Liqiang Song

John D Perpich

Chenggang Wu

Thierry Doan

Zuzanna Nowakowska

Jan Potempa

Peter J Christie

Eric Cascales
email: cascales@imm.cnrs.fr

Richard J Lamont
email: rich.lamont@louisville.edu

Bo Hu
email: bo.hu@uth.tmc.edu

A Unique Bacterial Secretion Machinery with Multiple Secretion Centers

Keywords: cryoelectron tomography, type IX secretion, protein transport, proton motive force

The Porphyromonas gingivalis type IX secretion system (T9SS) promotes periodontal disease by secreting gingipains and other virulence factors. By in situ cryoelectron tomography, we report that the P. gingivalis T9SS consists of eighteen PorM dimers arranged as a large, caged ring in the periplasm. Near the outer membrane, PorM dimers interact with a PorKN ring complex of ~52 nm in diameter. PorMKN translocation complexes of a given T9SS adopt distinct conformations energized by the proton motive force, suggestive of different activation states. At the inner membrane, PorM associates with a cytoplasmic complex that exhibits 12-fold symmetry and requires both PorM and PorL for assembly.

Activated motors deliver substrates across the outer membrane via one of eight Sov translocons arranged in a ring. The T9SSs are unique among known secretion systems in bacteria and eukaryotes in their assembly as supramolecular machines composed of apparently independently-functioning translocation motors and export pores.

Significance Statement

The newly described type IX secretion systems (T9SSs) translocate virulence factors and can mediate specialized gliding motility among bacterial pathogens of the Fibrobacteres-Chlorobi-Bacterioidetes superphylum. We visualized the spatial organization of the T9SS in its native context in the Porphorymonas gingivalis cell envelope by in situ cryoelectron tomography. The T9SS exhibits distinct symmetries across the bacterial cell envelope: i) a cytoplasmic complex requiring PorL and PorM for assembly exhibits 12-fold symmetry, ii) a periplasmic complex composed of PorM exhibits 18-fold symmetry and attaches to a continuous PorKN ring near the outer membrane, and iii) eight Sov translocons are arranged with 8-fold symmetry at the cell surface. The T9SS is the largest of the known bacterial secretion systems, and evidently arranges as multiple, independently-functioning translocation motors.

Introduction

Bacteria deploy at least eleven different translocation systems to export macromolecules across their cell envelopes [START_REF] Galan | Protein-injection machines in bacteria[END_REF][START_REF] Christie | The rich tapestry of bacterial protein translocation systems[END_REF][START_REF] Majewski | Secretins revealed: structural insights into the giant gated outer membrane portals of bacteria[END_REF] . Of these, the recently-discovered Type IX secretion systems (T9SSs) carried by many species of the Fibrobacteres-Chlorobi-Bacteroidetes (FCB) superphylum play various critical roles in pathogenesis and colonization [START_REF] Veith | Type IX secretion: the generation of bacterial cell surface coatings involved in virulence, gliding motility and the degradation of complex biopolymers[END_REF][START_REF] Lasica | The type IX secretion system (T9SS): Highlights and recent insights into its structure and function[END_REF][START_REF] Mcbride | Bacteroidetes gliding motility and the type IX secretion system[END_REF][START_REF] Sato | A protein secretion system linked to bacteroidete gliding motility and pathogenesis[END_REF] . Two bacterial models have emerged for detailed mechanistic and structural analyses of the T9SS. The Por T9SS in Porphyromonas gingivalis, a key contributor to human periodontitis and an emerging systemic pathogen, transports various heme-binding proteins and toxins aiding in colonization and invasion of gingival and other tissues [START_REF] O'brien-Simpson | Role of RgpA, RgpB, and Kgp proteinases in virulence of Porphyromonas gingivalis W50 in a murine lesion model[END_REF][START_REF] Nakayama | Porphyromonas gingivalis and related bacteria: from colonial pigmentation to the type IX secretion system and gliding motility[END_REF] . The Gld T9SS in Flavobacterium johnsoniae secretes adhesins SprB and RemA and dozens of other effector proteins, but also functions as a rotary motor to direct the movement of a surface-bound adhesin along helical tracks, resulting in a form of gliding motility [START_REF] Shrivastava | The screw-like movement of a gliding bacterium is powered by spiral motion of cell-surface adhesins[END_REF][START_REF] Nakane | Helical flow of surface protein required for bacterial gliding motility[END_REF] . Studies have shown that these T9SSs are large, cell envelope-spanning nanomachines composed of at least 14 essential subunits (Fig. 1A) [START_REF] Mcbride | Bacteroidetes gliding motility and the type IX secretion system[END_REF][START_REF] Sato | A protein secretion system linked to bacteroidete gliding motility and pathogenesis[END_REF] . They translocate substrates, exported to the periplasm by the general secretory (Sec) pathway, across the outer membrane (OM). At the cell surface, they release their substrates to the milieu or deliver them to an 'attachment complex', which covalently links the substrates to anionic lipopolysaccharides (A-LPSs) for surface display [START_REF] Lasica | The type IX secretion system (T9SS): Highlights and recent insights into its structure and function[END_REF][START_REF] Veith | Protein substrates of a novel secretion system are numerous in the Bacteroidetes phylum and have in common a cleavable C-terminal secretion signal, extensive post-translational modification, and cell-surface attachment[END_REF] .

Three stable subassemblies have been identified through biochemical or single-particle microscopy approaches: i) an inner membrane (IM) molecular motor composed of the PorL and PorM subunits [START_REF] Vincent | Characterization of the Porphyromonas gingivalis type IX secretion transenvelope PorKLMNP core complex[END_REF][START_REF] James | Structure and mechanism of the proton-driven motor that powers type 9 secretion and gliding motility[END_REF] with structural and functional homologies with the MotAB/PomAB flagellar and ExbBD/TolQR PMFdependent motors [START_REF] Santiveri | Structure and function of stator units of the bacterial flagellar motor[END_REF][START_REF] Deme | Structures of the stator complex that drives rotation of the bacterial flagellum[END_REF][START_REF] Celia | Cryo-EM structure of the bacterial Ton motor subcomplex ExbB-ExbD provides information on structure and stoichiometry[END_REF][START_REF] Zhu | In situ structure of the Vibrio polar flagellum reveals a distinct outer membrane complex and its specific interaction with the stator[END_REF] ; ii) a large ~50 nm ring-shaped complex composed of periplasmic PorN and outer membrane (OM)-associated lipoprotein PorK [START_REF] Gorasia | Structural insights into the PorK and PorN components of the Porphyromonas gingivalis type IX secretion system[END_REF][START_REF] Gorasia | In situ structure and organisation of the type IX secretion system[END_REF] ; and iii) an OM-spanning translocon, termed Sov or SprA, configured as a 36-strand -barrel with a central pore of ~70 Å [START_REF] Lauber | Type 9 secretion system structures reveal a new protein transport mechanism[END_REF] . Despite this progress, it is not yet known how these substructures interact or coordinate their activities, in part because there is no overarching view of the T9SS in the native context of the bacterial cell envelope. Here, we visualized the intact T9SS of P. gingivalis by in situ cryoelectron tomography (cryo-ET). Remarkably, the T9SS presents not as a single functional entity, as shown for all known bacterial and eukaryotic protein transport systems, but rather as a composite of apparently independent translocation motors and Sov translocons arranged symmetrically as distinct rings in the cytoplasm, periplasm, and cell surface.

Results and Discussion

The T9SS: A large, caged-ring structure. P. gingivalis cells are small (~0.5 m in diameter) and well suited for cryo-ET imaging (Fig. 1B, Fig. S1A). A typical three-dimensional (3D) reconstruction of a P. gingivalis cell revealed the IM and OM, as well as an electron dense surface layer (EDSL), which is known to consist of T9SS substrates attached at the cell surface (Fig. 1B -D) [START_REF] Veith | Type IX secretion: the generation of bacterial cell surface coatings involved in virulence, gliding motility and the degradation of complex biopolymers[END_REF] . In wild-type cells, two periplasmic rods attached to the OM and ~50 nm apart were visualized in side views (Fig. 1B-D, Fig. S1A, Movie S1). In fact, the rods constitute the peripheral edges of a large ring similar in size (~50 nm) to the previously visualized PorKN rings [START_REF] Gorasia | Structural insights into the PorK and PorN components of the Porphyromonas gingivalis type IX secretion system[END_REF][START_REF] Gorasia | In situ structure and organisation of the type IX secretion system[END_REF] as shown in tilt-and top-down views (Fig. S1A,B, Movies S1 and 2). The rods visualized in side view are missing in a strain lacking PorK (Fig. 1E,F), which is an essential T9SS subunit [START_REF] Sato | A protein secretion system linked to bacteroidete gliding motility and pathogenesis[END_REF][START_REF] Gorasia | Structural insights into the PorK and PorN components of the Porphyromonas gingivalis type IX secretion system[END_REF] . The porK mutant strain also lacks the EDSL (Fig. 1E,F) and fails to secrete T9SS effectors such as the Arg gingipain (Rgp) protease, whereas the trans-complemented strain is restored for Rgp secretion (Fig. S2). Similarly, mutants deleted of porN are devoid of the EDSL and OMassociated rods, and do not secrete Rgp protease (Figs. S1A and2). Previously, it was shown that two IM subunits PorM and PorL with large domains respectively in the periplasm and cytoplasm [START_REF] Vincent | Characterization of the Porphyromonas gingivalis type IX secretion transenvelope PorKLMNP core complex[END_REF] are also required for T9SS function, but not for assembly of the PorKN ring [START_REF] Gorasia | Structural insights into the PorK and PorN components of the Porphyromonas gingivalis type IX secretion system[END_REF] . Accordingly, here we determined that the porM and porL mutants lack the EDSL and do not secrete the Rgp effector, but elaborate the OM-associated rod structures (Figs. S1 and2). Taken together, these findings strongly indicate that the visualized structures correspond to in situ T9SS nanomachines and that the OM-associated rods and rings visualized in side-and top-down views correspond to the PorKN ring complexes characterized by single particle cryoelectron microscopy (cryo-EM) [START_REF] Gorasia | Structural insights into the PorK and PorN components of the Porphyromonas gingivalis type IX secretion system[END_REF] (Fig. 1L-O). The average number of T9SSs visualized by in situ cryo-ET was ~1.3 per cell with a range of 0 to 4, although this might underestimate the actual number of partially or fully assembled T9SSs per cell (Fig. 1D).

To determine the 3D structure of the entire T9SS machine, we imaged over 800 wild-type P. gingivalis cells, and picked 694 ring-shaped particles from 516 tomograms (Table S1). By sub-tomogram averaging, we generated an overall structure of the T9SS at a resolution of ~32 Å (Fig. 1H, Fig. S1C). The T9SS spans the entire cell envelope, with dimensions of ~57 nm in height and ~52 nm in width, and consists of three distinct substructures: i) a surface structure (SS) on the extracellular side of the OM, ii) a periplasmic structure designated here as the core complex (CC), and iii) a cytoplasmic complex (Fig. 1H-I). The SS is located ~6 nm above the OM and below the EDSL, and in top-down view presents as a ring with outer and inner diameters of 48 and 31 nm (Fig. 1H,J). During initial alignment and without symmetry applied, the SS ring appears to be a featureless ring. We applied 18-fold symmetry to smooth the density; however, upon local refinement and classification (see below), we determined that the SS in fact is composed of 8 discontinuous structural units. The CC consists of two parts: i) the PorKN rods/ring structure (side/top-down views) of ~8 nm in thickness and 52 nm in diameter (Fig. 1H,K), and ii) strips of densities presenting in side view as arches of ~17 nm in height that connect the OM-associated PorKN rods to the IM (Figs. 1H-I, 2A). The cytoplasmic complex consists of a small central knob flanked by two densities when visualized in side-view (Fig. 1H-I). In 3D surface renderings of the intact T9SS shown at 45 o and 90 o (top-down) angles, ring-shaped architectures of the SS, CC, and cytoplasmic complex are readily evident (Fig. L-O).

Besides these densities, we detected three layers of density in the periplasm and a fourth between the cell surface and the EDSL (Fig. 1H, Fig. S3A,B). A 3D map reconstructed by sub-tomogram averaging of randomly picked regions of the P. gingivalis cell envelope established that these four layers are constituents of the cell envelope and are not parts of the T9SS machine (Fig. S3C,D). Consequently, the T9SS presents as a large, caged-ring structure that encompasses a central region of the P. gingivalis cell envelope whose IM, periplasmic and OM features are largely intact.

