prion propagation[END_REF].

Full-length Sup35p and its N and NM domains spontaneously assemble into protein fibrils in vitro [9,23e25]. Fibrils formed by Sup35 N or Sup35NM exhibit the characteristics of amyloids in that they are unbranched, have increased resistance to proteolysis, bind Congo red and Thioflavin T, exhibit a 4.7 Å reflection in X-ray fiber diffraction images and exhibit Fourier Transform Infrared (FT-IR) spectra dominated by a peak at 1620 cm À1 [START_REF] Glover | Selfseeded fibers formed by Sup35, the protein determinant of [PSIþ], a heritable prion-like factor of S. cerevisiae[END_REF][START_REF] King | Prioninducing domain 2-114 of yeast Sup35 protein transforms in vitro into amyloid-like filaments[END_REF][START_REF] Serio | Nucleated conformational conversion and the replication of conformational information by a prion determinant[END_REF][START_REF] Uptain | Strains of [PSI(þ)] are distinguished by their efficiencies of prion-mediated conformational conversion[END_REF]. We previously showed that Sup35NM and full-length Sup35p assemblies are different [START_REF] Krzewska | Biochemical and functional analysis of the assembly of full-length Sup35p and its prion-forming domain[END_REF].

Sup35p NM domains primary structure are highly divergent among yeasts. Nonetheless, the ability of Sup35p to convert to a prion form appears conserved among distantly related yeasts [28e31]. In particular, when expressed in S. cerevisiae, the Yarrowia lipolytica Sup35p stood out with an exceptionally high spontaneous prion conversion rate of 10 À1 compared to 10 À5 -10 À7 for other yeast orthologs [START_REF] Nakayashiki | Yeast [PSIþ] "prions" that are crosstransmissible and susceptible beyond a species barrier through a quasi-prion state[END_REF]. Thus, it is of interest to understand how the behavior of Y. lipolytica Sup35p observed in a heterologous cellular context translates in vitro in terms of fibrillar assembly properties.

Here, we show that different Y.lipolytica strains contain one of two Sup35p orthologs that differ by the number of repeats within their NM domain. The Y. lipolytica Sup35 proteins are able to assemble into protein fibrils that we show to be of amyloid nature by FT-IR spectroscopy. Interestingly, contrary to S. cerevisiae Sup35p, fibrils made of full-length or NM domains of Y. lipolytica Sup35 proteins did not bind Thioflavin-T, a well-known marker of amyloid aggregates.

Materials and methods

Nomenclature note

To avoid confusion between orthologs, the Saccharomyces cerevisiae Sup35p and Sup35NM proteins are hereafter named Sc.Sup35p and Sc.Sup35NM throughout the manuscript. The corresponding Yarrowia lipolytica orthologs are hereafter named Yl.Sup35p and Yl.Sup35NM.

Strains and plasmids

Plasmids pET15b-SUP35 and pET15b-SUP35NM allowing the overexpression of hexahistidine-tagged Sc.Sup35p and Sc.Sup35NM in Escherichia coli were previously described [START_REF] Krzewska | Molecular chaperones and the assembly of the prion Sup35p, an in vitro study[END_REF][START_REF] Krzewska | Biochemical and functional analysis of the assembly of full-length Sup35p and its prion-forming domain[END_REF]. The Yl.SUP35-A and Yl.SUP35NM-A coding sequences were obtained by PCR using genomic DNA from the Yarrowia lipolytica strains 136,463 and PO1d [START_REF] He | Two genes encode 7SL RNAs in the yeast Yarrowia lipolytica[END_REF][START_REF] Nicaud | Protein expression and secretion in the yeast Yarrowia lipolytica[END_REF]. The Yl.SUP35eB and Yl.SUP35NM-B coding sequences were obtained by PCR using genomic DNA from the Yarrowia lipolytica strain IAM-4948 (a generous gift from Dr. Yoshikazu Nakamura, University of Tokyo) [START_REF] Nakayashiki | Yeast [PSIþ] "prions" that are crosstransmissible and susceptible beyond a species barrier through a quasi-prion state[END_REF]. PCR fragments were cloned at the NdeI-BamHI sites in pET15b, generating the pET15b-Yl.SUP35-A, pET15b-Yl.SUP35eB, pET15b-Yl.SUP35NM-A and pET15b-Yl.SUP35NM-B plasmids, allowing the overexpression of corresponding hexahistidine-tagged proteins. All constructs were verified by DNA sequencing.

Assembly of Sup35p into protein fibrils

The Sup35 proteins were overexpressed in Escherichia coli strain BL21-CodonPlus, purified by affinity chromatography as previously described [START_REF] Krzewska | Molecular chaperones and the assembly of the prion Sup35p, an in vitro study[END_REF][START_REF] Krzewska | Biochemical and functional analysis of the assembly of full-length Sup35p and its prion-forming domain[END_REF], and stored at À80 C. At least two independent preparations of each protein were used throughout the study. Sup35 proteins were dialyzed for 2 h at 4 C against the assembly buffer (50 mM Tris.Cl, pH 8.0, 200 mM NaCl, 5% glycerol, 5 mM bmercaptoethanol, 10 mM MgCl 2 , 2 mM EGTA) and centrifuged for 5 min at 15,000 g and at 4 C. The supernatant was recovered, the protein concentration was adjusted to the desired concentration (typically 10 mM) in assembly buffer and the samples incubated at 6 C under very gentle agitation (<100 rpm). At regular time intervals, aliquots were removed from the assembly reaction and mixed with Thioflavin T (10 mM). Fluorescence was recorded with a Quantamaster QM 2000-4 spectrofluorometer (Photon Technology International, NJ) using excitation and emission wavelengths set at 440 and 480 nm.

Electron microscopy

Fibril preparations were imaged in a Jeol 1400 transmission electron microscope on carbon-coated grids (200 mesh) following negative staining with 1% uranyl acetate. Images were recorded with a Gatan Orius CCD camera (Gatan Inc, Pleasanton, CA, USA) and processed with the ImageJ software (NIH).

Fourier transform InfraRed spectroscopy (FTIR)

FTIR spectra recording and analysis was performed as described previously [START_REF] Kabani | A mutation within the C-terminal domain of Sup35p that affects [PSIþ] prion propagation[END_REF]. Fibrils were centrifuged for 20 min at 16,000 g and at 4 C then extensively washed with D 2 O. The spectra were recorded on a JASCO 660 Plus FTIR spectrometer equipped with a nitrogen-cooled MTC detector. The background consisted of D 2 O and water vapor. A total of 512 interferograms were collected with a resolution of 2 cm À1 . The sample chamber was continuously purged with CO 2 -free dry air. All the spectra were baseline-corrected, smoothed and normalized prior to further data processing. The amide I (1600-1700 cm À1 ) band of the spectra was fitted using a Gaussian species model centered at 1624, 1640, 1652, 1664, and 1678 cm À1 (during the fitting procedure, the peak positions were free to vary inside an interval width that was limited to 25 cm À1 while peak height was free).

Results

The Yarrowia lipolytica Sup35p A and B orthologs display different NM domains

We found that different Y. lipolytica strains encode one of two different SUP35 orthologs. The SUP35 ortholog we cloned from the 136,463 and PO1d strains, which are related to the W29 strain used for the Y. lipolytica genome sequencing [START_REF] Dujon | Genome evolution in yeasts[END_REF], was named Yl.SUP35 A and encodes a 728 amino acids long Yl.Sup35p A protein (Fig. 1A). The SUP35 ortholog we cloned from the IAM-4948 strain previously used by Nakamura and colleagues [START_REF] Nakayashiki | Yeast [PSIþ] "prions" that are crosstransmissible and susceptible beyond a species barrier through a quasi-prion state[END_REF] was named Yl.SUP35 B and encodes a 742 amino acids long Yl.Sup35p B protein (Fig. 1A). The differences between the Y. lipolytica and S. cerevisiae Sup35 proteins were previously described [START_REF] Nakayashiki | Yeast [PSIþ] "prions" that are crosstransmissible and susceptible beyond a species barrier through a quasi-prion state[END_REF] and lie essentially in their NM domains which contain very different repeated motifs (Fig. 1B and Table 1). Yl.Sup35p A and Yl.Sup35p B differ by the number of each of the four repeats found in their NM domains (Fig. 1B and Table 1). Conversely, as expected from the essential nature of the function it fulfills, the C-terminal eRF3 domains of S. cerevisiae and Y. lipolytica Sup35p orthologs are highly conserved (Fig. 1B).

Yarrowia lipolytica Sup35 proteins assemble into thioflavin Tnegative amyloid fibrils

S. cerevisae and Y. lipolytica Sup35 proteins were overexpressed in E. coli and purified as previously described [START_REF] Krzewska | Molecular chaperones and the assembly of the prion Sup35p, an in vitro study[END_REF][START_REF] Krzewska | Biochemical and functional analysis of the assembly of full-length Sup35p and its prion-forming domain[END_REF]. Next, we assessed the ability of these purified proteins to spontaneously assemble into fibrils under physiological pH and salt concentration conditions [START_REF] Kabani | A mutation within the C-terminal domain of Sup35p that affects [PSIþ] prion propagation[END_REF][START_REF] Krzewska | Molecular chaperones and the assembly of the prion Sup35p, an in vitro study[END_REF][START_REF] Krzewska | Biochemical and functional analysis of the assembly of full-length Sup35p and its prion-forming domain[END_REF]. The assembly kinetics were followed by Thioflavin-T binding, as described [START_REF] Kabani | A mutation within the C-terminal domain of Sup35p that affects [PSIþ] prion propagation[END_REF][START_REF] Krzewska | Molecular chaperones and the assembly of the prion Sup35p, an in vitro study[END_REF][START_REF] Krzewska | Biochemical and functional analysis of the assembly of full-length Sup35p and its prion-forming domain[END_REF][START_REF] Mcparland | Partially unfolded states of beta(2)microglobulin and amyloid formation in vitro[END_REF]. As expected from our previous work, Sc.Sup35p formed Thioflavin T-positive fibrils after 2e3 days of incubation (Fig. 2A and Fig. 2B). No Thioflavin T fluorescence was detected upon incubation of Yl.Sup35p A and Yl.Sup35p B proteins under the same conditions for over 30 days, suggesting these proteins did not assemble into fibrils (Fig. 2A). Indeed, negative-stained electron microscopy of Y. lipolytica Sup35 proteins revealed only amorphous aggregates, oligomers and prefibrillar species during the first 7 days of incubation (Fig. 2C). However, longer incubations of 30 days allowed the Yl.Sup35p A and Yl.Sup35p B proteins to assemble into fibrils (Fig. 2C). Sc.Sup35p, Yl.Sup5p A and Yl.Sup35p B fibrils displayed very similar morphologies (Fig. 2B andC). Modifications in the experimental conditions of the assembly reactions (e.g. temperature, agitation speed) did not allow a faster assembly of Yl.Sup35p proteins nor did it result in the formation of Thioflavin T-positive fibrils (data not shown).

