Antonio Pérez

Santiago Uceda

Eric Boëda
email: boeda.eric@gmail.com

Edwin Silva

Lucénida Carrión
email: lucenida.carrion@leyendas.gob.pe

Rolando Romero

Marcos Paulo Ramos

Pilar Babot
email: pilarbabot@csnat.unt.edu.ar

Cobbles, tools, and plants: Techno-functional variability within lithic industries of complex societies in Central Coast, Peru ( 1800-400 BP)

Keywords: Techno-functional approach, lithic industries in complex societies, Peruvian Central Coast, plant micro-remain analysis, Caesalpinia spinosa

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

Introduction

The Maranga Archaeological Complex (MAC) is a diverse set of monumental archeological sites located in the urban heart of Lima, Peru (Fig. 1, 2), which sits in the lower valley of the Rimac River. The MAC has a vast archaeological record, dating from the Early Intermediate Period (~ 1800-1400 BP) to the Late Horizon . Within the MAC, artifacts from complex societies in this region, representing cultures both previous and contemporary to the Inca state, have been found [START_REF] Carrión | Descripción general de Maranga[END_REF].

Despite the material wealth of this complex, the lithic industry has not been an object of much archaeological interest. This is, in part, because "archaeologists have generally overlooked lithic artifacts from Andean complex societies" (Downey, 2010, p. 79), due to the presence of other evidence, such as monumental architecture, textiles, ceramics, and metals, among others. In addition, these industries are generally considered expedient and unsophisticated, without further evolution over time, so they are judged to be of little diagnostic value in techno-cultural studies (SM 1: Archaeological and archaeobotanical background).

The use of technological analysis to study the variability and technical evolution of complex societies in the Central Andes is still quite underutilized, and where it has been applied the objects being studied have not been examined with any overlapping methodologies [START_REF] Bencic | Obsidian technology at the Wari site of Conchopata in Ayacucho, Peru[END_REF][START_REF] Bencic | Lithic production, use and deposition at the Wari site of Conchopata in Ayacucho, Peru[END_REF][START_REF] Bencic | Industrias líticas de Huari y Tiwanaku[END_REF][START_REF] Bencic | Back to the Grind: Understanding lithic technology at Iwawi, Bolivia[END_REF][START_REF] Burger | Late Paracas obsidian tools from Animas Altas, Peru[END_REF][START_REF] Costin | The impact of Inka conquest on local technology in the Upper Mantaro Valley, Peru[END_REF][START_REF] Cruz | Especialización artesanal en la producción de las herramientas líticas del valle de Nepeña, Provincia del Santa-Ancash[END_REF][START_REF] Deleonardis | Paracas Cultural Contexts: New Evidence from the West Bank of Callango[END_REF][START_REF] Downey | Working with Expedient Lithic Technologies in the Northern Highlands[END_REF][START_REF] Downey | Catequil's lithics: Stone tools from an Andean Complex Society[END_REF][START_REF] Fortin | Wari lithic Networks: An examination of stone tool use at Cerro Baúl and Cerro Mejía[END_REF][START_REF] Fortin | From Bedrock to Bifaces: An Examination of Wari Lithic Technology within the Moquegua Valley of Southern Peru[END_REF][START_REF] Gero | Genderlithics: Women's Roles in Stone Tool Production[END_REF][START_REF] Gero | Assessing social information in material objects: How well do lithics measure up?[END_REF]Gero, , 1983aGero, , 1983b;;[START_REF] Giesso | La producción de instrumental lítico en Tiwanaku: el impacto del surgimiento y expansión del estado en las unidades domésticas locales[END_REF]Giesso, , 2003aGiesso, , 2003b[START_REF] Giesso | Stone Tool Production in the Tiwanaku Heartland.The Impact of State Emergence and Expansion on Local Households[END_REF][START_REF] Hu | War or peace? Assessing the rise of the Tiwanaku state through projectile point analysis[END_REF][START_REF] Lavallée | Industrias líticas del Periodo Huaraz, procedentes Chavín de Huantar[END_REF][START_REF] Lavallée | Curacazgo prehispánico de los Andes Centrales[END_REF][START_REF] León | Lascas de obsidiana de tipo kombewa de Waru Kayan, Paracas[END_REF][START_REF] Read | A Method for Taxonomic Typology Construction and an Example: Utilized Flakes[END_REF][START_REF] Russell | The Impact of Inka Policy on the Domestic Economy of the Wanka, Peru: Stone Tool Production and Use[END_REF][START_REF] Seddon | Lithic Artifacts[END_REF][START_REF] Silva | Las industrias líticas tardías alrededor del nevado Huandoy, Ancash, Perú[END_REF][START_REF] Silva | Los artefactos líticos del Parque de las Leyendas: del Horizonte Medio al Horizonte Tardío[END_REF][START_REF] Stone | The Socio-Economic Implications of Lithic Evidence from Huari, Peru[END_REF][START_REF] Vining | Social pluralism and lithic economy at Cerro Baúl, Peru. BAR[END_REF][START_REF] Yataco | Cadena Operativa y Tecnología Lítica Arqueológica de la Zona de Alto Chicama, Distrito de Quiruvilca[END_REF]. Although the objectives of these investigations are diverse, the vast majority have focused on the study of artisan specialization. Meanwhile, typological studies are scarce, with the most significant efforts having been published in the 1960s and 1970s [START_REF] Brown | On stone implements from Northwest Peru[END_REF][START_REF] Donnan | The Utilization of Flakes for Cleaning Fish[END_REF][START_REF] Hyslop | An Archaeology Investigation of the Lupaca Kingdom and Its Origins[END_REF][START_REF] Rivera | Análisis tipológico de las puntas de proyectil Wari[END_REF][START_REF] Taschini | L'industria litica Pre-incaica di Cajamar-quilla (Peru)[END_REF]. In both approaches there is a lack of integration between aspects of production and the functioning/function of the tools.

Within the framework of this problem, archaeobotanical studies at the macroscopic level have demonstrated a clear relationship between artisanal production and the processing of different plant resources since the Early Intermediate Period in the MAC [START_REF] Bellido | Una perspectiva arqueobotánica en los períodos tardíos[END_REF]. However, the role of knapped lithic production in this context remains unknown. In this sense, our hypothesis is that there is a close functional relationship between knapped lithic tools and plant use, and that, therefore, lithic production is conceived from the selection stage predominantly for the processing of plant resources.

Therefore, our dual approach combining technological and archaeobotanical analyses is the first such application to an Andean lithic industry, and is presented as a useful technique to understand cultural change based on the internal structure of these lithic tools.

The Maranga Archaeological Complex (MAC)

The sites that provided the complex lithic objects of this study are part of the MAC, located in the territory belonging to the theme park Parque de las Leyendas -Felipe Benavides Barreda (PATPAL). With a total area of approximately 3 km 2 , the PATPAL is located on the Central Coast of Peru in the lower valley of the Rimac River on its southern bank, 3.5 km from the Pacific Ocean (Fig. 1). This area is dominated by a relatively flat relief, varying between 48 and 68 m.a.s.l., and is ecologically defined by the Desert-Subtropical Formation (d-ST) (ONERN, 1986), generally characterized by a dispersed xerophytic vegetation. The nine MAC sites considered for this investigation were: the seven "public buildings" of Huaca El Rosal, Huaca Middendorf, Huaca 33, Huaca 34, Huaca 43, Huaca San Miguel, and Huaca 58B, and the two defensive walls: Wall 46A and Wall 55E (Fig. 2). These are the main sites that, at present, have provided lithic artifacts from within the complex [START_REF] Carrión | Descripción general de Maranga[END_REF].

Materials and methods

Sample composition

Nine lithic sets were selected, each associated with a different archaeological site, constituting a population of 1257 artifacts that included knapped material and macro-tools (i.e. tools worked on large supports) (Table 1). This paper presents only the analytical results concerning knapped artifacts (n=1108). On the other hand, it should be noted that these artifacts come from both primary and secondary contexts. In the case of MAC, secondary contexts correspond to architectural fillings. For the functional analysis, only 39 artifacts recovered in primary contexts, identified as activity areas where lithic and ceramic material were found in association, were selected. Because the materials coming from the architectural fillings could predate the building of the structures within which they were incorporated, and because their origins are unknown, they were not considered for technological or functional analysis. The relative chronology of both primary-and secondary-context lithic assemblages was assigned by taking into account the stylistic information provided by the associated ceramics, as well as the construction moments for primary context buildings.

Historically, the main archaeological purpose in studying MAC archaeological sites was the reconstruction of architectural stages and general chronology of those monumental sites [START_REF] Carrión | Descripción general de Maranga[END_REF]. Activity areas inside the buildings remain unstudied, this work being one of the first to discuss mobile items. As such, we have proposed general characterizations of the activity contexts of provenance of lithic artifacts as can be seen in Tables 1 and2.

Technological analysis

As part of a broad technological methodology [START_REF] Pérez | Étude techno-fonctionnelle des artefacts lithiques tardifs[END_REF], the total artifactual population (n=1108) was subject to macroscopic petrographic, taphonomic, technological production, and techno-functional analyses.

The work began with the macroscopic identification of the large rock families present in each collection. Then, the microscopic analysis of thin section allowed us to determine the specific types of raw material. As for the taphonomic analysis, a detailed observation of the surface of the lithic artifacts was performed, either by naked eye or via binocular microscope. Some of the variables described in Levi Sala (1986), [START_REF] Borrazo | Hacia una tafonomía lítica[END_REF] and [START_REF] Weitzel | Cuentan los fragmentos. Clasificación y causas de fractura de artefactos formatizados por talla[END_REF] were taken into account for the recording of the taphonomic observations.

