Juan Pablo Rodríguez

M A Díaz-Granados

M. S Rodríguez

S A Fonseca

G L Mestra
email: gmestra@yahoo.com

J C Penagos
email: jcpenagos@acueducto.com.co

Č Maksimović
email: c.maksimovic@imperial.ac.uk

N Mcintyre
email: n.mcintyre@imperial.ac.uk

Integrated management and modelling in urban drainage systems: the potentialities in a developing megacity

Keywords: Integrated management and modelling, developing countries, Bogotá city, pollutant and sediment loads, wrong connections

In developing countries, lack of sanitation coverage and continuously growing populations are increasing the pressures on receiving waters. In the context of Bogotá (Colombia), this paper presents recent and ongoing research towards improved management of urban drainage systems using an integrated framework. Research results have shown there is a need to assess the urban drainage system as one entity when considering pollution control objectives. This holistic approach offers the opportunity to investigate the interactions between sub-systems and the impact of the whole system on the river water quality. In Bogotá, now is the time to develop plans towards an efficient integrated system which maximises the benefits from the resources, supporting data and software tools available. It is needed the development and application of modelling tools at different levels of detail. As part of this, an integrated modelling toolbox named City Drain which operates under MATLAB/Simulink is being upgraded, customised and used as a research tool.

INTRODUCTION

Urban wastewater systems consist principally of three sub-systems as follows: the sewer system, the waste water treatment plant (WWTP) system, and the receiving body system. Up to now most sewer systems and WWTPs are designed, operated, and improved as separate entities as a consequence of the difference in their main functions [START_REF] Langeveld | Interactions within wastewater systems[END_REF][START_REF] Schroeder | Integrated simulation of the Berlin sewage system and evaluation of a global real-time control concept[END_REF][START_REF] Gill | Using integrated catchment modelling (ICM) to improve the operation of sewer networks and sewage treatment[END_REF]. However, research results point to the importance of the dynamic interactions between sewer and WWTP to assess performance of the urban water system [START_REF] Langeveld | Interactions within wastewater systems[END_REF]. The same remains true at the combined sewer overflow (CSO)/river and WWTP/river interfaces. For example, previous research showed that for a hypothetical case study prolonged hydraulic overloading of the WWTP, in order to reduce overflow spills, can in turn affect the treatment processes efficiency exerting considerable impact on the receiving system water quality (Rauch and Harremoes, 1997). Additionally, each receiving water body presents its own properties and must therefore be evaluated under a holistic framework with respect to the discharges from the urban catchment [START_REF] Solvi | Modelling the sewer-treatment-urban river system in view of the EU Water Framework Directive[END_REF].

The idea of an integrated urban drainage modelling is not new. [START_REF] Beck | Dynamic modelling and control applications in water quality maintenance[END_REF] discussed a "water quality system" which involves the water distribution network, the sewer system, the treatment plant and the river. It was not until the early 1990s that the concept of the holistic approach began to be adopted in academic studies [START_REF] Mitchell | State of the art in integrated urban water modelling. 6th international conference on sustainable techniques and strategies in urban water management Lyon[END_REF]. Currently, it is widely accepted that an integrated assessment of the emissions from sewer systems and WWTP is necessary when attempting to reduce the total impact of the urban drainage system on the receiving water body [START_REF] Langeveld | Interactions within wastewater systems[END_REF]. In this context, models can be used to gain better understanding of certain phenomena and to predict the spatial and temporal evolution of a system when looking towards an integrated management of urban drainage systems. Simulation models play a crucial role in environmental management plans due to they can apply best available scientific knowledge to predict responses to changing controls. Detailed integrated studies of the sewer network -WWTP -receiving water system are comparatively rare due to high cost (Rauch et al., 2002b;[START_REF] Ashley | Solids in Sewers: Characteristics, Effects and Control of Sewer Solids and Associated Pollutants[END_REF] and practical applications of the holistic approach are still limited. However, progress has been made in developing integrated modelling tools, allowing for application at full catchment-scale as presented for example in [START_REF] Freni | Catchment-scale modelling approach for a holistic urban water quality management[END_REF] and Devesa et al. (In Press). Furthermore, in order to increase the application of such holistic approach, the Central European Simulation Research Group (HSGSim) prepared a guideline document proposing a seven-step procedure to integrated modelling [START_REF] Muschalla | The HSG Guideline Document for Modelling Integrated Urban Wastewater Systems[END_REF].