The PorMKN core complex (CC). We next focused on solving refined structures of the CC, with emphasis on the side-view arches. Prior studies have shown that PorM has an N-proximal transmembrane domain (TMD) and a large ~480-residue C-terminal region that extends across the periplasm and interacts with the OM-associated PorKN complex [START_REF] Vincent | Characterization of the Porphyromonas gingivalis type IX secretion transenvelope PorKLMNP core complex[END_REF] . By X-ray crystallography, the periplasmic regions of PorM (PorMp) and the F. johnsoniae homolog GldM (GldMp) presented as elongated dimers with dimensions 180 Å in length and 50 and 35 Å in crosssections [START_REF] Leone | Type IX secretion system PorM and gliding machinery GldM form arches spanning the periplasmic space[END_REF][START_REF] Sato | a core component of bacterial type IX secretion system, forms a dimer with a unique kinked-rod shape[END_REF] . Through local refinements of periplasmic densities associated with T9SSs in WT cells, we determined that the periplasmic arches correspond to 18 distinct knobs arranged in a ring of 52 nm in diameter when visualized in top-down view (Table S1, S3, Fig. 2B, Fig. S4A). The porM mutant possesses the PorKN ring, but lacks the underlying knobbed ring and the cytoplasmic complex (Fig. 2C,F). In consideration of the PorMp and GldMp X-ray structures [START_REF] Leone | Type IX secretion system PorM and gliding machinery GldM form arches spanning the periplasmic space[END_REF][START_REF] Sato | a core component of bacterial type IX secretion system, forms a dimer with a unique kinked-rod shape[END_REF] , we propose that the side-view arches and top-down-view knobs correspond to 18 PorM dimers, whose N-proximal regions are anchored in the IM and C-terminal regions extend through the periplasm to the OM-associated PorKN ring (Fig. 2D,E).

The PorMKN complex exhibits two other noteworthy features. First, in the native T9SS, the PorKN ring is 8 nm thick and the PorM arches are 17 nm in length, whereas in the porM mutant, the PorKN ring is only 7 nm thick (Fig. 2A,C Fig. S5A,B). In the atomic structure, the PorMp dimers are 18 nm in length [START_REF] Sato | a core component of bacterial type IX secretion system, forms a dimer with a unique kinked-rod shape[END_REF] and composed of 4 domains, D1-D4, with domain D4 directly interacting with PorN [START_REF] Vincent | Characterization of the Porphyromonas gingivalis type IX secretion transenvelope PorKLMNP core complex[END_REF][START_REF] Leone | Type IX secretion system PorM and gliding machinery GldM form arches spanning the periplasmic space[END_REF] . To reconcile these in situ and in vitro findings, the PorKN ring visualized in situ might actually consist of PorK, PorN, and domain D4 of PorM. Alternatively, the interaction between PorM and PorKN might induce a structural change resulting in thickening of the PorKN ring.

Second, in the global averaged map, the PorKN ring forms a 60° angle with respect to the OM (Fig. 3A).

However, when the PorKN rings were locally refined at the junctions with the PorM arches, we detected three distinct structural Classes in which the PorMKN complex forms angles of 90°, 70° and 50° relative to the OM (Fig. 3B). We quantified the PorMKN complexes of each Class among the entire set of 12,492 complexes in the 694 imaged T9SSs (18 complexes per T9SS × 694 T9SSs). Among the 12,492 complexes, the Class 1 (90°) and Class 3 (50°) complexes were more abundant (35.5 and 37.5 %, respectively) than the Class 2 (70°) complexes (27 %) (Fig. S6A). Furthermore, among each of the 694 imaged T9SSs, the Class 3 complexes dominated with an average of 10 complexes and a range of 3 to 16 per T9SS (Fig. S6A). The Class 2 structures were least abundant, insofar as ~14 % of the T9SSs lacked any Class 2 complexes and the bulk of the remaining T9SSs had ≤ 6. The Class 1 complexes had an intermediate distribution with an average of 4 per T9SS and a range of 0-10. Importantly, in the absence of porM, there was little variation from the 60° global average at the PorKN ring -OM junctions (Fig. S5D), indicating that PorM contributes directly to the distinct PorKN -OM junction architectures in WT machines.

Recently, it was shown that PorL and PorM and the F. johnsoniae GldL and GldM homologs form structurally-similar proton-motive-force (PMF)-transducing motors [START_REF] James | Structure and mechanism of the proton-driven motor that powers type 9 secretion and gliding motility[END_REF][START_REF] James | Structures of the type IX secretion/gliding motility motor from across the phylum Bacteroidetes[END_REF] . Moreover, in the atomic structures, PorMp and GldMp dimers possess distinct bends in the hinge regions between the D1 -D2 or D2 -D3 domain interfaces [START_REF] James | Structure and mechanism of the proton-driven motor that powers type 9 secretion and gliding motility[END_REF][START_REF] Lauber | Type 9 secretion system structures reveal a new protein transport mechanism[END_REF][START_REF] Leone | Type IX secretion system PorM and gliding machinery GldM form arches spanning the periplasmic space[END_REF] . These finding have led to a model in which PMF-energy transduction regulates the activation states of the PorLM and GldLM translocation motors [START_REF] James | Structure and mechanism of the proton-driven motor that powers type 9 secretion and gliding motility[END_REF][START_REF] Vincent | Dynamic proton-dependent motors power type IX secretion and gliding motility in Flavobacterium[END_REF] . Upon sensing of the PMF, PorLM and GldLM transduce PMF energy in the form of conformational changes in the periplasmic domains of PorM and GldM as a prerequisite for substrate secretion or gliding motility [START_REF] Vincent | Dynamic proton-dependent motors power type IX secretion and gliding motility in Flavobacterium[END_REF][START_REF] Veith | The type IX secretion system and its role in bacterial function and pathogenesis[END_REF] . In this context, we propose that the visualized conformational Classes of the PorKMN complexes correspond to different activation states of the motors.

Evidence for PMF-induced structural changes. To further explore whether transduction of PMF energy impacts the conformational status specifically at the PorKN -OM junction, we treated P. gingivalis cells with the protonophore carbonyl cyanide m-chlorophenyl hydrazone (CCCP), which disrupts the PMF and inactivates the T9SS [START_REF] James | Structure and mechanism of the proton-driven motor that powers type 9 secretion and gliding motility[END_REF][START_REF] Pate | Evidence that gliding motility in prokaryotic cells is driven by rotary assemblies in the cell envelopes[END_REF] . Approximately 90 % of the visualized T9SSs from CCCP-treated cells possessed all of the structural features of machines from untreated cells, but the remaining 10 % lacked PorM arches and the cytoplasmic complex (Fig. 3A,D). Among the 90 % of T9SSs bearing WT features, we identified significant changes in the relative percentages of Classes with distinct PorKN-OM junction angles. Most notably, Class 2 structures (bend-angle 70 o ) increased significantly, from 27.0 to 56.6%, upon CCCP-treatment. Conversely, the Class 1 (bend-angle 90 o ) and Class 3 (bend-angle 50 o ) structures decreased from 35.5 to 20.4% and 37.5 to 23%, respectively (Fig. 3E). The significant increase in the percentage of Class 2 structures in PMF-collapsed cells supports a proposal that this Class corresponds to the quiescent or 'closed' state of the motor, whereas the Class 1 and 3 structures represent or activated or 'open' states. Why activated motors are represented by bend-angles of 90 o or 50 o could be explained by recent evidence that GldLM in the F. johnsoniae T9SS functions as a rotary motor, such that the rotating periplasmic domain of GldM is envisioned to form a large arc as it extends to the PorKN junction [START_REF] James | Structure and mechanism of the proton-driven motor that powers type 9 secretion and gliding motility[END_REF][START_REF] Leone | Type IX secretion system PorM and gliding machinery GldM form arches spanning the periplasmic space[END_REF][START_REF] Sato | a core component of bacterial type IX secretion system, forms a dimer with a unique kinked-rod shape[END_REF] . Recently, it was also shown that mutations in GldL have different impacts on T9SS activity. Whereas a GldL E49A mutation abolishes both substrate secretion and gliding motility, another mutation, E59A, prevents gliding but does not impact substrate secretion. These findings suggest that GldLM motors exist in a quiescent state and in two activated states, one supporting both gliding motility and substrate secretion and a second only secretion [START_REF] Vincent | Dynamic proton-dependent motors power type IX secretion and gliding motility in Flavobacterium[END_REF] . PorLM complex assembled across the IM [START_REF] James | Structures of the type IX secretion/gliding motility motor from across the phylum Bacteroidetes[END_REF] .

Whether the P. gingivalis

Among the small fraction (~10 %) of T9SSs from CCCP-treated cells that lacked the PorM arches and cytoplasmic complexes (Fig. 3D), the bend angle of the PorKN -OM junction approximates that observed for porM mutant machines. CCCP-induced disengagement of the PorM arches therefore phenocopies the porM mutation with respect to the overall PorKN ring architecture. Importantly, however, this subpopulation of CCCP-deenergized T9SSs retains the surface structures (SS) that are lacking in the porM or porL mutant machines (Fig. 2C,4D). In a small fraction of T9SSs de-energized by collapse of the PMF, PorM and PorL thus appear to disengage from the PorKN ring without destabilization of the previously docked SS.

We sought to gain further biochemical evidence for PMF-mediated activation of T9SSs, and to this end we capitalized on the availability of antibodies to assay for energy-mediated conformational changes in the PorM and PorN homologs, GldM and GldN, in F. johnsoniae cells. Reminiscent of previous findings for the PMF-responsive energy transducer TonB [START_REF] Ra Larsen | Assessing energy-dependent protein conformational changes in the TonB system[END_REF] , trypsin treatments of spheroplasts derived from CCCP-untreated or -treated cells yielded different GldM protease cleavage patterns (Fig. 3F). In the absence of CCCP treatment, GldM remained intact upon exposure to trypsin, whereas CCCP treatment resulted in generation of two GldM cleavage products. By LS-MS/MS spectrometry, we determined that both cleavage sites (downstream residues Lys252 and Arg322, respectively) reside within the linker region between domains D2 and D3 (Fig. 3F). We were unable to assess the effect of PMF energy depletion on the conformational status of GldN, as the GldN trypsin cleavage patterns were unaltered regardless of prior treatment of cells with CCCP (Fig. S7). Overall, these data confirm earlier predictions that GldLM transduces IM energy from the PMF in the form of a conformational change(s) in the periplasmic arms of GldM to activate the T9SS for transport or gliding motility [START_REF] Leone | Type IX secretion system PorM and gliding machinery GldM form arches spanning the periplasmic space[END_REF] . In view of the structural homology of GldM and PorM, we suggest that PMF sensing similarly induces conformational changes in the PorM translocation motor. With respect to the PMF-driven conformational changes we observed here for individual PorKMN complexes, it is interesting to note that complexes of a given bend-angle Class tend to spatially cluster within each T9SS (Fig. S6A-B). This clustering could imply that adjacent complexes are coordinately activated in response to PMF energy sensing.

Refinement of the cytoplasmic complex. To better understand the structural basis for energy transduction, we locally refined the cytoplasmic complex (Fig. 4A-C, Fig. S4B). Three distinct densities were evident: i) a central knobbed cylinder projecting ~14 nm from the IM into the cytoplasm, ii) a flanking ring of ~6 nm in thickness and ~20 nm in diameter, and iii) a spoked-ring of ~32 nm in diameter with 12-fold symmetry (Fig. 4B,C, Fig. S4B). PorL and PorM are the only essential T9SSs proteins identified to date with cytoplasmic domains (Fig. 1A). Indeed, the porL and porM mutant machines lack the periplasmic arches and cytoplasmic densities, showing a co-dependence on PorL and PorM for assembly of these structures (Figs. 2C,4D). PorM spans the IM once and its ~9-residue, N-terminal cytoplasmic domain is unlikely to contribute detectable density. By contrast, PorL has two N-terminal transmembrane helices (TMHs) and its large (~230-residue) C-terminal domain resides in the cytoplasm [START_REF] James | Structures of the type IX secretion/gliding motility motor from across the phylum Bacteroidetes[END_REF] . Recently, the 3D structures of the PorLM and GldLM complexes, were shown to consist of the 10 TMHs from five PorL subunits arranged in a ring surrounding the two TM helices of the PorM dimer [START_REF] James | Structure and mechanism of the proton-driven motor that powers type 9 secretion and gliding motility[END_REF][START_REF] James | Structures of the type IX secretion/gliding motility motor from across the phylum Bacteroidetes[END_REF] . Thus, the 5 cytoplasmic domains of PorL associated the recently visualized PorLM complex [START_REF] James | Structures of the type IX secretion/gliding motility motor from across the phylum Bacteroidetes[END_REF] might contribute to one or more of the cytoplasmic densities detected by in situ cryo-ET.