Subsequently, the technological analysis focused on the determination of the initialization and production methods used for the exploitation of the cores and flakes, using a diacritical analysis of the pieces and following a structural classification of, affordance, debitage, shaping, and confection (also called "retouch") systems [START_REF] Boëda | Techno-logique and Technologie[END_REF]Boëda and Ramos, 2017). These four terms refer to different modes of production. Affordance refers to the selection of techno-functional criteria naturally present in the initial block and which will remain in the finished product since they participate in the functionalization of the piece. Debitage denotes the fractioning of a volume of matter through a panoply of methods into different units of techno-functional criteria and volumes. Shaping refers to the configuration that is invested from the beginning in order to obtain the techno-functional criteria missing in the initial volume. Confection refers to the operation of transformation of supports into tools [START_REF] Boëda | Identification de chaînes opératoires lithiques du Paléolithique ancien et moyen[END_REF][START_REF] Boëda | Techno-logique and Technologie[END_REF]Boëda & Ramos, 2017).

The tools (n = 497) were subject to techno-functional analysis [START_REF] Boëda | Technogenèse de systèmes de production lithique au Paléolithique inférieur et moyen en Europe occidentale et au Proche-Orient[END_REF][START_REF] Boëda | Approche de la variabilité des systèmes de production lithique des industries du Paléolithique Inférieur et Moyen : chronique d'une variabilité attendue[END_REF][START_REF] Boëda | Techno-logique and Technologie[END_REF][START_REF] Lepot | Approche techno-fonctionnelle de l'outillage lithique moustérien : essai de classification des parties actives en termes d'efficacité technique[END_REF] in order to determine the techno-functional intentions that structure these artifacts. This analysis should not be understood as an integration between technological and traceological observations. The term "functional" is used in the sense of the artifact's structural organization according to its function. Thus, this analysis focused, first, on determining the techno-functional consequences of each technical operation, and second, on determining the schèmes opératoires applied to produce such techno-functional consequences.

Techno-functional consequences refer to the effects produced by the mentioned technical operations. The implications are technical at the level of structural configuration (surface types evidenced by each negative of removal), and functional, at the level of configuration of the transformative part (i.e. cutting edge, cutting wedge, cutting wedge angle, rake and flank face).

After observing these elements, we proceeded to identify hypothetical Techno-Functional Units (UTF): the transformative techno-functional unit (UTFt) or the active portion of the tool plus the edge; the transmitting techno-functional unit (UTFtr) or the intermediate portion transmitting the force from the handle to the UTFt; and the prehensive techno-functional unit (UTFp), i.e. the held portion. Generally, UTFtr and UTFp are difficult to separate, so here they were identified as UTFp. According to the UTF configuration of each tool, the artifacts were grouped into techno-types. A technotype is defined by pieces with the same number and characteristics of UTFt, and the same position and number of UTFp. Within these techno-types, there are sub-types which correspond to internal variations. In the objects studied, this variation corresponds to the delineation of the cutting edges. The determination of techno-types allows for the identification of groups of artifacts that share the same structural composition and that potentially fulfilled the same functions and functioning. Using this knowledge, the difference in proportions of these techno-types between the studied sites combined with the results of micro-remain analysis contributes to a functional perception of the Maranga toolkits.

Frequency analyses were carried out and a sampling simulation test was applied to evaluate the similarity or difference in the representation of the different techno-types in the studied sites and in different time periods.

Micro-remain analysis

In order to obtain direct functional information, one piece was selected from each techno-type identified in each of the sites, establishing a sample of 39 artifacts for micro-remain analysis (Table 2). Due to the greater variety of techno-types in the assemblages corresponding to the Early Intermediate Period, this era has a relatively greater representation than the other periods of occupation. All of these artifacts come from primary contexts, associated with walls and ceramic fragmentation. The analysis was carried out according to the "multiple analysis of microremains" [START_REF] Coil | Laboratory goals and considerations for multiple microfossil extraction in archaeology[END_REF], which considers the recovery and study of the total set of micro-remains rather than a specific type. This study focused especially on starch grains (SG); however, silicophytoliths (Si), calcium phytoliths (PCa), cellulose rings (C), microcarbons (Mic), diatoms (Di), spherulites (Sph), and silicified tissues (Pte) were also considered in terms of presence and absence.

In the case of our artifact sample, we used a "low impact protocol" composed of two major stages: (1) preparation of a herbarium and histological reference collection from modern plant material [START_REF] Babot | Tres técnicas de muestreo complementarias en colecciones de referencia de micropartículas[END_REF] and ( 2) recovery, analysis, and identification of the micro-remains preserved in the UTFt of the artefacts [START_REF] Babot | La cocina, el taller y el ritual. Explorando las trayectorias del procesamiento vegetal en el Noroeste Argentino[END_REF][START_REF] Babot | Starch grain damage as an indicator of food processing[END_REF][START_REF] Henry | Changes in starch grain morphologies from cooking[END_REF][START_REF] Icsn | The International Code for Starch Nomenclature[END_REF][START_REF] Madella | International Code for Phytolith Nomenclature 1.0[END_REF]. At the end of these two stages, the presence of the identified taxa was evaluated, depending on each archaeological period and in relation to the instrumental variability. This evaluation sought to determine the "ubiquity" [START_REF] Popper | Selecting quantitative measurements in paleoethnobotany[END_REF] of these taxa, or the number of times they appeared in the samples (SM 2: Methodology used in the preparation of reference collection and analysis of micro-remain in archaeological residues).

Results

Results of technological and techno-functional analysis 4.1.1. Procurement, management and taphonomy of raw materials

The microscopic analysis of thin sections allowed us to define a particular type of andesite, microporphyritic andesite, as the main raw material for artifactual manufacturing in the MAC (SM 3: Macro and micro petromineralogical analysis). Microporphyritic andesite is ubiquitous in the nine sets analyzed, in the form of ellipsoid cobbles between 50 and 150 mm long, 40 and 120 mm wide, and between 15 and 80 mm thick. Some experimental tests with this type of rock have allowed us to corroborate its adequate quality for knapping, especially due to its aphanitic texture and compact structure. On a relative quality scale of the rock used for knapping in MAC, microporphyritic andesite is the best followed by medium and coarse grain andesites.

At the petromineralogical level, the knapped lithic assemblage of Huaca El Rosal is made up exclusively of local microporphyritic andesite. The management of the andesite corresponds to a predominant selection of oblate ellipsoid cobbles, without fracture planes and with polished neocortex. It was possible to discern a technical adequacy between cores and flakes. Technical adequacy refers to the relationship between the techno-types of removals (inferred through diacritical analysis of the cores) and the flakes actually found at the site. At the taphonomic level, slightly abraded pieces predominate in Huaca El Rosal (n = 113, 59%). Likewise, the complete pieces predominate (n = 117, 93%), while among the fragmented pieces (n = 14, 7%) snap fractures are the most frequent, which are probably related to intentional direct percussion.

The data is not very different in the assemblage of Huaca Middendorf, which is also composed entirely of microporphyritic andesite. The variability here comes from a partial technical adequacy between cores and flakes. At the taphonomic level, the slightly abraded pieces also predominate (n = 25, 64%) and only two pieces have a snap fracture.

On the other hand, in Huaca 33, the variability is expressed in the exploited raw materials. On this site, microporphidic andesite (n = 123, 83%), tonalite (n = 13, 3%) and fine-grained quartzite (n = 12, 2%) were recorded. These rock types are found to form the same types of natural alluvial and coastal deposits as microporphyritic andesite. In general, the same proportions of microporphyritic andesite occur in Huaca 34 (n = 116, 69%), Huaca San Miguel (n = 159, 63%) and Huaca 58B (n = 111, 65%), increasing in Wall 46A (n = 30, 94%). On the contrary, at these last three sites, cobbles seem to have a unique alluvial origin, due to the presence of a slight smoothness on their surfaces, typical of cobbles that can currently be recovered on the river banks of the Rimac.

Similar proportions of microporphyritic andesite were recorded at Huaca 43 (n = 37, 75%) and Wall 55E (n = 25, 94%), with the only exception of an exploitation of microporphyritic andesite cores and diorite in the first, without counting on knapped tools on the last.

Debitage: Core initialization and production methods

The 350 cores analyzed allowed us to identify four initialization methods and three production methods in the nine sites studied (Fig. 3).

The first initialization method (Mi1) corresponds to the opening of a striking platform surface over the selected volume.

The second method (Mi2) consists in the selection of the lower face of a cobble cortical flake as a striking platform surface.

The third method (Mi3) corresponds exclusively to the selection of a convex striking platform, a flat flake-release surface, and a thickness (generally between 16 and 25 mm) that allowed for bipolar production.

The fourth method (Mi4) consists of the peripheral configuration of the flakerelease surface and the establishment of an initial striking platform through unipolar removals.

After the initialization phase, the schèmes opératoires are organized into relatively short series of removals, following any of three production methods: unipolar, bipolar, and alternate (Fig. 3).

Mp1 -Unipolar debitage. Linear and non-linear chronology. The removals are made into series in each plane of the volume, successively from left to right or viceversa, and in an intercropped way, following the guide rib created by the last negative and the adjacent surface. Four variants have been identified: Mp1a, Mp1b, Mp1c, and Mp1d. The technique used is always direct hard percussion. Mp2 -Bipolar debitage, produces simple or split type removals. The technique used is always direct hard percussion. Mp3 -Alternate debitage, produces alternating removals. The technique used is always direct hard percussion.

Debitage systems

The technical correspondence between the initialization and production methods allowed us to recognize the debitage systems used in the MAC: C1 (Fig. 4 and5), F3 (Fig. 6 and7) and D1 (Fig. 8 and9).

In Huaca El Rosal, C1 (on flakes and cobbles) and F3 (bipolar) systems were identified. C1 was identified across three different production methods (Mp1a-Mp1c). Likewise, two techno-types of flakes were recognized, corresponding to cortical and semi-cortical flakes. In Huaca Middendorf, the only recognized debitage system is C1, with two production methods (Mp1a and Mp1d). A single flake techno-type is present, which corresponds to 7 cortical flakes.

The production methods are diversified in Huaca 33, as C1, D1 (alternating production of quadrangular flakes), and F3 are recorded. C1 includes two production methods (Mp1a and Mp1c). Eight flakes techno-types are present, partially linked to the production methods.

