Jodelle Degois

Xavier Simon

Frederic Clerc

Cyril Bontemps

Pierre Leblond

Philippe Duquenne
email: philippe.duquenne@inrs.fr

One-year follow-up of microbial diversity in bioaerosols emitted in a waste sorting plant in France

Keywords: bioaerosol, biodiversity, 16S rDNA, Illumina sequencing, ITS rDNA, next-generation sequencing, occupational exposure, household waste sorting, determinants

Bioaerosols emitted in waste sorting plants (WSP) can induce some adverse health effects on the workers such as rhinitis, asthma and hypersensitivity pneumonitis. The composition of these bioaerosols is scarcely known and most of the time assessed using culture-dependent methods. Due to the wellknown limitations of cultural methods, these biodiversity measurements underestimate the actual microbial taxon richness. The aim of the study was to assess the airborne microbial biodiversity by using a sequencing method in a French waste sorting plant (WSP) for one year and to investigate the main factors of variability of this biodiversity. Static sampling was performed in five areas in the plant and compared to an indoor reference (IR), using closed-

Introduction

The management of waste-limiting landfill disposal is an important issue in our society. Thus, industrial structures and treatment channels are developing all over the world in order to collect, sort, recycle waste or produce energy from residual materials. Of these channels, the sorting of household, commercial and industrial solid waste is one of the challenges all countries face. For 2020, the European Union plans to increase the amount of domestic waste recycled, to represent 50% of overall waste mass (Council of the European Union, 2008). In France, the number of waste sorting plants increased from 265 to 410 installations between [START_REF] Nielsen | Seasonal variation in bioaerosol exposure during biowaste collection and measurements of leaked percolate[END_REF]and 2014(ADEME, 2015)). They treat several million tons of non-hazardous household or industrial waste every year and employ an increasing number of workers involved in sorting waste manually, cleaning and maintaining facilities, driving handling or loading machines, etc.

Published studies revealed that workers in waste sorting plants (WSP) were exposed to complex bioaerosols including dust, bacteria and fungi, as well as microbial compounds and metabolites [START_REF] Madsen | Waste workers' exposure to airborne fungal and bacterial species in the truck cab and during waste collection[END_REF][START_REF] Park | Airborne bacteria and fungi associated with waste-handling work[END_REF][START_REF] Schlosser | Extension of the sorting instructions for household plastic packaging and changes in exposure to bioaerosols at materials recovery facilities[END_REF][START_REF] Tolvanen | Airborne bio-aerosols and noise in a dry waste treatment plant in Pietarsaari, Finland[END_REF][START_REF] Van Tongeren | Exposure to organic dusts, endotoxins, and microorganisms in the municipal waste industry[END_REF]. The corresponding exposure has been associated to health disorders among workers, and respiratory, gastrointestinal or dermatologic symptoms have been reported for many situations that involve the handling of household waste [START_REF] Poole | Systematic Review: Occupational illness in the waste and recycling sector[END_REF][START_REF] Walser | Evaluation of exposure-response relationships for health effects of microbial bioaerosols -A systematic review[END_REF]. However, data provided by epidemiological studies is not sufficient to be able to establish a link between exposure and health issues, and there are no Occupational Exposure Limit values (OELs) available for airborne biological agents. As the health issues caused by bioaerosols are linked to the type of species and compounds to which workers are exposed, one of the reasons explaining the current lack of OELs may be linked to the complex composition of bioaerosols in the workplace atmosphere, such as WSP. Most of the existing published studies provide biodiversity data in WSP by culture and identification of microorganisms on nutrient media. They revealed that predominant airborne microorganisms in WSP belonged to Aspergillus and Penicillium for fungi and to Bacillus and Staphylococcus for bacteria (Breum et al., 1999b;[START_REF] Lehtinen | Occupational hygiene in terms of volatile organic compounds (VOCs) and bioaerosols at two solid waste management plants in Finland[END_REF][START_REF] Madsen | Expanded cardboard waste sorting and occupational exposure to microbial species[END_REF][START_REF] Rahkonen | Airborne contaminants at waste treatment plants[END_REF][START_REF] Viegas | Assessment of fungal contamination in waste sorting and incineration -Case study in portugal[END_REF]. A recent study has provided new data based on high-throughput sequencing (HTS) measurements, confirming previous culture-dependant findings, as well as highlighting new microbial taxa such as Wallemia, for fungi [START_REF] Degois | First metagenomic survey of the microbial diversity in bioaerosols emitted in waste sorting plants[END_REF]. Indeed, HTS has provided a new approach for environmental microbial studies over the last decade. The technique was already being used to characterize microbial communities in swine houses [START_REF] Kumari | Variations in abundance, diversity and community composition of airborne fungi in swine houses across seasons[END_REF], composting facilities (Mbareche et al., 2017b) or wastewater treatment plants [START_REF] Lin | Diversity and dynamics of microbial communities at each step of treatment plant for potable water generation[END_REF]. It provides a solution to some of the limitations of the culture method but it also introduces other types of biases such as DNA extraction efficiency or errors of sequencing [START_REF] Duquenne | On the identification of culturable microorganisms for the assessment of biodiversity in bioaerosols[END_REF]Mbareche et al., 2017a).

Nevertheless, the use of HTS in WSP has enabled a more complex bioaerosol composition to be revealed, with more microbial taxa detected, as opposed to culture [START_REF] Degois | First metagenomic survey of the microbial diversity in bioaerosols emitted in waste sorting plants[END_REF].

The evolution of microbial biodiversity over time in bioaerosols has also been investigated in different environments. Thus, geographical location, temperature, relative humidity and, more generally, the seasons were found to be potential determinants of biodiversity of airborne microbial communities for wastewater treatment plants [START_REF] Ding | Site-related and seasonal variation of bioaerosol emission in an indoor wastewater treatment station: level, characteristics of particle size, and microbial structure[END_REF], swine houses [START_REF] Kumari | Variations in abundance, diversity and community composition of airborne fungi in swine houses across seasons[END_REF] and outdoor air [START_REF] Bowers | Seasonal variability in airborne bacterial communities at a high-elevation site[END_REF]. As the impact of each of these factors on biodiversity varies from study to study, no conclusions can be drawn regarding the mode of action for environmental factors. To date, such an investigation has not yet been carried out for bioaerosols emitted in French

WSPs. Yet, a better understanding of factors affecting airborne communities would improve knowledge about microbial exposure for workers and would provide useful data for establishing further sampling strategies as well as for managing biological risk. Biodiversity measurements were taken in the working environment of a waste sorting plant (WSP) over the period of one year.

The aims of the present study were: 1) to characterize the bacterial and fungal biodiversity in bioaerosols in six sampling areas of a WSP using HTS, 2) to assess temporal variations of the microbial biodiversity in emitted bioaerosols and 3) to investigate the environmental parameters which may affect the microbial biodiversity in the bioaerosols emitted in a waste sorting plant.

Material and methods

Description of the waste sorting plant

The waste sorting plant (WSP) is in eastern France and employs about 50 workers for the sorting activities. The plant treats non-hazardous household and industrial waste collected from 44 towns (representing approximately 235,000 inhabitants). An overview of the general organization of the plant and the sorting activities is provided in Figure 1. Household waste comprised various materials (cardboard, different types of plastic packaging, cartons and metals) unloaded directly onto the ground by trucks in a waste unloading area (Figure 1 ). Waste was then selected at random to be fed into the sorting chain to then be manually sorted by workers in sorting cabins (Figure 1 and ) before being stored in pits. Industrial waste such as furniture or wood items was fed into a wood chipper (Figure 1 ) and sent to an incineration unit adjacent to the WSP along a conveyor belt to supply the hot water system of the neighbouring town. As mentioned above, incoming waste was treated at random but the mode of treatment was mostly based on a "last in, first out" approach. However, it was not possible to specify the storage time and the nature of the waste treated on the days of sampling.

The company daily recorded the amount of waste delivered to the unloading area, shredded or squeezed by the press. The median mass was 71,640 kg (59,200-89,920 kg) in the unloading area, 16,320 kg (5,040-24,500 kg) for the shredder, and 23,296 kg (18,800-39,151 kg) for the press. Even if it mainly included food packaging, most of the treated waste was of varying materials (between 59% and 86% of daily mass of waste) and the remaining fraction consisted of large cardboard items and papers/magazines processed through a separate sorting chain, which was not examined in this study.

Sampling methods and strategy

Stationary bioaerosol samples were collected at five sampling points positioned near significant working areas inside the WSP, known as "waste exposure areas" in the following sections of the paper: point 1 in the unloading area; point 2 in the first sorting cabin; point 3 in the third sorting cabin; point 4 near the shredder and point 5 near the press (Figure 1). In addition, a sample was taken from an IR (indoor reference) (point 6) in a meeting room in another administrative part of the building, hermetically separated from waste sorting activities. Samples were collected every month from January to December 2016 for all the sampling points in order to assess the variation of microbial biodiversity (12 samples collected per sampling location).

To perform bioaerosol sampling, three-piece 37 mm closed-face cassettes (CFC) with a polycarbonate membrane filter (Nuclepore ® , 0.8 μm pore size) and a cellulose support pad (Millipore ® , thick cellulose absorbent pad) were attached to constant flow pumps (Bravo, Tecora, France) calibrated at a flow rate of 10 L/min using a flow meter (Mass Flowmeter 4040, TSI Inc., USA).

Flow rates were measured before and after sampling. CFCs were placed at the approximate height of the human respiratory tract and sampling was performed for five hours during the work shift. In parallel to bioaerosol sampling, temperature and relative humidity (RH) of the air were measured using a real-time measuring device (Labguard ® 3D Mobiguard, Biomérieux, France) positioned near the sampling device. All samples were transported to the laboratory in a cold box, stored at 4°C and analyzed within 24 hours of being collected.

Pre-treatment of samples and DNA extraction

To elute the particles from the filter, a volume of 5 mL of a sterile solution (0.01 % Tween 80 in ultra-pure water) was introduced into each CFC through the inlet. Then, the cassette inlets and outlets were closed with caps and shaken (Heildolph ® , Multi-Reax shaker) at 2000 rpm for 20 min and at room temperature. DNA extraction was performed using a commercial kit (FastDNA ® SPIN kit for soil, MP Biomedicals, Illkirch, France), according to the manufacturer's instructions, with 500 μL of eluted particle solution. The DNA samples were stored at -20°C until sequencing.