The PorM dimers are in close juxtaposition with the outer most cytoplasmic knobs, suggesting the possibility that they are composed of PorL pentamers (see Fig. S4B, classes e-h). If so, it is necessary to account for the 18-vs 12-fold symmetry mismatch that exists between the PorM arches and putative PorL knobs. Conceivably, a PorM dimer engages stably with the PorL pentamer only when activated, for example, upon sensing of bound substrate in the periplasm. Such a model of substrate-induced PorLM engagement might allow for one PorL pentamer to interact with and energize more than one spatiallyjuxtaposed PorM dimer, which would account for our observation that PorKMN complexes of a given structural Class, or activation state, tend to spatially cluster within each T9SS (Fig. S6A-B). It is important to note, however, that PorL's cytoplasmic domain is disordered when purified and analyzed by single-particle cryo-EM [START_REF] James | Structure and mechanism of the proton-driven motor that powers type 9 secretion and gliding motility[END_REF] . If this domain is also too flexible for detection by in situ cryo-ET, the visualized densities might instead correspond to other Por components. One candidate is PorX, the response regulator of the PorXY two-component system that was shown to interact with PorL [START_REF] Vincent | The PorX response regulator of the Porphyromonas gingivalis PorXY two-component system does not directly regulate the type IX secretion genes but binds the PorL subunit[END_REF] , but further studies are needed to define subunit contributions to the observed cytoplasmic densities.

The Surface Structure (SS). Finally, we refined the structure of the SS visualized initially as an extracellular ring (Fig. 1J). Strikingly, the SS presents as eight distinct densities of 8 nm in diameter arranged in a ring of ~48 nm in diameter (Fig. 5A-B, Fig. S4C). Each of the eight densities has a lateral pore near the OM that faces the center of the T9SS supramolecular complex (Fig. 5E-F). The overall size and presence of a lateral pore are highly reminiscent of the 267 kDa SprA/Sov translocon solved at ~3.5 Å by single particle cryo-EM [START_REF] Gorasia | In situ structure and organisation of the type IX secretion system[END_REF] . This cryo-EM structure docks well on the SS densities (Fig. 5G-H, Fig. S8A,B). Moreover, a strain deleted of Sov retained all features of the T9SS except for the SS; this strain also lacked the EDSL and failed to secrete Rgp protease (Fig. 5C,D, Figs. S1,2). Considering the size of Sov protein, we propose that the SS densities are largely composed of Sov, with potential contributions from other components, e.g., the PorV shuttle protein [START_REF] Glew | PorV is an outer membrane shuttle protein for the type IX secretion system[END_REF] . In the cryo-EM structure, the location of detergent associated with the translocon was used to define the likely plane of the OM, giving rise to the proposal that the -barrel extends ~2 nm above the OM and an extracellular cap projects an additional 5 nm [START_REF] Lauber | Type 9 secretion system structures reveal a new protein transport mechanism[END_REF] . In line with those findings, the SS densities project ~6-7 nm above the OM (Fig. 5E). Interestingly, beneath each of the Sov densities, the OM is missing its inner leaflet (Fig. 5A, Fig. S8A). This gap or hole might be a point of entry for substrates, delivered by the PorLMKN translocation motor, into the Sov translocon. The importance of the translocation motor -Sov contact for a stable association of translocons with the T9SS is underscored by the fact that deletions of any of the translocation motor subunits abolished detectable Sov surface densities (Figs. 2C,4D). Intriguingly, despite numerous attempts, no studies have yet shown a direct interaction between the PorL/M/K/N proteins and the Sov translocon.

were confirmed by sequencing and introduced into P. gingivalis by electroporation, and mutant strains were selected on TSB pates containing erythromycin. Loss of mRNA from the target gene without disruption of expression of the other genes in the operon (por operon) was confirmed by RT-PCR (Fig. S10) using the primers listed in Table S2.

For complementation of porK in P. gingivalis ATCC33277, a 500-bp sequence upstream of PGN_1678 containing the promoter region was fused to the coding region of porK using PCR fusion and the primers listed in Table S2. The resulting PCR product was confirmed by sequencing and cloned into the pT-COW vector to create pT-COW-porK. The plasmid was transformed into chemically competent E. coli (strain S17-1) and selected using ampicillin. The plasmid was transferred into P. gingivalis ΔporK via conjugation with E. coli [START_REF] James | LuxS involvement in the regulation of genes coding for hemin and iron acquisition systems in Porphyromonas gingivalis[END_REF][START_REF] Park | Contact-dependent protein secretion in Porphyromonas gingivalis[END_REF] , and transconjugants were selected with gentamicin (50 μg ml -1 ) and tetracycline.

F. johnsoniae CJ1827 (UW101 rpsL, Str R ) and its gldM isogenic mutant have been previously described [START_REF] Shrivastava | GldL, GldM, and SprA are required for secretion of the cell surface gliding motility adhesins SprB and RemA[END_REF] . F. johnsoniae strains were grown in Casitone-Yeast extract (CYE) medium (10 g L -1 casitone, 5 g L -1 yeast extract, 8 mM MgSO 4 , 10 mM Tris-HCl pH7.6) at 28°C with agitation. The protonmotive force uncoupler carbonyl cyanide-m-chlorophenylhydrazone (CCCP, Sigma Aldrich) was solubilized in ethanol and was added in the culture 30 min prior to cell harvesting, at the final concentration of 10 M.

Spheroplasting and limited proteolysis. F. johnsoniae UW101 cells were grown in 5 mL of CYE medium to an A 600 =0.8, harvested by centrifugation and resuspended in 100 L of 20 mM Tris-HCl pH 8.0, 20% sucrose, 1 mM EDTA, and 100 μg ml -1 of lysozyme. After 30 min incubation at room temperature (20°C), 100 L of ice-cold sterile water was added, and the mixture was carefully mixed by three inversions. 50 L of the spheroplast suspension were treated with trypsin (100 μg ml -1 ). After 5 min on ice, 17 μL of boiling 4× Laemmli loading buffer were added, and immediately vortexed and boiled for 5 min prior to SDS-PAGE.

Immunoprecipitation and LC-MS/MS analyses. F. johnsoniae spheroplasts obtained from a 50-mL culture treated with 10 M CCCP were prepared and subjected to limited proteolysis using trypsin.

Proteolysed spheroplasts were resuspended in 1 mL of CelLytic (Sigma-Aldrich) in presence of aprotinin (Sigma-Aldrich), and incubated for 30 min on a wheel at room temperature. Unsolubilized material was discarded by centrifugation for 10 min at 20,000 × g, and subjected to immunoprecipitation using anti-GldM antibodies bound to Protein A Sepharose CL-4B (GE Healthcare). After 2 hours of incubation at room temperature on a wheel, the beads were washed three times with CelLytic B, resuspended in Laemmli loading buffer, and subjected to SDS-PAGE and InstantBlue staining (Expedeon). Bands corresponding to the proteolytic products were cut, washed with 100 mM acetonitrile/ammonium bicarbonate pH 7.5, reduced by 10 mM dithiothreitol in 100 mM ammonium bicarbonate pH 7.5, alkylated by 55 mM iodoacetamide in 100 mM ammonium bicarbonate pH 7.5 and then digested overnight at 37°C with 10 ng/ μl -1 of Trypsin/Lys-C Mix from Pseudomonas aeruginosa (Promega) in 25 mM ammonium bicarbonate pH 7.5, 0.025% (v/v) proteaseMAX surfactant (Promega). Tryptic peptides were extracted from gel bands by0.1% (v/v) trifluoroacetic acid, 0.01% (v/v) proteaseMAX, 50% (v/v) acetonitrile, and dried by speed vacuum. Samples solubilized in 0.05% (v/v) trifluoroacetic acid, 2% (v/v) acetonitrile were analysed on a ESI-Q-Exactive Plus (ThermoFisher) mass spectrometer coupled to a nanoliquid chromatography (Ultimate 3000, Dionex). Peptides were eluted from a C18 column (Acclaim PepMap RSLC, 75 μm × 150 mm, 2 μm, 100 Ǻ, Dionex) by a 6-40% linear gradient of mobile phase B (0.1% (v/v) formic acid/80% (v/v) acetonitrile) in mobile phase A (0.1% (v/v) formic acid) for 52 min.

The peptides were detected in the mass spectrometer in a positive ion mode using a Top 10 Data Dependent workflow. A single full scan acquisition in the Orbitrap at a resolution of 70,000 in a 350-1900 m/z range was followed by a fragmentation MS/MS step, at a resolution of 17,500, of the 10 top ions, in the Higher Energy Collisional Dissociation cell set at 27 eV. The spectra were processed by Proteome Discoverer software (ThermoFisher, version 2.1.0.81) using the Sequest HT algorithm with the search following parameters: F. johnsoniae UW101 GldM protein (GeneBank identifier (GI): 58531935); trypsin enzyme (maximum 2 missed cleavages); fixed modification: carbamidomethyl (Cys); variable modification: oxidation (Met); mass values specific for monoisotopic; precursor mass tolerance: ±10 ppm; fragment mass tolerance: ±0.02 Da.

Preparation of frozen-hydrated specimens.

Strains grown anaerobically at 37 o C were harvested in the late log phase by centrifugation at 9000 × g. The pellet was resuspended in phosphate-buffered saline (PBS). To prepare CCCP treated wild-type cells, CCCP was added to a final concentration of 20 µM and incubated for 20 min and then mixed with 10 nm diameter colloidal gold beads (used as fiducial markers in image alignment) and deposited onto freshly glow-discharged, holey carbon grids for 1 min. The grids were then blotted with filter paper and rapidly frozen in liquid ethane, using a gravity-driven plunger apparatus as previously described [START_REF] Hu | Structural bases for F plasmid conjugation and F pilus biogenesis in Escherichia coli[END_REF] .

CryoET data collection and 3-D reconstructions. Frozen-hydrated specimens were imaged at -170 °C using a Polara G2 electron microscope (FEI Company) equipped with a field emission gun and a direct detection device (Gatan K2 Summit). The microscope was operated at 300 kV with a magnification of ×9,400, resulting in an effective pixel size of 4.5 Å at the specimen level. We used SerialEM [START_REF] Mastronarde | Automated electron microscope tomography using robust prediction of specimen movements[END_REF] to collect low-dose, single-axis tilt series with dose fractionation mode at about 8 µm defocus. Tilt series were collected from -51° to 51° with increment of 3° and a cumulative dose of ~70 e-/Å 2 distributed over 35 stacks. Each stack contains ~8 images. We used Tomoauto 39 to facilitate data processing, which includes drift correction of dose-fractionated data using Motioncorr [START_REF] Li | Electron counting and beam-induced motion correction enable near-atomic-resolution single-particle cryo-EM[END_REF] and assembly of corrected sums into tilt series, automatic fiducial seed model generation, alignment and contrast transfer function correction of tilt series by IMOD [START_REF] Kremer | Computer visualization of three-dimensional image data using IMOD[END_REF] , and weighted back projection (WBP) reconstruction of tilt series into tomograms using Tomo3D [START_REF] Agulleiro | Tomo3D 2.0--exploitation of advanced vector extensions (AVX) for 3D reconstruction[END_REF] . Each tomographic reconstruction is 3,710 × 3,838 × 1,600 pixels and ~1.30 Gb in size. In total, 4,227 tomographic reconstructions from 6 different strains were generated.

Sub-tomogram averaging and correspondence analysis.

We used tomographic package I3 for subtomogram analysis as described previously [START_REF] Winkler | 3D reconstruction and processing of volumetric data in cryo-electron tomography[END_REF] . A total of 2936 type IX secretion machines (400 × 400 × 400 voxels) were visually identified and then extracted from 1947 tomographic reconstructions. Two of the three Euler angles of each secretion machine were estimated based on the orientation of each particle in the cell envelope. To accelerate image analysis, 4×4×4 binned sub-tomograms (100×100×100 voxels) were used for initial alignment. The alignment proceeds iteratively with each iteration consisting of three parts in which references and classification masks are generated, sub-tomograms are aligned and classified, and finally class averages are aligned to each other. After multiple cycles of alignment and classification for 4×4×4 binned sub-tomograms, we used 2×2×2 binned sub-tomograms for refinement.

The initial alignments were carried out without symmetry imposed. To determine the symmetry of the SS, CC and cytoplasmic complex, after the initial alignments of the intact machine, the rotation centers of the sub-volumes were moved to the corresponding centers of SS, CC and cytoplasmic complex, respectively, and masks were applied to include the SS, CC or cytoplasmic complex. Once the symmetries were determined through classification, one of the classes was used as reference to align other classes and the corresponding symmetry was applied to the final refinements. Fourier shell correlation (FSC) between the two independent reconstructions was used to estimate the resolution of the averaged structures.