The Huaca 58B site shares the same production characteristics as Huaca 33, with the exception of a smaller number of flake techno-types (n = 3). Huaca 34 has the same criteria, although here the flakes do not correspond to the first phases of exploitation of the cores, but rather to a full debitage.

In Huaca 43, the debitage systems are D1 and F3, with 4 techno-types of flakes corresponding to the initial and intermediate stages of production. In Wall 46A, only the C1 system is present, with two production methods (Mp1a and Mp1c). The flakes present correspond to retouch operations and not to the production stage. Similar characteristics are presented in Huaca San Miguel, where production was carried out exclusively using the C1 system, with a single production method (Mp1a).

However, it should be noted that at all sites in the MAC there is a correlation between the size of the cobbles selected and the debitage system used. For systems C1 (on flakes) and D1, cobles and flakes between 55 to 65 mm long, 50 to 70 mm wide, and 25 to 30 mm thick were selected. For C1 system (on cobbles), cobbles between 100 to 120 mm long, 140 to 150 mm wide, and 70 to 80 mm thick were used. Finally, in the case of the F3 system, smaller cobles were used, between 35 to 55 mm long, 30 to 40 mm wide and 15 to 25 mm thick.

Regarding the production objectives of the recognized systems, quadrangular flakes of flat-convex section predominate, between 30 to 80 mm long, 25 to 120 mm width, and 5 to 30 mm thick. These flakes do not present any evidence of shaping or confection. These flakes did not undergo any further configuration into potentially useful tools.

Affordance, Shaping and tool Confection

Prior to the shaping, cobble blanks of microporphyritic andesite with ellipsoid section were selected, generally between 60 to 80 mm long, 60 to 90 mm width, and between 30 to 50 mm thick. Only techno-type 3 (ut infra) is outside these size ranges, since smaller pebbles were selected for its preparation, with between 30 to 40 mm long, 20 to 25 mm width, and between 7 to 15 mm thick.

The techno-functional criteria of the future tools come from this selection time. These criteria reveal an almost completely concentrated affordance in the UTFp.

In general, shaping schèmes involved two or three partial removals on both sides of the cobble, in order to configure an asymmetric cutting dihedral composed of a flat bottom surface and a convex top surface. This was followed by a confection stage corresponding to the unifacial retouching of these dihedrals, forming abrupt angles (between 70° and 80°).

In all the tools analyzed (n = 497), 528 UTFt were identified, with denticulate, convex, concave, pointed, and rectilinear delineation. Among these shapes, denticulate delineation was predominant (n = 291, 58.55%). The UTFp identified (n = 502) have a great technical homogeneity, mostly corresponding to the cortical part of the worked cobbles, with an ellipsoid section composed of two rounded edges in a position opposite to the UTFt. The edges of this ellipsoid section were parallel, divergent, or convergent, although divergent edges predominated. In some cases (n = 102, 20.31%), the UTFp were subject to configuration, always through short removals by unifacial shaping.

By grouping the observed UTFt and UTFp, eight techno-types were determined (Fig. 10), of which the first six have only one UTFt, while the last two (techno-types 7 and 8), have two or more UTFt. It should also be noted that techno-types 7 and 8 were configured on thick flat-convex flakes using the C1 debitage system.

After the techno-type frequency was considered across the studied sites (SM 4: Frequency of techno-types by archaeological site), a predominance of techno-types 1 to 4 became evident, while techno-types 5 to 8 are present in low numbers. In addition, no techno-type was present across all sites, although techno-type 4 occurs at all sites with the exception of Wall 55E. In the stage of data exploration, we obtained 497 records referring to the sites and techno-types present in them. This classification was found with a non-normal behavior. Table 3 shows that this classification does not meet the assumption of the minimum expected frequencies of 5. Therefore, in order to conclude on the relationship between sites and techno-types, the bootstrap resampling method was conducted on a simulation based on 5000 samples (Table 4). The results reaffirmed the existing relationship between sites and techno-types, as observed with an association based on a Cramér's V value of 0.667. With these results, Fig. 11 shows a correspondence analysis made from the four sites with the most pieces. Huaca San Miguel has the greatest presence in relation with techno-type 2, Huaca 34 with techno-type 4, while Huaca el Rosal and the Huaca 58B are associated with techno-type 1. Amongst the identified techno-types, techno-type 8 is represented by a single piece (Fig. 12), recovered in Huaca El Rosal. The diacritical analysis of the piece allowed us to recognize that its production involved organized removals to shape the lateral and central surfaces of the flake support from an unmodified flat surface. The confection stage involved the modification of four isolated areas in order to establish four different UTFt. It is a tool matrix. Each of the UTFt presents evidence of reshaping and resharpening, as shown by large reflected removals.

As an example of the techno-functional analysis carried out in all the pieces, Fig. 13 is presented, where there is an instrumental coexistence between the three main principles of affordance, debitage via bipolar production, and shaping.

Results of the plant micro-remain analysis

In the 39 artifacts analyzed, five morphotypes of starch grains, two morphotypes of silica phytoliths, and one related to silicified plant tissues were identified. The total distribution of plant micro-remains and the corresponding taxonomic assignation is shown in Table 4 (SM 5: Description of morphotypes). 491 -* Percentage is calculated as the proportion of the artifacts analized that have plant micro-remains of the taxa. Ubiquity is calculated by dividing the number of artifacts in which the taxon is recorded by the total number of artifacts in the sample group, i.e. n=39.

Content of micro-remains at the assemblage level

The microfossil fertility was established as a variable character in the different artifact analyzed, ranging from less than 5 to more than 25 plant micro-remains counted per artifact (SM 6: Contents of plant micro-remains by techno-type in each assemblage analyzed). From those, diagnostic microfossils allowed for the identification of four plant taxa in the artifactual residues. Figure 14 shows the correspondence relationship between techno-types and taxa (SM 7: Taxa frequency by techno-type).

Concerning Huaca El Rosal, a greater amount of SG was observed, although the amount of Sip is equally important. Specifically, the SG of Caesalpinia spinosa was predominantly found in techno-types 1 and 2, the SG of Zea mays was found in low proportion in techno-types 2 and 5, and the SG of Cucurbita sp. have been recorded in significant quantities in techno-types 3 and 4. With respect to Sip, only Zea mays could be identified in techno-type 5. Sph was not present in any analyzed artifact, so fecal contamination is ruled out. The results of the plant micro-remains analysis of the UTFt identified from the techno-type 8 are presented in Fig. 15. Meanwhile, the assemblage from Huaca Middendorf has a greater amount of Sip, although the amount of SG is significant. Sph was not present in any analyzed artifact. In this assemblage, the SG of Caesalpinia spinosa predominate in the technotypes 1, 2, and 3. The SG of Cucurbita sp. are found in techno-type 4. On the other hand, the Sip and Ti of Zea mays, are present also in techno-type 4.

The assemblage from Huaca 33 has a greater amount of SG, followed by Sip. In this assemblage, the SG of Caesalpinia spinosa are predominant, and they are present in all techno-types except for 1 and 4. Only in two techno-types (2 and 3), Zea mays Sips are associated with the SG of Caesalpinia spinosa. The example is techno-type 7 (Fig. 16). The Huaca 58B assemblage has a predominance of SG, followed far behind by the Ti, Sip, and CR. In this assemblage, the SG of Caesalpinia spinosa are present in techno-types 1, 2, and 3. In Huaca 34, the distribution is similar to Huaca 58B, with the exception that only techno-type 1 has SG of Caesalpinia spinosa, in addition to Sip of Poaceae. On the other hand, techno-type 3 registers SG of Prosopis sp. In Huaca 43, there is a greater amount of Sip. It was only possible to identify SG of Caesapinia spinosa in techno-type 4 (Fig. 20).

On the other hand, the Wall 46A assemblage, represented only by techno-type 4, has 14 SG, all probably belonging to modified (fermented) Zea mays (Fig. 17). Likewise, the Wall 55E assemblage has a strong proportion of Sip (n = 31), all assignable to Zea mays, registered in both techno-type 1 and 2 (Fig. 18). Finally, the assemblage corresponding to Huaca San Miguel has a greater number of Ti (n = 17), followed by Sip (n = 12), GA (n = 10), Esf (n = 11) and CR (n = 3). Di and Mic are not present in any analyzed artifact. The SG of Caesalpinia spinosa and the Sip of Poaceae predominate.

The percentage and ubiquity analysis establishes a greater presence in the case of Caesalpinia spinosa (33% and 0.33, respectivelly), followed by Zea mays (17.9%, 0.18). Although Caesalpinia spinosa occurs in all the techno-types analyzed, its presence predominates in the techno-types 1 and 3, with different UTFt.

Discussion and Conclusions

From a diachronic perspective, it can be noted that during the earliest periods (Early Intermediate Period and initial Middle Horizon), Maranga artisans selected ellipsoid cobbles of very fine-grained andesite (microporphyritic andesite) from the alluvial deposits around the Rimac River and from the nearby Pacific coast. However, towards the final periods of occupation (Late Intermediate Period and Late Horizon), there is a diversification of raw materials, and a change in the use of sources.

The natural presentation of the raw material offers the necessary technical criteria to materialize the techno-functional intentions. In addition, the selection operated, in all cases, towards the collection of hard and solid rocks, among which microporphyritic andesite stands out for its conchoidal fracture and excellent knapping quality.

The industries analyzed are governed in a very marked way by the conceptions of affordance (concentrated in the UTFp), debitage (production of flakes of flat-convex section) and shaping (confection of UTFt). These assemblages are a clear example of the phenomenon of false homothety, that is, an apparent homogeneity of UTFp forms and volumes (Boëda, 2013, p. 92). The utility of quadrangular flakes produced by the identified debitage systems is still unclear, with the exception of the use of flat-convex and thick cortical flakes from the C1 system for techno-types 7 and 8.