Historically in Bogotá (Colombia), efforts have focused on analyzing and improving the individual performance of the urban water cycle components without taking into account the interactions between them. Bogotá city needs the development and implementation of measurement programs and modelling tools at different levels of detail, considering overall urban water fluxes and various treatment schemes, including their economic aspects. These efforts are considered to be useful for the development of best management practices. Within this framework, this paper presents recent research results in the context of Bogotá on developing software tools for estimating percentage of wrong connections in the separate sewer system [START_REF] Mestra | Identificación de conexiones erradas en alcantarillados separados mediante la aplicación de herramientas computacionales[END_REF], assessing CSOs performance [START_REF] Fonseca | Herramienta computacional para el análisis y evaluación del comportamiento de aliviaderos de alcantarillados combinados (Caso estudio[END_REF], and evaluating sediment accumulation rates in the sewer system [START_REF] Ph | Instrumentación y análisis ambiental de una subcuenca del sistema de alcantarillado de Bogotá -Informe Final[END_REF]. Additionally, relevant data on domestic and industrial wastewater loads and in-sewer sediments characteristics and properties which are now available as inputs for the integrated management and modelling in Bogotá are presented (Uniandes, 2008a).

SUPPORTING DATA AND TOOLS FOR INTEGRATED MODELLING -THE BOGOTÁ CASE

Context

Bogotá, the capital city of Colombia, is located at an approximate altitude of 2600 m.a.s.l with approximately 330 km 2 of urban area. The Bogotá Savanna is the most densely populated area of the country, being an important industrial and agricultural region. The population of Bogotá city increased by approximately 4 million in the last 30 years, from 2.9 million inhabitants in 1973 to 6.8 million in 2005 (DANE, 2005). It is estimated that the saturation population will be around 12 million inhabitants, accounting for about 20% of the total population of Colombia. Bogotá city is a prime example of a mega-city under stress due to unplanned and unregulated urban

Integrated management and modelling in urban drainage systems: potentialities in a developing megacity -Juan Pablo RODRÍGUEZ et al.

/ 13

developments, severe water quality problems in the water courses, lack of sanitation coverage and water treatment facilities, lack of institutional co-ordination, mismanagement of water resources, financial constraints, lack of wise expenditure on the required infrastructure, and a conventional fragmented wastewater management [START_REF] Rodríguez | Bogotá's urban drainage system: context, research activities and perspectives[END_REF]. A detailed description of the development and current state of the components of the urban water cycle of Bogotá city, their interactions and previous research under a holistic approach can be found in [START_REF] Rodríguez | Bogotá's urban drainage system: context, research activities and perspectives[END_REF]In preparation).

Domestic and industrial wastewater loads: a comparative assessment

Based on measurements from field campaigns in the sewer system sub-catchments, which were carried out in order to identify daily water quantity and quality patterns of dry weather flow [START_REF] Díaz-Granados | Patrones Intradiarios de Variación de Caudales y Concentraciones en Sistemas de Drenaje Urbano[END_REF], domestic pollutant loads were estimated (see Table 1). When comparing found ranges and average values for the Bogotá city, with typical values reported in specialized literature [START_REF] Butler | Urban Drainage[END_REF], it can be concluded that there are accordance as presented in Table 1.

Wrong connections: a GIS-based tool for identifying their likelihood

There are many wrong connections from the wastewater system flowing into the storm drainage system in Bogotá city. The separate sewer system acts more as a "dual" combined system rather than as a separate one. [START_REF] Mestra | Identificación de conexiones erradas en alcantarillados separados mediante la aplicación de herramientas computacionales[END_REF] presented a GIS-based computational tool in order to identify the likelihood of wrong connections presence in the Bogotá's separate sewer system, specifically misconnections from the wastewater system into the storm water system. Based on interviews and surveys to engineers and contractors in charge of households/properties' connection to the sewer system, it was found that main factors which have a relevant effect on the presence of wrong connections are urban densification processes, sewer system ageing level, construction gap between the storm water and wastewater systems, socioeconomic level and strata, land use, pipe depth and distance between property and the wastewater and storm water systems, pipe material, road type and property type. The mentioned computational tool uses 8 variables which take into account all these factors. Each of the variables has a numeric value ranging from 0 to 2, where 0 means the minimum likelihood of wrong connections and 2 the maximum likelihood. The sum of the 8 variables values allows qualifying the existence of wrong connections in three different ranges: 0 -4 low, 4.1 -8 medium, and 8.1 -13 high likelihood of misconnections. This GIS-based tool was tested using a catchment named Jaboque located in the Salitre sub-catchment in Bogotá. It was possible to identify properties with a high likelihood of wrong connection presence. By means of dye experiments and CCTV inspections in 69 properties situated in an area of the Jaboque catchment (see Figure 1a), a total number of 19 misconnections were identified (Figure 1b). [START_REF] Mestra | Identificación de conexiones erradas en alcantarillados separados mediante la aplicación de herramientas computacionales[END_REF] concluded that the developed tool appropriately predicted this condition. Additionally, the tool was applied to the entire Salitre subcatchment with an area of about 122 Km 2 (see Figure 2). It was possible to identify areas with high potentiality for misconnections in which field inspections should be focused. Regarding the entire city, it is planned to apply this GIS-based tool for assigning percentages of wrong connections in areas without any data from field inspections and measurement campaigns.