Visualization and molecular modeling. We used IMOD to visualize the maps. IMOD and EMAN2 [START_REF] Kremer | Computer visualization of three-dimensional image data using IMOD[END_REF][START_REF] Chen | A complete data processing workflow for cryo-ET and subtomogram averaging[END_REF] were used to generate 3D surface rendering of P. gingivalis cells and UCSF ChimeraX (http://www.rbvi.ucsf.edu/chimerax) to visualize sub-tomogram averages in 3D and molecular modelling.

Each of 18 PorM dimer structures (PDB 7CMG) was manually fitted onto the each of 18 arches of densities. We docked the Sov homolog SprA (PDB 6H3J) into the 8 pore densities on the cell surface.

Quantification and statistical analyses. Quantification and statistical analyses are integral parts of the algorithms and software used in our high throughput cryoelectron tomography pipeline. In particular, massive data enabled us to use multivariate statistical analysis and classification (which are implemented in tomographic package i3) for processing and interpretation of the sub-tomograms of T9SS extracted from P. gingivalis cell reconstructions.

  Por T9SS rotates is not yet known. If so, a PMF-energized rotating PorM dimer might impose large bend-angles at the PorKN junction, and two such activated (Class 1 and 3) structural states might have been captured by in situ cryo-ET imaging. By contrast, in the deenergized state, a nonrotating PorM dimer potentially fixes the PorKN junction at the intermediate bend-angle represented by the Class 2 structures. Even if PorM does not rotate, PMF sensing clearly induces large conformational changes in the PorM dimers that are likely mediated through the recently visualized

Figure 1 .

 1 Figure 1. Porphyromonas gingivalis type IX machine revealed by cryo-ET. A, Schematic

Figure 2 .

 2 Figure 2. Molecular architecture of PorM. A, Central section of the native T9SS (WT) showing that

Figure 3 .

 3 Figure 3. Proton-motive force-dependent conformational change of the P. gingivalis PorM protein

Figure 4 .

 4 Figure 4. Architecture of the cytoplasmic complex. A, Unrefined structure. B, The cytoplasmic

Figure 5 .

 5 Figure 5. Organization of secretion pore (Sov) at the OM. A and B, Central-section and top-down

Acknowledgements. We are grateful to members of the Hu, Lamont and Cascales labs for critical discussions. We are grateful to Dr. William Margolin for suggestions and critiques of the manuscript, Pascal Mansuelle (IMM proteomic platform, Marseille) for mass spectrometry analyses, and Mark McBride (University of Wisconsin, Milwaukee) for anti-GldM and -GldN antibodies. Funding: B.H. was supported in part by McGovern Medical School start-up funds, the Welch Foundation (AU-1953-20180324) and NIH 1R35GM138301. R.J.L. was supported by DE011111, DE012505 and DE023193. E.C. was supported by the Fondation Bettencourt-Schueller, and A*MIDEX grant A-M-AAP-ID-17-33-170301-07.22 and ANR grant ANR-20-CE11-0011. P.J.C. was supported by NIH grants R01GM48476 and R35GM131892. C.W. was supported by NIH grant DE030895. analyzed the data. L.S. and B.H. generated the figures, and P.J.C. wrote the manuscript with contributions from B.H., R.J.L., and E.C. Correspondence and requests for materials should be addressed to B.H. or R.J.L.

Data and materials availability: Density maps and coordinate data that support the T9SS structures determined by cryo-electron tomography have been deposited in the Electron Microscopy Data Bank (EMDB) as EMD-24227 (Intact machine: PorM arches/ring), EMD-24228 (Cytoplasmic complex) and EMD-24229 (Sov Surface Structure; SS). The authors declare that all other data supporting the findings of this study are available within the paper and its supplementary materials.

Recently, however, a protein interactome analysis identified PorW as a mutually interacting partner of Sov and the PorKN ring. PorW thus appears to constitute a critical link between these subassemblies [START_REF] Gorasia | Protein interactome analysis of the type IX secretion system identifies PorW as the missing link between the PorK/N ring complex and the Sov translocon[END_REF] .

Summary. We have shown that the P. gingivalis T9SS assembles as a large supramolecular machine composed of a cytoplasmic complex with 12-fold symmetry, a periplasmic complex with 18-fold symmetry, and a surface structure consisting of 8 symmetrically arranged putative pores (Fig. 5I). Taken together with previous findings, we propose a working model in which the major machine subunits assemble with an overall stoichiometry of (PorL 5 /PorM 2 /PorK 2 /PorN 2 ) 18 (Sov) 8 (Fig. 5J). Observed differences in the bend angles at PorMKN -OM junctions (Fig. 3B), coupled with the absence of a discernible central channel (Fig. 2A, Fig. S3A), support our model that the T9SS is a multi-port export apparatus composed of independently -but coordinately -functioning IM energy sensors, periplasmic translocation motors, and OM translocons (Fig. 4J-K, Movie 3). This proposed architecture and mechanism of action strikingly contrast with other known dedicated protein translocation systems functioning in bacteria [START_REF] Galan | Protein-injection machines in bacteria[END_REF][START_REF] Christie | The rich tapestry of bacterial protein translocation systems[END_REF][START_REF] Majewski | Secretins revealed: structural insights into the giant gated outer membrane portals of bacteria[END_REF][START_REF] Hu | In situ molecular architecture of the Salmonella Type III secretion machine[END_REF][START_REF] Hu | Structural bases for F plasmid conjugation and F pilus biogenesis in Escherichia coli[END_REF] (Fig. S9). Deciphering the molecular and structural details underlying IM energy activation, substrate recruitment, and OM export remain as major challenges for future research on this unique family of bacterial transporters.

Materials and Methods

P. gingivalis growth and mutant constructions. P. gingivalis ATCC 33277 (33277) and W83 and their isogenic mutants were cultured anaerobically at 37°C in Trypticase soy broth (TSB) supplemented with 1 mg ml -1 yeast extract, 5 μg ml -1 hemin, and 1 μg ml -1 menadione. When appropriate, erythromycin (10 μg ml -1 ) was added to the medium.

The PCR fusion technique was utilized to generate allelic exchange mutants ΔporK, ΔporL, ΔporM, ΔporN, (all 33277) and Δsov (W83) as described previously [START_REF] Simionato | Porphyromonas gingivalis genes involved in community development with Streptococcus gordonii[END_REF] . Upstream and downstream fragments of each gene were amplified by PCR and fused to ermF using the primers listed in Table S2. Constructs


Ying Jiao

Zero bias transformation and asymptotic expansions

Keywords: MSC 2000 subject classifications: 60G50, 60F05 normal approximation, zero bias transformation, Stein's method, asymptotic expansions, concentration inequality

whether they are published or not. The documents may come

Zero bias transformation and asymptotic expansions

Introduction

Zero bias transformation has been introduced by Goldstein and Reinert [START_REF] Goldstein | Stein's method and the zero bias transformation with application to simple random sampling[END_REF] in the framework of Stein's method. By the fundamental works of Stein [START_REF] Stein | A bound for the error in the normal approximation to the distribution of a sum of dependent random variables[END_REF][START_REF] Stein | Approximate Computation of Expectations[END_REF], we know that a random variable (r.v.) Z with mean zero follows the normal distribution N (0, σ 2 ) if and only if E[Zf (Z)] = σ 2 E[f ′ (Z)] for any Borel function f such that both sides of the equality are well defined. More generally, for any r.v. X with mean zero and finite variance σ 2 > 0, a r.v. X * is said to have the zero biased distribution of X if the equality

(1) E[Xf (X)] = σ 2 E[f ′ (X * )]
holds for any differentiable function f such that (1) is well defined. So combined with the Stein's equation xf (x)σ 2 f ′ (x) = h(x) -Φ σ (h) where h is a given function and Φ σ (h) denotes the expectation of h under N (0, σ 2 ), we have

E[h(X)] -Φ σ (h) = E[Xf h (X) -σ 2 f ′ h (X)] = σ 2 E[f ′ h (X * ) -f ′ h (X)]
where f h is the solution of Stein's equation given by

(2)

f h := 1 σ 2 φ σ (x) ∞ x
(h(t) -Φ σ (h))φ σ (t) dt. * Laboratoire de probabilités et modèles aléatoires, Université Paris 7, jiao@math.jussieu.fr.

1 An important remark is that X * need not be independent of X ( [START_REF] Goldstein | Stein's method and the zero bias transformation with application to simple random sampling[END_REF], see also [START_REF] Goldstein | L 1 bounds in normal approximation[END_REF]). In fact, let W = X 1 + • • • + X n be sum of independent mean zero random variables, Goldstein and Reinert proposed the construction W * := W (I) +X * I where, for any i ∈ {1, • • • , n}, W (i) := W -X i and X * i is independent of W (i) , and I is a random index valued in {1, • • • , n} which is independent of (X 1 , • • • , X n , X * 1 , • • • , X * n ) and satisfies P(

I = i) = σ 2 i /σ 2 W with σ 2
i being the variance of X i and σ 2 W that of W . We observe that the above construction of zero bias transformation is quite similar to Lindeberg method except that, in zero bias transformation, we consider an average of punctual substitutions of X i by X * i ; while in Lindeberg method, we substitute progressively X i by central normal distribution with the same variance.

The asymptotic expansion of expectations of the form E[h(W )] is a classical topic in central limit theorems. Using Stein's method, Barbour [START_REF] Barbour | Asymptotic expansions based on smooth functions in the central limit theorem[END_REF][START_REF] Barbour | Asymptotic expansions in the poisson limit theorem[END_REF] has obtained a full expansion of E[h(W )] for sufficiently regular function h. Compared to the classical Edgeworth expansion (see [START_REF] Petrov | Sums of Independent Random Variables[END_REF]ChapV], also [START_REF] Rotar | Stein's method, Edgeworth's expansions and a formula of Barbour[END_REF]), the results of [START_REF] Barbour | Asymptotic expansions based on smooth functions in the central limit theorem[END_REF] do not require the distribution of X i to be smooth; however, as a price paid, we need some suitable regularity conditions on the function h. The result of [START_REF] Barbour | Asymptotic expansions based on smooth functions in the central limit theorem[END_REF] can also be compared to those in [START_REF] Hipp | Edgeworth expansions for integrals of smooth functions[END_REF][START_REF] Götze | Asymptotic expansions in the central limit theorem under moment conditions[END_REF] using Fourier transform. The key point of Barbour's method is a Taylor type formula with cumulant coefficients, which allows to write the difference E[W f (W )]-σ 2 W E[f ′ (W )] as a series which involves cumulants of order ≥ 3 and to iterate the procedure of replacing W -expectations by normal expectations until the desired order. It has been pointed out in [START_REF] Rotar | Stein's method, Edgeworth's expansions and a formula of Barbour[END_REF] that the key formula of Barbour can also be obtained by Fourier transform.

Zero bias transformation have been used in [START_REF] Karoui | Stein's method and zero bias transformation for CDOs tranches pricing[END_REF] to obtain a first order correction term for the normal approximation of E[h(W )], where the motivation was to find a rapid numerical method for large-sized credit derivatives. The function of interest is the so-called call function in finance: h(x) = (xk) + where k is a real number. Since such h is only absolutely continuous, the function f h is not regular enough to have the third order derivative. To achieve the estimation, the authors have used a conditional expectation technique, together with a concentration inequality due to Chen and Shao [START_REF] Chen | A non-uniform Berry-Esseen bound via Stein's method[END_REF][START_REF] Chen | Stein's method for normal approximation[END_REF].

The main difficulty in generalizing the result in [START_REF] Karoui | Stein's method and zero bias transformation for CDOs tranches pricing[END_REF] to obtain a full expansion of E[h(W )] is that W and W * -W are not independent. In fact, if we consider the Taylor expansion of f ′ h (W * ) at W and then apply the expectation, there appear terms of the form E[f

(l) h (W )(W * -W ) k ],
where f (l) denotes the l th -order derivative of f . For the first order expansion in [START_REF] Karoui | Stein's method and zero bias transformation for CDOs tranches pricing[END_REF], the conditional expectation argument allows us to replace

E[f ′′ h (W )(W * -W )] by E[f ′′ h (W )]E[W * -W ]
and put the covariance in the error term. However, in higher order expansion, the error term could no longer contain such covariances. An alternative way is to consider the Taylor expansion of ) , there is no crossing term. However, the expectations of the form E[f (l) h (W (i) )] appear, which make it difficult to apply the recurrence procedure. To overcome this difficulty, we propose a so-called reverse Taylor formula which enables us to replace E[f (l) h (W (i) )] by expectation of functions of W , up to an error term. Let N be a positive integer, X and Y be two independent random variables such that Y has up to N th order moments, and f be an N th order differentiable function such that f (k) (X) and f (k) (X + Y ) are integrable for any k = 0, • • • , N . We define the notation m (k)

E[f ′ h (W * ) -f ′ h (W )] at W (i) . As X * i is independent of W (i
Y := E[Y k ]/k!. Denote by δ N (f, X, Y ) the error term in the N th order Taylor expansion of E[f (X + Y )]. Namely, (3) δ N (f, X, Y ) := E[f (X + Y )] - N k=0 m (k) Y E[f (k) (X)].
Recall that for any N ≥ 1,

(4) δ N (f, X, Y ) = 1 (N -1)! 1 0 (1 -t) N -1 E f (N ) (X + tY ) -f (N ) (X) Y N dt
provided that the term on the right side is well defined. This is a consequence of the classical Taylor formula in its integral form (e.g. [START_REF] Laudenbach | Calcul différentiel et intégral[END_REF]).