With respect to the debitage systems used, it can be seen that systems C1 (unipolar), D1 (alternating production of quadrangular flakes) and F3 (bipolar) are present in all periods of MAC cultural sequence, and in almost all of the sites. Possibly, the absence of systems D1 and F3 in Walls 46A and 55E can be explained by their particular functionality as defensive structures, rather than putative processing areas. Given a site's general functionality, we might expect that artifacts recovered from these sites are related to that purpose or perhaps with tasks occurred during its construction. However, no data allows us to conclude this. Moreover, the strong presence of D1-type cores in sites such as Huaca 34 and Huaca 43, allows us to state that this concept of production has its peak in the Late Intermediate Period (~1050-470 BP).

Regarding the techno-functional relationship without considering the archaeological periods, the statistical analyzes performed allow us to preliminarily indicate that in Maranga, techno-types 1 and 3 are more related to Caesalpinia spinosa processing, techno-type 2 to Prosopis sp., techno-types 1, 2, 4 and 6 to Zea mays L., and techno-types 3 and 4 to Cucurbita sp. Taking into account the diachronic dimension, during the Early Intermediate Period and the beginning of the Middle Horizon, the use of seven techno-types for the processing of a single resource is recorded, Caesalpinia spinosa (commonly called the tara plant): techno-types 1, 2, 3, 5, 6, 7, and 8. In the Late Intermediate Period, techno-type 1 was still used, but a new one was also used -technotype 4. In the Late Horizon, only techno-types 2, 5, 6, 7 and 8 were used to process this plant resource. The tara is an ornamental tree or shrub that grows wild normally between 800 and 2800 meters above sea level and also does so under cultivation (De la [START_REF] De La Cruz | Aprovechamiento integral y racional de la tara (Caesalpinia spinosa-Caesalpinia tinctoria)[END_REF]). In the cases studied, the tara record corresponds to the processing of pods and/or seeds where dehydrated, minimally birefringent, and fractured starches indicate a crushed or ground process [START_REF] Babot | Starch grain damage as an indicator of food processing[END_REF]. This becomes relevant because it is related to the two useful parts of the plant.

Modern uses of crushed or pulverized tara include integration into food processing chains, traditional medicines, and artisanal products [START_REF] Cordero | Respuesta ecofisiológica de Caesalpinia spinosa (Mol.) Kuntze a condicionantes abióticos, bióticos y de manejo, como referente para la restauración y conservación del bosque de nievlas de Atiquipa (Perú)[END_REF][START_REF] Melo | Efecto de la madurez en los componentes de valor commercial (tanitos y goma) de tara Caesalpinia spinosa (Molina) Kuntze[END_REF]. Normally, the pods are picked by hand to aid the drying process before they are transformed. Furthermore, tara gum is a yellowish white powder with no smell and no taste, fully soluble in hot water and partially soluble in cold water. This gum is composed of polysaccharides, mainly high molecular weight galactomannans, which is an additive or thickener and food stabilizer in the modern industry. It is obtained by reducing the seeds to flours. The presence of tannins in the pods has been exploited at least since colonial times in the elaboration of a black dye. They are also used in leather tanning. The artisanal process in both cases involves crushing the pods. Another component of the fruit is gallic acid, an active antimicrobial substance used in traditional medicine (De la [START_REF] De La Cruz | Aprovechamiento integral y racional de la tara (Caesalpinia spinosa-Caesalpinia tinctoria)[END_REF]. The infusion of mature pods is also used to heal wounds and reduce fever. Due to both the binding capacity of tara gum and the presence of tannins, these fruits constitute modern additives in the paint industry, which suggests a potential similar use in pre-Hispanic occupations.

There is also a (albeit smaller) recurrence of Prosopis sp. and Zea mays amongst the residues, corresponding at least in one case (Wall 46A) to the presence of starch. This starch shows damage compatible with the fermentation of the grain, in particular the occurrence of radial fissures and the presence of grains with a remaining annular thickening [START_REF] Gardiman | Vereda III: uma análise paleoetnobotânica (Tesis de graduación en Arqueología)[END_REF], a situation that indicates the maize chicha spill (Table 5). The evidence of Prosopis sp. could come from the processing of the pod for a variety of possible uses, with starches of this genre showing damage related to dehydration and mashing, such as cracks, loss of relief and shine, and structure damage evidenced in the partial loss of birefringence [START_REF] Babot | Starch grain damage as an indicator of food processing[END_REF]. On the other hand, the presence of graminoid silica phytoliths (Poaceae) is cautiously assignable here to the contribution of the natural vegetation cover at the site, both contemporary and ancient, mainly Stipa sp.

Of the microscopically identified families, Fabaceae, Cucurbitaceae, and Poaceae were also registered as plant macro-remain in the MAC [START_REF] Bellido | Una perspectiva arqueobotánica en los períodos tardíos[END_REF]. Both the macro-and the micro-remains from the residues in the lithic artifacts indicate that the plants were being processed, consumed, used, and discarded inside these monumental buildings, further indicating the activities that were carried out in them during the construction stages themselves or after them. These activities include the preparation and consumption of fermented foods and beverages. Also, these remains suggest the eventual use of these areas for medicinal preparations or the extraction of tannins and gums from the tara, and possibly the elaboration of other products from these substances. These products could be used for artisanal purposes, as in the case of tannins for leather tanning and the elaboration of black dyes, or for constructive ends in the case of gums as binder in pigment mixtures for wall paintings or plasters. They could also have been used in the food itself, in the case of gum as a thickening or stabilizing additive.

Various recurrent plant macro-remains come from Huacas 34, 43, and San Miguel [START_REF] Manrique | Arquitectura y secuencia cultural prehispánica en Huaca 43, Complejo Arqueológico Maranga[END_REF][START_REF] Manrique | Informe Final del Proyecto de Investigación en las Huacas 344, 43 y la Muralla 46 A del Complejo Arqueológico Maranga[END_REF]. Most of the identified taxa correspond to a variety of cultivated foods that include fruits, grains, legumes, vegetables, roots, rhizomes, tubers, and condiments. Within the condiments we have mainly Arachis hypogaea, Capsicum sp., Canna indica and Cucurbita sp. Wild foods have also been identified: fruits such as pacae (Inga sp.), lúcuma (Pouteria lucuma), and algarrobo (Prosopis juliflora); algae such as Condracanthus chamissoi, among others. Similarly, plant remains that would be used in construction and craft work such as woods, reeds (Gynerium sagittatum, Phragmites australis), juncos (Cyperus sp.), totoras (Typha sp.), calabazas (Lagenaria siceraria) and cotton (Gossypium barbadense) compose the archaeobotanical ensembles of MAC [START_REF] Bellido | Una perspectiva arqueobotánica en los períodos tardíos[END_REF].

In relation to the results of this work, we highlight that all the taxa recovered as micro-remains in the lithic tools were also recovered as plant macro-remains at the MAC sites. In particular, Caesalpinia sp. macro-and micro-remains were found in the same building, that is Huaca San Miguel [START_REF] Bellido | Una perspectiva arqueobotánica en los períodos tardíos[END_REF].

The registration of these taxa is consistent with the information provided by the chroniclers of the Indies who indicated that the algarrobos and the tara were among the trees that made up the riparian forests in the coastal zone upon the arrival of the Spanish [START_REF] Rostworowski | Recursos naturales y renovables y pesca. Siglos XVI y XVII. Curacas y sucesiones, costa norte., Obras Completas IV[END_REF], and that they constituted a local resource, like the maize that was cultivated under irrigation in the area [START_REF] Rostworowski | Señoríos indígenas de Lima y Canta[END_REF].

It becomes evident, then, that as time went by, Maranga's inhabitants continued to process the same main resource (Caesalpinia spinosa), but with less variability in techno-types and, consequently, with greater standardization in techno-types, particularly techno-types 2, 5, and 8. In other words, it is likely that an instrumental standardization occurred in the MAC towards c. XV A.D.

Finally, the techno-functional and plant micro-remain analyses taken together have allowed us to question previous approaches that, only from a partial typological or technological perspective, categorized these assemblages as expedient, archaic, poorly elaborated, and without greater evolution [START_REF] Taschini | L'industria litica Pre-incaica di Cajamar-quilla (Peru)[END_REF]. Our research, however, has made evident the significant techno-functional variability in the MAC's archaeological sequence. The results show that the lithic artifacts studied respond to a thorough selection of technical criteria, that is a volumetric and functional configuration specific to certain processes, and, therefore, were manufactured following a logical and highly effective project.

Additionally, toward c. XV AD political or social constraints -as cultural preferences-could condition the manufacturing of lithic tools by driving into a technological standardization, at the same time that plant processing become more diverse. This makes sense when considering that the MAC area was under the influence of the Inca State during the Late Horizon.

This study has proposed one of the first approaches to mobile items present inside the monumental sites of the Peruvian Central Coast. Beyond specifying some of the artifact classes and plants that were being used, as well as changes in usage trends over the course of more than a dozen centuries in the MAC sites, this research raises new questions about the cultural processes and preferences that conditioned the technological choices we have recorded and the need to explore in detail the activity areas and the materials inside the monumental buildings.

Fig. 1 .

 1 Fig. 1. Location of the Archaeological Complex Maranga (MAC), in the lower part of the Rimac river valley, left bank (Modified from maps-for-free.com).

Fig. 2 .

 2 Fig. 2. Left: Location map of the nine selected sites. Dashed white line: perimeter of the MAC in the PATPAL. Yellow: the sites from which the lithic sets come (Source: Modified from Google Earth Pro satellite image). Right: General photographs of three of the archaeological sites (Source: Parque de las Leyendas, 2014).

Fig. 3 .Fig. 4 .

 34 Fig. 3. Core initialization (Mi) and production methods (Mp) used in the Maranga Archaeological Complex.

Fig. 7 .

 7 Fig. 7. Core reduction system type F3 (simple bipolar): examples from Huaca 34

Fig. 9 .

 9 Fig. 9. Core reduction system type D1 (alternating production of quadrangular flakes), example from Huaca San Miguel: (a) traditional drawing, (b) structural analysis, (c) technological analysis. (Modified from Silva, 2014).