Sewer sediments: their properties and a GIS-based tool for estimating accumulation rates

Sewer sediment is defined by [START_REF] Butler | Sediment transport in sewers Part 1: Background[END_REF] as: "any type of settleable particulate material that is found in storm water or foul sewage and is able to form deposits in sewers and associated hydraulic structures". Types, sizes and quantities of sewer sediment can vary widely according to the sewer type, geographical location, catchment type, catchment slope and local procedures. Particles transported by domestic sewage are mainly organic, while particles in stormwater presents mineral properties [START_REF] Bertrand-Krajewski | Sewer sediment production and transport modelling: a literature review[END_REF]. There are five main sources of solids entering sewer systems (Ashley et al., 2004a): the atmosphere, the surface of the catchment, domestic sewage, processes inside the sewer system, and industrial and commercial effluents. Sewer solids are of importance not only in terms of transport phenomena but also for wastewater quality processes in urban drainage networks due to the solids providing a transport matrix for different pollutants [START_REF] Ristenpart | Solids transport by flushing of combined sewers[END_REF]. It has been observed that total suspended solids (TSS) are the main vector for many pollutants such as COD, hydrocarbon, heavy metals, micro-pollutants, etc. [START_REF] Chebbo | The transfer of solids in combined sewer networks[END_REF][START_REF] Gong | Stormnet: a connectionist model for dynamic management of wastewater treatment plants during storm events[END_REF][START_REF] Mark | Prediction of locations with sediment deposits in sewers[END_REF][START_REF] Arthur | Sediment transport in sewers -a step towards the design of sewers to control sediment problems[END_REF][START_REF] Otfinowska | Design of a conceptual model of suspended sediments with validation on sewage-system data from ten urban catchments[END_REF]. As a consequence, the management of sewer solids is a key component in developing a holistic approach to the design and operation of wastewater systems (Ashley et al., 2004a).