Next, we assessed the secondary structure content of fibrillar Sc.Sup35p, Yl.Sup35p A and Yl.Sup35p B by Fourier transform infrared (FTIR) spectroscopy (Fig. 3A). The FT-IR spectra showed very similar amide I bands for all fibrils, with a shoulder at 1624 cm À1 indicative of a cross-b-sheet amyloid structure (Fig. 3A). Fourier deconvolution and curve fitting of the spectra showed no significant differences in the secondary structure content of the three kinds of fibrils, with ~62% of b-sheet structures (Fig. 3B), in agreement with previous results [START_REF] Kabani | A mutation within the C-terminal domain of Sup35p that affects [PSIþ] prion propagation[END_REF].

The NM domains of Yarrowia lipolytica Sup35p A and B assemble into thioflavin T-negative fibrils.

We previously showed that the C-terminal domain of Sc.Sup35p affects its assembly properties into fibrils, and that fibrillar Sc.Sup35p and Sc.Sup35NM show significant structural differences [START_REF] Kabani | A mutation within the C-terminal domain of Sup35p that affects [PSIþ] prion propagation[END_REF][START_REF] Krzewska | Biochemical and functional analysis of the assembly of full-length Sup35p and its prion-forming domain[END_REF][START_REF] Luckgei | The conformation of the prion domain of Sup35 p in isolation and in the full-length protein[END_REF]. We then asked whether the Yl.Sup35NM and B domains would behave differently than their full-length counterparts. The NM domains of S. cerevisiae and Y. lipolytica Sup35 proteins were overexpressed in E. coli, purified, and their ability to assemble into fibrils assessed by Thioflavin T binding and negative-stain electron microscopy (Fig. 4). While the Sc.Sup35NM domain readily assembled into thioflavin T-positive fibrils [START_REF] Krzewska | Biochemical and functional analysis of the assembly of full-length Sup35p and its prion-forming domain[END_REF], Yl.Sup35NM A and Yl.Sup35NM B, like their full-length counterparts, were only able to assemble into fibrils that remained Thioflavin T-negative after 30 days of incubation (Fig. 4).

Sc.Sup35p PQGGYQQ-YN R1 [41e49] R2 [56e64] R3 [65e74] R4 [75e83] R5 [84e93] Yl.Sup35p A FVPGQS R1 [34e39] R2 [40e44] R3 [46e51] QGGYQGGYQGGY R1 [65e76]] R2 [77e94] R3 [99e110] R4 [118e130] GGALKIGGDKP R1 [169e179] R2 [183e193] R3 [197e207] R4 [213e222] KESTP R1 [236e240] R2 [245e249] R3 [254e258] R4 [263e267] Yl.Sup35p B FVPGQS R1 [40e45] R2 [46e51] R3 [52e57] QGGYQGGYQGGY R1 [70e81] R2 [82e93] R3 [98e109] R4 [116e127] R5 [134e145] GGALKIGGDKP R1 [180_190] R2 [193e203] R3 [207e217] R4 [221e231] R5 [234e243] KESTP R1 [260e264] R2 [265e269] R3 [274e278]

Discussion

In the present paper, we show that two phylogenetically divergent Y. lipolytica Sup35p orthologs assemble into amyloid fibrils devoid of thioflavin T binding capacity. Y. lipolytica IAM-4948 strain expresses the Yl.Sup35p B ortholog [START_REF] Nakayashiki | Yeast [PSIþ] "prions" that are crosstransmissible and susceptible beyond a species barrier through a quasi-prion state[END_REF], while the W29 reference strain (and its derivatives) expresses the Yl.Sup35p A ortholog [START_REF] Dujon | Genome evolution in yeasts[END_REF] (Fig. 1 and Table 1). Despite the slight differences observed between these two orthologs, mostly within their NM domains (Table 1), purified Yl.Sup35p A and Yl.Sup35p B behaved similarly in our test tube assays (Figs. 2e4).

In a S. cerevisiae reporter system, Y. lipolytica Sup35p showed an unusually high propensity to convert to the prion form [START_REF] Nakayashiki | Yeast [PSIþ] "prions" that are crosstransmissible and susceptible beyond a species barrier through a quasi-prion state[END_REF]. From these data, we expected the assembly properties of Yl.Sup35p in vitro to mirror its behavior in vivo, meaning fast assembly kinetics. Contrary to this initial assumption, the assembly of Yl.Sup35p and Yl.Sup35NM, realized under the same experimental conditions as those used for Sc.Sup35p and Sc.Sup35NM [START_REF] Kabani | A mutation within the C-terminal domain of Sup35p that affects [PSIþ] prion propagation[END_REF][START_REF] Krzewska | Molecular chaperones and the assembly of the prion Sup35p, an in vitro study[END_REF], was very slow and it took up to 30 days to obtain homogenous fibrillar preparations (Fig. 2A, C and4). Oligomers and amorphous or prefibrillar aggregates, which could correspond to intermediates en route to assembly, were observed before Yl.Sup35p fibrils formed (Fig. 2C).

Thioflavin T, as well as other fluorescent dyes (e.g. Congo red, primuline), are often used as markers of amyloids. Yl.Sup35p and Yl.Sup35NM fibrils did not bind Thioflavin T in our experimental conditions (Figs. 2A and4A) despite their amyloid nature, as demonstrated by FT-IR spectroscopy (Fig. 3). We recently reported that amyloid fibrils made of dipeptide-repeat proteins do not bind Thioflavin T albeit their amyloid nature [START_REF] Brasseur | Dipeptide repeat derived from C9orf72 hexanucleotide expansions forms amyloids or natively unfolded structures in vitro[END_REF]. Similarly, human and fish islet amyloid polypeptide (IAPP), which differ by 11 non conservative substitutions, form thioflavin T-positive and thioflavin Tnegative amyloid fibrils, respectively [START_REF] Wong | Analysis of the amyloidogenic potential of pufferfish (Takifugu rubripes) islet amyloid polypeptide highlights the limitations of thioflavin-T assays and the difficulties in defining amyloidogenicity[END_REF]. We also showed in the past not only that non-amyloid and amyloid Ure2p yeast prion fibrils bind Thioflavin T [START_REF] Bousset | The native-like conformation of Ure2p in fibrils assembled under physiologically relevant conditions switches to an amyloid-like conformation upon heat-treatment of the fibrils[END_REF] but also that non fibrillar Ure2p assemblies obtained in the presence of Hsp104p bind Thioflavin T to a higher extent than fibrillar assemblies [START_REF] Savistchenko | Molecular chaperones and the assembly of the prion Ure2p in vitro[END_REF]. Finally, we demonstrated that Congo red binding to fibrillar assemblies and the associated yellow-green birefringence in polarized light are not indicative of an amyloid nature [START_REF] Bousset | Structural characterization of the fibrillar form of the yeast Saccharomyces cerevisiae prion Ure2p[END_REF]. Thus, cautions need be taken when using fluorescent dyes to assess the assembly and structural characteristics of prions and prion-like proteins.

Several hypotheses can be made to account for the marked differences we report for the behavior of Yl.Sup35p in vitro compared to what was previously known from in vivo assays [START_REF] Nakayashiki | Yeast [PSIþ] "prions" that are crosstransmissible and susceptible beyond a species barrier through a quasi-prion state[END_REF]. These could lie in fundamental differences between our experimental assembly conditions and the unique environment of the cytosol. The unusually high prion conversion rate of Yl.Sup35p could also result from its specific or stochastic interactions with partner proteins within the S. cerevisiae reporter cells [START_REF] Nakayashiki | Yeast [PSIþ] "prions" that are crosstransmissible and susceptible beyond a species barrier through a quasi-prion state[END_REF], such as components of the translation machinery, molecular chaperones or the cytoskeleton, that are likely to affect its assembly properties. The importance of the cellular context will need to be determined by assessing whether Yl.Sup35p can form prions, and at which frequency, in Yarrowia lipolytica.

Fig. 1 .

 1 Fig. 1. Alignment of Sc.Sup35, Yl.Sup35p A and Yl.Sup35p B proteins. The S. cerevisiae Sc.Sup35p coding sequence (Uniprot P05453) was obtained from the 74-D694 strain. The Yl.Sup35p A coding sequence was obtained from Y. lipolytica strains 136,463 and PO1d (Uniprot Q6CFC9). The Yl.Sup35p B coding sequence was obtained from Y. lipolytica strain SS397A (accession number AB039752). (A) Primary structure and domain organization of Sc.Sup35p, Yl.Sup35p A and Yl.Sup35p B proteins (see text for details). (B) Sequence alignment of Sc.Sup35p, Yl.Sup35p A and Yl.Sup35p B proteins (the alignment was made using T-COFFEE and BOXSHADE).

Fig. 2 .

 2 Fig. 2. Assembly of full-length Y. lipolytica and S. cerevisiae Sup35p proteins. (A) Full-length Y. lipolytica and S. cerevisiae Sup35 proteins were allowed to assemble at a final concentration of 10 mM and at 6 C under very low agitation (<100 rpm). Aliquots were removed at regular time intervals and assayed using the Thioflavin T binding assay described in the Materials and Methods section. (B, C) Assembly reactions were imaged at the indicated time points using negative-stained electron microscopy (scale bars, 1 mm).

Fig. 3 .

 3 Fig. 3. FT-IR spectra of fibrillar Y. lipolytica and S. cerevisiae Sup35 proteins. (A) The recorded FT-IR spectra (black lines) and curve fit data (grey lines) are shown for fibrillar Y. lipolytica and S. cerevisiae Sup35 proteins. The spectra were fitted using a Gaussian species model centered at 1624, 1640, 1652, 1664 and 1678 cm-1. (B) Secondary structure content of S. cerevisiae and Y. lipolytica Sup35p fibrils derived from the deconvolution of the FTIR data presented in (A).

Fig. 4 .

 4 Fig. 4. Assembly of Y. lipolytica and S. cerevisiae Sup35NM proteins. (A) The indicated purified proteins were allowed to assemble at a final concentration of 10 mM at 6 C under very low agitation (<100 rpm). Aliquots were removed at regular time intervals and assayed using the Thioflavin T binding assay described in the Materials and Methods section. (B) Sc.Sup35NM (7 days), Yl.Sup35NM A and Yl.Sup35NM B (30 days) fibril preparations were imaged using negative-stained electron microscopy (scale bars, 1 mm).

Table 1

 1 Consensus sequences and positions of the repeat motifs (R) in S. cerevisiae and Y. lipolytica Sup35 proteins.

	protein	consensus motif	motif residues

Acknowledgements

We thank Mathieu Chauleau for participating to the early stages of this study. MK and RM are supported by the Centre National de la Recherche Scientifique. This work has benefited from the Imagerie-Gif core facility of the Institute for Integrative Biology of the Cell (I2BC), which is supported by the Agence Nationale de la Recherche (ANR-11-EQPX-0029/Morphoscope, ANR-10-INBS-04/FranceBioImaging; ANR-11-IDEX-0003-02/Saclay Plant Sciences).