Fig. 10 .

 10 Fig. 10. Description of techno-types in the Maranga Archaeological Complex.

Fig. 11 .

 11 Fig. 11. Diagram of correspondence between sites and techno-types.

Fig. 12 .

 12 Fig. 12. Techno-type 8, Huaca El Rosal: (a) Photograph of the piece, (b) diacritical analysis, (c) techno-functional analysis.

Fig. 14 .

 14 Fig. 14. Correspondence analysis between techno-types and identified plant taxa.

Fig. 15 .

 15 Fig. 15. Huaca el Rosal, Techno-type 8, plant micro-remains analysis by UTFt. The legend for the tool confection also applies to the following figures 16, 17, and 18.

Fig. 16 .

 16 Fig. 16. Huaca 33, techno-type 7, plant micro-remains recovered from both UTFt; taxonomic assignment (a-c): SG of Caesalpinia spinosa. (d-f) Modern SG of the same species. The white color of the tool corresponds to a modification of the original, in order to better observe the removal negatives.

Fig. 17 .

 17 Fig. 17. Wall 46A, Techno-type 4, (a-f) probable modified SG from Zea mays, recovered in the single UTFt.

Fig. 18 .

 18 Fig. 18. Muralla 55E, Techno-type 2, plant micro-remains recovered in the single UTFt, possibly of Zea mays ("maize"): (a, b) rondel, flat rondel and killed rondel; (d) tetra-lobate (crosses) and rondel silica short cell, (b, e) tetra and poli-lobate silica short cells, (c) hair cell mesophyll.

Table 1 .

 1 Lithic assemblages studied in the technological analysis and relative chronological association

										No. of artifacts
	N°	Site	Provenance	Context of provenance	Relative chronology	Period (sensu Rowe, 1962)		Cores		Knapped Flakes	Tools	Total Knapped tools	Macro-tools	Total artifacts per lithic assemblage
							n	%	n	%	n	%
		Huaca El Rosal	UE8, 9 and 10. Occupation floors associated with wall	Indeterminate activity area	~1800-1400 BP	Early Period Intermediate	85	24.29	6	2.30	79	15.90	170	21	191
						Early
		Huaca Middendorf	Layers A and B	Building fill	~1800-1250 BP	Intermediate Period -early	8	2.29	7	2.68	18	3.62	33	6	39
						Middle Horizon
		Huaca 33	UE30 and 31. Occupation floors	Indeterminate activity area	~1800-1400 BP	Early Period Intermediate	73	20.86	53	20.31	22	4.43	148	33	181
		Huaca 58B	Occupation floor associated with a adobitos wall	Indeterminate activity area	~1800-1400 BP	Early Intermediate Period	43	12.29	25	9.58	10 2	20.52	170	3	173
			Occupation floor
		Huaca 34	associated with ceramic	Indeterminate activity area	~1050-470 BP	Late Intermediate Period	26	7.43	86	32.95	11 4	22.94	226	12	238
			fragmentation
			Occupation floor
		Huaca 43	associated with ceramic	Indeterminate activity area	~1050-470 BP	Late Intermediate Period	20	5.71	16	6.13	13	2.62	49	4	53
			fragmentation
		Wall 46A	Tamped floor-seal	Indeterminate activity area	~1050-470 BP	Late Intermediate Period	14	4.00	8	3.07	10	2.01	32	2	34
		Wall 55E	Building fill covering wall	Building fill	~470-400 BP	Late Horizon	8	2.29	5	1.92	14	2.82	27	1	28
		Huaca San Miguel	EU 2 and 7: Occupation floor associated with wall	Indeterminate activity area	~470-400 BP	Late Horizon	73	20.86	55	21.07	12 5	25.15	253	67	320
				Total			350 100.00 261 100.00 497 100.00	1108	149	1257

Table 2 .

 2 Corpus of artifacts analyzed by micro-remains analysis.

	Period (sensu Rowe, 1962)	Site	Provenance	Techno-type 1	Techno-type 2	Techno-type 3	Techno-type 4	Techno-type 5	Techno-type 6	Techno-type 7	Techno-type 8	No. of pieces	%
		Huaca El Rosal	UE8, 9, and 10 -associated with wall Occupation floors	1	1	1	1	1	1	1		8	20.5
	Early Intermediate Period	Huaca 33 (H33)	UE30 and 31 -Occupation floors	1	1	1	1	1	0	1		6	15.4
			Occupation floor
		Huaca 58B	associated with an	1	1	1	1	1	1	0		6	15.4
			adobitos wall
	Early Intermediate Period -early Middle Horizon	Huaca Middendorf	Layers A and B: Building fill	1	1	1	1	1	0	0		5	12.8
			Occupation floor
		Huaca 34	associated with ceramic	1	1	1	1	0	0	0		4	10.3
	Late Intermediate Period	Huaca 43	fragmentation Occupation floor associated with ceramic	0	1	0	1	0	0	0		2	5.1
			fragmentation
		Wall 46A	Tamped floor-seal	0	0	0	1	0	0	0		1	2.6
		Wall 55E	Building fill covering wall	1	1	0	0	0	0	0		2	5.1
	Late Horizon	Huaca San Miguel	EU 2 and 7: Occupation wall floor associated with	1	1	1	1	1	0	0		5	12.8
		Total		7	8	6	8	5	2	2		39	100. 0

Table 3 .

 3 Test of relationship between sites and techno-types before resampling.

			Chi-squared test
		value	degrees of freedom (df)	asymptotic significance (bilateral)
	Pearson's chi-squared test	221.339 a	56	.000
	Likelihood ratio	230.114	56	.000
	Linear-by-Linear Association	.066	1	.798
	Number of Valid Cases	497

a. 51 cells (70.8%) have expected a count less than 5. The minimum expected count is .02

Table 4 .

 4 Degree of association between sites and techno-types after simulation.

	Sampling simulation for symmetric measurements
				sampling simulation a
	Nominal-by-nominal	Value Bias	Dev. error	Inferior Superior
	Phi correlation	.667	.045	.040	.	.
	coefficient (phi)
	Cramér's V	.252	.026	.018	.	.
	Contingency	.555	.024	.021	.	.
	coefficient
	Number of Valid	497	0	0	.	.
	Cases
	a. Unless otherwise indicated, sample simulation results are based on 5000 sample simulation samples.

Table 4 .

 4 Total plant micro-remains and taxonomic assignation in the artifactual sample.

		Taxonomic assignation			Type and amount of micro-remain			TO	Freque	Ubiqui
	Order	Family	Genre / Species	Common name	SG	Sip	PCa	Ti	Mic	Sph	CR	Di	TA L	ncy (%)*	ty*
	Fabales	Fabaceae	Caesalpini a spinosa (Molina) Kuntze	Tara	73	0	0	0	0	0	0	0	73	33.0	0.33 (N=13 )
			Prosopis sp.	Carob (Algarro bo)	25	0	0	0	0	0	0	0	15	5.1	0.05 (N=2)
	Poales	Poaceae	Zea mays L. Indetermi nate	Maize (Maíz) -	28 6	58 6	0 0	9 0	0 0	0 0	0 0	0 0	95 12	17.9 5.1	0.18 (N=7) 0.05 (N=2)
				Wax
				gourd
	Cucurbita les	Cucurbitac eae	Cucurbita sp.	(Calabaz a blanca,	13	5	0	0	0	0	0	0	18	7.7	0.08 (N=3)
				chiclayo,
				lacayote)
		N.I. (Not identified)		32	71	10	84	17	16	20	18	278	-
		TOTAL		177	140	10	93	17	16	20	18

Acknowledgements

We would like to gratefully acknowledge Daniel Campi, Carlos Aschero, Salomón Hocsman, Julieta Zapatiel, and Mónica Burgos for their support in the Archaeobotany Laboratory of the Institute of Archaeology and Museum of the National University of Tucumán and in the Superior Institute of Social Studies of CONICET during archaeobotanical analysis. Special thanks also to Rafael Vega-Centeno for allowing access to the lithic collections of his project. Finally, our thanks to the engineers Oscar Bernuy, Edwin Loayza and Miguel Chumbe for their collaboration in the petromineralogical analysis at the Geological Mining and Metallurgical Institute of Peru (INGEMMET). This work was supported by the National Council of Science, Technology and Technological Innovation (CONCYTEC), 2016-2018 Grant of A. Pérez.


Jean Pinquet
email: pinquet@parisnanterre.fr

Positivity properties of the ARFIMA(0, d, 0) specifications and credibility analysis of frequency risks

Keywords: C02, C14, C18, C23, C35 ARFIMA(0, d, 0) specifications, Poisson mixtures, Semiparametric analysis, Linear credibility, Spectral measure of stationary random effects

Allowing for the seniority of claims and of risk exposure in the prediction of frequency risks necessitates dynamic random effects in Poisson mixtures. Non-life insurance data show evidence of long memory in stationary random effects. This paper proves that the ARFIMA(0, d, 0) mixtures of Poisson distributions ensure nonnegative credibilities per period in the affine prediction of frequency risks. This is true regardless of the risk exposure. This property is maintained if the random effect is the product of a time-invariant component (which provides the highest level of memory in the data) and of a component that follows an ARFIMA(0, d, 0) specification. The proof uses approximations of the ARFIMA(0, d, 0) time series by AR(p) time series, which result from truncations of the filtering equations that define the former ones. Every given ARFIMA(0, d, 0) specification inherits the positivity properties of the truncations because the supremum of the spectral densities of these truncations is integrable on the frequency domain. These semiparametric specifications are easily estimated from longitudinal count data, with the generalized method of moments.