The Environmental research centre (Centro de Investigaciones en Ingeniería Ambiental -CIIA) at the Universidad de los Andes in cooperation with the sewer system managers (Water utilities of Bogotá, EAAB) carried out an extensive sewer sediment characterisation program. Sampling stations included sewer pipes and manholes, gully pots and storm channels. They were selected based on experts' knowledge (by means of surveys), customers claims data base, and a GISbased prioritizing matrix which includes aspects such as road type, land use, transport capacity, ageing of the system, network material, and population density (Uniandes, 2008b). A total number of 2293 simple samples were characterized, including 2121 manholes, 460 gully pots and 3 storm channels. Main sediment characteristics and properties in the Bogotá's urban drainage system are presented in Table 3. Additionally, a GIS-based tool named SIGTASED was developed for quantifying the amount of sediments which are accumulated in the sewer system at a sub-catchment scale (Uniandes, 2008b). The main formulation used in the tool, based on regression analysis of field data surveyed from Cleveland (OH), is known as the Cleveland simplest model [START_REF] Fan | Sewer-Sediment Control: Overview of an EPA Wet Weather Flow Research Program[END_REF] which estimates the sediment accumulation rate based on the sewer system length, per capita flow including infiltration and sewer system average slope. The software tool uses information such as sewer network characteristics (pipe length, diameter and slope), address points and the bi-monthly consumption rate (m 3 ) for estimating the accumulation rates. Figure 3 presents the GIS tool user interface in which a coloured map represents the expected accumulation rate in each of the sub-catchments. , 2008a). The CCME WQI is a tool for simplifying the reporting of water quality data and gives a broad overview of environmental performance. Water quality can be ranked by relating it to one of the following categories: excellent, good, fair, marginal and poor. In the Bogotá case, The CCME WQI clearly demonstrates that the water quality in the receiving water courses is frequently threatened or impaired and conditions often depart from natural or desirable levels (see Table 4). In order to improve water quality conditions in the Bogotá's urban rivers, the local environmental authority (SDA) and the environmental research centre (CIIA) at Universidad de los Andes set gradual water quality objectives (WQO) for each of the four reaches in which each one of the main receiving water courses in Bogotá (Salitre, Fucha, Tunjuelo and Torca rivers) were divided. Four different temporal stages were established as follows: 4, 10, 20 and 40 years. These objectives were defined using monitoring records from the water quality monitoring network in the urban receiving water courses in Bogotá (between July 2006 and May 2007) to assess current state and modelling results using the QUAL2K software for prospective scenarios [START_REF] Raciny | Modelación de la Calidad del Agua de los Ríos Urbanos Salitre, Fucha y Tunjuelo. XXIII Latinamerican Congress on Hydraulic (IARH)[END_REF]. Temporal stages are: WQO to be achieved in 4 years (Water quality objectives take in to account expected sanitation infrastructure developments), To be achieved in 10 years (It is based on the planned water quality objectives for the Bogotá River by 2020), and To be achieved in 20 and 40 years (The main goal behind these objectives is to preserve aquatic environments. As a healthy ecosystem relies on appropriate quality of the sediments, sediment quality objectives were defined regarding heavy metals, polycyclic aromatic hydrocarbon (PAH) and organo-chlorine pesticides and polychloride biphenyl (PCB)).

THE APPLICATION OF AN INTEGRATED MODELLING TOOL IN THE CONTEXT OF BOGOTÁ CITY

Different modelling approaches can be applied, where the main difference is the amount of data required, the information that can be obtained from the model, the analysis performed and the simulation period. There are a variety of modelling approaches to describe water motion as well as the transport and conversion of matter. [START_REF] Schuetze | Suitable integrated modelling -based on simplified models[END_REF] concluded that the combination of sub-models with different complexity and models with a modular building structure facilitate the integrated modelling. Many different tools have been developed such as GEMINI [START_REF] Guderian | Evaluation of total emissions from treatment plants and combined sewer overflows[END_REF]; SYNOPSIS [START_REF] Schuetze | Integrated simulation and optimun control of the urban wastewater system[END_REF][START_REF] Schütze | Modelling, Simulation and Control of Urban Wastewater Systems[END_REF], WEST [START_REF] Meirlaen | Immission based real-time control of the integrated urban wastewater system[END_REF], REBEKA (Rauch et al., 2002a;[START_REF] Fankhauser | REBEKA II -a stochastic software tool for assessing impacts of urban drainage on receiving waters[END_REF], SEWSYS [START_REF] Ahlman | Modelling of Substance Flows in Urban Drainage Systems[END_REF], SIMBA (ifak), Mannina et al. model (Mannina et al., 2004a;Mannina et al., 2004b), CITY DRAIN [START_REF] Achleitner | Modular conceptual modelling in urban drainage -Development and application of City Drain[END_REF][START_REF] Achleitner | CITY DRAIN © -An open source approach for simulation of integrated urban drainage systems[END_REF] and SMUSI [START_REF] Muschalla | Optimization of integrated urban wastewater systems using multiobjective evolution strategies[END_REF] among others.

Achleitner ( 2006) developed an open source toolbox based on the European Water Framework Directive (WFD) requirement. This model, named CITY DRAIN, has been developed using Matlab/Simulink©. The key aspect of using this modelling environment is that the user can choose and freely adapt from a block library representing the elements of the total system [START_REF] Rauch | Simulating integrated urban wastewater systems -methodology and examples[END_REF]. The software is designed for the integrated modelling of urban drainage systems aiming to provide a flexible and adjustable tool for different scenarios. One of the main advantages of this tool is the possibility to modify the code behind it or even to implement and add new blocks according to specific needs [START_REF] Vojinovic | Integrated urban water systems modelling with a simplified surrogate modular approach[END_REF]. Overall the computation is based on a fixed discrete time step approach where each subsystem uses the same time increments, usually being predetermined by the temporal resolution of the rain data used.