DNA sequencing and sequence data analysis

The DNA samples were purified, amplified and sequenced by the INRA Transfert Environnement (Narbonne, France) sequencing platform. The bacterial biodiversity was examined by sequencing the 16S rDNA (v3-v5 variable region) using primers F-5'-GTGYCAGCMGCCGCGGTA-3' and R-5'-CCCCGYCAATTCMTTTRAGT-3'. The primers used to amplify the specific fungal barcode ITS2 were w404 5'-CTTTCCCTACACGACGCTCTTCCGATCTTGTGARTCATCGAATCTTTG-3' and w40 5'-GGAGTTCAGACGTGTGCTCTTCCGATCTTCCTCCGCTTATTGATATGC-3'.

For the DNA amplification by PCR, the mixture contained 5 μL of DNA template, 4 μL of dNTPs (25 mM), 1 μL of each primer (100 ng/μL), and 0.5 μL of Taq polymerase in a total volume of 50 μL. The PCR cycle was 2 min at 95°C (denaturation step), 1 min at 56°C or 65°C for ITS2 or 16S rDNA amplification respectively (annealing step) and 1 min at 72°C (elongation step). The two genetic targets were sequenced using paired-end sequencing with Miseq technology (Illumina, San Diego, USA). Sequence analysis was performed according to Mothur's SOP [START_REF] Kozich | Development of a dual-index Sequencing strategy and curation pipeline for analyzing amplicon sequence data on the MiSeq Illumina sequencing platform[END_REF]. Simply put, the reads were trimmed; barcodes, primers and homopolymers longer than 8 pb were removed using Mothur (v.1.37.4) software [START_REF] Schloss | Introducing mothur: open-source, platform-independent, community-supported software for describing and comparing microbial communities[END_REF].

Chimera were detected and removed using UNCHIME [START_REF] Edgar | UCHIME improves sensitivity and speed of chimera detection[END_REF].

After alignment, the reads with 100% identity were clustered into a single sequence. The bacterial reads were assigned by the Wang method (Wang et al., 2007) that was executed in Mothur with SILVA database. Operational taxonomic units (OTUs) were defined using the traditional method. The ITS2 reads were processed in the same way as bacterial sequences. Fungal OTUs were assigned at the genus level using the massBLASTer algorithm in the UNITE database (https://unite.ut.ee/analysis.php) with an e-value of over 10 -60 and more than 95% identity.

Statistical analysis

Sample labelling

The names of the samples follow a letter code (B or F for bacterial or fungal samples respectively) with a first digit to indicate the month of sampling and a second digit that represents the sampling location. For example, the bacterial analysis for the sample collected in February in the sorting cabin is named B23 in figures, which represents Bayesian's results.

Diversity analysis

Alpha diversity, which estimates the diversity within a sample, was calculated according to the Shannon index and Specific Richness (SR) to measure variations in the first sorting cabin, the sorting cabin, and the shredder. To assess the statistical differences, Mann-Whitney and Kruskal-Wallis tests were performed using Statgraphics Centurion XVI software (Statistical Graphics Corp.,USA).

The Sorensen-Dice coefficient (QS) [START_REF] Tuomisto | A consistent terminology for quantifying species diversity? Yes, it does exist[END_REF] was calculated to compare the beta diversity between measurements and the similarity in composition between the bioaerosols emitted inside the plant and in the IR.

The more the microbial compositions between two samples have in common, the closer the index value is to 1 (identical composition). This index was calculated using the Mothur software for bacterial and fungal biodiversity, over a period of four months: January (winter), April (spring), August (summer) and October (fall). These months were chosen because of the success of sequencing for the two genetic targets (ITS and 16S rDNA) at all the sampling points.

Correlation calculation

To establish a relation between the QS and the temperature or the RH, the value of the Spearman' rs statistic was calculated using Statgraphics Centurion XVI software (Statistical Graphics Corp.,USA).

Bayesian networks analysis

Bayesian networks were used to identify the similarities between measurements [START_REF] Pearl | Bayesian Networks[END_REF]. The software used for this technique is BayesiaLab (www.bayesia.com). For each measurement, the biodiversity data is summarized into the presence (value 1) or absence (value 0) of the genus. Each measurement is then a vector whereby each coordinate symbolizes the presence or absence of a specific genus (Table S1). The data table used displays measurements as columns and genera as rows.

Based on this data, the idea is to automatically build a network that highlights the statistical correlations between the measurements, based on the frequency of shared genus between samples. Each measurement becomes a node and whenever a statistical correlation is strong enough (p value < 0.0001), then a link is established between the nodes (Figure S1).

In a second step of the statistical analysis, a data clustering method is applied to form groups of nodes with strong enough similarity. In Figure S1, the nodes bearing the same color belong to the same group. This analysis can be viewed through a dendogram (Figure S2), the highest level of which shows the best compromise between the overall number of groups formed and the average similarity within the groups. It is expected that the standard statistical settings for forming the groups will identify patterns of biodiversity between measurements, particularly when taking into account the season in which measurement was carried out.

Results

Sampling conditions

During the sampling campaign (twelve days in the year 2016), 72 bioaerosol samples were collected (representing 5 sampling points and an IR for each day) for microbial biodiversity analysis. The number of workers in the WSP was representative of the previous years but the plant had recorded an increase in the amount of waste sorted. No production dysfunctions were observed during sampling periods except a halt in production in November with no activity for the two sorting cabins. The temperature was monitored at the different sampling sites over the period of the campaign. It varied from 0°C to 27°C (Figure S2 A andB). Temperatures measured in waste management areas were at the maximum in summer (about 27°C in July) and the minimum in winter (about 0°C in the hall and 12°C inside the cabins in January and February). Measured values were similar between areas located in the hall (unloading waste area, shredder and press/storage) and varied between cabins. The temperature monitored at the IR was more stable (about 20°C) and significantly higher than in waste management areas. The recorded values of relative humidity (RH) varied from 28% to 60% and were similar between the two cabins and the IR; they were higher in areas located in the hall especially during fall and winter (Figure S3 A and B).

General sequencing data

A total of 63 samples (87% success rate) and 68 samples (94% success rate) were successfully sequenced for bacterial and fungal biodiversity, respectively. Samples with DNA concentrations lower than 3 ng.µL -1 were not sequenced (winter samplings or IR). The number of reads (a read is a sequence analyzed by the sequencing machine) after sequencing varied extensively depending on the location and on the month of sampling. The median of the reads was approximately 25,000 except for 16S rDNA reads for the IR, which contained less reads than the other sampling locations (Table 1). Two samples with a very low number of bacterial reads were not considered.

Bacterial biodiversity and monthly variation

Dominant bacterial phyla and their variations

Thirty bacterial phyla were observed but 96% of the reads represented the five most prominent phyla in all the samples (Figure 2A, B, andC): Actinobateria, Bacteroidetes, Cynaobacteria, Firmicutes and Proteobacteria.

Among them, Firmicutes and Proteobacteria were the two most dominant specimens in samples collected in waste treatment areas (median=41.5% and 24.6% respectively) and in the IR (median= 20.0% and 48.6% respectively), followed by Actinobacteria and Bacteroidetes. The variation of the main bacterial phyla over the course of the study is illustrated in Figure 2A, B, and C for three sampling locations: unloading waste area, in the first-sorting cabin and in the IR (for the other sampling points, see Figure S4A, B, and C in the supplementary materials). A high proportion of Cyanobacteria was recorded in February in the unloading waste area with 65.7% of the bacterial reads attributed to this phylum. This result was not observed at any other sampling site or at any other time during the sampling campaign.

Dominant bacterial genera and their variations

At the genus level, 605 bacterial genera were observed in our samples. The more abundant bacteria are known as core microbiome in biodiversity studies.

In the current project, the core microbiome was composed of 45 genera with 6 Actinobacteria, 7 Bacteroidetes, 15 Firmicutes, 1 Nitrospinae, and 16 Proteobacteria genera. These main genera represented between 9.3% (waste unloading area in February) and 83.8% (IR in September) of the bacterial reads.

As for phyla, the description of the results is focused on three sampling points: the unloading waste area, the first-sorting cabin and the IR (Figure 2D, E, andF). Results from the other sampling plants are presented in Figure S4 of the supplementary materials. One of the most dominant bacterial genera found at each sampling point, including the IR, was Staphylococcus (Figure 2D, E andF) with a median of approximately 9% of relative abundance in all sampling points, except near the press in March. Prevotella and Pseudomonas were present throughout the whole year in substantial proportion in the two waste treatment locations (about 3%) and in lower proportion in the IR (less than 0.6%). Streptococcus was also found every month but in lower proportions with a maximal value of 11.3%, observed in the IR in January. Other bacterial genera were frequently found during the year in the bioaerosols collected in the unloading area and in the first sorting cabin: Lactobacillus (3%), Bacillus (1%), Leuconostoc (2%) which was absent in the IR, as well as Aerococcus, Lactococcus, Acinetobacter, and Corynebacterium.

Fungal biodiversity and its variation

Dominant fungal phyla and their variations over the year

Among the 6 phyla detected in the bioaerosols, Ascomycota, Basidiomycota, and Mucoromycota were the dominant ones (Figure 3A, B, andC and Figure S5 A, B and C in the supplementary materials). Ascomycota reads represented more than 50% of the fungal reads (56.3% in the waste treatment areas and 53.2 % in the IR). The main fungal phyla were present at each period and at each sampling point with varied proportions depending on the location and the time of the sampling.

Dominant fungal genera and their variations over the year

A total of 592 fungal genera were identified when all the collected bioaerosol samples were considered. The fungal core microbiome was composed of 45 genera. This core microbiome varied from 58.7% (in October near the press) to 91.2 % (in January in the first-sorting cabin) with a median value of 77.2%.