Introduction

Allowing for the seniority of claims and of risk exposure in the prediction of frequency risks necessitates dynamic random effects in Poisson mixtures. Stationary random effects with a short memory are used in this setting by [START_REF] Pinquet | Allowance for the age of claims in bonus-malus systems[END_REF], [START_REF] Bolancé | Time-varying credibility for frequency risk models: estimation and tests for autoregressive specifications on the random effects[END_REF], [START_REF] Brouhns | Bonus-malus scales in segmented tariffs with stochastic migration between segments[END_REF] and [START_REF] Lu | Dynamic frailty count process in insurance: a unified framework for estimation, pricing, and forecasting[END_REF]. However, these specifications do not provide a good fit to non-life insurance data in the framework of the generalized method of moments. An example is detailed in Section 2.3 (see Table 1). The unconstrained estimation of the autocovariances of random effects is not well fitted to a vanishing function with an exponential decay. The ARFIMA(0, d, 0) specifications [START_REF] Granger | An introduction to long-memory time series models and fractional differencing[END_REF]) have a power-law decay and a long memory (see Equation ( 11)). They provide a better fit to the data investigated in Section 2.3.

A major issue in the credibility analysis of frequency risks is the nonnegativity of the credibilities per period, a condition which must be fulfilled regardless of the risk exposure in order to avoid negative predictors. A sufficient condition for nonnegative credibilities per period is given by [START_REF] Pinquet | Poisson models with dynamic random effects and nonnegative credibilities per period[END_REF]. Consider the inverse variance-covariance matrices of the random effects, which are also referred to as precision matrices: the sufficient condition is the nonpositivity of their off-diagonal entries. A statistical interpretation is that the generalized partial autocorrelation coefficients of the random effects are nonnegative. This sufficient condition is specified for AR(p) time series in [START_REF] Pinquet | Poisson models with dynamic random effects and nonnegative credibilities per period[END_REF], and is verified numerically -but not proven -for the ARFIMA(0, d, 0) specifications. The verifications use the hereditarity of the sufficient condition for nonnegative credibilities. Hereditarity here means that if the condition is fulfilled for a given length of the history, then it also holds for shorter lengths.

This paper proves that the ARFIMA(0, d, 0) mixtures of Poisson distributions ensure nonnegative credibilities regardless of the risk exposure. The proof uses approximations of the ARFIMA(0, d, 0) time series by AR(p) time series, which result from truncations of the filtering equations that define the former ones. Every given ARFIMA(0, d, 0) specification inherits the positivity properties of the truncations because the supremum of the spectral densities of these truncations is integrable on the frequency domain.

Section 2 presents Poisson mixtures with a second-order stationary specification for the random effects. The specifications encompass the three levels of memory in the mixing distribution and they ensure nonnegative credibilities regardless of the risk exposure. A case study addresses the goodness of fit between the unconstrained estimations of the autocovariances of random effects reported by [START_REF] Pinquet | Allowance for the age of claims in bonus-malus systems[END_REF] and the values related to the semiparametric models. The results motivate the choice of a specification with a long memory for the random effects. The proof of the aforementioned positivity property for the ARFIMA(0, d, 0) specifications is provided in Section 3. Concluding remarks are given in Section 4, whereas technicalities are relegated to an appendix.

2 Poisson mixtures that encompass the three levels for the length of memory in the random effects and that ensure nonnegative credibilities per period 2.1 A sufficient condition for nonnegative credibilities

We consider a longitudinal dataset. The number of events per year (e.g., claims of a given type) is the dependent variable and we write

N i,t ∼ P (λ i,t u i,t ) ; i = 1, . . . , m; t = 1, . . . , T i . (1)
Equation ( 1) specifies a Poisson distribution on the count variable, which is observed on an unbalanced panel dataset. The frequency risk based on the observable information is denoted by λ and can be estimated ad libitum from the regression components (e.g., log-linear specification for the expectation, index model, neural network, regression tree, or random forest). These regression components cannot include the individual history in a mixture model in order to restrict experience rating to the second component of the expectation. The Poisson distributions in (1) are conditional on U i,t = u i,t , where U i,t is a positive random effect. A second-order stationary framework is retained for the random effects. The use of stationary random effects in Poisson mixtures is linked with a plausible invariance assumption: the predictive ability of events on frequency risk depends on their seniority but not on calendar time. The autocovariance function of U is denoted by γ U .

The expectation of the random effects is set equal to one. This is assumed without loss of generality because the intercept is a regression component. The individual index i is removed in the prediction. Linear credibility is related to an affine probabilistic regression of U T +1 with respect to N 1 , . . . , N T in the model with random effects [START_REF] Bühlmann | Experience rating and credibility[END_REF]).

The predictor is

L (U T +1 | N 1 , . . . , N T ) = 1 - T h=1 α T,h + T h=1 α T,h N T +1-h λ T +1-h , (2)
where the index h refers to the lag between the forecast period and the risk exposure period. The credibility for period T + 1 -h is denoted by α T,h . The experience rated frequency premium for period T + 1 is the product of this predictor and λ T +1 , with estimated parameters. Linear credibility predictors are also obtained from INAR models (Al-Osh and Alzaid (1987), [START_REF] Bermúdez | Allowing for time and cross dependence assumptions between claim counts in ratemaking models[END_REF]). The credibility weights are the probability masses of the Bernoulli distributions used in the INAR specifications. These credibility weights do not depend on the risk exposure, as is the case for Poisson mixtures.

The variance-covariance matrix of the random effects U 1 , . . . , U T is denoted by V T U and is assumed to be positive definite, regardless of the length T of the history. The vector of stacked credibilities per period is

v T α def = vec 1≤ h ≤T (α T,h ) = Λ -1 T + V T U -1 v T γ U , with: Λ T = diag 1≤ h ≤T (λ T +1-h ) ; v T γ U = vec 1≤ h ≤T (γ U (h)) .
A sufficient condition for nonnegative credibilities regardless of the risk exposure (i.e., [START_REF] Pinquet | Poisson models with dynamic random effects and nonnegative credibilities per period[END_REF]). The proof uses results referring to nonnegative matrices (i.e., the class of M -matrices and the Perron-Frobenius theorem-see [START_REF] Berman | Nonnegative Matrices in the Mathematical Sciences[END_REF]). The proof also uses the vocabulary of affine and projective geometry.

v T α ∈ (R + ) T ∀T ∈ N * , ∀λ 1 , . . . , ∀λ T > 0) is that the off-diagonal entries of V T U -1 are nonpositive ∀T ∈ N * (
The inverse of a variance-covariance matrix is termed a precision matrix. In this paper, autocovariance functions for which the off-diagonal entries of the precision matrices are nonpositive are said to reach the positivity level S. Poisson mixtures derived from autocovariance functions that reach level S ensure nonnegative credibilities per period regardless of the risk exposure. Autocovariance functions that reach level S are nonnegative entrywise, and entail nonnegative filtering coefficients in the affine probabilistic regression of the random effects with respect to past consecutive values.

The total credibility (i.e., T h=1 α T,h in Equation ( 2)) is always less than one if the random effects reach level S (Pinquet ( 2020)). Therefore, the intercept of the affine predictor is positive, and nonnegative credibilities per period entail positive predictors of frequency risks.

Autocovariance functions that reach level S

The autocovariance functions given in this section are sorted by increasing length of memory. Autoregressive specifications have a short memory. Those that reach level S are characterized by Proposition 1.

Proposition 1 An AR(p) specification (p ∈ N * ) on the random effect U defined by

U c t = p h=1 ϕ h U c t-h + I U t , ∀t ∈ Z ,
(where U c = U -1 is a centered sequence and where I U is a white noise process, termed the innovation of U ), reaches level S if and only if

ϕ h ≥ 0, ∀h = 1, . . . , p; p h=1 ϕ h < 1; ϕ h ≥ 1≤τ ≤p-h ϕ τ ϕ τ +h , ∀h = 1, . . . , p -1. (3)
The last condition is obviously fulfilled if ϕ is decreasing.

The proof is obtained from the derivation of the precision matrices of the random effects (see [START_REF] Pinquet | Poisson models with dynamic random effects and nonnegative credibilities per period[END_REF]). These autoregressive specifications are applied to positive random effects in a credibility analysis of frequency risks. The compatibility between the autocovariance functions and positive random effects is not an issue for practitioners using a second-order semiparametric approach. The estimation by the generalized method of moments and the derivation of positive linear credibility predictors do not need a solution to this compatibility issue. However this problem is of interest from a mathematical point of view. The compatibility between exponentials of Gaussian sequences and AR(p) specifications that follow the conditions of Proposition 1 is verified numerically for p = 1, 2, 3 by [START_REF] Pinquet | Poisson models with dynamic random effects and nonnegative credibilities per period[END_REF]. The verification strategy uses the Levinson-Durbin recursion. This verification is performed on large grids in the parameter space, at the horizon of a century. The compatibility between log-Gaussian sequences and these AR(p) specifications was never falsified in the verifications, and likewise for the ARFIMA(0, d, 0) autocovariance functions.

The ARFIMA(0, d, 0) specifications are defined by the equation (I -L) d U c = I U . The lag operator is denoted by L and the fractional differencing parameter d belongs to the interval 0, 1 2 . The filtering equation is

U c t = +∞ h=1 ϕ h U c t-h + I U t , ∀t ∈ Z, (4)
with

ϕ 1 = d; ϕ h+1 ϕ h = h -d h + 1 , ∀h ∈ N * .
The filtering coefficients are positive and decreasing. The equality +∞ h=1 ϕ h = 1 reflects the long memory of the ARFIMA(0, d, 0) specifications (see Equation ( 11) and Appendix A.1). Due to this equality, U c can be replaced by U in Equation (4).