For allowing long term simulations the blocks implemented are based on simple conceptual models for hydraulics (frequently denoted as hydrological models [START_REF] Durchschlag | Total emissions from combined sewer overflow and wastewater treatment plants[END_REF]). Mass transport of pollutants is implemented for conservative matter/tracer substances.

An application of the City Drain toolbox in the context of Bogotá can be found in [START_REF] Fonseca | Herramienta computacional para el análisis y evaluación del comportamiento de aliviaderos de alcantarillados combinados (Caso estudio[END_REF], where this toolbox was coupled with fuzzy logic techniques in order to assess CSOs performance. The assessment is based on operational parameters, design standards, receiving bodies' water quality regulations and experts' knowledge. There are four variables (operative dilution factors, CSO setting, dry weather CSO spills and receiving bodies' water quality impact) which collectively give an assessment value ranked between 0 and 10 according with the fuzzy logic rules in each of the calculation time steps. The City Drain toolbox coupled with the fuzzy logic module was applied to the sub-catchment "El Virrey" (see Figure 4). Nine CSOs structures were dynamically assessed for two day simulation period with two hypothetical rain events. Results from CSO No. 1 indicate a value of 6.29 as the minimum and 9.25 as the maximum in the dynamic CSO assessment (average value of 7.76 with a standard deviation of 0.43). In contrast, downstream CSO No. 7 has a low assessment value of about 4 in dry weather period due to the occurrence of CSO spills even during this hydrological condition. As during rainfall events the operative dilution factor increases and if there are not a considerable impact on the water quality in the receiving body, the CSO assessment can considerable improve. It was concluded that a CSO assessment should be based on not only their typological characteristics but on operative parameters, upstream CSOs structures performance, water quality state and impact on the receiving watercourse.

ONGOING WORK TOWARDS AN INTEGRATED MANAGEMENT AND MODELLING IN BOGOTÁ: PERSPECTIVES AND CONCLUSIONS

This paper has provided an overview of recent research and development for an integrated approach to urban drainage system management in Bogotá which pretends to maximise the benefits from the resources, supporting data and software tools available. Complementarily, ongoing research in the context of Bogotá plans to contribute to the development of an integrated urban drainage system modelling framework. Modelling work is planned at two different scales: sub-catchment modelling and macromodelling scale. The subcatchment modelling scale aims to perform comparative models of sewer systems using two study cases, including one experimental subcatchment in Bogotá (Colombia) (Uniandes, 2001) and one subcatchment in Linz (Austria) [START_REF] Hochedlinger | Online UV-VIS measurements -The basis for future pollution based sewer real time control in Linz[END_REF]. The main goal is to test and implement computational routines for improved representation of sediment transport processes and water quality transformations in the sewer system compartment. The gained results will serve as a basis for identifying the modelling scheme to use when looking at the macro-modelling scale. Comparing results and performance of different case studies, with different characteristics in their boundary conditions and data sets, offers the opportunity to perform more robust analysis and tests on the developed modelling approaches. Modelling of sediment transport becomes useful for the prediction of the water quality in the receiving waters and for the assessment of the efficiency of different measures to reduce pollution loads [START_REF] Berlamont | Modelling(partly) cohesive sediment transport in sewer systems[END_REF]. In addition, models allowing for the simulation of solid production and transport during storm events can be very useful for dynamic management and operation of WWTPs. point source pollution can lead to implementation of ineffective and inappropriate measurements [START_REF] Vaze | Experimental study of pollutant accumulation on an urban road surface[END_REF].

Integrated modelling is planned at a macromodelling scale aiming to increase the understanding about the comparative performance of different types of sewer systems (combined and separate) by means of an analysis of Bogotá's urban drainage system, in which separate systems are used under a high presence of wrong connections. Obtained results could improve the knowledge about how to manage and/or operate, and how to prioritize investments on individual parts of the urban drainage system under such specific conditions but allowing transferability to other developing cities. Additionally, this modelling scale is planned to be used to perform a modelbased effectiveness evaluation if implementing structural best management practices (BMPs) in the stormwater system of Bogotá. It is proposed that this task be based on the BMPs' comparative assessment methodology proposed by [START_REF] Scholes | A systematic approach for the comparative assessment of stormwater pollutant removal potentials[END_REF], and it aims to develop a model-based framework to support decision-making processes to implement BMPs in Bogotá and other cities in developing countries. It is planned that macromodelling analysis be based on the ongoing implementation of the Bogotá city urban drainage system using the City Drain model initially presented by Rodríguez et al. (2008a). From this implementation, it is clear the complexity of the system and how the interactions between the different elements (i.e. rural catchments, combined and separated urban catchments, combined sewer overflows, channels, interceptors, pumping stations, etc.) determine the water quality of the receiving water courses.