The relative abundance of the fungal microbiome found in the unloading waste area, the first-sorting cabin and the IR and its variation over the year is described in Figure 3. The fungal biodiversity found in the sorting cabin, near the shredder and near the press is illustrated in Figure S5 of the supplementary materials. Among the core microbiome, 8 genera were found in huge proportions throughout the whole year: Cladosporium, Alternaria, Debaryomyces, Penicillum, Candida, Wallemia, Cryptococcus, and Mucor (Figure 3 and Figure S5 of the supplementary materials). Wallemia was the predominant genus in the two occupational sampling points (Figure 3 D, E, andF). It represented 17.3% (up to 66.4% in February) and 21.4% (up to 54.8% in November) of the ITS2 reads in the unloading waste area and in the first sorting cabin respectively. In the IR, Wallemia was also recorded in substantial proportion (6.5%). Cladosporium and Debaryomyces were also dominant genera found at the three sampling locations throughout the year.

Cladosporium was the predominant genus in the IR. In the unloading waste area and in the first-sorting cabin, about 5% of fungal reads were attributed to Penicillium and Candida (5.2% and 6.1% in the unloading waste area; 5.2% and 4.5% in the first-sorting cabin). They only represented about 1% in the IR.

Variation of alpha biodiversity

The sampling location had no statistical impact on Shannon indexes for bacteria and fungi. The Shannon index values were higher for the bacterial biodiversity than for the fungal biodiversity. The number of microbial genera did not depend on the period of sampling, neither for bacteria nor fungi.

Season and humidity as the main factors of microbial diversity

The similarity of microbial (fungal and bacterial) composition between the IR and the sampling points inside the WSP was measured by calculating the Sorensen-Dice coefficient (QS). For bacterial biodiversity, this coefficient varied from 0.07 in winter to 0.6 in spring in the sorting cabin (Figure 4A).

There was more similarity between bacterial communities in waste treatment areas and IR in fall-summer than in winter-spring. A positive correlation was observed between the QS values and temperature values (Spearman rs p-value=0.51, n=20). The higher the temperature was, the closer the bacterial composition of bioaerosols was to that of IR.. The sampling location had an impact on bacterial biodiversity regarding Sorensen-Dice coefficient (Kruskall-Wallis test: p=0.003). For fungal biodiversity, the values varied from 0.12 to 0.64 (Figure 4B). The coefficient was minimal (about 0.15) in summer and maximal in spring with values superior to 0.5. The fungal composition in bioaerosols collected at the IR and at the sampling points in the WSP showed closer similarities in spring and in fall. The fungal biodiversity was negatively correlated with the relative humidity of the air (Spearman rs p-value=-0.67, n=20), the higher the RH, the more the composition of fungal bioaerosols differed, as compared to the IR.

Results from Bayesian networks statistics

The Bayesian networks were used to group bioaerosol samples according to the similarities in their microbial composition at the genus level. This operation is called "clustering" and is only based on statistical data. The groups were then analyzed to determine whether the composition of bioaerosol can be attributed to an environmental parameter (e.g. a given pattern of bioaerosol is more often present during a specific season). A first clustering provided 20 clusters with statistical similar composition for fungal biodiversity and 17 for bacterial biodiversity (Figure 5 and Figure S6 in the supplementary materials).

The clusters represent the sets of bacterial or fungal genus that are frequently identified together in a sample. After clustering, the composition of each cluster was analyzed to determine the common patterns between clusters: the day or location of sampling, the intensity of activity (amount of treated waste), the number of workers, the monitored climatic conditions at each sampling point, the atmospheric conditions during and three days before the collection.

The clusters were named according to the parameters (bacterial diversity Figure S6 in the supplementary materials; fungal biodiversity Figure 5A). For fungi, 18 factors were attributed to a parameter that was most of time associated with the day of sampling. After the first clustering, a second Bayesian analysis was performed to establish relationships between the renamed clusters (Figure 5B).This analysis shows that the composition of microbial aerosols is related with the seasons of the year. For bacteria, no relation with identified parameters was found to link the samples. In a same group, they were bioaerosols sampled at different locations and at different time (Figure S6 in supplementary materials).

Discussion

Overview of the sampling campaign

During the sampling campaign, the climatic conditions were those of the semicontinental climate usually found in the North East of France. The year 2016 in particular was characterized by very marked seasons, with very low temperatures in winter and rather high temperatures in summer. The production as well as working activities were considered representative for the investigated WSP.

Dominant airborne microorganisms in the WSP

The biodiversity data revealed that some microbial taxa was present in substantial proportions throughout the whole year and at all the sampling locations. For bacteria, the Firmicutes was the dominant bacterial phylum followed by Proteobacteria and Bacteroidetes, with an exception for the unloading waste area in February (Figure 2). About 605 bacterial genera were found but only few of them (45) account for most of the 16S rDNA reads in collected bioaerosol samples. Among these genera, Staphylococcus, Streptococcus, Lactobacillus and Lactococcus but also Prevotella, Pseudomonas, Acinetobacter and Bacillus, were present and dominant in all samples. For fungi, the dominant phyla were Ascomycota and Basidiomycota.

Among the 592 fungal genera, only five genera (Wallemia, Cladosporium, Debaryomyces, Candida, and Penicillium) constituted more than half of the ITS2 reads.

The present study corroborates our previous findings in another WSP located in France, using HTS technologies [START_REF] Degois | First metagenomic survey of the microbial diversity in bioaerosols emitted in waste sorting plants[END_REF]. In this previous study, the same genera (Staphylococcus, Streptococcus, Penicillium) were also found to be in abundance. Staphylococcus was isolated from bioaerosol samples collected in truck cabs for waste collection [START_REF] Madsen | Waste workers' exposure to airborne fungal and bacterial species in the truck cab and during waste collection[END_REF] and during cardboard sorting [START_REF] Madsen | Expanded cardboard waste sorting and occupational exposure to microbial species[END_REF]. Staphylococcus and Bacillus were found by culture in bioaerosols emitted during paper sorting (Breum et al., 1999). In the present study, the genus Prevotella was also found in high relative abundance throughout the whole year. To our knowledge, this genus was not reported in studies measuring the bacterial composition in bioaerosol emitted in waste sorting plant. The fungal genera Penicillium and Aspergillus were also isolated by culture from samples collected in truck cab samples [START_REF] Madsen | Waste workers' exposure to airborne fungal and bacterial species in the truck cab and during waste collection[END_REF] and in a waste treatment plant (Breum et al., 1999;[START_REF] Lehtinen | Occupational hygiene in terms of volatile organic compounds (VOCs) and bioaerosols at two solid waste management plants in Finland[END_REF][START_REF] Tolvanen | Airborne bio-aerosols and noise in a dry waste treatment plant in Pietarsaari, Finland[END_REF][START_REF] Viegas | Assessment of fungal contamination in waste sorting and incineration -Case study in portugal[END_REF]. In addition to Penicillium, Cladosporium was the predominant fungal genus in a sorting paper plant (Breum et al., 1999). Candida and Alternaria were also found in high proportion, using HTS, in a domestic compost heap and in the bioaerosols emitted from it (Mbareche et al., 2017a;[START_REF] Wéry | Bioaerosols from composting facilities-a review[END_REF].

The results from our study indicate that Wallemia was one of the predominant genera in the bioaerosols emitted in the investigated WSP. This genus belongs to the order Wallemiales (Wallemiomycotina, Basidiomycota) and includes several species (W. ichthyophaga, W. muriae, W. sebi, W. mellicola, W. tropicalis, W. Canadensis, W. hederae and W. peruviensis). These species are salty environments [START_REF] Zalar | Taxonomy and phylogeny of the xerophilic genus Wallemia (Wallemiomycetes and Wallemiales, cl. et ord. nov.)[END_REF] and are known to be spoilage organisms of food items with low water activity (aw) [START_REF] Zajc | The Genus Wallemia-From Contamination of Food to Health Threat[END_REF]. But some are also found in the airborne state and in settled dust in occupational and indoor environments, as well as dwellings [START_REF] Fröhlich-Nowoisky | High diversity of fungi in air particulate matter[END_REF][START_REF] Li | Characterization of residential household dust from Shanghai by particle size and analysis of organophosphorus flame retardants and metals[END_REF][START_REF] Zeng | Detection and quantification of Wallemia sebi in aerosols by real-time PCR, conventional PCR, and cultivation[END_REF]. They usually differ in their xerotolerance and tolerance to solutes (NaCl, MgCl2), growth rate and temperature, enzymes and metabolites production profiles as well as in their size of conidia [START_REF] Jančič | A Taxonomic Revision of the Wallemia sebi Species Complex[END_REF][START_REF] Jančič | Halophily reloaded: new insights into the extremophilic life-style of Wallemia with the description of Wallemia hederae sp[END_REF]. The predominance of Wallemia in bioaerosols from the WSP can be explained by the presence of food packaging among waste (trays, yoghurt pots and sauce jars, beverage bottles, etc.) on which food residue remains, creating favorable conditions for their growth and survival. This residue may come from food items with the water activity reduced initially or that has been lowered by dehydration during the period prior to sorting. Similar hypotheses can be put forward to explain the predominance of other fungal genera such as Cladosporium and Penicillium, as several species belonging to these genera were found to be able to grow in the presence of low water activity [START_REF] Gunde-Cimerman | Extremophilic fungi in arctic ice: a relationship between adaptation to low temperature and water activity[END_REF]. Their growth and survival on paper and cardboard have also been shown [START_REF] Das | Endoglucanase production by paperdegrading mycoflora[END_REF].

The presence of Wallemia in bioaerosols from WSP has already been demonstrated in our previous published study carried out using molecular methods in the same working environment [START_REF] Degois | First metagenomic survey of the microbial diversity in bioaerosols emitted in waste sorting plants[END_REF]. This result contrasts with the results of studies conducted using culture-based methods for which this genus has not been identified [START_REF] Tolvanen | Airborne bio-aerosols and noise in a dry waste treatment plant in Pietarsaari, Finland[END_REF][START_REF] Viegas | Assessment of fungal contamination in waste sorting and incineration -Case study in portugal[END_REF]. Fungal species belonging to Wallemia have a very low growth rate on usual culture media and its numeration is usually impaired due to masking by faster growing species. Thus, its presence in different environments is underestimated while using culture-based methods. On the other hand, the Aspergillus genus was poorly detected in the present study as compared to previous published studies [START_REF] Degois | First metagenomic survey of the microbial diversity in bioaerosols emitted in waste sorting plants[END_REF][START_REF] Tolvanen | Airborne bio-aerosols and noise in a dry waste treatment plant in Pietarsaari, Finland[END_REF]. This underrepresentation may be due to specific factors at our site (undetermined environmental or ecological factors) or due to a methodological bias. Indeed, we have already shown that Aspergillus might be under-represented in HTS analyses compared to culture methods, probably due to an inefficient DNA extraction [START_REF] Degois | Characterization of experimental complex fungal bioaerosols: Impact of analytical method on fungal composition measurements[END_REF]. These specific examples refer to the complementarity of the two approaches whose respective advantages and limitations have already been widely discussed [START_REF] Degois | First metagenomic survey of the microbial diversity in bioaerosols emitted in waste sorting plants[END_REF][START_REF] Degois | Characterization of experimental complex fungal bioaerosols: Impact of analytical method on fungal composition measurements[END_REF][START_REF] Duquenne | On the identification of culturable microorganisms for the assessment of biodiversity in bioaerosols[END_REF]Mbareche et al., 2017).