Let U tr(p) be the AR(p) sequence defined by a p-truncation of Equation ( 4):

U tr(p) t def = h=p h=1 ϕ h U tr(p) t-h + I U t , ∀t ∈ Z. (5)
The AR(p) sequence U tr(p) is an approximation of U c , with truncated filtering coefficients and with the same innovation. From Proposition 1, the time series U tr(p) p∈N * reach level S if U is of type ARFIMA(0, d, 0). Section 3 provides a proof that this positivity property extends from the sequence U tr (p) p∈N * to U in the ARFIMA(0, d, 0) case. The previously specified autocovariance functions vanish at infinity. However, this weak ergodicity property is not a desirable assumption (see Table 1,Section 2.3). The highest level of memory in the mixing distribution is reached if we place side by side a time-invariant random effect and an ergodic random effect. The dynamic random effects (U t ) t∈N * applied to Poisson distributions in the credibility analysis are decomposed multiplicatively:

U t def = P Q t , ∀t ∈ N * . ( 6
)
The variables P and (Q t ) t∈N * are assumed to be positive and uncorrelated, with

E(P ) = E(Q t ) = 1 ∀t ∈ N * .
The dynamic random effect Q is also assumed to follow either an AR(p) specification that reaches level S, or an ARFIMA(0, d, 0) specification. If

σ 2 P is the variance of P, then γ U (h) = σ 2 P + (1 + σ 2 P ) γ Q (h) , ∀h ∈ Z (7) if P 2 and Q t Q t+h are assumed to be uncorrelated ∀t, h. Indeed, E(U t U t+h ) = E(P 2 ) E(Q t Q t+h ) = (1+σ 2 P ) (1+γ Q (h))
, which leads to Equation ( 7). Then, lim

h→+∞ γ Q (h) = 0 ⇒ lim h→+∞ γ U (h) = σ 2 P .
Therefore, the non-ergodic specifications correspond to σ 2 P > 0. If U has a short memory (i.e., if γ U is summable), then the function

s U (θ) = h∈Z e -iθh γ U (h)
is nonnegative. This function is defined on the frequency domain R/2πZ (identified with [-π, π[). The function s U is the density of a spectral measure S U with respect to the Lebesgue measure restricted to [-π, π[. The autocovariance function is obtained from the spectral measure by

γ U (h) = 1 2π π -π e ihθ dS U (θ), ∀h ∈ Z. (8)
Equation ( 8) is also valid for the two higher levels of memory in the random effects. The spectral measure of U is still absolutely continuous with respect to Lebesgue if U follows an ARFIMA(0, d, 0) specification, but the spectral density tends to infinity at 0. If σ 2 P > 0 (see ( 7)), then the memory in the mixing distribution reaches the highest level, and the spectral measure S U has a Dirac mass located at 0.

In Equation ( 6), the random effect U reaches level S if Q reaches level S and if

V T Q -1 1 T ∈ R + T , ∀ T ∈ N * . ( 9
)
The intercept is denoted by 1 T in (9). This condition is fulfilled if Q follows an AR(p) specification that reaches level S (Pinquet (2020)). In Section 3, it is shown that (9) also holds for the ARFIMA(0, d, 0) specifications.

A case study

In this case study, the ARFIMA(0, d, 0) specifications on the random effects provide a better fit to the data than those of type AR(1). We use the estimated autocovariances of random effects reported by [START_REF] Pinquet | Allowance for the age of claims in bonus-malus systems[END_REF]. To avoid selection bias, the working sample is an unbalanced panel dataset. The lengths of the histories range between one and seven years. The unconstrained estimators are derived with a method of moments in a Poisson mixture model with regression components. The estimated autocovariances are given in Table 1. The semiparametric specifications of the random effects are estimated by a generalized method of moments -GMM - [START_REF] Hansen | Large sample properties of generalized method of moments estimators[END_REF]). A rough estimation strategy is used, which avoids derivations at the individual level. The estimated parameters minimize the sum of squared adjustment errors (denoted by SSE) between the unconstrained estimations of Table 1 and the fitted values.

The autocovariance functions of type ARFIMA(0, d, 0) are generated from the recursive formula

γ(h + 1) = h + d h + 1 -d γ(h), ∀h ∈ N. ( 10
)
This formula entails the following equivalence in the neighborhood of infinity

γ(h) ∼ γ(0) A(d) h 2d-1 , with: A(d) > 0 ∀d ∈]0, 1/2[. (11)
Hence, the ARFIMA(0, d, 0) specifications are weakly ergodic, they have a power-law decay and they also have a long memory (i.e., h=+∞ h=0 γ(h) = +∞).

Ergodic specifications are estimated first (i.e., σ 2 P = 0, U = Q in ( 6)). Then

If Q is AR(1) : γ Q (0) = 1.12; ϕ 1 = 0.82; SSE = 0.0700;

If Q is ARFIMA(0, d, 0) : γ Q (0) = 1.28; d = 0.37; SSE = 0.0067.

The autocovariance specification with a long memory strongly outperforms its counterpart with a short memory. An autocovariance function with an exponential decay can provide a good fit to the estimations of Table 1 if it does not vanish at infinity. In what follows, a time-invariant component P is included in the random effect. If Q is of type AR(1), then we obtain σ 2 P = 0.47; γ Q (0) = 0.54; ϕ 1 = 0.45; SSE = 0.0073.

The fitted specification reaches the highest level of memory.

The estimation is unchanged if Q is of type ARFIMA(0, d, 0) (hence, σ 2 P = 0). This result confirms that this specification of the ergodic component provides a good fit to the data. The adjustment errors related to the three estimations are displayed in Figure 1. The results of this case study motivate the choice of ARFIMA(0, d, 0) mixtures of Poisson distributions in the credibility analysis of frequency risks. The efficiency ranking of the three specifications is not modified if the rough GMM estimation used in this section is replaced by a GMM procedure that takes into account the full information.

The following section provides a proof that the ARFIMA(0, d, 0) specifications reach level S. Therefore, the credibilities per period derived from the related Poisson mixtures are nonnegative regardless of the risk exposure.

3 The ARFIMA(0, d, 0) specifications reach level S

The AR(p) sequences that reach level S have been characterized in Section 2.2. If a random effect U follows an ARFIMA(0, d, 0) specification, the autoregressive processes U tr (p) p∈N * defined in (5) reach level S. Proposition 2 gives a condition that extends the property of reaching level S from the sequence U tr (p) p∈N * to U .

Proposition 2 Let U be a stationary random effect, defined by the filtering equation

U c t = h∈N * ϕ h U c t-h + I U t , ∀t ∈ Z.
The variance of the innovation I U is assumed to be positive. It is also assumed that

ϕ h ≥ 0, ∀h ∈ N * ; p h=1 ϕ h < 1, ∀p ∈ N * ; ϕ h ≥ τ ∈N * ϕ τ ϕ τ +h , ∀h ∈ N * (12)
(the last condition is fulfilled if ϕ is decreasing). Suppose that the spectral densities of the processes U tr(p) p∈N * are dominated on the frequency domain [-π, π[ by an integrable function. Then U reaches level S. This positivity property is maintained if a time-invariant random effect is placed side by side with U.

Sketch of the proof: The conditions ( 12) are the extension of conditions (3) to all p ∈ N * . Therefore, the autoregressive processes U

γ U tr(p) = γ U entrywise on Z. Therefore, lim p→+∞ V T U tr(p) = V T U , ∀T ∈ N * .
The same result holds for the inverses of these matrices, and U reaches level S. The last result of Proposition 2 is obtained from ( 9), with U tr (p) p∈N * and U in place of Q. A detailed proof is given in Appendix A.1. The domination condition holds if and only if the supremum of the spectral densities of the truncations is integrable on the frequency domain.

An integrable function that dominates the spectral densities of the truncations is easily obtained if +∞ h=1 ϕ h < 1, in which case U has a short memory (see Appendix A.2). If +∞ h=1 ϕ h = 1, obtaining such a function is not obvious in the neighborhood of 0. Proposition 3 provides a condition on the filtering coefficients that implies the existence of an integrable and dominating function.

Proposition 3 Let U be a stationary sequence that fulfills the conditions given in ( 12), together with: +∞ h=1 ϕ h = 1. The spectral densities of the time series U tr (p) p∈N * are dominated by an integrable function if the following conditions are fulfilled.

ϕ h ≥ c h -α , ∀h ∈ N * ; c > 0 ; 1 < α < 3/2. ( 13
)
The proof is given in Appendix A.2. The main result now follows because the conditions given in ( 13) are fulfilled by the ARFIMA(0, d, 0) specifications.

Proposition 4

The ARFIMA(0, d, 0) specifications fulfil the conditions given in (13), with

α = 1 + d, 0 < d < 1/2; c = d h∈N * 1 - d h 1 + 1 h d , 0 < c < d.
Therefore, the ARFIMA(0, d, 0) mixtures of Poisson distributions ensure nonnegative credibilities regardless of the risk exposure.

The filtering coefficients of an ARFIMA(0, d, 0) specification are defined by

ϕ 1 = d; ϕ h+1 ϕ h = h -d h + 1 = 1 - 1 + d h + 1 ∀h ∈ N * .
The proof uses the auxiliary sequence

ψ h = d × h -(1+d) (h ∈ N * ). Then, ψ 1 = ϕ 1 and log(ϕ h+1 ) -log(ϕ h ) -(log(ψ h+1 ) -log(ψ h )) = log(h -d) -log(h + 1) + (1 + d) × log 1 + 1 h = log 1 - d h + d log 1 + 1 h def = e h .
We have that e h < 0 ∀h ∈ N * from the strict concavity of the logarithm, and e h ∼ -d+d 2 2h 2 in the neighborhood of infinity. Hence, the sequence e is negative entrywise and absolutely summable. Proposition 4 results from ϕ h+1 = ψ h+1 exp

τ =h τ =1 e τ ∀h ∈ N * , with c = d × exp
h∈N * e h . The power-law property of the ARFIMA(0,d,0) specifications (see ( 11)) is derived from the recursive generation of the autocovariance functions (see ( 10)), using a proof similar to the one given in Proposition 4 for the filtering coefficients. Appendix A.3 links the spectral densities of the ARFIMA(0, d, 0) specifications and the dominating functions obtained in Appendix A.2 for the spectral densities of the truncations.