Author contributions

MK designed and performed the experiments. MK and RM analyzed the data and wrote the paper.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


.05

Social system

Coffee market (1990 crisis)

Case study:

Coffee value chain in Central America

Agroecosystem management

Fram. 3 :

 3 Using an ES lens to highlight social interdependencies and reflect on collective action Barnaud et al. (submitted) Ecology & Society

management •Restoration of natural habitats

	Lescourret, Magda et al. (2015) Current
	Opinion in Environm. Sustainab.
	MetaProgram ECOSERV	.06

•Landscape planning •Farming systems •Semi-natural habitat Structure: functional biodiversity, coffee Processes: organic matter dynamics, microbial biomass dynamics, coffee growth, … Ecosystem Structure: NGOs, companies, organic farmers, consumer groups Processes: certification, design of technical systems, compliance control procedures, distribution of premium, coordination between actors in the territory

Create a new local SSE system

	and functioning Better diversity Providers	Ecosystem services		Improvement of regulating services but lower coffee productivity (organic) Beneficiaries
	on certified
	farms
		Ecosystem
	Co-produce, preserve, degrade, manage	(dis) service A Antagonism , synergy	Benefit, harm	• Provisioning: coffee productivity • Regulating: water conservation, pest control, soil quality
	A focus on		Balance
	shade			between
	management		multiple ES
	ACTION ARENA : Potential or existing collective actions

• Landscape: planning and management • Farming systems: shade structure and management

Multiple services

Ecosystem services

Providers

Landowners Farmers

Beneficiaries

Farmers Consumers

Biological pest control services

Pest pressure dis-services

Intermediarie s

Institutions


Bernard Bercu
email: bernard.bercu@math.u-bordeaux.fr.

AND S ÉBASTIEN GADAT Manon Costa
email: manon.costa@math.univ-toulouse.fr

STOCHASTIC APPROXIMATION ALGORITHMS FOR SUPERQUANTILES ESTIMATION

Keywords: 62L20, Secondary : 60F05, 62P05, Stochastic approximation, Quantile and Superquantile, Limit theorems

This paper is devoted to two different two-time-scale stochastic approximation algorithms for superquantile estimation. We shall investigate the asymptotic behavior of a Robbins-Monro estimator and its convexified version. Our main contribution is to establish the almost sure convergence, the quadratic strong law and the law of iterated logarithm for our estimates via a martingale approach. A joint asymptotic normality is also provided. Our theoretical analysis is illustrated by numerical experiments on real datasets.

Introduction

Estimating quantiles has a longstanding history in statistics and probability. Except in parametric models where explicit formula are available, the estimation of quantiles is a real issue. The most commun way to estimate quantiles is to make use of order statistics, see among other references [START_REF] Bahadur | A note on quantiles in large samples[END_REF][START_REF] Ghosh | A new proof of the Bahadur representation of quantiles and an application[END_REF]. Another strategy is to make use of stochastic approximation algorithms and the pioneering work in this vein is the celebrated paper by Robbins and Monro [START_REF] Robbins | A stochastic approximation method[END_REF].

Let X be an integrable continuous random variable with strictly increasing cumulative distribution function F and probability density function f . For any α ∈]0, 1[, the quantile θ α of order α of F is given by (1.1) F (θ α ) = P(X ≤ θ α ) = α, whereas the superquantile ϑ α of order α is defined by

(1.2) ϑ α = E[X | X ≥ θ α ] = E[XI {X≥θα} ] P(X ≥ θ α ) = E[XI {X≥θα} ] 1 -α .
One can observe that the superquantile provides more information on the tail of the distribution of the random variable X. Our goal in this paper is to simultaneously estimate quantiles and superquantiles, also respectively known as values at risk and conditional values at risk, which have become increasingly popular as measures of risk in finance [START_REF] Rockafellar | Optimization of conditional value-at-risk[END_REF][START_REF] Rockafellar | Conditional value-at-risk for general loss distributions[END_REF].

The paper is organized as follows. Section 2 is devoted to a brief overview of the previous literature on the recursive estimation of quantiles and superquantiles. The main results of the paper are given in Section 3. We propose the almost sure convergence of two-time-scale stochastic approximation algorithms for superquantile estimation. The quadratic strong law (QSL) as well as the law of iterated logarithm (LIL) of our stochastic algorithms are also provided. Moreover, we establish the joint asymptotic normality of our estimates. Numerical experiments on real data are given in Section 7. All technical proofs are postponed to Appendices A and B.

Overview of existing literature

A wide range of literature exists already on the recursive estimation of quantiles [START_REF] Robbins | A stochastic approximation method[END_REF]. However, to the best of our knowledge, only a single paper is available on the recursive estimation of superquantiles [START_REF] Bardou | Computing var and cvar using stochastic approximation and adaptive unconstrained importance sampling[END_REF]. In many practical situations where the data are recorded online with relatively high speed, or when the data are simply too numerous to be handled in batch systems, it is more suitable to implement a recursive strategy where quantiles and superquantiles are sequentially estimated with the help of stochastic approximation algorithms [START_REF] Duflo | Random iterative models[END_REF], [START_REF] Kushner | Stochastic approximation and recursive algorithms and applications[END_REF]. We also refer the reader to [START_REF] Cardot | Online estimation of the geometric median in Hilbert spaces: Nonasymptotic confidence balls[END_REF][START_REF] Cardot | Efficient and fast estimation of the geometric median in hilbert spaces with an averaged stochastic gradient algorithm[END_REF][START_REF] Godichon-Baggioni | Estimating the geometric median in hilbert spaces with stochastic gradient algorithms : Lp and almost sure rates of convergence[END_REF][START_REF] Godichon-Baggioni | Online estimation of the asymptotic variance for averaged stochastic gradient algorithms[END_REF] for the online estimation of geometric medians and variances.

Bardou et al. [START_REF] Bardou | Computing var and cvar using stochastic approximation and adaptive unconstrained importance sampling[END_REF] have previously studied the averaged version [START_REF] Polyak | Acceleration of stochastic approximation by averaging[END_REF][START_REF] Ruppert | Efficient estimations from a slowly convergent robbins-monro process[END_REF] of a one-timescale stochastic algorithm in order to estimate θ α and ϑ α . Here, we have chosen to investigate a two-time-scale stochastic algorithm [START_REF] Borkar | Stochastic approximation with two time scales[END_REF][START_REF] Gadat | Stochastic heavy ball[END_REF][START_REF] Konda | Convergence rate of linear two-time-scale stochastic approximation[END_REF][START_REF] Mokkadem | Convergence rate and averaging of nonlinear two-time-scale stochastic approximation algorithms[END_REF] which performs pretty well and offers more flexibility than the one-time-scale algorithm. Let (X n ) be a sequence of independent and identically distributed random variables sharing the same distribution as X. We shall extend the statistical analysis of [START_REF] Bardou | Computing var and cvar using stochastic approximation and adaptive unconstrained importance sampling[END_REF] by studying the two-time-scale stochastic algorithm given, for all n ≥ 1, by (2.1)

     θ n+1 = θ n -a n I {X n+1 ≤θn} -α , ϑ n+1 = ϑ n + b n X n+1 1 -α I {X n+1 >θn} -ϑ n ,
where the initial values θ 1 and ϑ 1 are square integrable random variables which can be arbitrarily chosen and the steps (a n ) and (b n ) are two positive sequences of real numbers strictly smaller than one, decreasing towards zero such that We shall also investigate the asymptotic behavior of the convexified version of algorithm (2.1), based on the Rockafellar-Uryasev's identity [START_REF] Rockafellar | Optimization of conditional value-at-risk[END_REF] and given, for all n ≥ 1, by

     θ n+1 = θ n -a n I {X n+1 ≤θn} -α ϑ n+1 = ϑ n + b n θ n + (X n+1 -θ n ) 1 -α I {X n+1 >θn} -ϑ n , (2.3)
where as before the initial values θ 1 and ϑ 1 are square integrable random variables which can be arbitrarily chosen. We also refer the reader to the original contribution [START_REF] Ben-Tal | Expected utility, penalty ffunctions, and duality in stochastic nonlinear programming[END_REF] where this convexification first appeared. The almost sure convergence

(2.4) lim n→∞ θ n = θ α a.s.
is a famous result that was established by Robbins and Monro [START_REF] Robbins | A stochastic approximation method[END_REF], Robbins and Siegmund [START_REF] Robbins | A convergence theorem for non negative almost supermartingales and some applications[END_REF]. Moreover, the asymptotic normality is due to Sacks, see Theorem 1 in [START_REF] Sacks | Asymptotic distribution of stochastic spproximation procedures[END_REF]. It requires the additional assumption that the probability density function f is differentiable with bounded derivative in every neighborhood of the quantile θ α . More precisely, if the step a n = a 1 /n where a 1 > 0 and 2a 1 f (θ α ) > 1, we have the asymptotic normality

(2.5) √ n θ n -θ α L -→ N 0, a 2 1 α(1 -α) 2a 1 f (θ α ) -1 .
One can observe that in the special case where the value f (θ α ) > 0 is known, it is possible to minimise the previous limiting variance by choosing a 1 = 1/f (θ α ) and to obtain from (2.5) the asymptotic efficiency

√ n θ n -θ α L -→ N 0, α(1 -α) f 2 (θ α ) .
Some useful refinements on the asymptotic behavior of the sequence (θ n ) are also well-known. The LIL was first proved by Gaposhkin and Krasulina, see Theorem 1 in [START_REF] Gaposkin | On the law of the iterated logarithm in stochastic approximation processes[END_REF] and Corollary 1 in [START_REF] Kersting | Almost sure approximation of the robbins-monro process by sums of independent random variables[END_REF]. More precisely, if the step a n = a 1 /n where 2a

1 f (θ α ) > 1, we have the LIL lim sup n→∞ n 2 log log n 1/2 θ n -θ α = -lim inf n→∞ n 2 log log n 1/2 θ n -θ α = a 2 1 α(1 -α) 2a 1 f (θ α ) -1 1/2 a.s. (2.6)
In particular, it follows from (2.6) that

(2.7) lim sup n→∞ n 2 log log n θ n -θ α 2 = a 2 1 α(1 -α) 2a 1 f (θ α ) -1 a.s.
which is the limiting variance in (2.5). The QSL is due to Lai and Robbins, see Lemma 1 and Theorem 2 in [START_REF] Lai | Adaptive design and stochastic approximation[END_REF] as well as Theorem 3 in [START_REF] Pelletier | On the almost sure asymptotic behaviour of stochastic algorithms[END_REF]. More precisely, they proved that

(2.8) lim n→∞ 1 log n n k=1 θ k -θ α 2 = a 2 1 α(1 -α) 2a 1 f (θ α ) -1 a.s.
Besides the classical choice a n = a 1 /n where a 1 > 0, slower step-size a n = a 1 /n a where a 1 > 0 and 1/2 < a < 1 have been studied in depth. We refer the reader to pioneer work of Chung [START_REF] Chung | On a stochastic approximation method[END_REF] and to Fabian [START_REF] Fabian | On asymptotic normality in stochastic approximation[END_REF] who obtained that the asymptotic normality still holds for the Robbins-Monro algorithm. More precisely, if f (θ α ) > 0, they showed that

(2.9) √ n a θ n -θ α L -→ N 0, a 1 α(1 -α) 2f (θ α ) .
In addition, it follows from Lai and Robbins [START_REF] Lai | Adaptive design and stochastic approximation[END_REF] or Pelletier [START_REF] Pelletier | On the almost sure asymptotic behaviour of stochastic algorithms[END_REF] that

lim sup n→∞ n a 2(1 -a) log n 1/2 θ n -θ α = -lim inf n→∞ n a 2(1 -a) log n 1/2 θ n -θ α = a 1 α(1 -α) 2f (θ α ) 1/2 a.s. (2.10)
In particular, (2.11) lim sup

n→∞ n a 2(1 -a) log n θ n -θ α 2 = a 1 α(1 -α) 2f (θ α ) a.s.
Moreover, we also have from [START_REF] Lai | Adaptive design and stochastic approximation[END_REF], [START_REF] Pelletier | On the almost sure asymptotic behaviour of stochastic algorithms[END_REF] that

(2.12) lim n→∞ 1 n 1-a n k=1 θ k -θ α 2 = a 1 α(1 -α) 2(1 -a)f (θ α )
a.s.