The so-called reverse Taylor formula gives an expansion of E[f (X)] in terms of expectations of functions of X + Y and of moments of Y . We would like to note that, in the expansion formula [START_REF] Karoui | Stein's method and zero bias transformation for CDOs tranches pricing[END_REF], the variables X + Y and Y are not independent. We specify some notation and conventions. First of all, N * := N \ {0} denotes the set of strictly positive integers. For any integer d ≥ 1 and any

J = (j l ) d l=1 ∈ N d * , |J| is defined as j 1 + • • • + j d , and m (J) Y := m (j 1 ) Y • • • m (j d ) Y .
(5) E[f (X)] = d≥0 (-1) d J∈N d * , |J|≤N m (J) Y E[f (|J|) (X + Y )] + ε N (f, X, Y )
holds, where ε N (f, X, Y ) is defined as

(6) ε N (f, X, Y ) = - d≥0 (-1) d J∈N d * , |J|≤N m (J) Y δ N -|J| (f (|J|) , X, Y ).
The main result of this paper is an expansion formula for the sum of independent random variables. We present below its formal form without giving precise conditions on the function and on the summand variables (this will be done in Section 3). The methodology appeals to the zero bias transformation. From now on, we consider a family of independent random variables X i (i = 1, • • • , n), with mean zero and finite variance

σ 2 i > 0. Let W = X 1 +• • •+X n and σ 2 W = Var(W ).
Denote by X * i a random variable which follows the zero-biased distribution of X i and which is independent of

W (i) := W -X i . Theorem 1.2 Assume that X 1 , • • • , X n
and the function h are sufficiently good (in a sense that we shall precise later). Then, for any integer N ≥ 0, E[h(W )] can be written as the sum of two terms C N (h) and e N (h), with C 0 (h) = Φ σ W (h) and e 0 (h

) = E[h(W )] - Φ σ W (h), and recursively for N ≥ 1, (7) C N (h) = C 0 (h) + n i=1 σ 2 i d≥1 (-1) d-1 J∈N d * , |J|≤N m (J • ) X i m (J † ) X * i -m (J † ) X i C N -|J| (f (|J|+1) h ), e N (h) = n i=1 σ 2 i d≥1 (-1) d-1 J∈N d * , |J|≤N m (J • ) X i m (J † ) X * i -m (J † ) X i e N -|J| (f (|J|+1) h ) + N k=0 ε N -k (f (k+1) h , W (i) , X i )m (k) X * i + δ N (f ′ h , W (i) , X * i ) , (8)
where for any integer d ≥ 1, and any J ∈ N d * , J † ∈ N * denotes the last coordinate of J, and J • denotes the element in N d-1 * obtained from J by omitting the last coordinate.

In view of the classical formula relating the cumulants and moments, our principal term C N (h) is similar to that obtained by Barbour. Note that in C N (h), there appear normal expectations of iteration of operators which are of the form g → f (l) g acting on h. As pointed out by Barbour [1,p.294], such expectation can be expressed as expectation of h multiplied by a Hermite polynomial.

The proof of the equality E[h(W )] = C N (h) + e N (h) is based on the reverse Taylor formula and the zero bias transformation. It is important to precise the conditions under which all terms in the formal expansion are well defined. Moreover, we also need to show that e N (h) is "small' enough as an error term. In our results, the error term e N (h) is expressed in a recursive way so that it is actually a linear combination of remainders of Taylor and reverse Taylor formulas and can be thus estimated. A key ingredient in the estimation is a concentration inequality which provides upper bound for P(a ≤ W ≤ b) involving exponent ≤ 1 of the interval length (ba), i.e. (ba) α with 0 < α ≤ 1. This allows to us to obtain, under relatively mild moment conditions on X i 's than those in [START_REF] Barbour | Asymptotic expansions based on smooth functions in the central limit theorem[END_REF], estimations for the Taylor and reverse Taylor remainders. For example, as a consequence of Theorem 1.2 and the remainder estimations, we recover a classical result, initially obtained by using Fourier transform, asserting that if X 1 , • • • , X n are i.i.d. random variables with mean zero and variance σ 2 > 0, which admit (2 + α) th order moments, then the law of (X

1 + • • • + X n )/
√ n converges to N (0, σ 2 ) and that the convergence speed is of order

(1/ √ n) α .
The rest of the paper is organized as follows. We firstly prove the reverse Taylor formula and the formal expansion in Section 2. In Section 3, we introduce the admissible function space and discuss the conditions on h and on X i 's; this is inspired by ideas in [START_REF] Barbour | Asymptotic expansions based on smooth functions in the central limit theorem[END_REF] and we can in addition include some more irregular functions. We then restate the main expansion result in this context. Section 4 is devoted to error estimations. Finally, some technical proofs are left in Appendix.

Reverse Taylor formula and formal expansion

To prove Proposition 1.1, the main point is to replace E[f (|J|) (X + Y )] by its classical Taylor expansion of (N -|J|) th order, so that all summand terms are of the same order and some of them can be cancelled off progressively.

Proof of Proposition 1.1 We replace E[f (|J|) (X + Y )] on the right side of (5) by

N -|J| k=0 m (k) Y E[f (|J|+k) (X)] + δ N -|J| (f (|J|) , X, Y )
and observe that the sum of terms containing δ vanishes with ε N (f, X, Y ). Hence we obtain that the right side of (5) equals

d≥0 (-1) d J∈N d * , |J|≤N m (J) Y N -|J| k=0 m (k) Y E[f (|J|+k) (X)]
If we split the last sum for k = 0 and for 1 ≤ k ≤ N -|J| respectively, the formula above can be written as

(9) d≥0 (-1) d J∈N d * , |J|≤N m (J) Y E[f (|J|) (X)] + d≥0 (-1) d J∈N d * , |J|≤N m (J) Y N -|J| k=1 m (k) Y E[f (|J|+k) (X)].
We make the index changes J ′ = (J, k) and u = d + 1 in the second part, we find that it is just u≥1 (-1) u-1

J ′ ∈N u * , |J ′ |≤N m (J ′ ) Y E[f (|J ′ |) (X)].
Thus, the terms in the first and the second parts of (9) cancel out except the one of index d = 0 in the first part, which proves the proposition.

, E[h(W )] -C 0 (h) is equal to σ 2 W E[f ′ h (W * ) -f ′ h (W )] = n i=1 σ 2 i E[f ′ h (W (i) + X * i )] -E[f ′ h (W )] .
Consider the following Taylor expansion

E[f ′ h (W (i) + X * i )] = N k=0 m (k) X * i E[f (k+1) h (W (i) )] + δ N (f ′ h , W (i) , X * i ).
By replacing E[f

(k+1) h (W (i)
)] in the above formula by its (N -k) th reverse Taylor expansion, we obtain that

E[f ′ h (W (i) + X * i )] equals N k=0 m (k) X * i d≥0 (-1) d J∈N d * |J|≤N -k m (J) X i E[f (|J|+k+1) h (W )]+ε N -k (f (k+1) h , W (i) , X i ) +δ N (f ′ h , W (i) , X * i ).
Note that the term with indexes k = d = 0 in the sum inside the bracket is

E[f ′ h (W )]. Therefore E[f ′ h (W (i) + X * i )] -E[f ′ h (W )
] can be written as the sum of the following three parts

n k=1 m (k) X * i d≥0 (-1) d J∈N d * , |J|≤N -k m (J) X i E[f (|J|+k+1) h (W )], ( 10
) d≥1 (-1) d J∈N d * , |J|≤N m (J) X i E[f (|J|+1) h (W )], ( 11
) N k=0 m (k) X * i ε N -k (f (k+1) h , W (i) , X i ) + δ N (f ′ h , W (i) , X * i ). ( 12
)
By interchanging summations and then taking the index changes K = (J, k) and u = d+1, we obtain

(10) = u≥1 (-1) u-1 K∈N u * , |K|≤N m (K • ) X i m (K † ) X * i E[f (|K|+1) h (W )].
As the equality m (J)

X i = m (J • ) X i m J † X i holds for any J, (10)+(11) simplifies as d≥1 (-1) d-1 J∈N d * , |J|≤N m (J • ) X i m (J † ) X * i -m (J † ) X i E[f (|J|+1) h (W )].
By the hypothesis of induction, we have E[f

(|J|+1) h (W )] = C N -|J| (f (|J|+1) h )+e N -|J| (f (|J|+1) h
), so the equality E[h(W )] = C N (h) + e N (h) follows from [START_REF] Goldstein | Stein's method and the zero bias transformation with application to simple random sampling[END_REF] and [START_REF] Götze | Asymptotic expansions in the central limit theorem under moment conditions[END_REF]. 2

Admissible function space

In this section, we describe the function set for which we can make the N th order expansion in Theorem 1.2. We need conditions on regularity and on the increasing speed at infinity of the function h. Actually, from ( 7) and ( 8), we are concerned with the (N -k) th order expansion of f

(k+1) h for k = 1, • • • , N . So it would be natural to expect that f ′
h still belongs to this set. Then by a recursive procedure, all terms will be well defined.

Recall ([11], Chapter VI) that any function g on R which is locally of finite variation can be uniquely decomposed into the sum of a function of pure jump and a continuous function locally of finite variation and vanishing at the origin. That is, g = g c + g d where g c is called the continuous part of g and g d is the purely discontinuous part.

Let α ∈ (0, 1] and p ≥ 0 be two real numbers. For any function f on R, the following quantity has been defined by Barbour in [START_REF] Barbour | Asymptotic expansions based on smooth functions in the central limit theorem[END_REF]: [START_REF] Petrov | Sums of Independent Random Variables[END_REF] f α,p := sup

x =y |f (x) -f (y)| |x -y| α (1 + |x| p + |y| p )
.

The finiteness of this quantity implies that the function f is locally α-Lipschitz, and the increasing speed of f at infinity is at most of order |x| α+p . All functions f such that f α,p < +∞ forms a vector space over R, and • α,p is a norm on it. We list below several properties of • α,p , which will be useful afterwards and we leave the proofs in the Appendix A. Lemma 3.1 Let f be a function on R, α, β ∈ (0, 1] and p, q ≥ 0.

1) If p ≤ q, then f α,p < +∞ implies f α,q < +∞. 2) If α ≤ β, then f β,p < +∞ implies f α,p+β-α < +∞. 3) If P is a polynomial of degree d, then f α,p < +∞ implies P f α,p+d < +∞. 4) Assume that F is a primitive function of f , then f α,p < +∞ implies F 1,p+α < +∞.
(Hence F α,p+1 < +∞ by 2).)

Inspired by [START_REF] Barbour | Asymptotic expansions based on smooth functions in the central limit theorem[END_REF], we introduce the following function space.

Definition 3.2 Let N ≥ 0 be an integer, and α ∈ (0, 1], p ≥ 0 be two real numbers.

Denote by H N α,p the vector space of all Borel functions h on R verifying the following conditions: a) h has N th order derivative which is locally of finite variation and which has finitely many jumps, b) the continuous part of h (N ) satisfies h

(N ) c α,p < +∞.
Condition a) implies that the pure jump part of h (N ) is bounded. Condition b) implies that h (N ) c has at most polynomial increasing speed at infinity, therefore also is h. These conditions allow us to include some irregular functions such as indicator functions. Let k be a real number and I k (x) = 1 1 {x≤k} . Then I k α,p is clearly not finite. However, I k,c α,p = 0, which means that for any α ∈ (0, 1] and any p ≥ 0, I k (x) ∈ H 0 α,p . Note that any function h in H 0 α,p can be decomposed as h = h c +h d , where h c satisfies h c α,p < +∞, the discontinuous part h d is a linear combination of indicator functions of the form 1 1 {x≤k} plus a constant (so that h c (0) = 0). Proposition 3.3 Let N ≥ 0 be an integer, α, β ∈ (0, 1] and p, q ≥ 0 be real numbers. Then the following assertions hold:

1) when N ≥ 1, h ∈ H N α,p if and only if h ′ ∈ H N -1 α,p ; 2) if p ≤ q, then H N α,p ⊂ H N α,q ; if α ≤ β, then H N β,p ⊂ H N α,p+β-α ; 3) when N ≥ 1, H N α,p ⊂ H N -1 1,α+p ⊂ H N -1 α,p+1 ; 4) if h ∈ H N α,p and if P is a polynomial of degree d, then P h ∈ H N α,p+d .
Proof. 1) results from the definition. 2), 3) and 4) are consequences of Lemma 3.1. 2

The following result on the operator h → f h is fundamental. It shows that compared to h, the solution of Stein's equation f h has one more order in regularity and its derivative has the same order in increasing speed at infinity. The proof of this proposition, which is rather technical, is postponed to Appendix B.