Concluding remarks

The ARFIMA(0, d, 0) mixtures of Poisson distributions ensure nonnegative credibilities regardless of the risk exposure: this result is verified numerically in [START_REF] Pinquet | Poisson models with dynamic random effects and nonnegative credibilities per period[END_REF], and has been proven in the present paper. The other positivity issue is the compatibility of the ARFIMA(0, d, 0) specifications with positive random effects. The results provided by the author remain at the level of numerical verifications.

The innovation I U is white noise, which implies the last equality of (15). From ( 12), the sequence (Φ p ) p∈N * converges uniformly on the closed unit disk of C to a limit Φ. The spectral density of U is obtained from

1 -Φ(e -iθ ) 2 s U (θ) = γ I U (0), ∀θ ∈ [-π, π[-{0}. (16)
Hence: lim

p→+∞ s U tr(p) (θ) = s U (θ), ∀θ ∈ [-π, π[-{0}.
If h∈N * ϕ h < 1, the spectral density of U is defined at θ = 0, and

1 -Φ(e -iθ ) ≥ 1 -Φ(e -iθ ) ≥ 1 -Φ(1) = 1 - h∈N * ϕ h , ∀θ ∈ [-π, π[.
The second inequality results from the nonnegativity of the filtering coefficients. Then the spectral density s U is bounded above by the constant

γ I U (0)
(1-h∈N * ϕ h) 2 . This upper bound also holds for the spectral densities (s U tr(p) ) p∈N * .

As s U (0) = h∈Z γ U (h), the condition h∈N * ϕ h < 1 implies a short memory for U . The ARFIMA(0, d, 0) specifications have a long memory, hence h∈N * ϕ h = 1 in (4).

Suppose that the spectral densities (s U tr(p) ) p∈N * are dominated by an integrable function f , i.e. -1 = V T U -1 . Indeed, the map V -→ V -1 is continuous on the set of positive definite matrices with any given order, and the matrices V T U are assumed to be positive definite ∀T ∈ N * . Level S is related to the set of positive definite matrices with nonpositive off-diagonal entries. Matrices with nonpositive off-diagonal entries are a closed set, and the positivity level S is reached by U .

s U tr(p) (θ) ≤ f (θ), ∀p ∈ N * , ∀θ ∈ [-π, π[-{0}, with
The sequence U tr(p) is an AR(p) process that reaches level S. It can be multiplied by a time-invariant random effect without loss of the positivity properties, because

V T U tr(p) -1 1 T ∈ (R + ) T , ∀p, T ∈ N * (see Pinquet (2020)). As lim p→+∞ V T U tr(p) -1 = V T U -1
, the same property holds for U .

A.2 Proof of Proposition 3

From (15), the spectral densities of the truncations are dominated by an integrable function if

1 -Φ p (e -iθ ) ≥ D(θ), ∀p ∈ N * , ∀θ ∈ [-π, π[, with +π -π dθ D 2 (θ) < +∞.
The function θ -→ D(θ) is a pointwise lower bound for the distance between 1 and the set {Φ p (e -iθ )} p∈N * .

A pointwise lower bound on the frequency domain is obtained from

1 -Φ p (e -iθ ) = 1 -ϕ 1 e -iθ - h=p h=2 ϕ h e -ihθ ≥ 1 -ϕ 1 e -iθ - h=p h=2 ϕ h ⇒ 1 -Φ p (e -iθ ) ≥ 1 -ϕ 1 e -iθ -(1 -ϕ 1 ) = D 1 (θ), ∀p ∈ N * , ∀θ ∈ [-π, π[. We have that D 1 (0) = 0; D 1 (θ) > 0, ∀θ ∈ [-π, π[-{0}; D 1 is continuous on [-π, π[.
Therefore, the function 1/D 2 1 is integrable on [-π, π[ outside any neighborhood of 0. The following derivations provide a solution in a neighborhood of 0.

From 1 -Φ p (e -iθ ) = [1 -p h=1 ϕ h cos(h θ)] + i p h=1 ϕ h sin(h θ), we obtain 1 -Φ p (e -iθ ) ≥ 1 - p h=1 ϕ h cos(h θ) = 1 - p h=1 ϕ h cos(h θ).
The last equality results from ϕ h ≥ 0, ∀h ∈ N * ;

p h=1 ϕ h < 1, ∀p ∈ N * . Then +∞ h=1 ϕ h = 1 ⇒ 1 - p h=1 ϕ h cos(h θ) = p h=1 ϕ h (1 -cos(h θ)) + h>p ϕ h ⇒ 1 - p h=1 ϕ h cos(h θ) ≥ +∞ h=1 ϕ h min(1, 1 -cos(h θ)), ∀p ∈ N * .
The latter function is even, and we restrict to θ > 0. Then

+∞ h=1 ϕ h min(1, 1 -cos(h θ)) ≥ h=[ 3π 2θ ] h=[ π 2θ ]+1 ϕ h . The conditions ϕ h ≥ c h -α (c > 0, 1 < α < 3/2) imply h=[ 3π 2θ ] h=[ π 2θ ]+1 ϕ h ≥ c h=[ 3π 2θ ] h=[ π 2θ ]+1 h -α ≥ c [ 3π 2θ ]+1 [ π 2θ ]+1 x -α dx. Then, c [ 3π 2θ ]+1 [ π 2θ ]+1 x -α dx = c α -1 × π 2θ + 1 1-α - 3π 2θ + 1 1-α = c α -1 × π 2θ + 1 1-α ×   1 - 3π 2θ + 1 π 2θ + 1 1-α   ⇒ c [ 3π 2θ ]+1 [ π 2θ ]+1 x -α dx ≥ c α -1 × π 2θ + 1 1-α ×   1 - 3π 2θ + 1 π 2θ + 1 1-α   .
From the inequalities

3π 2θ + 1 π 2θ + 1 > 3π 2θ π 2θ + 1 = 3π π + 2θ ≥ 2 if θ ∈ 0, π 4 ; π 2θ + 1 1-α ≥ 3π 4θ 1-α if θ ∈ 0, π 4

Figure 1 :

 1 Figure 1: Adjustment errors (unconstrained -fitted autocovariances of random effects).

e

  ihθ s U tr(p) (θ) dθ = lim p→+∞ γ U tr(p) (h), ∀h ∈ Z.The third equality results from the dominated convergence theorem ase ihθ s U tr(p) (θ) ≤ f (θ), ∀θ ∈ [-π, π[-{0} , ∀p ∈ N * . Therefore, lim p→+∞ V T U tr(p) = V TU holds for every length T. This limit also holds for the precision matrices, i.e. lim p→+∞ V T Utr(p)

,×

  the function D 2 : θ (|θ| ≤ π/4) -→ c×(1-2 1-α ) |θ| α-1 is a pointwise lower bound for the distance between 1 and the set {Φ p (e -iθ )} p∈N * . A lower bound for the distance between 1 and the set{Φ p (e -iθ )} p∈N * ∀θ ∈ [-π, π[ is the function θ -→ D(θ), defined by • D(θ) = D 1 (θ) = 1 -ϕ 1 e -iθ -(1 -ϕ 1 ) if π/4 < |θ| ≤ π. • D(θ) = D 2 (θ) = c×(1-2 1-α ) |θ| α-1 if |θ| ≤ π/4; The function 1/D 2 is integrable on [-π, π[ if 1 < α < 3/2.This completes the proof.A.3 The spectral densities of the ARFIMA(0, d, 0) specifications and the dominating functions for the spectral densities of the truncationsIf U follows an ARFIMA(0, d, 0) specification, the spectral density s U is defined from (16) by1 -e -iθ 2d s U (θ) = s I U (θ) = γ I U (0), ∀θ ∈ [-π, π[-{0} ⇒ s U (θ) γ I U (0) = 1 -e -iθ -2d = [2 (1 -cos(θ))] a function that dominates the spectral densities on -π 4 , π 4 -{0} is f (θ) = γ I U (0) D 2 (θ) = γ I U (0) c×(1-2 1-α ) α-1 3π 4 1-α |θ| α-1 2 , with α = 1 + d. 2 -d ) -2 |θ| -2d γ I U (0).From (17), f ∈ O(s U ) in the neighborhood of 0. As an example, consider d = 1/3. Thenc = d h∈N * 1 -(1 -2 -1/3 ) -2 = 1.834 × 1.771 × 23.497 = 76.288.The dominating function derived from Appendix A.2 is much greater than the supremum of the spectral densities of the truncations. This result also holds for the other definition of the dominating function that is retained for[-π, π[ -[-π 4 , π4]. This is not surprising, as the proof of Appendix A.2 uses rough arguments. A much lower dominating function -obtained numerically -is shown in Figure2for d = 1/3.

Figure 2 :

 2 Figure 2: Plots of s U , f = 1.31 × s U (the dominating function), s U tr(1) and s U tr(2) (denoted by s1 and s2 in the figure) if U is of the ARFIMA(0, d, 0) type, with d = 1/3 and γ I U (0) = 1.

Table 1

 1 Estimated autocovariances of the multiplicative random effect U.

	h (lag)	0	1	2	3	4	5	6
	γ U (h) 1.269 0.802 0.615 0.586 0.553 0.457 0.442

  tr(p) p∈N * reach level S. Let us denote the spectral densities of U and of U tr(p) p∈N * by s U and by (s U tr(p) ) p∈N * , respectively. It is easily seen that lim p→+∞ s U tr(p) = s U pointwise on [-π, π[-{0}. From the dominated convergence theorem, the assumption on the dominating function made in Proposition 2 implies that lim

	p→+∞

A Appendix

A.1 Proof of Proposition 2

From Proposition 1, the conditions given in (12) imply that U tr(p) reaches level S for all p ∈ N * . Equation ( 5) is reformulated as

The lag operator is denoted by L, and Φ p is a polynomial function defined on Banach algebras, the elements of which are either operators such as L or complex numbers. From the conditions given in (12), the roots of Φ p lie outside the unit disk if the polynomial is applied to C. Equation ( 14) implies 1 -Φ p (e -iθ ) 2 s U tr(p) (θ) = s I U (θ) = γ I U (0), ∀θ ∈ [-π, π[. (15)