The biodiversity of airborne bacterial communities emitted in waste sorting centers is different, compared to that recorded at composting or anaerobic digestion facilities where bacteria belonging to Saccharopolyspora, Saccharomonospora, Thermomonospora, Thermobifida, or Legionella can be found. These bacteria are thermophilic and/or are involved in the degradation process of organic matter, as seen in composting or anaerobic digestion processes and related to the type of waste treated in this sort of facility [START_REF] Dubuis | Bioaerosols concentrations in working areas in biomethanization facilities[END_REF][START_REF] Wéry | Bioaerosols from composting facilities-a review[END_REF]. The airborne fungal core microbiome found in the WSP was similar, as described before, as other facilities treating dry and solid waste.

The present study provides additional biodiversity data about bioaerosols emitted in WSP and reveals their actual complex composition throughout the whole year.

Reported health disorders cause by the detected microbial taxa

Several of the dominant microbial genera found in the present study include species that are involved in health disorders among general and occupational populations. For example, some Prevotella species have been involved in human health disorders such as pulmonary diseases and chronic otitis and sinusitis [START_REF] Tanaka | The Relationship of Prevotella intermedia, Prevotella nigrescens and Prevotella melaninogenica in the Supragingival Plaque of Children, Caries and Oral Malodor[END_REF]. The effects of airborne fungal particles are well documented in the literature and the risk associated with exposure to airborne fungal particles is better documented than the bacterial risk. Fungi can induce different types of health issues including respiratory allergies (asthma, rhinitis or hypersensitivity pneumonitis), toxicity, and irritation or infection. Exposure to airborne fungal species belonging to Cladosporium, Aspergillus and Penicillium has been associated with allergic skin diseases such as contact dermatitis [START_REF] Kallawicha | Exposure to ambient bioaerosols is associated with allergic skin diseases in Greater Taipei residents[END_REF]. In addition, these fungi produce volatile organic compounds that may cause several symptoms such as a dry cough, pharyngitis or headache. Furthermore, the wall cell of Aspergillus and Candida ssp. are composed of mannan, which is a known allergic compound [START_REF] Dyląg | Fungi present in home and their impact on human health-A short review[END_REF]. In addition, Wallemia sebi is involved in allergic disorders such as farmer's lung disease and produces a toxic metabolite, wallemiol, which has the same effect as the mycotoxin penicillic acid [START_REF] Wood | Studies on a toxic metabolite from the mould Wallemia[END_REF].

Because of the high number of genus, only the main genera are represented in the present paper. However, as the exposure-response relationship was not established, the minor and rare taxa could also affect worker health. Furthermore, the pathogenic property is not the only aspect to consider and some airborne bacteria present in minority proportions can carry endotoxins or β-D-glucans. These components have been reported to induce several health effects such as irritation of the upper airways or mucous membranes, inflammation [START_REF] Bønløkke | Upperairway inflammation in relation to dust spiked with aldehydes or glucan[END_REF][START_REF] Lawniczek-Walczyk | Endotoxins and beta-glucans as markers of microbiological contamination -characteristics, detection, and environmental exposure[END_REF] and irritation of the respiratory system, fever and diarrhea [START_REF] Duquenne | Measurement of endotoxins in bioaerosols at workplace: A critical review of literature and a standardization issue[END_REF][START_REF] Liebers | Health effects due to endotoxin inhalation (review)[END_REF]. Exposure to these compounds was reported in other plants treating solid and dry waste [START_REF] Lehtinen | Occupational hygiene in terms of volatile organic compounds (VOCs) and bioaerosols at two solid waste management plants in Finland[END_REF][START_REF] Tolvanen | Airborne bio-aerosols and noise in a dry waste treatment plant in Pietarsaari, Finland[END_REF].

On the other hand, the biodiversity data provided by the present study needs to be expanded for an eligible risk assessment in the WSP. Firstly, the relationship between airborne microorganism exposure and human health disorders must be specified [START_REF] Walser | Evaluation of exposure-response relationships for health effects of microbial bioaerosols -A systematic review[END_REF]. Secondly, personal samplings are required for the assessment of occupational exposure. Indeed, stationary sampling does not take into account the movement of workers during their shift, nor the professional gestures that can affect exposure.

Finally, the HTS technologies do not allow the identification at the species level due to the short reads provided by Illumina sequencing, the sequencing depth, the sequenced regions and the errors of sequence annotations in databases [START_REF] Nilsson | Taxonomic reliability of DNA sequences in public sequence databases: a fungal perspective[END_REF]Yamamoto and Bibby, 2014a). The genus level does not allow for a precise risk assessment. Indeed, a genus can gather a large number of species (e.g. more than a hundred for the genus Bacillus) and within a genus, only some species can be pathogenic [START_REF] Kwok | Species identification and phylogenetic relationships based on partial HSP60 gene sequences within the genus Staphylococcus[END_REF]. Nevertheless, the HTS constitutes a powerful screening and characterization tool at the genus level, enabling to focus on potential pathogenic species, in a second and more targeted process.

Sorting activities compared to the indoor reference

The Sorensen-Dice coefficient is a good index to measure a difference in the composition of two samples. In the present study, a significant difference (index value <0.5) was found between samples collected in areas with ongoing sorting activities and the IR, and this was the case for all seasons. This result suggests specific bioaerosols exposure for workers handling waste compared to the administrative staff. For airborne bacterial communities, the difference in biodiversity between waste-exposed sampling points and the IR was higher in winter and in spring when the highest differences in temperature between the two sampling locations were taken into consideration. The cold climatic conditions were apparently less favorable to bacterial growth of most bacteria encountered during warmer seasons. Despite the low temperatures measured, the specific richness was higher at waste-exposed locations than in IR. This suggested that the waste contributes significantly to bacterial biodiversity. For fungal communities, the difference in composition between waste-exposed sampling points and the reference was less significant, with higher differences observed when the temperatures were extreme (in winter and in summer). The IR was heated during the winter and cooled during the summer with an air conditioning system (air drying). Moreover, the air in the administrative building was renewed with the outdoor air that surrounded the waste treatment building. As a consequence, the indoor air may have been polluted by the outdoor air. Such a phenomenon has already been described [START_REF] Adams | Dispersal in microbes: fungi in indoor air are dominated by outdoor air and show dispersal limitation at short distances[END_REF] and could partly explain the similarities in fungal biodiversity between the two locations in winter and in summer. In the present study, no differences in microbial biodiversity were observed between the sampling points inside the waste treatment buildings, whatever statistical method was used (Bayesian network, Shannon index). This could be explained by the temperature and the relative humidity, which were similar between sampling points, especially in the hall (sampling points 1, 4 and 5).

Waste represented the main source of bioaerosol emissions. With the exception of sampling point number 4, the type of waste was identical for all sampling points, and the only difference was the duration of storage at each point (several days, weeks or months at the locations 1 and 5 versus just a few minutes at locations 2 and 3). In addition, the workers opened the two doors located on either side of the box in order to cool the first-sorting cabin during the hot season. However, the first-sorting cabin was located in front of the shredder which produced large amounts of airborne dust and which may have contaminated the air of the first-sorting cabin. All these observations could explain the similarities of microbial composition detected between the waste-exposed locations.

Seasonal variations of microbial biodiversity in bioaerosol

The Bayesian network approach was used for clustering the fungal bioaerosol samples depending on their composition similarities. The results showed correlation according to the month or the season of sampling (consecutive months) regardless of the locations of the waste-exposed samples. The period of sampling did not explain correlation of the airborne bacterial samples. No other parameters (number of workers, sampling locations, amount of waste) could provide an explanation for how samples were clustered by the Bayesian network analysis for both bacterial and fungal communities.

Seasonal variations in the concentration of airborne microorganisms were investigated in studies carried out on outdoor air from different countries [START_REF] Bezerra | Diversity and dynamics of airborne fungi in Sao Luis, State of Maranhao, Brazil[END_REF][START_REF] Bowers | Seasonal variability in bacterial and fungal diversity of the near-surface atmosphere[END_REF][START_REF] Crandall | Meteorological factors associated with abundance of airborne fungal spores over natural vegetation[END_REF] and on indoor air [START_REF] Chao | Populations and determinants of airborne fungi in large office buildings[END_REF][START_REF] Frankel | Seasonal variations of indoor microbial exposures and their relation to temperature, relative humidity, and air exchange rate[END_REF][START_REF] Wang | Size and seasonal distributions of airborne bioaerosols in commuting trains[END_REF]. The seasonal variations were also investigated in occupational environments such as a wastewater treatment plant [START_REF] Ding | Site-related and seasonal variation of bioaerosol emission in an indoor wastewater treatment station: level, characteristics of particle size, and microbial structure[END_REF][START_REF] Oppliger | Influence of seasons and sampling strategy on assessment of bioaerosols in sewage treatment plants in Switzerland[END_REF], swine confinement buildings [START_REF] Nehme | Cultureindependent approach of the bacterial bioaerosol diversity in the standard swine confinement buildings, and assessment of the seasonal effect[END_REF], composting plants [START_REF] Fracchia | The assessment of airborne bacterial contamination in three composting plants revealed site-related biological hazard and seasonal variations[END_REF] and biowaste collection plants [START_REF] Nielsen | Seasonal variation in bioaerosol exposure during biowaste collection and measurements of leaked percolate[END_REF].