The restrictive assumption 2a 1 f (θ α ) > 1, which involves the knowledge of f (θ α ), is no longer needed. However, the convergence rate n a is always slower than n, which means that the choice a n = a 1 /n theoretically outperforms the one of a n = a 1 /n a , at least asymptotically.

In the special case of the one-time-scale stochastic algorithm where a n = b n , Bardou et al. [START_REF] Bardou | Computing var and cvar using stochastic approximation and adaptive unconstrained importance sampling[END_REF] [START_REF] Bardou | Computing var and cvar using stochastic approximation and adaptive unconstrained importance sampling[END_REF] the joint asymptotic normality of the averaged version [START_REF] Polyak | Acceleration of stochastic approximation by averaging[END_REF][START_REF] Ruppert | Efficient estimations from a slowly convergent robbins-monro process[END_REF] of their one-time-scale stochastic algorithm

(2.15) √ n θ n -θ α ϑ n -θ α L -→ N 0, Σ
where the asymptotic covariance matrix Σ is explicitly calculated,

θ n = 1 n n k=1 θ k and ϑ n = 1 n n k=1 ϑ k .
We will show that our two-time-scale stochastic algorithms given by (2.1) and (2.3) allow us to avoid the Ruppert and Polyak-Juditsky averaging principle. Moreover, they perform pretty well both from a theoretical and a practical point of view and offer more flexibility than the one-time-scale stochastic algorithm.

Main results

In order to state our main results, it is necessary to introduce some assumptions.

(A 1 ) The probability density function f is differentiable with bounded derivative in every neighborhood of θ α .

(A 2 ) The function Φ defined, for all θ ∈ R, by Φ(θ) = f (θ) + θf (θ) is bounded in every neighborhood of θ α . Our first result concerns the basic almost sure convergence of the two-time-scale stochastic algorithms (2.1) and (2.3) to the superquantile ϑ α . Theorem 3.1. Assume that (A 1 ) holds and that the random variable X is square integrable. Then, we have the almost sure convergences

(3.1) lim n→∞ ϑ n = ϑ α a.s. (3.2) lim n→∞ ϑ n = ϑ α a.s.
Our proof is slightly different from that of Bardou et al. [START_REF] Bardou | Computing var and cvar using stochastic approximation and adaptive unconstrained importance sampling[END_REF] established for the one-time-scale stochastic algorithm where a n = b n . It can be found in Appendix A for sake of completeness. We now focus our attention on the almost sure rates of convergence of the sequences ( ϑ n ) and ( ϑ n ). We divide our analysis into two parts depending on the step size (b n ) in the superquantile recursive procedure. First of all, we shall consider the optimal step b n = b 1 /n. Then, we shall study the case where b n = b 1 /n b with 1/2 < b < 1. For all θ ∈ R, denote

(3.3) σ 2 α (θ) = 1 (1 -α) 2 Var(XI {X>θ} ) and τ 2 α (θ) = 1 (1 -α) 2 Var((X -θ)I {X>θ} ).
It follows from straightforward calculation that

τ 2 α (θ α ) = σ 2 α (θ α ) - αθ α 1 -α (2ϑ α -θ α ).
Consequently, as soon as θ α ≥ 0, we always have τ 2 α (θ α ) ≤ σ 2 α (θ α ) since ϑ α ≥ θ α . Theorem 3.2. Assume that (A 1 ) and (A 2 ) hold and that the random variable X has a moment of order > 2. Moreover, suppose that f (θ α ) > 0 and that the step sequences (a n ) and (b n ) are given by

a n = a 1 n a and b n = b 1 n
where a 1 > 0, b 1 > 1/2 and 1/2 < a < 1. Then, ( ϑ n ) and ( ϑ n ) share the same QSL

(3.4) lim n→∞ 1 log n n k=1 ϑ k -ϑ α 2 = b 2 1 2b 1 -1 τ 2 α (θ α ) a.s.
In addition, they also share the same LIL

lim sup n→∞ n 2 log log n 1/2 ϑ n -ϑ α = -lim inf n→∞ n 2 log log n 1/2 ϑ n -ϑ α = b 2 1 2b 1 -1 1/2 τ α (θ α ) a.s. (3.5)
In particular,

lim sup n→∞ n 2 log log n ϑ n -ϑ α 2 = b 2 1 2b 1 -1 τ 2 α (θ α ) a.s.
Remark 3.1. In the special case where the step sequence (b n ) is given by

b n = 1 n + 1 ,
it is easy to see that ϑ n and ϑ n both reduce to

ϑ n = 1 n n k=1 X k 1 -α I {X k >θ k-1 }
and

ϑ n = 1 n n k=1 θ k-1 + 1 n n k=1 X k -θ k-1 1 -α I {X k >θ k-1 } .
In this setting, we immediately obtain from Theorem 3.2 that

lim n→∞ 1 log n n k=1 ϑ k -ϑ α 2 = τ 2 α (θ α ) a.
a n = a 1 n a and b n = b 1 n b
where a 1 > 0, b 1 > 0 and 1/2 < a < b < 1. Then, ( ϑ n ) and ( ϑ n ) share the same QSL

(3.6) lim n→∞ 1 n 1-b n k=1 ϑ k -ϑ α 2 = b 1 2(1 -b) τ 2 α (θ α ) a.s.
In addition, they also share the same LIL

lim sup n→∞ n b 2(1 -b) log n 1/2 ϑ n -ϑ α = -lim inf n→∞ n b 2(1 -b) log n 1/2 ϑ n -ϑ α = b 1 2 1/2 τ α (θ α ) a.s. (3.7)
In particular,

lim sup n→∞ n b 2(1 -b) log n ϑ n -ϑ α 2 = b 1 2 τ 2 α (θ α ) a.s.
Remark 3.2. Similar computations in the case where 1/2 < b < a < 1 would lead to the same results for the convexified algorithm ( ϑ n ). However, for the standard algorithm ( ϑ n ), it is necessary to replace the asymptotic variance τ 2 α (θ α ) by σ 2 α (θ α ). This emphasizes the interest of using the convexified algorithm.

We now focus our attention on the asymptotic normality of our two-time-scale stochastic algorithms (2.1) and (2.3). Theorem 3.4. Assume that (A 1 ) and (A 2 ) hold and that the random variable X has a moment of order > 2/a. Moreover, suppose that f (θ α ) > 0 and that the step sequences (a n ) and (b n ) are given by

a n = a 1 n a and b n = b 1 n b
where

a 1 > 0, b 1 > 0 and 1/2 < a < b ≤ 1 with b 1 > 1/2 if b = 1.
Then, ( ϑ n ) and ( ϑ n ) share the same joint asymptotic normality

(3.8) √ n a θ n -θ α √ n b ϑ n -ϑ α L -→ N 0, Γ θα 0 0 Γ ϑα
where the asymptotic variances are given by

Γ θα = a 1 α(1 -α) 2f (θ α ) and Γ ϑα =        b 2 1 τ 2 α (θ α ) 2b 1 -1 if b = 1, b 1 τ 2 α (θ α ) 2 if b < 1.
Remark 3.3. One can observe that the asymptotic covariance matrix in (3.8) is diagonal. It means that, at the limit, the two algorithms for quantile and superquantile estimation are no longer correlated. This is due to the fact that we use two different time scales contrary to Bardou et al. [START_REF] Bardou | Computing var and cvar using stochastic approximation and adaptive unconstrained importance sampling[END_REF]. Moreover, in the special case where b = 1, we also recover the same asymptotic variance as the one obtained in [START_REF] Bardou | Computing var and cvar using stochastic approximation and adaptive unconstrained importance sampling[END_REF] for the averaged version of their one-time-scale stochastic algorithm.

Remark 3.4. The asymptotic variance τ 2 α (θ α ) can be estimated by

τ 2 n = 1 n n k=1 X k -θ k-1 1 -α 2 I {X k >θ k-1 } - 1 n n k=1 X k -θ k-1 1 -α I {X k >θ k-1 } 2 .
Via the same lines as in the proof of the almost sure convergences (3.1) and (3.2), one can verify that τ 2 n → τ 2 α (θ α ) a.s. Therefore, using Slutsky's Theorem, we deduce from (3.8) that ( ϑ n ) and ( ϑ n ) share the same asymptotic normality

(3.9) √ n b ϑ n -ϑ α τ n L -→ N (0, ν 2 )
where

ν 2 =        b 2 1 2b 1 -1 if b = 1, b 1 2 if b < 1.
Convergence (3.9) allows us to construct asymptotic confidence intervals for the superquantile ϑ α .