Proposition 3.4 Assume that h ∈ H N α,p . Then f h ∈ H N +1 α,p .
We now restate Theorem 1.2 in the function space context.

Theorem 3.5 Let N ≥ 0 be an integer, α ∈ (0, 1] and p ≥ 0. Assume that h ∈ H N α,p . Let X 1 , • • • , X n be zero-mean random variables which have (N +max(α+p, 2)) th order moment. Then all terms in [START_REF] Goldstein | Stein's method and the zero bias transformation with application to simple random sampling[END_REF] and (8) are well defined, and the equality

E[h(W )] = C N (h) + e N (h) holds. Proof. When N = 0, h ∈ H 0 α,p and then h(x) = O(|x| α+p ). Hence E[h(W )
] and Φ σ W (h) are well defined. Assume that we have proved the theorem for 0,

• • • , N -1. Let h ∈ H N α,p . Then by Proposition 3.3, h(x) ∈ H N α,p ⊂ H N -1 α,p+1 • • • ⊂ H 0 α,p+N , so h(x) = O(|x| α+p+N ). By Proposition 3.4, f h ∈ H N +1
α,p and by Proposition 3.3 1), for any

|J| = 1, • • • , N , f (|J|+1) h ∈ H N -|J| α,p
. So the induction hypothesis implies that C N -|J| (f

(|J|+1) h
) and e N -|J| (f

E[(X * i ) k ] = E[X k+2 i ] σ 2 i (k + 1)
, all moments figuring in ( 7) and ( 8) exist. Thus all terms are well defined, and the formal proof in the previous section shows that

E[h(W )] = C N (h) + e N (h). 2
4 Error estimations

Concentration inequalities

We shall prove some concentration inequalities similar to several results in [START_REF] Chen | A non-uniform Berry-Esseen bound via Stein's method[END_REF][START_REF] Chen | Stein's method for normal approximation[END_REF], which give upper bounds for probabilities of the form P(a W b) with a and b being two real numbers. We shall take into consideration the parameter α and give some variants where appear certain lower order moments if α < 1. When α = 1, we recover some estimations in [START_REF] Karoui | Stein's method and zero bias transformation for CDOs tranches pricing[END_REF]. These concentration inequalities will be useful to estimate the approximation error terms and the proof is based on the zero bias transformation. Lemma 4.1 Let α ∈ (0, 1] be a real number and X be a r.v. with mean zero, finite variance σ 2 > 0 and up to (α + 2) th order moments. Let X * have the zero biased distribution of X and be independent of X. Then, for any ε > 0,

P(|X -X * | > ε) ≤ 1 2ε α (α + 1)σ 2 E[|X s | α+2 ],
where X s = X -X and X is an independent copy of X.

Proof. Similar to the Markov inequality, the following inequality holds:

P(|X -X * | > ε) ≤ 1 ε α E[|X -X * | α ].
Moreover, since X and X * are independent, the definition of the zero bias transformation (see [START_REF] Karoui | Stein's method and zero bias transformation for CDOs tranches pricing[END_REF]Pro2.3]) implies that

E[|X -X * | α ] = 1 2(α + 1)σ 2 E[|X s | α+2 ]. 2 Proposition 4.2 Let X i (i = 1, • • • , n) be independent random variables with mean zero and variance σ 2 i > 0. Let W = X 1 + • • • + X n
and denote its variance by σ 2 W . For a, b ∈ R such that a ≤ b and any real number α ∈ (0, 1], we have

(14) P(a ≤ W ≤ b) ≤ 2 b -a 2σ W α + 2 α + 1 n i=1 E X s i σ W α+2 + 1 2σ 2 W n i=1 σ 4 i 1 2 . Proof. Let I [a,b] (x) = 1 if x ∈ [a, b] and I [a,b] (x) = 0 otherwise. Its primitive function f (x) := x (a+b)/2 I [a,b] (t)dt satisfies |f (x)| ≤ (b -a)/2. Then E(I [a,b] (W * )) = 1 σ 2 W E(W f (W )) ≤ min b -a 2σ W , 1 .
Note that for any u ≥ 0 and any α ∈ (0, 1], min(u, 1) ≤ u α . Then for any ε > 0,

P(a -ε ≤ W * ≤ b + ε) ≤ b -a + 2ε 2σ W α ≤ b -a 2σ W α + ε σ W α
where the last inequality is because for any u and v positive, one always has (u + v) α ≤ u α + v α . On the other hand, by using a conditional expectation technique,

P(a -ε ≤ W * ≤ b + ε) ≥ P(a ≤ W ≤ b, |X I -X * I | ≤ ε) ≥ P(a ≤ W ≤ b)P(|X * I -X I | ≤ ε) - 1 4 n i=1 σ 4 i σ 4 W 1 2 .

9

We recall that W * = W (I) + X * I where I is a random variable taking values in {1, • • • , n} with P(

I = i) = σ 2 i /σ 2 W , W (i) := W -X i , X *
i has the zero biased distribution of X i and is independent of W (i) . In this proof exceptionally, we assume that X * i is also independent of X i . By Lemma 4.1,

P(|X * I -X I | ≤ ε) = 1 - n i=1 σ 2 i σ 2 W P(|X * i -X i | > ε) ≥ 1 - 1 2σ 2 W (α + 1)ε α n i=1 E[|X s i | α+2 ].
Finally, the inequality ( 14) follows by taking

ε = 1 σ 2 W (α + 1) n i=1 E[|X s i | α+2 ] 1 α . 2 Corollary 4.3 Let a, b ∈ R such that a ≤ b and i ∈ {1, • • • , n}, then P(a ≤ W (i) ≤ b) ≤ 4 b -a 2σ W α + 4 α + 1 n j=1 E X s j σ W α+2 + 1 σ 2 W n j=1 σ j 4 1 2 + 4 2σ i σ W α
where

W (i) = W -X i .
Proof. Let ε > 0 be a real number, then

P(a ≤ W (i) ≤ b, |X i | ≤ ε) ≤ P(a -ε ≤ W ≤ b + ε).
Note that W (i) and X i are independent and

P(|X i | ≤ ε) = 1 -P(|X i | > ε) ≥ 1 - E[|X i |] ε .
By Proposition 4.2 and taking ε = 2E[|X i |], we obtain the inequality. 2

Estimations of error terms

In this section, we shall estimate the error term e N (h) in Theorem 3.5. The recursive formulas ( 8) and ( 6) permit us to reduce the problem to the estimation of classical Taylor expansion errors.

For any positive random variable Y and any real number β ≥ 0, we introduce the notation m (β)

Y := E[Y β ]/Γ(β + 1), where Γ is the Gamma function Γ(x) = ∞ 0 t x-1 e -t dt.
This notation generalizes the one introduced in §1 since when β ∈ N, Γ(β + 1) = β!. Proposition 4.4 Let N ≥ 0 be an integer, α ∈ (0, 1] and p ≥ 0 be two real numbers. Let X be a random variable which has up to (N + α + p) th moments and satisfies the following concentration inequality

P(a ≤ X ≤ b) ≤ c(b -a) α + r, ∀a, b ∈ R, a ≤ b,
where c and r are two constants. Let Y be a random variable which is independent of X and has up to (N + α + p) th moments. Then, for any function g ∈ H N α,p and any

k = 0, • • • , N , δ N -k (g (k) , X, Y ) ≤ V (g (N ) d ) cm (N -k+α) |Y | + rm (N -k) |Y | + g (N ) c α,p u α,p,X m (N -k+α) |Y | + v α,p m (N -k+α+p) |Y | , ( 15
)
where V (g

(N ) d ) denotes the total variation of g (N )
d , the coefficients u α,p,X and v α,p are defined as u α,p,X = 1 + (1 + 2 p )E[|X| p ] Γ(α + 1) and v α,p = 2 p Γ(α + p + 1).

Proof. We have by ( 4) that when k < N ,

δ N -k (g (k) , X, Y ) = 1 (N -k -1)! 1 0 (1 -t) N -k-1 E g (N ) (X + tY ) -g (N ) (X) Y N -k dt. Since g ∈ H N α,p , the function g (N ) d
can be written as

g (N ) d (x) = g (N ) d (0) + 1≤j≤M ε j 1 1 x≤K j - 1≤j≤M K j ≥0 ε j . Therefore, g (N ) d (X + tY ) -g (N )
d (X) = M j=1 ε j 1 1 K j -tY + <X≤K j -tY -, where Y + = max(Y, 0) and Y -= min(Y, 0). Thus the concentration inequality hypothesis implies that

E |g (N ) d (X + tY ) -g (N ) d (X)| Y ≤ M j=1 |ε j | ct α |Y | α + r .
Moreover, one has

1 0 (1 -t) N -k-1 (N -k -1)! E M j=1 |ε j | ct α |Y | α + r |Y | N -k dt = M j=1 |ε j | cm (N -k+α) |Y | + rm (N -k) |Y |
by using the following equality concerning Beta function

B(x, y) := 1 0 t x-1 (1 -t) y-1 dt = Γ(x)Γ(y) Γ(x + y) , x, y > 0.
On the other hand, by definition of the norm • α,p , we have

g (N ) c (X + tY ) -g (N ) c (X) ≤ g (N ) c α,p |tY | α (1 + |X + tY | p + |X| p ) ≤ g (N ) c α,p |tY | α 1 + (2 p + 1)|X| p + 2 p |tY | p ,
where the last inequality results from (a + b) p ≤ 2 p (a p + b p ). Note that

1 0 (1 -t) N -k-1 (N -k -1)! E |tY | α 1+(2 p +1)|X| p +2 p |tY | p |Y | N -k dt = u α,p,X m (N -k+α) |Y | +v α,p m (N -k+α+p) |Y |
.

Thus we obtain the estimation [START_REF] Stein | A bound for the error in the normal approximation to the distribution of a sum of dependent random variables[END_REF]. Finally, it remains to check the case when k = N . Consider the continuous and discontinuous parts of δ 0 (g (N ) , X, Y ) = E g (N ) (X + Y )g (N ) (X) respectively. By using similar method as above, we obtain that E |g

(N ) d (X +Y )-g (N ) d (X)| ≤ V (g (N ) d ) cm (1)
|Y | +r and E |g (N ) c (X +Y )-g (N ) c (X)| ≤ g (N ) c α,p E[|Y | α ] 1 + (2 p + 1)E[|X| p ] + 2 p E[|Y | α+p ] ,
which implies [START_REF] Stein | A bound for the error in the normal approximation to the distribution of a sum of dependent random variables[END_REF].

2

By Proposition 1.1 and Proposition 4.4, we obtain the error estimation for the reverse Taylor expansion.

Corollary 4.5 With the previous notation, we have |ε

N (g, X, Y )| ≤ d≥0 J∈N d * , |J|≤N m (J) |Y | V (g (N ) d ) cm (N -|J|+α) |Y | + rm (N -|J|) |Y | + g (N ) c α,p u α,p,X m (N -|J|+α) |Y | + v α,p m (N -|J|+α+p) |Y | , (16)
Combining the concentration inequality (Corollary 4.3) and the above estimations (Proposition 4.4 and Corollary 4.5), we obtain upper bounds for the Taylor and reverse Taylor remainders δ N -k (f

(k) h , W (i) , X i ) and ε N -k (f (k)
h , W (i) , X i ), where the summand variables X 1 , • • • , X n are independent. This allows us, together with the recursive formula [START_REF] Götze | Asymptotic expansions in the central limit theorem under moment conditions[END_REF], to obtain an upper bound for the asymptotic expansion remainder e N (h).

In particular, we give in the following the order estimation of e N (h) when X 1 , • • • , X n are in addition i.i.d. random variables. Proposition 4.6 Suppose that X 1 , • • • , X n are i.i.d. random variables with mean zero and up to (N + 2 + α + p) th order moments, normalized such that each X i has the same distribution as Z/

√ n where Z is a fixed random variable with mean zero and finite nonzero variance. Then for any function g ∈ H N α,p and any k = 0, • • • , N , we have

δ N -k (g (k) , W (i) , X * i ) = O 1 √ n N -k+α+p , (17) ε N -k (g (k) , W (i) , X i ) = O 1 √ n N -k+α+p , ( 18
)
where i) . The implied constants depend on g

W (i) = W -X i and X * i is independent of W (
(N ) c α,p , V (g (N )
d ) and up to (Nk + 2 + α + p) th order moments of Z.