The corresponding studies reported contrasting results regarding to the seasonal effect on concentrations of airborne microorganisms.

Several studies found a seasonal variability of microbial biodiversity in bioaerosols but the impact of the season on microbial communities depended on the study and the investigated place [START_REF] Bowers | Seasonal variability in airborne bacterial communities at a high-elevation site[END_REF][START_REF] Ding | Site-related and seasonal variation of bioaerosol emission in an indoor wastewater treatment station: level, characteristics of particle size, and microbial structure[END_REF][START_REF] Fröhlich-Nowoisky | High diversity of fungi in air particulate matter[END_REF][START_REF] Kumari | Variations in abundance, diversity and community composition of airborne fungi in swine houses across seasons[END_REF][START_REF] Nehme | Cultureindependent approach of the bacterial bioaerosol diversity in the standard swine confinement buildings, and assessment of the seasonal effect[END_REF]. No general tendency was observed with regards the published data and the impact of environmental parameters is difficult to predict [START_REF] Liao | Use of temporal/seasonal-and size-dependent bioaerosol data to characterize the contribution of outdoor fungi to residential exposures[END_REF]. To our knowledge, the impact of environmental parameters on airborne biodiversity has not been assessed in a waste sorting plant and therefore, our results cannot be compared with any other published data.

Other determinants of biodiversity in bioaerosols

Temperature and relative humidity of the air may partly explain our results, as these parameters were shown to greatly influence microbial growth [START_REF] Brown | Fate and Transport of Microorganisms in Air, Manual of Environmental Microbiology, Fourth Edition[END_REF], as well as the aerosolization of fungal spores from contaminated matters [START_REF] Madsen | Effects of airflow and changing humidity on the aerosolization of respirable fungal fragments and conidia of Botrytis cinerea[END_REF]. In the present study, the meteorological conditions for sampling were very different throughout the year. However, for a given sampling day, the temperature and humidity conditions were different between the hall and the two cabins, and showed correlation in the same cluster. Thus, temperature and relative humidity were not the only parameters impacting the airborne fungal biodiversity in the WSP. This hypothesis was confirmed by an extended statistical analysis performed on the Shannon index values, combining multilinear regression analysis and general linear models. This analysis considered the temperature and the relative humidity at the sampling point but also the temperature, the relative humidity and the rainfall during the day of the sampling and three days before.

The best-fit model took into account all the previous parameters however it could explain only 40% of the Shannon index values (data not shown).

The impact of the environment on microbial biodiversity in bioaerosols is multifactorial and complex and combines not only the climatic conditions but also others parameters such as the amount and nature of waste or other pollutant sources. This confirms that microbial biodiversity and more generally, bioaerosol composition would be multifactorial and difficult to characterize. So far, other parameters have also been proven to impact microbial concentrations such as human activity [START_REF] Maron | Temporal variability of airborne bacterial community structure in an urban area[END_REF], particles size and density of urban area [START_REF] Fang | Culturable airborne bacteria in outdoor environments in Beijing, China[END_REF], air pollution [START_REF] Gao | Seasonal size distribution of airborne culturable bacteria and fungi and preliminary estimation of their deposition in human lungs during non-haze and haze days[END_REF][START_REF] Gao | Variation of correlations between factors and culturable airborne bacteria and fungi[END_REF], CO2 concentrations [START_REF] Chao | Populations and determinants of airborne fungi in large office buildings[END_REF], operations of a process [START_REF] Tolvanen | Airborne bio-aerosols and noise in a dry waste treatment plant in Pietarsaari, Finland[END_REF], indoor air pollutants [START_REF] Wang | Size and seasonal distributions of airborne bioaerosols in commuting trains[END_REF], the site investigated [START_REF] Fracchia | The assessment of airborne bacterial contamination in three composting plants revealed site-related biological hazard and seasonal variations[END_REF] and physical, environmental and ecological factors [START_REF] Emerson | High temporal variability in airborne bacterial diversity and abundance inside singlefamily residences[END_REF].

The age of the stored waste has also been shown to have an impact on both airborne bacteria and fungi exposure levels [START_REF] Gladding | A study of the potential release of bioaerosols from containers as a result of reduced frequency residual waste collections[END_REF][START_REF] Schlosser | Extension of the sorting instructions for household plastic packaging and changes in exposure to bioaerosols at materials recovery facilities[END_REF].

Limitations of the study

A large number of samples were collected, at different workstations and at different times of the year. This provided a satisfactory overview of the airborne microbial biodiversity in the WSP. Stationary samples were adequate in providing information about the exposure of workers in the cabins, but less apt regarding machine operators who moved around during their shift.

Samples near the press, the shredder and the unloading area therefore did not allow for the characterization of the bioaerosol exposure for these employees. However, sampling on filters and analysis by molecular biology allowed for long-term sampling (5 hours) for a better representation of exposure assessment than short-term sampling with Andersen type impactors used for the analyzes by culture. Molecular biology methods and particularly HTS are limited due to biases for DNA extraction, PCR or sequence analysis [START_REF] Haugland | Evaluation of different methods for the extraction of DNA from fungal conidia by quantitative competitive PCR analysis[END_REF]Mbareche et al., 2017b;Yamamoto et al., 2014).

These techniques are based on DNA analysis, which does not provide information on the viability and pathogenicity of airborne microorganisms and the associated health risk for workers. In addition, short sequences that are generated by sequencing limit the identification of microorganisms at the genus level, which may hamper a sound health risk assessment. However, HTS provides an overview of microbial biodiversity in poorly characterized fields. Finally, exposure to viruses was not assessed in this study.

Further prospects

Most studies in waste treatment plants focus on fungi. Since bacteria were proven to be present in higher abundance than fungi, and to be the main colonizers of household waste bins [START_REF] Gladding | A study of the potential release of bioaerosols from containers as a result of reduced frequency residual waste collections[END_REF], more investigations of bacterial biodiversity in WSP would add to knowledge about risk assessment in this field. To improve this knowledge, it would be relevant to couple HTS sequencing with other techniques such as sequencing longer sequences or an analysis by culture followed by an identification by MALDITOF-MS, which would provide identification at the species level and/or allow the viability of microorganisms to be preserved. An analysis of the viral risk in WSP would also be relevant. Thanks to the present study, airborne microbial biodiversity was measured over a period of one year in one WSP. It would be interesting to compare with other WSPs with a similar or different organization.

Evaluating the impact of environmental factors on airborne biodiversity in workplaces is not easy especially due to the significant amount of unknown or uncontrolled factors. A pilot study would be an accurate way of assessing all the potential parameters (in particular the nature of waste) that could have an effect on biodiversity and evaluating how these parameters modify the composition of airborne microorganisms. Even if such studies describe more simple conditions as compared to the occupational environment, they would contribute to a more accurate understanding of such a complex phenomenon.

The biodiversity results obtained at the genus level also represent a good basis for the definition of taxa to be monitored in this sector for occupational health purposes, if similar measurements confirm their presence and abundance in other WSPs.

Conclusions

The study investigated the microbial diversity in bioaerosols emitted over one year in different representative areas of a French WSP.

The observed biodiversity was variable and dependant on the season, and was also found to be more complex than in studies using culture-based methods. Among the large number of genera observed, the genera Staphylococcus, Streptococcus, Prevotella, Pseudomonas, Cladosporium, Wallemia, Penicillium, and Candida were recorded in high proportions throughout the year. The season seemed to have an impact on fungal biodiversity. The parameters affecting bacterial biodiversity were probably multifactorial and are yet to be identified. Workers handling waste were exposed to specific bioaerosols, different from those emitted in the offices.

Because of the high bioaerosol complexity in WSP, a long-term monitoring survey would be more appropriate in providing an accurate assessment of bioaerosol composition.

Analysis of the composition of bioaerosols by HTS provided a good overview of the microorganisms workers may be exposed to. This has improved knowledge in terms of exposure measurements in this occupational environment. This technique should be coupled with methods of identification at the species level and/or conservation of viability for better risk assessment.

Figure 1 :

 1 Figure 1: Schematic organization of the household waste sorting plant and location of the sampling points.

Figure 2 :

 2 Figure 2: Bacterial biodiversity at the phylum (A, B and C) and at the genus (D,E and F) levels in bioaerosols collected in the unloading waste area, in the first-sorting cabin and in the indoor reference. To facilitate the reading, only the 45 main genera found in a proportion superior than 0.3% were represented.

Figure 3 :

 3 Figure 3: Fungal biodiversity at the phylum (A, B and C) and at the genus (D,E and F) levels in bioaerosols collected in the unloading waste area, in the firstsorting cabin and in the indoor reference. To facilitate the reading, only the 45 main genera found in a proportion superior than 0.3% were represented.

Figure 4 :

 4 Figure 4: Sorensen-Dice coefficient for A. bacterial and B. fungal biodiversity. This coefficient represents a distance of community composition between a sampling point and the indoor reference. A coefficient near 1 means that the two samples had a similar composition.

Even if the study is not exhaustive, it has produced an overview of airborne biodiversity over a period of one year for a standard facility, which will assist us in making future studies more relevant.

Acknowledgments

The authors would like to kindly acknowledge Cécile Dziurla, Catherine Coulais, Véronique Koehler and Alexis Boivin for their invaluable technical assistance. Special thanks to Véronique Ritz and Jérôme Schmitt from the company for making it possible to carry out the measurement campaigns and for making the facilities available.

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or non-profit sectors.