Our martingale approach

All our analysis relies on a decomposition of our estimates as sum of a martingale increment and a drift term. More precisely, it follows from (2.1) and (2.3) that for all n ≥ 1, (4.1)

   ϑ n+1 = (1 -b n ) ϑ n + b n Y n+1 ϑ n+1 = (1 -b n ) ϑ n + b n Z n+1 where Y n+1 = X n+1 1 -α I {X n+1 >θn}
and

Z n+1 = θ n + (X n+1 -θ n ) 1 -α I {X n+1 >θn} .
Let H α (θ) and L α (θ) be the functions defined, for all θ ∈ R, by

(4.2) H α (θ) = 1 1 -α E[XI {X>θ} ] and L α (θ) = θ + 1 1 -α E[(X -θ)I {X>θ} ].
We clearly have that almost surely

E[Y n+1 |F n ] = H α (θ n ) and E[Z n+1 |F n ] = L α (θ n ).
It allows use to split Y n+1 and Z n+1 as sum of a martingale increment and a drift term,

Y n+1 = ε n+1 + H α (θ n ) and Z n+1 = ξ n+1 + L α (θ n ). One can also verify that E[ε 2 n+1 |F n ] = σ 2 α (θ n ) and E[ξ 2 n+1 |F n ] = τ 2 α (θ n )
where the two variances are given by (3.3). Then, we immediately deduce from (4.1) that for all n ≥ 1,

(4.3)    ϑ n+1 = (1 -b n ) ϑ n + b n (ε n+1 + H α (θ n )) ϑ n+1 = (1 -b n ) ϑ n + b n (ξ n+1 + L α (θ n )).
Hereafter, assume for the sake of simplicity that for all n ≥ 1, b n < 1, since this is true for n large enough. Let (P n ) be the increasing sequence of positive real numbers defined by (4.4)

P n = n k=1 (1 -b k ) -1
with the convention that P 0 = 1. Since (1 -b n )P n = P n-1 , we obtain from (4.3) that   

P n ϑ n+1 = P n-1 ϑ n + P n b n (ε n+1 + H α (θ n )) P n ϑ n+1 = P n-1 ϑ n + P n b n (ξ n+1 + L α (θ n ))
which implies the martingale decomposition (4.5)

       ϑ n+1 = 1 P n ϑ 1 + M n+1 + H n+1 ϑ n+1 = 1 P n ϑ 1 + N n+1 + L n+1 where (4.6) M n+1 = n k=1 b k P k ε k+1 , N n+1 = n k=1 b k P k ξ k+1 (4.7) H n+1 = n k=1 b k P k H α (θ k ), L n+1 = n k=1 b k P k L α (θ k ).
Our strategy is to establish the asymptotic behavior of the two martingales (M n ) and (N n ) as well as to determine the crucial role played by the two drift terms (H n ) and (L n ). Several results in our analysis rely on the following keystone lemma which concerns the convexity properties of the functions H α and L α defined in (4.2).

Lemma 4.1. Assume that (A 1 ) and (A 2 ) hold. Then, L α is a convex function such that L α (θ α ) = ϑ α , L α (θ α ) = 0 and that for all θ ∈ R,

(4.8) 0 ≤ L α (θ) -L α (θ α ) ≤ ||f || ∞ 2(1 -α) θ -θ α 2 .
In addition, we also have H α (θ α ) = ϑ α and that for all θ ∈ R,

(4.9) H α (θ) -H α (θ α ) + θ α f (θ α ) 1 -α (θ -θ α ) ≤ ||Φ|| ∞ 2(1 -α) θ -θ α 2 .
Proof. It follows from (4.2) that for all θ ∈ R,

L α (θ) = F (θ) -α 1 -α and L α (θ) = f (θ) 1 -α .
Consequently, L α is a convex function such that L α (θ α ) = 0. Hence, we deduce from a Taylor expansion with integral remainder that for all θ ∈ R,

L α (θ) = L α (θ α ) + (θ -θ α ) 2 1 0 (1 -t)L α (θ t )dt
where θ t = θ α +t(θ-θ α ), which immediately leads to (4.8) using (A 1 ). Unfortunately, H α is not a convex function. However, we obtain from (4.2) that for all θ ∈ R,

H α (θ) = - θf (θ) 1 -α and H α (θ) = - Φ(θ) 1 -α
where Φ(θ) = f (θ)+θf (θ). Finally, (4.9) follows once again from a Taylor expansion with integral remainder together with (A 2 ).

We have just seen that the function H α is not convex. Consequently, in order to prove sharp asymptotic properties for the sequence ( ϑ n ), it is necessary to slightly modify the first martingale decomposition in (4.5). For all θ ∈ R, let (4.10)

G α (θ) = F (θ) -α -f (θ α )(θ -θ α ), R α (θ) = H α (θ) -ϑ α + C α (θ -θ α )
where

C α = -H α (θ α ) = θ α f (θ α ) 1 -α .
We deduce from (2.1), (4.3) and (4.10) that for all n ≥ 1, (4.11)

   θ n+1 -θ α = (1 -a n f (θ α ))(θ n -θ α ) -a n V n+1 + G α (θ n ) , ϑ n+1 -ϑ α = (1 -b n )( ϑ n -ϑ α ) + b n ε n+1 +R α (θ n )-C α (θ n -θ α ) with V n+1 = I {X n+1 ≤θn} -F (θ n ).
Hereafter, we shall consider a tailor-made weighted sum of our estimates given by ∆ 1 = 0 and, for all n ≥ 2, (4.12)

∆ n = ( ϑ n -ϑ α ) -δ n (θ n -θ α )
where (δ n ) is a deterministic sequence, depending on (a n ) and (b n ), which will be explicitly given below. It follows from (4.11) together with straightforward calculation that for all n ≥ 2,

(4.13) ∆ n+1 = (1 -b n )∆ n + b n W n+1 + R α (θ n ) + a n b n δ n+1 G α (θ n ) + ν n+1 (θ n -θ α )
where (4.14)

W n+1 = ε n+1 + a n b n δ n+1 V n+1 and (4.15) ν n+1 = 1 b n (1 -b n )δ n -C α b n -δ n+1 (1 -a n f (θ α )) .
We have several strategies in order to simplify the expression of ν n+1 . A first possibility that cancels several terms in (4.15) is to choose

δ n+1 = C α b n f (θ α )a n .
It clearly reduces ν n+1 to

ν n+1 = (1 -b n )δ n -δ n+1 b n .
Another more sophisticated choice, which only works if

a n f (θ α ) -b n = 0, is to take (4.16) δ n+1 = C α b n a n f (θ α ) -b n .
It implies that (4.17)

ν n+1 = (1 -b n )(δ n -δ n+1 ) b n .
The two choices are quite similar in the special case where b n = b 1 /n. However, in the case where the step b n = b 1 /n b with a < b < 1, the second choice outperforms the first one as ν n goes faster towards zero as n grows to infinity. Throughout the sequel, we shall make use of the second choice given by (4.16). We deduce from (4.13) the new martingale decomposition

(4.18) ∆ n+1 = 1 P n M n+1 + H n+1 + R n+1 where (4.19) M n+1 = n k=1 b k P k W k+1 , H n+1 = n k=1 b k P k ν k+1 θ k -θ α and (4.20) R n+1 = n k=1 b k P k R α (θ k ) + a k b k δ k+1 G α (θ k ) .
5. Proofs of the almost sure convergence results

5.1.

The basic almost sure properties. The starting point in our analysis of the almost sure convergence of our estimates is the following lemma.

Lemma 5.1. Assume that (A 1 ) holds and that the random variable X is square integrable. Then, we have the almost sure convergences

(5.1) lim n→∞ M n+1 P n = 0 and lim n→∞ N n+1 P n = 0 a.s.
Proof. Let (Σ ε n ) and (Σ ξ n ) be the two locally square integrable martingales

Σ ε n = n-1 k=1 b k ε k+1 , Σ ξ n = n-1 k=1 b k ξ k+1 .
Their predictable quadratic variations [START_REF] Duflo | Random iterative models[END_REF] are respectively given by

Σ ε n = n-1 k=1 b 2 k σ 2 α (θ k ) and Σ ξ n = n-1 k=1 b 2 k τ 2 α (θ k ).
It follows from convergence (2.4) and the continuity of the variances σ 2 α (θ) and τ 2 α (θ) given by (3.3)

that σ 2 α (θ n ) -→ σ 2 α (θ α ) and τ 2 α (θ n ) -→ τ 2 α (θ α ) a.
s. Consequently, we get from the right-hand side of (2.2) that

(5.2) lim n→∞ Σ ε n < +∞ and lim n→∞ Σ ξ n < +∞ a.s
Therefore, we obtain from the strong law of large numbers for martingales given e.g. by theorem 1.3.24 in [START_REF] Duflo | Random iterative models[END_REF] that (Σ ε n ) and (Σ ξ n ) both converge almost surely. The rest of the proof proceeds in a standard way with the help of Kronecker's lemma. As a matter of fact, we can deduce from the left-hand side of (2.2) that the sequence (P n ), defined in (4.4), is strictly increasing to infinity. In addition, we just showed the almost sure convergence of the series which is exactly what we wanted to prove.

Proof of Theorem 3.1. We recall from (4.5) that for all n ≥ 1,

       ϑ n+1 = 1 P n ϑ 1 + M n+1 + H n+1 ϑ n+1 = 1 P n ϑ 1 + N n+1 + L n+1 .
We have from (2.4) together with the continuity of the functions H α and L α that

(5.3) lim n→∞ H α (θ n ) = H α (θ α ) a.s and (5.4) lim n→∞ L α (θ n ) = L α (θ α ) a.s One can observe that H α (θ α ) = L α (θ α ) = ϑ α .
ϑ n = ϑ α a.s.
which completes the proof of Theorem 3.1.

5.2.

A keystone lemma. The QSL as well as the LIL for our estimates require the sharp asymptotic behavior of the sequence (P n ) defined in (4.4). Surprisingly, to the best of our knowledge, the following keystone lemma is new. It involves the famous Euler-Riemann zeta function.

Lemma 5.2. Assume that for some 0 < b 1 < 1, (5.8)

P n = n k=1 1 - b 1 k -1
.

Then, we have

(5.9) lim n→∞ 1 n b 1 P n = Γ(1 -b 1 )
where Γ stands for the Euler gamma function. Moreover, suppose that (5.10)

P n = n k=1 1 - b 1 k b -1
where 1/2 < b < 1. Then, we have where ζ stands for the Riemann zeta function.

Remark 5.1. The link between the first case b = 1 and the second case 1/2 < b < 1 is given the following formula due to Euler. For all |x| < 1,

log Γ(1 -x) = γx + ∞ n=2 x n n ζ(n)
where γ is the Euler-Mascheroni constant.

Remark 5.2. The case b 1 ≥ 1 can be treated in the same way. For example, concerning the first part of Lemma 5.2, it is only necessary to replace P n defined in (5.8) by

P n = n k=1+ b 1 1 - b 1 k -1
where b 1 is the integer part of b 1 . Then, we obtain that

lim n→∞ 1 n b 1 P n = Γ(1 -{b 1 }) Γ(1 + b 1 )
where

{b 1 } = b 1 -b 1 stands for the fractional part of b 1 .
Proof. In the first case b = 1, we clearly have (5.12)

P n = n k=1 1 - b 1 k -1 = Γ(n + 1)Γ(1 -b 1 ) Γ(n + 1 -b 1 )
.