Proof. By Corollary 4.3, we have for any a ≤ b and any α ∈ (0, 1] that

P(a ≤ W (i) ≤ b) ≤ c(b -a) α + r(n)
where the coefficients are given by

c = 2 2-α σ α , r(n) = 4 σ 2+α (α + 1) E[|Z s | α+2 ] √ n α + 1 √ n + 8 √ n α .
By Proposition 4.4, we obtain an upper bound of δ N -k (g (k) , W (i) , X * i ) which is determined by a linear combination of terms (with coefficient not depending on n):

(19) m (N -k+α) |X * i | , r(n)m (N -k) |X * i | , E[|W (i) | p ]m (N -k+α) |X * i |
and m

(N -k+α+p) |X * i | . Note that r(n) = O((1/ √ n) α ). For any k = 0, • • • , N , E[|X * i | k ] equals E[|X i | k+2 ]/(σ 2 i (k + 1)
) and is of order (1/ √ n) k . So the first three terms in (19) are of order (1/ √ n) N -k+α and the last term is of order (1/ √ n) N -k+α+p , which implies the first assertion. The second assertion then follows by Corollary 4.5.

2

Remark 4.7 According to ( 15) and ( 16), the implicit constants in ( 17) and ( 18) can be explicitly calculated. Proposition 4.8 Let N ≥ 0 be an integer, α ∈ (0, 1] and p ≥ 0 be two real numbers. Let h be a function in H N α,p , and X 1 , • • • , X n be as in Proposition 4.6. Then the error term e N (h) defined in [START_REF] Götze | Asymptotic expansions in the central limit theorem under moment conditions[END_REF]

satisfies e N (h) = O 1 √ n N +α+p
, where the implied constant depends on up to (N + 2 + α + p) th order moment of Z.

Proof. We prove the theorem by induction on N . When N = 0,

e 0 (h) = n i=1 σ 2 i δ 0 (f ′ h , W (i) , X * i ) + ε 0 (f ′ h , W (i) , X i ) . Since h ∈ H 0 α,p , f h ∈ H 1 α,p . Then by Proposition 4.6, e 0 (h) = O((1/ √ n) α+p ).
Assume that we have already proved the theorem for 0, • • • , N -1. Consider h ∈ H N α,p and e N (h) defined as in [START_REF] Götze | Asymptotic expansions in the central limit theorem under moment conditions[END_REF]. For any J such that 1

≤ |J| ≤ N , e N -|J| = O((1/ √ n) N -|J|+α+p ).
In addition, since

|J • | + |J † | = |J|, we have that m (J • ) X i m (J † ) X * i -m (J † ) X i is of order (1/ √ n) |J| .
On the other hand,

f ′ h ∈ H N α,p , so δ N (f ′ h , W (i) , X * i ) = O((1/ √ n) N +α+p ). Moreover, for any k = 0, • • • , N , f (k+1) h ∈ H N -k α,p . So ε N -k (f (k+1) h , W (i) , X * i ) = O((1/ √ n) N -k+α+p ).
Finally we have m

(k) X * i = O((1/ √ n) k ).
(h) = O((1/ √ n) α )
, where the coefficient depends on up to (2 + α) th moment of the summand variables. This is similar to a result (Theorem 6) in [13, §V.3]. When α = 1, it corresponds to the order estimation in the classical Berry-Esseen inequality.

Let h(x) = (xk) + be the call function discussed in [START_REF] Karoui | Stein's method and zero bias transformation for CDOs tranches pricing[END_REF]. As a primitive function of the indicator function, we know that h ∈ H 1 α,0 . So the call function admits a first order expansion given by ( 7) as:

C 1 (h) = Φ σ W (h) + n i=1 σ 2 i E[X * i ]Φ σ W (f ′′ h ). Moreover, since σ 2 W Φ σ W (f ′′ h ) = Φ σ W (xf ′ h ) = 1 σ 2 W Φ σ W ( x 2 3σ 2 W -1)xh(x) .
We recover the correction term in [START_REF] Karoui | Stein's method and zero bias transformation for CDOs tranches pricing[END_REF].

A Proof of Lemma 3.1

Proof. For the first two assertions, it suffices to prove respectively the boundness of the following two functions

1 + |x| p + |y| p 1 + |x| q + |y| q , |x -y| β-α 1 + |x| p + |y| p 1 + |x| p+β-α + |y| p+β-α .
These functions are both continuous on R, therefore are bounded on any compact subset of R 2 . Thus we may assume without loss of generality that r = x 2 + y 2 ≥ 1. In this case, max{|x|, |y|} ≥ r/ √ 2, so

1 + |x| p + |y| p 1 + |x| q + |y| q ≤ 1 + 2r p 1 + (r/ √ 2) q ≤ 3 • 2 q/2 , |x -y| β-α 1 + |x| p + |y| p 1 + |x| p+β-α + |y| p+β-α ≤ (2r) β-α 1 + 2r p 1 + (r/ √ 2) p+β-α ≤ 3 • 2 (p+3β-3α)/2 .
3) One has

|P (x)f (x) -P (y)f (y)| |x -y| α (1 + |x| p+d + |y| p+d ) ≤ (1 + |x| p + |y| p )P (x) 1 + |x| p+d + |y| p+d f α,p + |f (y)| • |P (x) -P (y)| |x -y| α (1 + |x| p+d + |y| p+d )
.

By using the argument as in the proof of 1) and 2), we obtain that the first term in the right-hand side is bounded. Since P is a polynomial of degree d, there exists a polynomial Q(x, y) in two variables and of degree d -1, such that Q(x, y) = (P (x) -P (y))/(xy). Therefore, the second term equals

|Q(x, y)| • |x -y| 1-α • |f (y)| 1 + |x| p+d + |y| p+d
which is bounded by a similar argument as for proving 1) and 2).

4) Since f α,p < +∞, |f (t)| ≪ 1 + |t| α+p . Therefore, for x, y ∈ R, x ≤ y, one has |F (x) -F (y)| ≤ y x |f (t)| dt ≪ y x 1 + |t| α+p dt ≤ (1 + |x| α+p + |y| α+p )|y -x|. Hence |F (x) -F (y)| |x -y|(1 + |x| p+α + |y| p+α ) is bounded. 2 B Proof of Proposition 3.4
We now prove the Proposition 3.4. Let h ∈ H N α,p . The function f h is one more order differentiable than h and is hence N + 1 times differentiable. Taking N th order derivative on both sides of Stein's equation, we get

(20) (xf h (x)) (N ) -σ 2 f (N +1) h (x) = h (N ) (x). The function (xf h (x)) (N ) is continuous, so f (N +1) h
is locally of finite variation and has finitely many jumps as h (N ) (x) does. In the following, we shall prove f

(N +1) h,c α,p < +∞.
Definition B.1 Let A be an interval in R and f be a Borel function on A. For any α ∈ (0, 1] and p ≥ 0, we define

(21) f A α,p := sup x =y x,y∈A |f (x) -f (y)| |x -y| α (1 + |x| p + |y| p ) .
This definition is analogous to [START_REF] Petrov | Sums of Independent Random Variables[END_REF], restricted to an interval. When A avoids an open neighborhood of 0, then the finiteness of f A α,p is equivalent to that of sup x =y

x,y∈A

|f (x)-f (y)|
|x-y| α (|x| p +|y| p ) . This property does not hold for the norm . α,p defined in [START_REF] Petrov | Sums of Independent Random Variables[END_REF]. As a consequence, we have the following result.

Lemma B.2 Let A ⊂ (-∞, -1] ∪ [1,
+∞) be an interval, α ∈ (0, 1] and p 0. Let q be a real number such that 0 q p. Then for any Borel function f defined on A, f A α,p < +∞ if and only if f (x)/x p-q A α,q < +∞.

Proof. If f (x)/x p-q A α,q < +∞, then by similar arguments as for proving Lemma 3.1, we have f A α,p < +∞. We now consider the converse assertion. Firstly, there exists a constant C > 0 such that |f (x)| C|x| α+p for any x ∈ A. For any x, y

∈ A, |x| < |y|, |f (x)x q-p -f (y)y q-p | |x -y| α (1 + |x| q + |y| q ) |f (x)| |x q-p -y q-p | |x -y| α (1 + |x| q + |y| q ) + |y| q-p |f (x) -f (y)| |x -y| α (1 + |x| q + |y| q ) .
The second term is finite since

|f (x) -f (y)| |x -y| α (|y| p-q + |x| q |y| p-q + |y| p ) ≤ |f (x) -f (y)| |x -y| α (1 + |x| p + |y| p ) = f A α,p .
By the mean value theorem, the first term is bounded by

C|x| α+p |x -y| • |q -p| • |x| q-p-1 |x -y| α (1 + |x| q + |y| q )
and thus by C|q -p| if we assume in addition that |y| < 2|x|. When |y| 2|x|, one has

|f (x)| |x q-p -y q-p | |x -y| α (1 + |x| q + |y| q ) C |x| α+p |x| q-p |x| α+q C. 2
The following lemma allows us to consider the estimations on several disjoint intervals respectively.

Lemma B.3 If

A = A 1 ∪ A 2 where A 1 and A 2 are two intervals such that A 1 ∩ A 2 = ∅, then sup{ f A 1 α,p , f A 2 α,p } ≤ f A α,p ≤ 2( f A 1 α,p + f A 2 α,p ).
Proof. The first inequality is obvious. For the second inequality, we only need to prove for any x ∈ A 1 and y ∈ A 2 that

|f (x) -f (y)| |x -y| α (1 + |x| p + |y| p ) ≤ 2( f A 1 α,p + f A 2 α,p ).
Without loss of generality, we may suppose that A 1 ∩ A 2 contains a single point z. Proof. Firstly, for any bounded interval A and any Borel function g, g A α,p < +∞ if and only if g is α-Lipschitz on A. We examine f

Then |f (x) -f (y)| ≤ |f (x) -f (z)| + |f (y) -f (z)|. In
(N +1) h,c using (20). Since h ∈ H N α,p , h (N ) c is locally α-Lipschitz. The function (xf h (x)) (N +1) = xf (N +1) h (x) + (N + 1)f (N )
h (x) has finitely many jumps. Hence (xf h (x)) (N ) is a primitive function of a locally bounded function, thus is locally 1-Lipschitz. So by (20), f

(N +1) h,c
is locally α-Lipschitz, which implies the lemma. 2

Let A 1 = [-1, 1], A 2 = (-∞, -1] and A 3 = [1, +∞). Lemma B.3 shows that to prove the finiteness of f

(N +1) h,c
, it suffices to prove respectively the finiteness of f

(N +1) h,c A i α,p , (i = 1, 2, 3). Lemma B.4 shows that f (N +1) h,c [-1,1] α,p
< +∞. So it remains to deal with f

(N +1) h,c
on the set A 2 ∪ A 3 = R \ (-1, 1). To this end, we introduce a "modified" Stein's equation as in [START_REF] Karoui | Stein's method and zero bias transformation for CDOs tranches pricing[END_REF]Appendix]:

(22) x f h (x) -σ 2 f ′ h (x) = h(x), x ∈ R \ (-1, 1)
whose solution is given by

(23) f h (x) := 1 σ 2 φσ(x) ∞ x h(t)φ σ (t) dt, x ≥ 1, 1 σ 2 φσ(x) x -∞ h(t)φ σ (t) dt x ≤ -1.
Working with (23), it will be easier to treat the derivative functions. In fact, in (2), the integrand function h -Φ σ (h) is centralized under the normal expectation. However, it is not the case when taking derivatives. This is one reason why we introduce (22). Note that in general, the right-hand side of (23) can not be extended as a continuous function on R, except in the special case Φ σ (h) = 0 where we recover the solution of classical Stein's equation.

To study f h , we introduce the function space E σ : for any σ > 0, let E σ be the space of all Borel functions h on R \ (-1, 1) such that |x|≥1 |h(x)P (x)|φ σ (x)dx < ∞ for any polynomial P . Note that E σ is a vector space which contains all Laurent polynomials (that is, polynomials in x and x -1 ) and is stable by multiplication by Laurent polynomials. Furthermore, as shown by the lemma below, it is invariant by the operator h → f h . Lemma B.5 Let h ∈ E σ . Then the function f h is well defined and f h ∈ E σ . Furthermore, if H is a primitive function of h, then H ∈ E σ .

Proof. Let P be an arbitrary polynomial on R.