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

Lucile Laulin
email: lucile.laulin@math.u-bordeaux.fr

ON THE CENTER OF MASS OF THE ELEPHANT RANDOM WALK

Keywords: 2010 Mathematics Subject Classification. Primary: 60G50 Secondary: 60G42, 60F05 Elephant random walk, Center of mass, Multi-dimensional martingales, Almost sure convergence, Asymptotic normality

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Let (S n ) be a standard random walk in R d . The center of mass G n of S n is defined by

(1.1) G n = 1 n n k=1 S k .
The question of the asymptotic behavior of G n was first raised by Paul Erdös. Very recently, Lo and Wade [START_REF] Lo | On the centre of mass of a random walk[END_REF] extended the results of Grill [START_REF] Grill | On the average of a random walk[END_REF] by studying the asymptotic behavior of (G n ). More precisely, let S n = X 1 + • • • + X n where the increments (X n ) are independent and identically distributed square integrable random vectors of R d with mean µ and covariance matrix Γ. They proved the strong law of large numbers

(1.2) lim n→∞ 1 n G n = 1 2 µ a.s.
together with the asymptotic normality,

(1.3) 1 √ n G n - n 2 µ L -→ N 0, 1 3 Γ .
Curiously, no other references are available on the asymptotic behavior of the center of mass. The proofs of many results on G n rely on independence and exchangeability of the increments of the walk. For example, one can observe that (1.4)

G n = 1 n n k=1 S k = 1 n n k=1 n -k + 1 X k
shares the same distribution as

Σ n = 1 n n k=1 kX k .
A natural question concerns the asymptotic behavior of G n in other situations where the increments of the walk are not independent and not identically distributed. Moreover, geometrical features of the random walk paths of (S n ) is a subject of ongoing interest. For example, the convex hull C n = Conv(S 1 , . . . , S n ) of the n first steps of (S n ) have recently received renewed attention [START_REF] Kabluchko | Convex hulls of random walks, hyperplane arrangements, and Weyl chambers[END_REF], [START_REF] Vysotsky | Convex hulls of multidimensional random walks[END_REF]. More particularly, for the random walk in R 2 , the strong law of large numbers and the asymptotic normality of the perimeter and the diameter of C n were established in [START_REF] Mcredmond | The convex hull of a planar random walk: perimeter, diameter, and shape[END_REF], [START_REF] Wade | Convex hulls of planar random walks with drift[END_REF].

In this paper, we investigate the asymptotic behavior of the center of mass of the multidimensional elephant random walk. It is a fascinating discrete-time random walk on Z d where d ≥ 1, which has a complete memory of its whole history. The increments depend on all the past of the walk and they are not exchangeable. The elephant random walk (ERW) was introduced by Schütz and Trimper [START_REF] Schütz | Elephants can always remember: Exact long-range memory effects in a non-markovian random walk[END_REF] in the early 2000s, in order to investigate how long-range memory affects the random walk and induces a crossover from a diffusive to superdiffusive behavior. It was referred to as the ERW in allusion to the traditional saying that elephants can always remember where they have been before. The elephant starts at the origin at time zero, S 0 = 0. At time n = 1, the elephant moves in one of the 2d directions with the same probability 1/(2d). Afterwards, at time n + 1, the elephant chooses uniformly at random an integer k among the previous times 1, . . . , n. Then, it moves exactly in the same direction as that of time k with probability p or in one of the 2d-1 remaining directions with the same probability (1-p)/(2d-1), where the parameter p ∈ [0, 1] stands for the memory parameter of the ERW [START_REF] Bercu | On the multi-dimensional elephant random walk[END_REF]. One can observe that the special case p = 1/(2d) reduces back to the simple random walk on Z d . Therefore, the position of the elephant at time n + 1 is given by

(1.5) S n+1 = S n + X n+1
where X n+1 is the (n + 1)-th increment of the random walk. The ERW shows three differents regimes depending on the location of its memory parameter p with respect to the critical value p d lying between 1/(2d) and 3/4, (1.6)

p d = 2d + 1 4d .
A wide literature is now available on the ERW in dimension d = 1 where p d = 3/4. A strong law of large numbers and a central limit theorem for the position S n , properly normalized, were established in the diffusive regime p < 3/4 and the critical regime p = 3/4, see [START_REF] Baur | Elephant random walks and their connection to Pólya-type urns[END_REF], [START_REF] Coletti | Central limit theorem and related results for the elephant random walk[END_REF], [START_REF] Coletti | A strong invariance principle for the elephant random walk[END_REF], [START_REF] Schütz | Elephants can always remember: Exact long-range memory effects in a non-markovian random walk[END_REF] and the recent contributions [START_REF] Bercu | Hypergeometric identities arising from the elephant random walk[END_REF], [START_REF] Bertoin | Scaling exponents of step-reinforced random walks[END_REF], [START_REF] Businger | The shark random swim (Lévy flight with memory)[END_REF], [START_REF] Coletti | Asymptotic analysis of the elephant random walk[END_REF], [START_REF] Fan | Some limit theorems for the elephant random walk[END_REF], [START_REF] Miyazaki | Limit theorems for the 'laziest' minimal random walk model of elephant type[END_REF], [START_REF] Vázquez | On the almost sure central limit theorem for the elephant random walk[END_REF]. The superdiffusive regime p > 3/4 is much harder to handle. Bercu [START_REF] Bercu | A martingale approach for the elephant random walk[END_REF] proved that the limit of the position of the ERW is not Gaussian. Quite recently, Kubota and Takei [START_REF] Kubota | Gaussian fluctuation for superdiffusive elephant random walks[END_REF] showed that the fluctuation of the ERW around its limit in the superdiffusive regime is Gaussian. Finally, Bercu and Laulin in [START_REF] Bercu | On the multi-dimensional elephant random walk[END_REF] extended all the results of [START_REF] Bercu | A martingale approach for the elephant random walk[END_REF] to the multi-dimensional ERW where d ≥ 1.

Our strategy for proving asymptotic results for the center of mass of the elephant random walk (CMERW) is as follows. On the one hand, the behavior of position S n is closely related to that of the sequence (M n ) defined, for all n ≥ 0, by M n = a n S n with a 1 = 1 and, for all n ≥ 2,

(1.7)

a n = n-1 k=1 k k + a = Γ(a + 1)Γ(n) Γ(n + a)
where Γ stands for the Euler Gamma function and a is the fundamental parameter of the ERW defined by

(1.8) a = 2dp -1 2d -1 .
We assume throughout all the paper that a > -1 inasmuch as a = -1 only appears in the singular case where d = 1 and p = 0. It follows from the very definition of the ERW that at any time n ≥ 1,

(1.9) E [X n+1 |F n ] = pI d - 1 -p 2d -1 I d 1 n n k=1 X k = 1 n 2dp -1 2d -1 S n = a n S n a.s.
Hence, we obtain from (1.5) and (1.9) that for any n ≥ 1,

(1.10)

E [S n+1 |F n ] = 1 + a n S n a.s.
Therefore, we deduce from (1.7) and (1.10) that

(1.11) E [M n+1 |F n ] = a n+1 1 + a n S n = a n S n = M n a.s.
It means that (M n ) is a locally square-integrable martingale adapted to the filtration (F n ) where F n = σ(X 1 , . . . , X n ). It can be rewritten [4, page 7] in the additive form (1.12)

M n = n k=1 a k ε k
where ε 1 = S 1 and, for all n ≥ 2,

(1.13)

ε n = S n - a n-1 a n S n-1 = S n -1 + a n -1 S n-1 .
On the other hand, an analogue of equation (1.4) is given by

G n = 1 n n k=1 S k = 1 n n k=1 1 a k M k = 1 n n k=1 1 a k k =1 a ε = 1 n n k=1 a k ε k n =k 1 a , = 1 n n k=1 a k (b n -b k-1 )ε k (1.14)
where the sequence (b n ) is given by b 0 = 0 and, for all n ≥ 1,

(1.15) b n = n k=1 1 a k .
In the particular case of the simple random walk on Z d , one can notice that a = 0 so a n = 1/n and b n = n(n + 1)/2. Hereafter, denoting (1.16)

N n = n k=1 a k b k-1 ε k , it is straightforward to see that E [N n+1 |F n ] = N n a.s. since E [ε n+1 |F n ] = 0. Hence, (N n )
is also a locally square-integrable martingale adapted to the filtration (F n ). We deduce from (1.14) that (1.17)

G n = 1 n (b n M n -N n ).
Relation (1.17) allows us to establish the asymptotic behavior of the CMERW via an extensive use of the strong law of large numbers and the central limit theorem for multidimensional martingales [START_REF] Chaabane | Théorèmes limites avec poids pour les martingales vectorielles[END_REF], [START_REF] Duflo | Random iterative models[END_REF], [START_REF] Hall | Martingale limit theory and its application[END_REF], [START_REF] Touati | Sur la convergence en loi fonctionnelle de suites de semimartingales vers un mélange de mouvements browniens[END_REF].

The paper is organized as follows. The main results are given in Section 2. We first investigate the diffusive regime p < p d and we establish the almost sure convergence, the law of iterated logarithm and the quadratic strong law for the CMERW. The asymptotic normality of the CMERW, properly normalized, is also provided. Next, we prove similar results in the critical regime p = p d . Finally, we establish a strong limit theorem in the superdiffusive regime p > p d . Our martingale approach is described in Section 3 while all technical proofs are postponed to Appendices A, B and C.

Main results

2.1. The diffusive regime. Our first result deals with the strong law of large numbers for the CMERW in the diffusive regime where 0 ≤ p < p d . The following strong law for the CMERW will be deduced as a simple consequence of the strong law for (S n ). We recall that we assume a > -1 as a = -1 only occurs in the singular case where d = 1 and p = 0.

Theorem 2.1. We have the almost sure convergence

(2.1) lim n→∞ 1 n G n = 0 a.s.
More precisely, for any α > 1/2, The almost sure rates of convergence for CMERW are as follows.

(2.2) lim n→∞ 1 n α G n = 0 a.s.
Theorem 2.2. We have the quadratic strong law

(2.3) lim n→∞ 1 log n n k=1 1 k 2 G k G T k = 2 3(1 -2a)(2 -a)d I d a.s.
where I d stands for the identity matrix of order d. In particular,

(2.4) lim n→∞ 1 log n n k=1 G k 2 k 2 = 2 3(1 -2a)(2 -a)
a.s.

Moreover, we also have the upper-bound in the law of iterated logarithm

(2.5) lim sup n→∞ G n 2 2n log log n ≤ √ 3 + √ 1 -2a 2 3(a + 1) 2 (1 -2a)d a.s.
Remark 2.1. The law of iterated logarithm for G n involves the difference between b n M n and N n both satisfying a law of iterated logarithm with the same speed. It is not possible to make the addition of the two limits superior. Consequently, we were only able to obtain an upper-bound in (2.5). However, it is worth saying that for the simple random walk on Z d for which a = 0, Strassen's invariance principle [START_REF] Strassen | An invariance principle for the law of the iterated logarithm[END_REF] implies that

lim sup n→∞ G n 2 2n log log n = 1 3d a.s.
which is not very far from the upper-bound (1 + √ 3) 2 /(3d) given in (2.5).