It is well-known that for any c > 0, (

Γ(n + c) Γ(n)n c = 1. 5.13) lim n→∞
Hence, we obtain from (5.12) and (5.13) that

(1 -x) = - ∞ =1 x that log(P n ) = - n k=1 log 1 - b 1 k b = n k=1 ∞ =1 1 b 1 k b = ∞ =1 n k=1 1 b 1 k b , = b 1 n k=1 1 k b + ∞ =2 b 1 n k=1 1 k b . (5.15) It is well-known that lim n→∞ 1 n 1-b n k=1 1 k b = 1 1 -b .
In

M k P k-1 2 = b 2 1 2b 1 -1 τ 2 α (θ α ) a.s.
In addition, they also share the same LIL

lim sup n→∞ n 2 log log n 1/2 M n P n-1 = -lim inf n→∞ n 2 log log n 1/2 M n P n-1 = b 2 1 2b 1 -1 1/2 τ α (θ α ) a.s. (5.18)
Proof. We first focus our attention on the martingale (M n ) defined by

M n+1 = n k=1 b k P k W k+1 where W n+1 = ε n+1 + a n b n δ n+1 V n+1 with V n+1 = I {X n+1 ≤θn} -F (θ n ). We clearly have E[V n+1 |F n ] = 0, E[W n+1 |F n ] = 0, and E[V 2 n+1 |F n ] = F (θ n )(1 -F (θ n )), E[W 2 n+1 |F n ] = τ 2 n (θ n )
where for all θ ∈ R,

(5.19) τ 2 n (θ) = σ 2 α (θ) + a n δ n+1 b n 2 F (θ)(1 -F (θ)) - 2a n δ n+1 b n F (θ)H α (θ).
We obtain from (4.16) that

(5.20) lim n→∞ a n δ n+1 b n = θ α 1 -α .
Consequently, we infer from (2.4), (5.19) and (5.20) that

(5.21) lim n→∞ τ 2 n (θ n ) = σ 2 α (θ α ) - αθ α 1 -α (2ϑ α -θ α ) = τ 2 α (θ α ) a.s
Hereafter, assume for the sake of simplicity that 1/2 < b 1 < 1 inasmuch as the proof follows exactly the same lines for b 1 ≥ 1. On the one hand, the predictable quadratic variation of (M n ) is given by

M n = n-1 k=1 b 2 k P 2 k τ 2 k (θ k ).
On the other hand, as b 1 > 1/2, we obtain from convergence (5.9) in Lemma 5.2 that

lim n→∞ 1 n 2b 1 -1 n k=1 b 2 k P 2 k = b 2 1 2b 1 -1 Γ 2 (1 -b 1 ).
Then, we deduce from (5.21) and Toeplitz's lemma that

(5.22) lim n→∞ 1 n 2b 1 -1 M n = b 2 1 2b 1 -1 Γ 2 (1 -b 1 )τ 2 α (θ α ) a.s.
Denote by f n the explosion coefficient associated with the martingale (M n ),

f n = M n -M n-1 M n .
We obtain from (5.22) that (5.23) lim

n→∞ nf n = 2b 1 -1 a.s.
It means that f n converges to zero almost surely at rate n. In addition, we already saw from (5.21) that

lim n→∞ E[W 2 n+1 |F n ] = τ 2 α (θ α ) a.s.
Furthermore, the random variable X has a moment of order > 2. It implies that for some real number p > 2,

sup n≥0 E[|W n+1 | p |F n ] < ∞ a.s.
Consequently, we deduce from the QSL for martingales given in theorem 3 of [START_REF] Bercu | On the convergence of moments in the almost sure central limit theorem for martingales with statistical applications[END_REF] that

(5.24) lim n→∞ 1 log M n n k=1 f k M 2 k M k = 1 a.s.
Hence, we obtain from the conjunction of (5.22) and (5.24) that (5.25) lim

n→∞ 1 log n n k=1 M 2 k P 2 k-1 = b 2 1 2b 1 -1 τ 2 α (θ α ) a.s.
We shall now proceed to the proof of the LIL given by (5.18). We find from (5.23) that the explosion coefficient

f n satisfies ∞ n=1 f p/2 n < +∞ a.s.
Therefore, we deduce from the LIL for martingales [START_REF] Stout | A martingale analogue of kolmogorov's law of the iterated logarithm[END_REF], see also corollary 6.4.25 in [START_REF] Duflo | Random iterative models[END_REF] that

lim sup n→∞ 1 2 M n log log M n 1/2 M n = -lim inf n→∞ 1 2 M n log log M n 1/2 M n = 1 a.s. (5.26)
Hence, it follows from the conjunction of (5.14), (5.22) and (5.26) that lim sup

n→∞ n 2 log log n 1/2 M n P n-1 = -lim inf n→∞ n 2 log log n 1/2 M n P n-1 = b 2 1 2b 1 -1 1/2 τ α (θ α ) a.s.
which is exactly what we wanted to prove. Finally, concerning the martingale (N n ) given by

N n+1 = n k=1 b k P k ξ k+1 , the only minor change is that E[ξ 2 n+1 |F n ] = τ 2 α (θ n ).
However, we already saw that τ 2 α (θ n ) -→ τ 2 α (θ α ) a.s. Consequently, (M n ) and (N n ) share the same QSL and the same LIL, which completes the proof of Lemma 5.3.

Proof of Theorem 3.2. We shall only prove Theorem 3.2 in the special case where b n = b 1 /n with 1/2 < b 1 < 1 inasmuch as the proof in the case b 1 ≥ 1 follows essentially the same lines. First of all, we focus our attention on the standard estimator ϑ n .

• Our strategy is first to establish the QSL for the sequence (∆ n ) given by (4.12) and then to come back to ϑ n . We recall from (4.18) that for all n ≥ 2,

∆ n+1 = 1 P n M n+1 + H n+1 + R n+1 .
We claim that the weighted sequence (∆ n ) satisfies the QSL (5.27) lim

n→∞ 1 log n n k=1 ∆ 2 k = b 2 1 2b 1 -1 τ 2 α (θ α ) a.s.
As a matter of fact, we already saw from (5.17) that

lim n→∞ 1 log n n k=1 M k P k-1 2 = b 2 1 2b 1 -1 τ 2 α (θ α ) a.s.
Hence, in order to prove (5.27), it is necessary to show that (5.28) lim

n→∞ 1 log n n k=1 H k P k-1 2 = 0 and lim n→∞ 1 log n n k=1 R k P k-1 2 = 0 a.s.
On the one hand, it follows from (2.11) that for n large enough and for all k ≥ n,

(5.29) θ k -θ α 2 ≤ 2D a log k k a a.s.
where

D a = 2a 1 (1 -a)α(1 -α) f (θ α ) .
Consequently, we obtain from (4.19) and (5.29) that (5.30)

|H n+1 | = O n k=1 b k P k |ν k+1 | √ log k k a/2
a.s.

Furthermore, one can easily check from (4.16) and (4.17) that

lim n→∞ n 1-a ν n+1 = (1 -a)C α a 1 f (θ α ) .
In addition, we also recall from convergence (5.9) in Lemma 5.2 that lim n→∞

1 n b 1 P n = Γ(1 -b 1 )
. Hence, we deduce from (5.30) that (5.31)

|H n+1 | = O n k=1 √ log k k 2-b 1 -a/2
a.s.

It follows from (5.31) that (5.32)

∞ n=1 H n P n-1 2 < +∞ a.s.
As a matter of fact, let

d = 2 -b 1 -a/2. If d > 1 that is b 1 < 1 -a/2,
H n P n-1 2 = O ∞ n=1 (log n) 3 n 2b 1 = O(1) a.s. Moreover, if d < 1 that is b 1 > 1-a/2, we get from (5.31) that |H n | = O (log n) 1/2 n 1-d a.s. leading to ∞ n=1 H n P n-1 2 = O ∞ n=1 (log n) n 2b 1 +d-1 = O ∞ n=1 (log n) n 2-a = O(1) a.s.
On the other hand, (4.20) together with (4.9) and (4.10) imply that (5.33)

|R n+1 | = O n k=1 b k P k θ k -θ α 2 a.s.
It follows from (5.29) and (5.33) that (5.34)

|R n+1 | = O n k=1 log k k 1+a-b 1 a.s.
which clearly leads to (5.35)

∞ n=1 R n P n-1 2 < +∞ a.s.
Therefore, we obtain from (5.32) and (5.35) that the two convergences in (5.28) hold true, which immediately implies (5.27). Hereafter, one can notice from (4.12) that (5.36)

n k=1 ϑ k -ϑ α 2 = n k=1 ∆ 2 k + n k=1 δ 2 k θ k -θ α 2 + 2 n k=1 δ k ∆ k θ k -θ α .
Hence, in order to prove (3.4), it is only necessary to show that

lim n→∞ 1 log n n k=1 δ 2 k θ k -θ α 2 = 0 a.s.
and to make use of the Cauchy-Schwarz inequality. Denote

Λ n = n k=1 θ k -θ α 2 .
We have from (2.12) that as soon as f (θ α ) > 0,

(5.37) lim

n→∞ 1 n 1-a Λ n = a 1 α(1 -α) 2(1 -a)f (θ α ) a.s.
Furthermore, we obtain from a simple Abel transform that (5.38)

n k=1 δ 2 k θ k -θ α 2 = δ 2 n Λ n + n-1 k=1 (δ 2 k -δ 2 k+1 )Λ k .
We obtain from (4.16) that (5.39) lim

n→∞ n 1-a δ n = b 1 C α a 1 f (θ α )
.

Then, we deduce from (5.37) that

lim n→∞ n 1-a δ 2 n Λ n = b 2 1 C 2 α α(1 -α) 2a 1 (1 -a)f 3 (θ α ) a.s. which implies that (5.40) lim n→∞ 1 log n δ 2 n Λ n = 0 a.s.
In addition, we also have from (4.16) that

lim n→∞ n 3-2a δ 2 n -δ 2 n+1 = 2(1 -a) b 1 C α a 1 f (θ α ) 2 .
It clearly ensures via (5.37) that (5.41) lim

n→∞ 1 log n n-1 k=1 (δ 2 k -δ 2 k+1 )Λ k = 0 a.s.
Then, it follows from (5.38) together with (5.40) and (5.41) that (5.42)

lim n→∞ 1 log n n k=1 δ 2 k θ k -θ α 2 = 0 a.s.
Consequently, we obtain from (5.27) together with (5.36), (5.42) and the Cauchy-Schwarz inequality that ( ϑ n ) satisfies the QSL

lim n→∞ 1 log n n k=1 ϑ k -ϑ α 2 = b 2 1 2b 1 -1 τ 2 α (θ α ) a.s.
• The proof of the QSL for the convexified estimator ( ϑ n ) is much more easier. We infer from (4.5), (4.7), (5.5) and the identity L α (θ α ) = ϑ α that for all n ≥ 1,

ϑ n+1 -ϑ α = 1 P n ϑ 1 + N n+1 + L n+1 -P n ϑ α , = 1 P n N n+1 + R n+1 (5.43) where (5.44) R n+1 = ϑ 1 -ϑ α + n k=1 b k P k L α (θ k ) -L α (θ α ) .
It follows from Lemma 5.3 that

(5.45) lim n→∞ 1 log n n k=1 N k P k-1 2 = b 2 1 2b 1 -1 τ 2 α (θ α ) a.s.
Hence, in order to prove (3.4), it is only necessary to show that (5.46) lim

n→∞ 1 log n n k=1 R k P k-1 2 = 0 a.s.
We shall prove the stronger result (5.47)

∞ n=1 R k P k-1 2 < +∞ a.s.
We obtain from (4.8) and (5.44) that for all n ≥ 1,

(5.48) R 2 n+1 ≤ 2 ϑ 1 -θ α 2 + M f n k=1 b k P k θ k -θ α 2 2
where

M f = ||f || 2 ∞ 2(1 -α) 2 .
As before, we obtain from a simple Abel transform that (5.49)

n k=1 b k P k θ k -θ α 2 = b n P n Λ n + n-1 k=1 (b k P k -b k+1 P k+1 )Λ k .
It is easy to see that

b n P n -b n+1 P n+1 = b n P n 1 -b 1 n + 1 -b 1 . It implies that 0 < n(b n P n -b n+1 P n+1 ) < (1 -b 1 )b n P n .
Hence, as 1/2 < a < 1 and 1/2 < b 1 < 1, we find from (5.14) and (5.37) that (5.47) holds true. Consequently, we deduce from (5.43) together with (5.45) and (5.46) that

lim n→∞ 1 log n n k=1 ϑ k -ϑ α 2 = b 2 1 2b 1 -1 τ 2 α (θ α ) a.s.
which is exactly the QSL given by (3.4).