Figure 2 :

 2 Figure 1: (a) 69 properties in which dye experiments and CCTV inspections were performed (b) 19 of 69 properties with presence of wrong connections[START_REF] Mestra | Identificación de conexiones erradas en alcantarillados separados mediante la aplicación de herramientas computacionales[END_REF]

8th

  World Wide Workshop for Young Environmental Scientists WWW-YES 2009: Urban waters: resource or risks? (2-5 June 2009)

Figure 3 :

 3 Figure 3: SIGTASED user interface (Uniandes, 2008b)

Figure 4 :

 4 Figure 4: City Drain application to the El Virrey Sub-catchment -obtained dynamic results from CSO No. 1 and No. 7 (Fonseca et al., 2008)

Table 1 : Measured domestic pollutant loads for the Bogotá city (Uniandes, 2008a) Per capita pollutant loads (g/inhabitant-day) Total pollutant loads (Ton/day) Parameter

 1

						Reported in
			Minimum Maximum	Average	specialized	Average
						literature*
	Ammonium (NH 4	+ )	6.76	13.46	9.42	8	66.4
	BOD 5		48.41	77.49	59.82	60	421.7
	COD		108.87	194.59	151.59	110	1068.7
	Soluble Phosphorus	1.06	1.97	1.53		10.8
	Total Phosphorus	2.04	2.79	2.45	3	17.3
	Nitrates (NO 3 )		0.08	1.55	0.58		4.1
	Nitrites (NO 2 )		0.02	0.28	0.10		0.7
	Total Kjeldahl Nitrogen (TKN)	10.74	18.64	15.40	12	108.5
	Total suspended solids (TSS)	43.54	171.91	92.84	60	654.6
	Volatile suspended solids (VSS)	39.96	85.95	55.36	48	390.3
	Volatile total solids (VTS)	63.24	150.84	99.60		702.2
	Total solids (TS)	143.76	493.94	268.50		1893
	Sulphates (SO 4 )	12.69	32.50	21.10	20	148.7
	Sulphurs (S = )		0.53	1.44	0.94		6.6

*

[START_REF] Butler | Urban Drainage[END_REF]

The secretary of district for the environment (Secretaría Distrital de Ambiente -SDA) carried out a detailed monitoring program between 2003 and 2007 in order to assess industrial wastewater loads in Bogotá city. Nearly 600 industries were monitored including 148 related with food industry, 107 with leather industry, 109 with oil and gasoline stations, 11 with printing industry, 52 with metal-mechanics industry, 58 with chemical industry, 55 with textile industry and 43 with service and health industry. Table 2 presents measured pollutant loads from industrial 8th World Wide Workshop for Young Environmental Scientists WWW-YES 2009: Urban waters: resource or risks? (2-5 June 2009) 4 / 13 activities of selected parameters in Bogotá and their contribution in comparison with domestic pollutant loads.

Table 2 : Comparison between measured domestic and industrial pollutant loads in the Bogotá city (Uniandes, 2008a)

 2

	Parameters	Total pollutant load from domestic activities (Ton/day)	Total pollutant (Ton/day) load from industrial activities	Industrial contribution
	BOD 5	421.7	43.1	9.27%
	COD	1068.7	121.5	10.21%
	TSS	654.6	19	2.81%
	Sulphates (SO 4 )	148.7	0.5	0.37%
	Sulphurs (S = )	6.6	0.15	2.17%

Table 3 : Sediment characteristics and properties in the Bogotá's urban drainage system (Uniandes, 2008b)