We are now interested in the asymptotic normality of the CMERW.

Theorem 2.3. We have the joint asymptotic normality

(2.6) 1 √ n S n G n L -→ N 0, Γ d
where Γ d stands for the covariance matrix

Γ d = 1 (1 -2a)d 1 1 (2-a) 1 (2-a) 2 3(2-a) ⊗ I d .
In particular,

(2.7) 1 √ n G n L -→ N 0, 2 3(1 -2a)(2 -a)d I d .
Remark 2.2. One can observe from Theorem 3.3 in [START_REF] Bercu | On the multi-dimensional elephant random walk[END_REF] that the ratio of the asymptotic variances between the CMERW and the ERW is given by

R(a) = 2 3(2 -a) .
In the diffusive regime, this ratio lies between 2/9 and 4/9 and it is always smaller than 1, as one can see in Figure 1. Moreover, in the special case where the elephant moves in one of the 2d directions with the same probability p = 1/(2d) < p d , it follows from (1.8) that the fundamental parameter a = 0. Consequently, we deduce from (2.7) that

1 √ n G n L -→ N 0, 1 3d I d .
We find again the asymptotic normality (1.3) where the mean value µ = 0 and the covariance matrix Γ = 1 d I d . An alternative approach to prove the asymptotic normality (2.7) via a functional central limit theorem for the multi-dimensional ERW can be found in [START_REF] Bertenghi | Functional limit theorems for the multi-dimensional elephant random walk[END_REF]. One can observe from Theorem 1 in [1] that (2.7) also holds in the singular case a = -1 which only appears when d = 1 and p = 0. The almost sure rates of convergence for the CMERW are as follows.

Theorem 2.5. We have the quadratic strong law

(2.10) lim n→∞ 1 log log n n k=2 1 (k log k) 2 G k G T k = 4 9d I d a.s.
In particular,

(2.11) lim n→∞ 1 log log n n k=2 G k 2 (k log k) 2 = 4 9
a.s.

Moreover, we also have the law of iterated logarithm

(2.12) lim sup n→∞ G n 2 2n log n log log log n = 4 9d a.s.
Our next result concerns the asymptotic normality of the CMERW.

Theorem 2.6. We have the joint asymptotic normality

(2.13) 1 √ n S n G n L -→ N 0, Σ d
where Σ d stands for the covariance matrix

Σ d = 1 d 1 2 3 2 3 4 9 ⊗ I d .
In particular,

(2.14) 1 √ n log n G n L -→ N 0, 4 9d I d .
Remark 2.3. In the critical regime, the ratio of the asymptotic variances between the CMERW and the ERW is 4/9.

2.3. The superdiffusive regime. Finally, we focus our attention on the superdiffusive regime where p > p d . The almost sure convergence of (S n ), properly normalized by n a , yields the following strong limit theorem for the CMERW.

Theorem 2.7. We have the almost sure convergence

(2.15) lim n→∞ 1 n a G n = G a.s.
where the limiting value G is a non-degenerate random vector of R d . Moreover, we also have the mean square convergence

(2.16) lim n→∞ E 1 n a G n -G 2 = 0.
Remark 2.4. The expected value of G is zero and its covariance matrix is given by

E GG T = 1 d(a + 1) 2 (2a -1) 2 Γ(2a -1) I d .
The distribution of G is far from being known.

A multi-dimensional martingale approach

We already saw from (1.17) that the CMERW can be rewritten as

G n = 1 n (b n M n -N n ).
In order to investigate the asymptotic behavior of (G n ), we introduce the multi-dimensional martingale (M n ) defined by

(3.1) M n = M n N n
where (M n ) and (N n ) are the two locally square-integrable martingales given by (1.12) and (1.16). The main difficulty we face here is that the predictable quadratic variation of (M n ) and (N n ) increase to infinity with two different speeds. A matrix normalization is necessary to establish the asymptotic behavior of the CMERW. Let (V n ) be the sequence of positive definite diagonal matrices of order 2d given by

(3.2) V n = 1 n √ n b n 0 0 1 ⊗ I d
where A ⊗ B stands for the Kronecker product of the matrices A and B.

Lemma 3.1. The sequence (M n ) is a locally square-integrable martingale of R 2d . Its predictable quadratic variation M n satisfies in the diffusive regime where a < 1/2,

(3.3) lim n→∞ V n M n V T n = V a.s.
where the limiting matrix V is given by

(3.4) V = 1 d(a + 1) 2 1 1-2a 1 2-a 1 2-a 1 3 ⊗ I d .
Remark 3.1. Via the same lines as in the proof of Lemma 3.1, we find that in the critical regime a = 1/2, the sequence of normalization matrices (V n ) has to be replaced by

(3.5) W n = 1 n √ n log n b n 0 0 1 ⊗ I d .
Moreover, the limiting matrix in (3.3) must be changed by

(3.6) W = 4 9d 1 0 0 0 ⊗ I d .
Proof. The increments of the ERW are bounded by 1, that is for any time n ≥ 1, X n = 1. Hence, it follows from (1.5) that S n ≤ n and G n ≤ n which imply that M n ≤ na n and N n ≤ na n b n + n 2 . We already saw in Section 1 that (M n ) is a locally squareintegrable martingale. Denote ∆M n = M n -M n-1 , and similarly for other processes. It follows from (1.12), (1.13) and (1.16) that the predictable quadratic variation associated with (M n ) is the square matrix of order 2d given, for all n ≥ 1, by

(3.7) M n = n k=1 E ∆M k ∆N k ∆M k ∆N k T F k-1 = n k=1 a 2 k 1 b k-1 b k-1 b 2 k-1 ⊗ E ε k ε T k F k-1 .
Moreover, we deduce from formulas (A.7) and (B.3) in [START_REF] Bercu | On the multi-dimensional elephant random walk[END_REF] that for all n ≥ 1,

(3.8) E ε n+1 ε T n+1 |F n = 1 d I d + a 1 n Σ n - 1 d I d - a n 2 S n S T n a.s.
where Σ n is a random positive definite matrix of order d satisfying (3.9) lim n→∞

1 n Σ n = 1 d I d a.s.
Consequently, we obtain from (3.7) together with (3.8) that

(3.10) M n = 1 d n k=1 a 2 k 1 b k-1 b k-1 b 2 k-1 ⊗ I d + a n-1 k=1 a 2 k+1 1 b k b k b 2 k ⊗ 1 k Σ k - 1 d I d -ξ n
where

ξ n = a 2 n-1 k=1 a k+1 k 2 1 b k b k b 2 k ⊗ S k S T k .
According to Theorem 3.1 in [START_REF] Bercu | On the multi-dimensional elephant random walk[END_REF], the remainder ξ n plays a negligible role as

(3.11) lim n→∞ S n n = 0 a.s.
Hereafter, it is not hard to see that

V n n k=1 a 2 k 1 b k-1 b k-1 b 2 k-1 ⊗ I d V T n = 1 n 3 b 2 n n k=1 a 2 k b n n k=1 a 2 k b k-1 b n n k=1 a 2 k b k-1 n k=1 a 2 k b 2 k-1 ⊗ I d .
Furthermore, from a well-known property of the Euler Gamma function, we have

(3.12) lim n→∞ Γ(n + a) Γ(n)n a = 1.
Hence, we obtain from (1.7), (1.15) and (3.12) that (3.13) lim n→∞ n a a n = Γ(a + 1) and lim

n→∞ b n n a+1 = 1 Γ(a + 2)
.

Consequently, as soon as a < 1/2, we immediately find from (3.13) that

lim n→∞ b 2 n n 3 n k=1 a 2 k = 1 (1 -2a)(a + 1) 2 , lim n→∞ b n n 3 n k=1 a 2 k b k-1 = 1 (2 -a)(a + 1) 2 , lim n→∞ 1 n 3 n k=1 a 2 k b 2 k-1 = 1 3(a + 1) 2 .
Therefore, (3.14) lim

n→∞ V n n k=1 a 2 k 1 b k-1 b k-1 b 2 k-1 ⊗ I d V T n = 1 (a + 1) 2 1 1-2a 1 2-a 1 2-a 1 3 ⊗ I d .
Finally, it follows from the combinaison of (3.9), (3.10), (3.11) and (3.14) that (3.15) lim

n→∞ V n M n V T n = 1 d(a + 1) 2 1 1-2a 1 2-a 1 2-a 1 3 ⊗ I d . a.s.
which is exactly what we wanted to prove.

Proof of Theorem 2.2. Our goal is to check that all the hypotheses of Theorem A.2 are satisfied. Thanks to Lemma 3.1, hypothesis (H.1) holds almost surely. In order to verify that Lindeberg's condition (H.2) is satisfied, we have from (3.1) together with (1.12), (1.16) and V n given by (3.2) that for all 1 ≤ k ≤ n

V n ∆M k = a k n √ n b n ε k b k-1 ε k , which implies that (B.2) V n ∆M k 2 ≤ 2a 2 k b 2 n n 3 ε k 2 .
Consequently, we obtain that for all ε > 0,

n k=1 E V n ∆M k 2 I { Vn∆M k >ε} F k-1 ≤ 1 ε 2 n k=1 E V n ∆M k 4 F k-1 , ≤ 4b 4 n ε 2 n 6 n k=1 a 4 k E ε k 4 F k-1 , ≤ 4b 4 n ε 2 n 6 sup 1≤k≤n E ε k 4 F k-1 n k=1 a 4 k . (B.3)
However, it follows from the right-hand side of formula (4.11) in [START_REF] Bercu | On the multi-dimensional elephant random walk[END_REF] that

(B.4) sup 1≤k≤n E ε k 4 F k-1 ≤ 4 3 a.s.
Therefore, we infer from (B.3) that for all ε > 0, (B.5) Consequently, (B.5) together with (B.6) ensure that Lindeberg's condition (H.2) holds almost surely, that is for all ε > 0,

n k=1 E V n ∆M k 2 I { Vn∆M k >ε} F k-1 ≤ 16
(B.7) lim n→∞ n k=1 E V n ∆M k 2 I { Vn∆M k >ε} F k-1 = 0 a.s.
We will now check that condition (H.3) is satisfied in the special case β = 2, that is

(B.8) ∞ n=1 1 log(det V -1 n ) 2 2 E V n ∆M n 4 F n-1 < ∞ a.s. We have from (3.2) that (B.9) det V -1 n = n 3/2 b n d .
Hence, we find from (3.13) and (B.9) that

(B.10) lim n→∞ log(det V -1 n ) 2 log n = d(1 -2a).
Consequently, we can replace log(det V -1 n ) 2 by log n in (B.8). Hereafter, we obtain from (B.2) and (B.4) that Therefore, (B.11) together with (B.12) immediately lead to (B.8). We are now in a position to apply the quadratic strong law given by Theorem A.2. We have from (A.2) and (B.10) that (B. [START_REF] Fan | Some limit theorems for the elephant random walk[END_REF] lim

∞ n=2 1 (log n) 2 E V n ∆M n 4 F n-1 = O ∞ n=1 1 (log n) 2 a 4 n b 4 n n 6 E ε n 4 F n-1 , = O ∞ n=1 1 (log n) 2
n→∞ 1 log n n k=1 (det V k ) 2 -(det V k+1 ) 2 (det V k ) 2 V k M k M T k V T k = d(1 -2a)V a.s.
where the limiting matrix V is given by (3.4). However, it follows from (1.14), (3.1) and (3.2) that

(B.14) 1 √ n G n = v T V n M n where v = 1 -1 ⊗ I d .
Consequently, we deduce from (B.