• It only remains to establish the LIL for our estimates ( ϑ n ) and ( ϑ n ). We start by proving the LIL for the sequence (∆ n ). We immediately obtain from (4.18) that (5.50)

n 2 log log n 1/2 ∆ n = n 2 log log n 1/2 M n + H n + R n P n-1 .
We already saw in Lemma 5.3 that the martingale (M n ) satisfies the LIL given by (5.18). In addition, it is easy to see from (5.

∆ n = -lim inf n→∞ n 2 log log n 1/2 ∆ n = b 2 1 2b 1 -1 1/2 τ α (θ α ) a.s. (5.51)
Hereafter, one can observe from (4.12) that (5.52)

n 2 log log n 1/2 ϑ n -ϑ α = n 2 log log n 1/2 ∆ n + n 2 log log n 1/2 δ n θ n -θ α .
It follows from (2.10) and (5.39) that

n 2 log log n δ n (θ n -θ α ) 2 = O log n n 1-a log log n a.s.
which clearly leads to

(5.53) lim n→∞ n 2 log log n 1/2 δ n θ n -θ α = 0 a.s.
Consequently, we obtain (3.5) from (5.51), (5.52) and (5.53). The proof for the convexified estimator ( ϑ n ) is straightforward. We obtain from (5.43) that

(5.54) n 2 log log n 1/2 ϑ n -ϑ α = n 2 log log n 1/2 N n + R n P n-1 .
We already saw in Lemma 5.3 that the martingale (N n ) satisfies the LIL given by (5.18). In addition, it is easy to see from (5.14), (5.48) and (5.49) that lim

n→∞ n R n P n-1 2 = 0 a.s. which clearly implies (5.55) lim n→∞ n 2 log log n 1/2 R n P n-1 = 0 a.s.
Finally, we deduce (3.5) from (5.18), (5.54) and (5.55), which completes the proof of Theorem 3.2.

5.4. The slow step size case. In order to prove Theorem 3.3, it is necessary to establish the following QSL and LIL for the martingales (M n ) and (N n ).

Lemma 5.4. Assume that the step sequences (a n ) and (b n ) are given by

a n = a 1 n a and b n = b 1 n b
where a 1 > 0, b 1 > 0 and 1/2 < a < b < 1. Then, (M n ) and (N n ) share the same QSL

(5.56) lim n→∞ 1 n 1-b n k=1 M k P k-1 2 = b 1 2(1 -b) τ 2 α (θ α ) a.s.
In addition, they also share the same LIL

lim sup n→∞ n b 2(1 -b) log n 1/2 M n P n-1 = -lim inf n→∞ n b 2(1 -b) log n 1/2 M n P n-1 = b 1 2 1/2 τ α (θ α ) a.s. (5.57)
Proof. We recall that the martingale (M n ) and its predictable quadratic variation are given by

M n+1 = n k=1 b k P k W k+1 and M n+1 = n k=1 b 2 k P 2 k τ 2 k (θ k )
where, thanks to (5.21),

lim n→∞ τ 2 n (θ n ) = τ 2 α (θ α ) a.s
It is not hard to see via a comparison series integral together with convergence (5.11) in Lemma 5.2 that (5.58) lim

n→∞ 1 b n P 2 n n k=1 b 2 k P 2 k = 1 2 .
Hence, we deduce from (5.58) and Toeplitz's lemma that (5.59) lim

n→∞ 1 b n P 2 n M n+1 = τ 2 α (θ α ) 2 a.s.
Denote by f n the explosion coefficient associated with the martingale (M n ),

f n = M n -M n-1 M n
It follows from the very definition of P n given by (4.

f k M 2 k M k = 1 a.s.
Therefore, we obtain from (5.16) and (5.59) together with (5.60) and (5.61) that (5.62) lim

n→∞ 1 n 1-b n k=1 M 2 k P 2 k-1 = b 1 2(1 -b) τ 2 α (θ α ) a.s.
Hereafter, we focus our attention on the proof of the LIL given by (5.57). Since b > 1/2, we obtain from (5.60) that the explosion coefficient

f n satisfies ∞ n=1 f p/2 n < +∞ a.s.
Therefore, we deduce from the LIL for martingales [START_REF] Stout | A martingale analogue of kolmogorov's law of the iterated logarithm[END_REF], see also corollary 6.4.25 in [START_REF] Duflo | Random iterative models[END_REF] that lim sup

n→∞ 1 2 M n log log M n 1/2 M n = -lim inf n→∞ 1 2 M n log log M n 1/2 M n = 1
a.s. (5.63) Hence, we find from (5.16), (5.59) and (5.63) that lim sup

n→∞ n b 2(1 -b) log n 1/2 M n P n-1 = -lim inf n→∞ n b 2(1 -b) log n 1/2 M n P n-1 = b 1 2 1/2 τ α (θ α ) a.s.
The proof for the martingale (N n ) is left to the reader inasmuch as it follows exactly the same lines than those for the martingale (M n ).

Proof of Theorem 3.3. We shall proceed as in the proof of Theorem 3.2. We already saw from (5.56) that

lim n→∞ 1 n 1-b n k=1 M k P k-1 2 = b 1 2(1 -b) τ 2 α (θ α ) a.s.
Our goal is to prove that the sequence (∆ n ) given by (4.12) satisfies the QSL (5.64) lim

n→∞ 1 n 1-b n k=1 ∆ 2 k = b 1 2(1 -b) τ 2 α (θ α ) a.s.
On the one hand, we have from (4.19) and (5.29) that

|H n+1 | = O n k=1 b k P k |ν k+1 | √ log k k a/2
a.s.

In addition, one can easily check from (4.16) and (4.17) that

lim n→∞ n 1-a ν n+1 = (b -a)C α a 1 f (θ α ) .
Hence, we obtain from convergence (5.11) in Lemma 5.2 together with a comparison series integral as previously done in the proof of Theorem 3.2 that (5.65)

|H n+1 | = O n k=1 P k √ log k k 1+b-a/2 = O P n √ log n n 1-a/2
a.s.

Consequently, we deduce from (5.65) that (5.66)

∞ n=1 H n P n-1 2 < +∞ a.s.
On the other hand, we already saw from (5.33) that

|R n+1 | = O n k=1 b k P k θ k -θ α 2 a.s.
which implies that (5.67)

|R n+1 | = O n k=1 P k log k k a+b = O P n log n n a a.s.
Then, as a > 1/2, we find from (5.67)

(5.68)

∞ n=1 R n P n-1 2 < +∞ a.s.
Therefore, we obtain from (5.66) and (5.68) that the QSL (5.64) holds true. In order to prove (3.6), it only remains to show via (5.36) that (5.69) lim

n→∞ 1 n 1-b n k=1 δ 2 k θ k -θ α 2 = 0 a.s.
We recall from (5.38) that

n k=1 δ 2 k θ k -θ α 2 = δ 2 n Λ n + n-1 k=1 (δ 2 k -δ 2 k+1 )Λ k .
We obtain from (4.16) that (5.70) lim

n→∞ n b-a δ n = b 1 C α a 1 f (θ α )
.

Then, it follows from (5.37) and (5.70) that (5.71) lim

n→∞ 1 n 1+a-2b δ 2 n Λ n = b 2 1 C 2 α α(1 -α) 2a 1 (1 -a)f 3 (θ α ) a.s.
Consequently, as a < b, we deduce from (5.71) that (5.72) lim

n→∞ 1 n 1-b δ 2 n Λ n = 0 a.s.
By the same token, we also find from (5.37) and (5.70) that (5.73) lim

n→∞ 1 n 1-b n-1 k=1 (δ 2 k -δ 2 k+1 )Λ k = 0 a.s.
Then, we clearly obtain from (5.72) and (5.73) that convergence (5.69) holds true.

As before, the proof of the QSL for the convexified estimator ( ϑ n ) is much more easier and left to the reader. We now focus our attention on the LIL for our estimates ( ϑ n ) and ( ϑ n ). We start by proving the LIL for the sequence (∆ n ) given by (4.12). We immediately obtain from (4.18) that (5.74)

n b 2(1 -b) log n 1/2 ∆ n = n b 2(1 -b) log n 1/2 M n + H n + R n P n-1 .
We already saw in Lemma 5.4 that the martingale (M n ) satisfies the LIL given by (5.57). In addition, as b < 1 < 2a, we get from (5.65) and (5.

b 2(1 -b) log n 1/2 R n P n-1 = 0 a.s.
Consequently, we find from (5.57) and (5.74) that (∆ n ) satisfies the LIL lim sup

n→∞ n b 2(1 -b) log n 1/2 ∆ n = -lim inf n→∞ n b 2(1 -b) log n 1/2 ∆ n = b 1 2 1/2 τ α (θ α ) a.s. (5.75)
Hereafter, we clearly have from (4.12) that (5.76)

n b 2(1 -b) log n 1/2 ϑ n -ϑ α = n b 2(1 -b) log n 1/2 ∆ n + δ n θ n -θ α .
It follows from (2.10) and (5.70) that

n b 2(1 -b) log n δ n (θ n -θ α ) 2 = O 1 n b-a a.s.
Since a < b, it clearly implies that (5.77) lim

n→∞ n b 2(1 -b) log n 1/2 δ n θ n -θ α = 0 a.s.
Therefore, we obtain (3.7) from (5.75), (5.76) and (5.77). The proof of the LIL for the convexified estimator ( ϑ n ) is straightforward and left to the reader, which achieves the proof of Theorem 3.3.

Proofs of the asymptotic normality results

The proof of Theorem 3.4 relies on the central limit theorem for the two-timescale stochastic algorithm given in Theorem 1 of Mokkadem and Pelletier [START_REF] Mokkadem | Convergence rate and averaging of nonlinear two-time-scale stochastic approximation algorithms[END_REF]. It is a sophisticated application of this result for the standard estimator ( ϑ n ), while it is a direct application for the convexified estimator ( ϑ n ).