Then ∞ 1 |P (x) f h (x)|φ σ (x) dx ≤ 1 σ 2 ∞ 1 dx |P (x)| ∞ x |h(t)|φ σ (t) dt = 1 σ 2 ∞ 1 dt |h(t)|φ σ (t) t 1 |P (x)| dx.
There exists a polynomial Q such that

t 1 |P (x)| dx ≤ Q(t) for any t ≥ 0. Therefore, the fact that h ∈ E σ implies that ∞ 1 |P (x) f h (x)|φ σ (x) dx < +∞.
The finiteness of the integral on (-∞, 1] is similar. The second assertion can be proved by integration by part.

(25) f ′ h (x) = x f Λ(h) (x).
Proof. By definition, Λ(h)(x) = h ′ (x)/xh(x)/x 2 , so Λ(h) ∈ E σ . To prove the equality, it suffices to verify that the function u(x) := x -1 f ′ h (x) satisfies the equation ( 22) for Λ(h) (see Remark above). In fact, if we divide the both side of the equation x f h (x)-σ 2 f ′ h (x) = h(x) by x and then take the derivative, we obtain xu

(x) -σ 2 u ′ (x) = Λ(h)(x). 2 Lemma B.10 If h ∈ E σ and if l is a real number such that h(x) = O(|x| l ), then f h (x) = O(|x| l-1 ).
Proof. Recall that ([5, LemA.1]) if |h(x)| ≤ g(x) and if g(x)/|x| is decreasing when x > 0 and is increasing when x < 0, then | f h (x)| ≤ g(x)/|x|. Hence, we prove the lemma for the cases where l < 1. By Lemma B.9, one has

f |x| l (x) = x -1 (|x| l + σ 2 f ′ |x| l (x)) = sgn(x)|x| l-1 + σ 2 f Λ(|x| l ) (x) = sgn(x)|x| l-1 + σ 2 (l -1) f |x| l-2 (x). Thus, f |x| l-2 = O(|x| l-3 ) implies f |x| l = O(|x| l-1
). Hence by induction on l, we obtain the result.

2

Remark B.11 With the notation of Barbour [START_REF] Barbour | Asymptotic expansions based on smooth functions in the central limit theorem[END_REF], the equivalent expectation form of f h is given by

(26) f h (x) = √ 2π σ E h(Z + x)e -Zx σ 2 1 1 {Z>0} , x > 0 - √ 2π σ E h(Z + x)e -Zx σ 2 1 1 {Z<0} , x < 0
where Z ∼ N (0, σ 2 ). So the above lemma can be interpreted as : the function

1 x l E 1 1 {Z>0} (Z + x) l+1 e -Zx σ 2
is bounded on [1, +∞). We can then deduce easily the following assertion : for all l ∈ R and m ∈ R + , the function

1 x l E 1 1 {Z>0} (Zx) m (Z + x) l+1 e -Zx σ 2
is bounded on [1, +∞) by using the fact that the function u m e -u 2σ 2 is bounded on [0, ∞).

We give below the relationship between the derivatives of f h and of h. In the following two formulas, the first one computes f (N ) h using the operator (24) and the second one expresses Λ N (h) using derivatives of h. Their proofs are by induction, which we omit in this article (interest readers may refer to [10, p.144-145]). We only remind that the first formula is a generalization of (25).

Lemma B.12 If h ∈ E N

σ with N being a strictly positive integer, then

f (N ) h (x) = ⌊N/2⌋ k=0 N 2k (2k -1)!!x N -2k f Λ N-k (h) (x); (27) Λ N (h)(x) = N k=0 (-1) k (2k -1)!! N + k 2k h (N -k) (x) x N +k . ( 28
)
where we have used the convention (-1)!! = 1 and ⌊N/2⌋ denotes the largest integer not exceeding N/2. Remark B.13 1) For any function h ∈ E N σ , the above results also hold for f (m) h (x) and Λ m (h) where 1 ≤ m ≤ N . As the operator h → f h is linear on h, the above lemma enables us to write the derivatives of f h as a linear combination of derivatives of h with Laurent polynomial coefficients and then to deduce their increasing speed at infinity.

2) The derivative function f (N +1) h has to be treated differently. In fact, we can no longer apply (28) to Λ N +1 (h) since h (N +1) does not necessarily exist. We separate the first term where k = 0 in (27) from the others and then take the derivative to obtain

f (N +1) h = x N f ′ Λ N (h) (x) + ⌊N/2⌋ k=0 N 2k (2k -1)!!(N -2k)x N -2k-1 f Λ N-k (h) (x) + ⌊N/2⌋ k=1 N 2k (2k -1)!!x N -2k f ′ Λ N-k (h) (x) = x N f ′ Λ N (h) (x) + ⌊(N +1)/2⌋ k=1 N + 1 2k (2k -1)!!x N +1-2k f Λ N+1-k (h) (x).
This will be a crucial point in the proof of Proposition B.15.

Lemma B.14 Let h be a Borel function defined on A 2 ∪A 3 = R\(-1, 1). If h A α,p < +∞ where A = A 2 or A 3 , then, for any integer n 0, one has

x n+1 f h/x n A α,p < +∞, x n f ′ h/x n A α,p < +∞.
Proof. We only prove for A 2 and the case for A 3 is by symmetry. Let g(x) = h(x)/x n . Assume that x and y are two real numbers such that 1 x < y. By Lemma B.10, this quantity is bounded. We then consider the upper bound of (30) under the supplementary condition that y 2x. As h A 2 α,p < +∞, there exists a constant C > 0 such that h(x) C|x| α+p . By applying the mean value theorem on the function Proposition B.15 Let N 0 be an integer, α ∈ (0, 1] and p 0. Let h be a function defined on A 2 ∪ A 3 = R \ (-1, 1) which is N times differentiable and such that h (N ) is locally of finite variation, having finitely many jumps and verifying h itself is also a linear combination of x N f ′ h (N-i) /x N+i where i = 0, • • • , N . As h (N ) A α,p < ∞, we have, similar as in in Lemma 3.1 4), that h (N -i) A α,p+i < +∞. Hence h (N -i) /x i A α,p < ∞ by Lemma B.2 and Lemma B.14 then implies that x N f ′ h (N-i) /x N+i A α,p < ∞. The terms x N +1-2k f Λ N+1-k (h) (x) are also, by (28) again, linear combinations of the functions of the form x N +1-2k f h (N+1-k-i) /x N+1-k+i . By a similar argument as above using Lemma B.2, h (N +1-k-i) /x 1+k+i A α,p < ∞. Finally, we apply Lemma B.14 to obtain x N +1-2k f h (N+1-k-i) /x N+1-k+i A α,p < ∞, which completes the proof. 2

Y = 1 .

 1 By convention, N 0 * denotes the set {∅} of the empty vector, |∅| = 0 and m (∅) Proposition 1.1 (Reverse Taylor formula) With the above notation, the equality

2 Using Proposition 1 . 1 ,

 211 we prove below the formal equality E[h(W )] = C N (h) + e N (h) by induction on N . Proof of Theorem 1.2 (formal part) The equality E[h(W )] = C 0 (h) + e 0 (h) holds by definition. In the following, we assume that the equality E[h(W )] = C k (h) + e k (h) has been verified for any k ∈ {0, • • • , N -1} and for any good enough function h. By Stein's equation

  , for the terms ε N -k and δ N in (8), since f (k+1) h (x) = O(|x| α+p+N -k ) for any k = 0, • • • , N , they are well defined. Finally, combined with the equality

( 2 (n + 1 ) 2 + 2 2 ≪ 2 + E I {Z>0} Zx σ 2 ( 2 , 2 α 2 + 2 α 2 , 2 20

 2122222222222 Z+x) n e -Zx σ 2 and the fact that |x -y| α-1 (1 + |x| p + |y| p ) ≥ |x| α-1+p where α < 1,(30) √ 2π σ |x| 1-α-p E I {Z>0} C(Z + x) α+p e -Zx σ (2x) n (Z + x) n + n(2x) n+1 (Z + x) n+1 + Z(2x) n+1 σ 2 (Z + x) n C √ 2π σ |x| 1-α-p 2 n • (n + 1) + 2 n+1 • n E I {Z>0} (Z + x) α+p e -Zx σ n+1 E I {Z>0} Zx σ 2 (Z + x) α+p e -Zx σ |x| 1-α-p E I {Z>0} (Z + x) α+p e -Zx σ Z + x) α+p e -Zx σwhich is bounded (see Remark B.11). In the case where y > 2x, one has x n+1 /(Z +x) n ≤ x when Z ≥ 0, and |x -y| α (1 + |x| p + |y| p ) ≥ max(|x| α+p , y • y p ), so(30) C √ 2π σ 1 2|x| α+p-1 E I {Z>0} (Z + x) α+p e -Zx σ |y| α+p-1 E I {Z>0} (Z + y) α+p e -Zy σwhich is bounded.We now give the final part of the proof.

(

  < +∞ where A = A 2 or A 3 . Then the function f h is N + 1 times differentiable, f (N +1) h is locally of finite variation, having finitely many jumps and verifying f(N +1) h,c A α,p < +∞.Proof. The functionf h is N + 1 times differentiable by (20), f (N +1) his locally of finite variation, having only finitely many jumps. By virtue of Lemmas B.3 and B.4, it suffices to prove max{ f for sufficiently positive number b. Therefore, without loss of generality, we may assume that h (N ) is continuous and hence f(N +1) his also continuous. By Remark B.13 2), the function f(N +1) h can be written as a linear combination of x N f ′ Λ N (h)and terms of the formx N +1-2k f Λ N+1-k (h) (x) where k = 1, • • • , ⌊ N +1 2 ⌋. By (28), x N f ′ Λ N (h)

  Consider now several examples. Let I k (x) = 1 1 {x≤k} be the indicator function. As mentioned before, I k ∈ H 0 α,0 . By Proposition 4.8, we know that if X 1 , • • • , X n are i.i.d. random variables with up to (2+α) th order moment, then e 0

		Combining all the above estimations, we prove the
	proposition.	2

  addition, since z is between x and y, we have |x -y| ≥ max(|x -z|, |y -z|) and |x| p + |y| p ≥ 1 2 max(|x| p + |z| p , |y| p + |z| p ). So |f (x)f (y)| |x -y| α (1 + |x| p + |y| p )

	≤ 2	|f (x) -f (z)| |x -z| α (1 + |x| p + |z| p )	+	|f (z) -f (y)| |z -y| α (1 + |z| p + |y| p )	,
	which implies the second inequality.			2
	Lemma B.4 If h ∈ H N α,p , then f	(N +1) h,c	A α,p < +∞ for any bounded interval A.

2

  Remark B.6 Note that f h is the only solution of (22) in E σ , provided that h ∈ E σ .More generally, for the derivatives of f h , we consider, for any integer N ≥ 1, the set E N σ which contains all functions h such that h is N times differentiable on R \ (-1, 1) and that h(N ) ∈ E σ . It is not difficult to observe that h ∈ E N σ if and only if it is a primitive function of an element in E N -1 If h ∈ H N α,p , then the restriction of h on R \ (-1, 1) is in E N σ .Proof. It suffices to show that the restriction of h (N ) on R\(-1, 1) is in E σ . This is obvious since h If h ∈ E 1 σ , then Λ(h) ∈ E σ . Furthermore, we have the following equality:

	Lemma B.7 (N ) c has at most polynomial increasing speed at infinity.	2
	Definition B.8 For any derivable function h on R \ (-1, 1), define the operator
	(24)	Λ(h)(x) :=	h(x) x	′	.
	Lemma B.9
	σ	. The relationship between E N σ and H N α,p is as follows.

  Then one has|x n+1 f g (x)y n+1 f g (y)| |x -y| α (1 + |x| p + |y| p ) Z+x) n e -Zx/σ 2 -h(Z+y)y n+1 (Z+y) n e -Zy/σ 2 |x -y| α (1 + |x| p + |y| p ) ,which can be bounded from above by the sum of the following two terms√ 2π σ E I {Z>0} |h(Z + x)h(Z + y)| |x -y| α (1 + |x| p + |y| p ) • y n+1 (Z + y) n e -Zy

					=	√ σ 2π	E I {Z>0}	h(Z+x)x n+1
	(29)					σ 2
	(30)	√ σ 2π	E I {Z>0} |h(Z + x)|	x n+1 (Z+x) n e -Zx σ 2 -y n+1 (Z+y) n e -Zy σ 2 |x -y| α (1 + |x| p + |y| p )
	Note that (29) is bounded from above by
		h A 2 α,p	√ σ 2π	y n+1 E I {Z>0}	1 (Z + y) n e -Zy σ 2	= h A 2

(α,p y n+1 f 1 |x| n (y).