(det V k ) 2 -(det V k+1 ) 2 (det V k ) 2 1 k G k G T k = d(1 -2a)v T V v a.s.
Furthermore, we obtain from (B.9) and (3.13) that

lim n→∞ n (det V n ) 2 -(det V n+1 ) 2 (det V n ) 2 = d(1 -2a).
Hence, (B.15) clearly leads to convergence (2.3), (B. [START_REF] Kabluchko | Convex hulls of random walks, hyperplane arrangements, and Weyl chambers[END_REF])

lim n→∞ 1 log n n k=1 1 k 2 G k G T k = v T V v = 2 3(1 -2a)(2 -a)d I d a.s.
By taking the trace on both sides of (B.16), we also obtain that (B.17) lim

n→∞ 1 log n n k=1 G k 2 k 2 = 2 3(1 -2a)(2 -a)
a.s.

Finally, we shall proceed to the proof of the upper-bound (2.5) in the law of iterated logarithm. Denote (B.18)

τ n = n k=1 a 2 k b 2 k-1 .
We already saw from (B.12) that a Consequently, we deduce from the law of iterated logarithm for martingales due to Stout [START_REF] Stout | Maximal inequalities and the law of the iterated logarithm[END_REF], see also Corollary 6.4.25 in [START_REF] Duflo | Random iterative models[END_REF], that (N n ) satisfies for any vector u ∈ R d , lim sup

1/2 1 n u, b n M n -N n ≤ lim sup n→∞ 1 2n log log n 1/2 1 n u, b n M n + lim sup n→∞ 1 2n log log n 1/2 1 n u, -N n ≤ lim sup n→∞ 1 2n log log n 1/2 1 n u, b n M n -lim inf n→∞ 1 2n log log n 1/2 1 n u, N n ≤ u √ d(a + 1) 1 √ 1 -2a + 1 √ 3 a.s. (B.23)
By the same token, we also find that for any vector

u of R d , lim inf n→∞ 1 2n log log n 1/2 u, G n = lim inf n→∞ 1 2n log log n 1/2 1 n u, b n M n -N n ≥ - u √ d(a + 1) 1 √ 1 -2a + 1 √ 3
S n √ n(log n) α = 0 a.s.
Hence, we obtain (2.9) using once again Toeplitz lemma.

Proof of Theorem 2.5. The proof of the quadratic strong law (2.10) is left to the reader as it follows essentially the same lines as that of (2.3). The only minor change is that the matrix V n has to be replaced by the matrix W n defined in (3.6). We shall now proceed to the proof of the law of iterated logarithm given by (2.12). On the one hand, it follows from (B.21) with a = 1/2 that for any vector u ∈ R d , lim sup where the limiting value L is a non-degenerate random vector of R d . Hence, (B.29) together with the Toeplitz lemma imply (2.15) where the limiting value

G = 1 a + 1 L.
Moreover, we have from (1.17) that

E 1 n a G n -G 2 = E 1 n a+1 (b n M n -N n ) -G 2 , ≤ 2E a n b n n a+1 S n -G 2 + 2E 1 n a+1 N n 2 .
On the one hand, we already saw from (3.13) that lim n→∞ a n b n n = 1 a + 1 .

Figure 1 .

 1 Figure 1. The 2-dimensional ERW in blue, the CMERW in black and the convex hull in red, for n = 10 6 steps and a diffusive memory parameter p = 1/2.

2. 2 .

 2 The critical regime. Hereafter, we investigate the critical regime where the memory parameter p = p d . Theorem 2.4. We have the almost sure convergence(2.8) lim n→∞ 1 √ n log n G n = 0 a.s.More precisely, for any α > 1n) α G n = 0 a.s.

Figure 2 .

 2 Figure 2. The 2-dimensional ERW in blue, the CMERW in black and the convex hull in red, for n = 10 6 steps and a critical memory parameter p = 5/8.

Figure 3 .

 3 Figure 3. The 2-dimensional ERW in blue, the CMERW in black and the convex hull in red, for n = 10 6 steps and a superdiffusive memory parameter p = 3/4.

b 4 n 3 ε 2 n 6

 436

  τ n is equivalent to n 3 /3(a + 1) 2 , (B.20) immediately lead to lim sup was already shown by formula (5.17) in[START_REF] Bercu | On the multi-dimensional elephant random walk[END_REF] that for any vector u ∈ R d , deduce from (1.17) and (3.13) together with (B.21) and (B.22) that for any vector u of R d ,

n 2 2nn 2 2n log n log log log n ≥ lim sup n→∞ u, G n 2 2n log n log log log n = 4 B. 3 .

 22243 hand, we obtain from the law of iterated logarithm for S n given in Theorem 3.5 of[START_REF] Bercu | On the multi-dimensional elephant random walk[END_REF] that for any vector u ∈ R d , lim sup n→∞ 1 2n log n log log log n clearly deduce from (B.27) that for any vector u ∈ R d , taking all rational points on the unit sphere S d-1 in R d , the bound in (B.28) holds simultaneously for all of them, which implies that lim sup n→∞ G log n log log log n ≤ sup u∈Q d ∩S d-1 lim sup n→∞ u, G n 2 2n log n log log log n = 4 9d a.s. In addition, for any single u ∈ S d-1 , we also obtain the reverse inequality lim sup n→∞ G to (2.12) which achieves the proof of Theorem 2.5. The superdiffusive regime. Proof of Theorem 2.7. It follows from Theorem 3.7 in [4] that (B.29) lim n→∞ 1 n a S n = L a.s.

  Finally, by taking all rational points on the unit sphere S d-1 in R d , the bound in (B.[START_REF] Vázquez | On the almost sure central limit theorem for the elephant random walk[END_REF] holds simultaneously for all of them, which implies that

	One can observe that the upper-bound in (B.25) is close to the optimal bound
				(vu) T V vu =	2 u 2 3(1 -2a)(2 -a)d	.
	lim sup n→∞	G n 2n log log n 2	≤ sup u∈Q d ∩S d-1 lim sup n→∞	u, G n 2n log log n 2	≤	√ 3(a + 1) 2 (1 -2a)d 3 + √ 2 1 -2a	a.s.
	completing the proof of Theorem 2.2.
	B.2. The critical regime.
	Proof of Theorem 2.4. We have from Theorem 3.4 in [4] that
	(B.26)			lim n→∞	√	1 n log n	S n = 0		a.s.
	Hence, (2.8) clearly follows from (B.26) together with the Toeplitz lemma. Moreover, we
	also have from Remark 3.3 in [4] that for any α > 1/2,
				lim n→∞
	(B.24)												a.s.
	Consequently, we obtain from (B.23) and (B.24) that for any vector u of R d ,
	(B.25)	lim sup n→∞	1 2n log log n	u, G n	2 ≤	u 2 d(a + 1) 2	√	1 1 -2a	+	1 √ 3	2	a.s.

Acknowledgments

The authors warmly thank the two anonymous referees for their helpful comments and constructive suggestions which helped to improve the paper substantially. In particular, the joint asymptotic normality for (S n , G n ) was suggested by an insightful comment from one of them.

Appendix A. Two non-standard results on martingales

The proofs of our main results rely on two non-standard central limit theorem and quadratic strong law for multi-dimensional martingales. A simplified version of Theorem 1 of Touati [START_REF] Touati | Sur la convergence en loi fonctionnelle de suites de semimartingales vers un mélange de mouvements browniens[END_REF] is as follows.

Theorem A.1. Let (M n ) be a locally square-integrable martingale of R δ adapted to a filtration (F n ), with predictable quadratic variation M n . Let (V n ) be a sequence of nonrandom square matrices of order δ such that V n decreases to 0 as n goes to infinity. Assume that there exists a symmetric and positive semi-definite matrix V such that

Moreover, assume that Lindeberg's condition is satisfied, that is for all ε > 0, (H.2)

where ∆M n = M n -M n-1 . Then, we have the asymptotic normality

The quadratic strong law requires more restrictive assumptions. The following result is a simplified version of Theorem 2.1 of Chaabane and Maaouia [START_REF] Chaabane | Théorèmes limites avec poids pour les martingales vectorielles[END_REF] where the normalization matrices (V n ) are diagonal.

Theorem A.2. Let (M n ) be a locally square-integrable martingale of R δ adapted to a filtration (F n ), with predictable quadratic variation M n . Let (V n ) be a sequence of nonrandom positive definite diagonal matrices of order δ such that its diagonal terms decrease to zero at polynomial rates. Assume that (H.1) and (H.2) hold almost surely. Moreover, suppose that there exists β ∈]1, 2] such that

Then, we have the quadratic strong law

Consequently, we obtain that

The asymptotic covariance matrix Γ d = U T V U where V is given by (3.4). It clearly leads to (2.6) as Proof of Theorem 2.6. The proof follows exactly the same lines as that of Theorem 2.3 replacing V n by W n . The details are left to the reader.