Proof of Theorem 3.4. We start with the proof for the standard estimator ϑ n . As it was previously done in Section 5, our strategy is first to establish the joint asymptotic normality for the couple (θ n , ∆ n ) where ∆ n is given by (4.12), and then to deduce the joint asymptotic normality for the couple (θ n , ϑ n ). We have from (2.1) together with (4.13) that for all n ≥ 1, (6.1)

θ n+1 = θ n + a n X n+1 ∆ n+1 = ∆ n + b n Y n+1 where X n+1 = f (θ n , ∆ n ) + ψ (θ) n + V n+1 Y n+1 = g(θ n , ∆ n ) + ψ (∆) n + W n+1 with f (θ, ∆) = α -F (θ), ψ (θ) n = 0, V n+1 = F (θ n ) -I {X n+1 ≤θn} and g(θ, ∆) = -∆, ψ (∆) n = R α (θ n ) + a n b n δ n+1 G α (θ n ) + ν n+1 (θ n -θ α ), W n+1 = ε n+1 + a n b n δ n+1 V n+1 .
By denoting ∆ α = 0, we clearly have f (θ α , ∆ α ) = 0 and g(θ α , ∆ α ) = 0. To be more precise

f (θ, ∆ α ) g(θ, ∆ α ) = -f (θ α ) 0 0 -1 θ -θ α ∆ -∆ α + O ||θ -θ α || 2 0 .
On the one hand, it follows from the conjunction of (4.9), (4.10) and (4.16) that

ψ (∆) n = r (∆) n + O ||θ n -θ α || 2
where r (∆) n

= nu n+1 (θ n -θ α ). On the other hand, we infer from (2.11) and (4.17) that

r (∆) n = O √ n a log n n = o b n a.s. Furthermore, E[V n+1 |F n ] = 0, E[W n+1
|F n ] = 0, and we already saw in Sections 4 and 5 that

E[V 2 n+1 |F n ] = F (θ n )(1 -F (θ n ) and E[W 2 n+1 |F n ] = τ 2 α (θ n ). One can also check that E[V n+1 W n+1 |F n ] = F (θ n ) H α (θ n ) - a n b n δ n+1 1 -F (θ n ) . It clearly implies that lim n→∞ E[V 2 n+1 |F n ] E[V n+1 W n+1 |F n ] E[V n+1 W n+1 |F n ] E[W 2 n+1 |F n ] = α(1 -α) α(ϑ α -θ α ) α(ϑ α -θ α ) τ 2 α (θ α ) a.s.
Consequently, all the conditions of Theorem 1 in [START_REF] Mokkadem | Convergence rate and averaging of nonlinear two-time-scale stochastic approximation algorithms[END_REF] are satisfied with

Σ θα = α(1 -α) 2f (θ α )
and

Σ ϑα =        b 1 τ 2 α (θ α ) 2b 1 -1 if b = 1, τ 2 α (θ α ) 2 if b < 1.
Therefore, as ∆ α = 0, we obtain from [START_REF] Mokkadem | Convergence rate and averaging of nonlinear two-time-scale stochastic approximation algorithms[END_REF] the joint asymptotic normality (6.2)

√ n a θ n -θ α √ n b ∆ n L -→ N 0, Γ θα 0 0 Γ ϑα where Γ θα = a 1 Σ θα and Γ ϑα = b 1 Σ ϑα .
Hereafter, in order to prove the joint asymptotic normality for the couple (θ n , ϑ n ), it is only necessary to show from the very definition of ∆ n given in (4.12) that

(6.3) lim n→∞ √ n b δ n θ n -θ α = 0 a.s.
We already saw from (5.39) and (5.70) that (6.4) lim

n→∞ n b-a δ n = b 1 C α a 1 f (θ α )
.

Hence, we deduce from (2.11) and (6.4) that

√ n b δ n (θ n -θ α ) = O √ n a log n √ n b a.s.
which ensures that (6.3) holds true. Consequently, (3.8) clearly follows from (6.2) and (6.3) . The proof for the convexified estimator ( ϑ n ) is much more easy to handle. We have from (2.3) that for all n ≥ 1, (6.5)

θ n+1 = θ n + a n X n+1 ϑ n+1 = ϑ n + b n Y n+1 where X n+1 = f (θ n , ϑ n ) + ψ (θ) n + V n+1 Y n+1 = g(θ n , ϑ n ) + ψ (ϑ)
n + W n+1 with f (θ, ϑ) = α -F (θ), ψ (θ) n = 0, V n+1 = F (θ n ) -I {X n+1 ≤θn} and g(θ, ϑ) = ϑ α -ϑ, ψ

(ϑ) n = L α (θ n )-ϑ α , W n+1 = Z n+1 -L α (θ n )
, where we recall that E[Z n+1 |F n ] = L α (θ n ) with L α (θ) given by (4.2). We clearly have f (θ α , ϑ α ) = 0 and g(θ α , ϑ α ) = 0. To be more precise,

f (θ, ϑ) g(θ, ϑ) = -f (θ α ) 0 0 -1 θ -θ α ϑ -ϑ α + O ||θ -θ α || 2 0 .
In addition, we deduce from (4.8) that ψ

(ϑ) n = L α (θ n ) -L α (θ α ) = O ||θ n -θ α || 2 . Furthermore, E[V n+1 |F n ] = 0, E[W n+1
|F n ] = 0, and we already saw in Sections 4 and 5 that

E[V 2 n+1 |F n ] = F (θ n )(1 -F (θ n ) and E[W 2 n+1 |F n ] = τ 2 α (θ n ). One can also verify that E[V n+1 W n+1 |F n ] = F (θ n ) L α (θ n ) -θ n . It clearly implies that lim n→∞ E[V 2 n+1 |F n ] E[V n+1 W n+1 |F n ] E[V n+1 W n+1 |F n ] E[W 2 n+1 |F n ] = α(1 -α) α(ϑ α -θ α ) α(ϑ α -θ α ) τ 2 α (θ α )
a.s.

Consequently, our two-time-scale stochastic algorithm satisfies all the conditions of Theorem 1 in [START_REF] Mokkadem | Convergence rate and averaging of nonlinear two-time-scale stochastic approximation algorithms[END_REF] where the asymptotic variances Σ θα and Σ ϑα have been previously defined. Finally, we obtain the joint asymptotic normality (3.8) where Γ θα = a 1 Σ θα and Γ ϑα = b 1 Σ ϑα , which completes the proof of Theorem 3.4.

Numerical experiments on real data

We briefly illustrate the asymptotic behavior of our two stochastic algorithms ( ϑ n ) and ( ϑ n ) with different tuning of parameters. Since we have several elements of variability in the parameters, we have chosen typical setups even if our presentation is not exhaustive. In our synthetic benchmark, we shall consider Exponential and Gamma distributions, even though explicit formula may be found for the pair (θ α , ϑ α ).

First of all, we wish to point out that our recursive procedure is very fast for both algorithms since a set of 1000 observations is handled in less than 0.1 second with a standard laptop. Next, Figure 1 illustrates the good almost sure behavior of the standard and convexified algorithms both on Exponential and Gamma distributions. Here, we consider the E(1/10) and G(4, 3) distributions. One can also use our method to estimate online 95% confidence intervals for the superquantile ϑ α as explained in Remark 3.4. This is illustrated in Figure 3 with the Exponential and Gamma distributions with a = 2/3 and b = 1.

7.1. Real data. We finally illustrate, as a proof of concept, the use of our two algorithms on financial real-data that are freely available on the R-package tseries (EuStockMarkets dataset). Some more recent ressources may also be downloaded on the Yahoo! Finance website. We consider the four time series of the financial stock-markets DAX, CAC40, SMI, FTSE between 2014 and 2018 and compute the CVaR of the weekly log-returns, that are common indicators in the analysis of financial markets. It is commonly admitted as a reasonnable approximation that in non-exceptionnal situations, the log-returns are not far from an independent and identically distributed set of observations. As a major interest in finance, we compute the negative CVaR at the level 10% and some 95% confidence intervals as well. Our results are presented in Figure 4

b

  k P k ξ k+1 = 0 a.s.

  1-b ) P n = exp(Λ) with c = b 1 /(1 -b) and the limiting value

n b 1 P

 1 n = Γ(1 -b 1 ). The second case 1/2 < b < 1 is much more difficult to handle. It follows from the Taylor expansion of the natural logarithm log

3 .Lemma 5 . 3 .

 353 addition, as b > 1/2, we always have for all ≥ 2, b > 1. Consequently, ζ(b ) where ζ is the Riemann zeta function. Therefore, we obtain from (5.15) that (5.16) lim n→∞ 1 exp(cn 1-b ) P n = exp(Λ) where c = b 1 /(1 -b) and the limiting value The fast step size case. The proof of Theorem 3.2 relies on the following lemma which provides the QSL and the LIL for the martingales (M n ) and (N n ). Assume that the step sequences (a n ) and (b n ) are given by a n = a 1 n a and b n = b 1 n where a 1 > 0, b 1 > 1/2 and 1/2 < a < 1. Then, (M n ) and (N n ) share the same QSL

Figure 1 .

 1 Figure 1. Almost sure convergence of our algorithms for α = 0.5 and b n = 1/n.

Figure 2

 2 represents the histogram of the rescaled algorithms for several values of a and b. One can check that the convexified algorithm outperforms the standard algorithm as soon as b < a.

Figure 2 .

 2 Figure 2. Distribution of the rescaled algorithms in different situations: top-left (a = 2/3 < b = 4/5 < 1), top-right (b = 2/3 < a = 4/5 < 1), bottom (a = 2/3 < b = 1). One can verify the asymptotic normality with larger variance for the standard rescaled algorithm (top-right).

Figure 3 .

 3 Figure 3. Online confidence interval with the convexified algorithm for the Exponential and Gamma distributions.

  for the convexified algorithm tuned with the parameters a = 2/3, a 1 = 5 and b = 1, b 1 = 3/4.

Figure 4 .

 4 Figure 4. Convexified algorithm on Yahoo! Finance datasets.

  Assume that (A 1 ) and (A 2 ) hold and that the random variable X has a moment of order > 2. Moreover, suppose that f (θ α ) > 0 and that the step sequences (a n ) and (b n ) are given by

				s.
	and
	lim sup n→∞	n 2 log log n	ϑ n -ϑ α	2 = τ 2 α (θ α )	a.s.
	Theorem 3.3.

  Moreover, it is easy to see that for all n ≥ 1, b n P n = P n -P n-1 . Hence, we obtain by a telescoping argument that

				n
	(5.5)					b k P k = P n -P 0 ,
				k=1
	which leads to		lim n→∞	1 P n	n k=1	b k P k = 1.
	Therefore, it follows from Toeplitz's lemma that
	(5.6)	lim n→∞	H n+1 P n	= ϑ α	and		lim n→∞	L n+1 P n	= ϑ α	a.s.
	Finally, we find from (4.5), (5.1) and (5.6) that
	(5.7)	lim

n→∞ ϑ n = ϑ α and lim n→∞

  It means that f n converges to zero almost surely at rate n b where 1/2 < b < 1. Furthermore, the random variable X has a moment of order > 2. It implies that for some real number p > 2,

					4) together with (5.59) that
	(5.60)			lim n→∞	n b f n = 2b 1	a.s.
		sup	E[|W n+1 | p |F n ] < ∞	a.s.
		n≥0
	Consequently, we deduce from the QSL for martingales given in theorem 3 of [4]
	that
	(5.61)	lim n→∞	1 log M n	n k=1