 3

	Parameter	Sewer pipes and Manholes Residential Industrial Commercial	Gully pots	Storm Channels
	pH	6.39 -10.77 6.52 -8.42	7.1 -8.57	7.1 -11.29	7
	Granulometry (mm)	Fine sediment 0.00775-0.09922 (D50) -Coarse sediment 0.074-4.57 (D60)*
	Density (kg/m 3 )	1262.9 -2007.9 (average 1583.88; standard deviation 162.3)*
	Viscosity (cps)	250 -276000 (average 24844.6; standard deviation 55198.1)*
	%TS	41.7 -67.6	45.1 -68.5	45.7 -71.7	61.9 -78.1	54 -80.6
	%VS	5.3 -16.9	5.6 -24.7	5.6 -19.5	3.9 -12.6	1.9 -10.5
	COD (g/kg)	34.5 -176.5	60 -1125	65 -133	24 -190	16 -116
	Benthic demand (g/m 2 *day)			1.2 -12.17*
	Total Kjeldahl Nitrogen -TKN (%)	0.18 -0.88	0.185 -2.6	0.17 -0.42	0.006 -0.86	0.31 -0.32
	Amoniacal nitrogen 0.003 -0.21 0.004 -0.06 0.007 -0.14	0 -0.21	0.006 -0.01
	Phosphorus	0.08 -1.1	0.17 -0.4	0.14 -0.23	0.06 -0.26	0.04 -0.17
	Fat oil and grease (%)			0.42 -9.47*
	Faecal coliforms (MPN)	1.2x10 3 -10.4x10 7	4.8x10 3 -1.5x10 6	5x10 3 -2.4x10 6 <2.6-5.4x10 4	3.3 x10 3 -5.1x10 4
	*Values are representative for sewer pipes, manholes, gully pots and storm channels

Table 4 : CCME WQI for Bogotá's urban rivers (Uniandes, 2008a)

 4

		Reach 1	Reach 2	Reach 3	Reach 4
	Salitre River	50 (marginal)	31 (poor)	31 (poor)	59 (marginal)
	Fucha River	100 (excellent)	34 (poor)	45 (marginal)	27 (poor)
	Tunjuelo River	80 (good)	27 (poor)	31 (poor)	23 (poor)
	Torca River	64 (marginal) 71 (acceptable)	-	-


ADAPTIVE DENSITY ESTIMATION UNDER WEAK DEPENDENCE GANNAZ IR ÈNE (1) AND WINTENBERGER OLIVIER (2)

Keywords: 1991 Mathematics Subject Classification. Primary 62G07; Secondary 60G10, 60G99, 62G20 adaptive estimation, cross validation, hard thresholding, near minimax results, nonparametric density estimation, soft thresholding, wavelets, weak dependence

come

Adaptive density estimation under weak dependence

Introduction

Let (X t ) t∈Z be a real valued time series admitting a common marginal density f that is compactly supported. The general purpose of this paper is to estimate f by wavelet estimators f n constructed from n observations (X 1 , . . . , X n ). In their seminal paper [START_REF] Donoho | Density estimation by wavelet thresholding[END_REF] [START_REF] Donoho | Density estimation by wavelet thresholding[END_REF] showed that projection-like linear estimators are not optimal: introduction of nonlinearity via thresholds of wavelet coefficients is investigated. Wavelets thresholding provides estimators which adapt themselves to the unknown smoothness of f , we refer to [START_REF] Vannucci | Nonparametric density estimation using wavelets[END_REF] [START_REF] Vannucci | Nonparametric density estimation using wavelets[END_REF] for a survey of the use of wavelet bases in density estimation. The present work extends near minimax results of soft and hard-threshold estimators from the independent and identically distributed (iid for short) framework, see Theorem 5 in [START_REF] Donoho | Density estimation by wavelet thresholding[END_REF] [START_REF] Donoho | Density estimation by wavelet thresholding[END_REF], to cases where weak dependence between variables occurs.

Our main assumptions give bounds for covariance terms as decreasing sequences which tend to zero when the gap between the past and the future of the time series goes to infinity. In order to give examples satisfying these conditions, we introduce coefficients that give bounds of covariance terms and that are computable for a large class of models. Weak dependent coefficients, introduced by Doukhan and Louhichi (1999) [START_REF] Doukhan | A new weak dependence condition and applications to moment inequalities[END_REF], as mixing ones, are well-adapted to that purpose. Using β-mixing coefficients Tribouley and Viennet (1998) [START_REF] Tribouley | Lp-adaptive density estimation in a β-mixing framework[END_REF] proposed minimax estimators with respect to the Mean Integrated Square Error (MISE for short). [START_REF] Comte | Adaptive estimation of the stationary density of discrete and continuous time mixing processes[END_REF] [START_REF] Comte | Adaptive estimation of the stationary density of discrete and continuous time mixing processes[END_REF] obtained near-minimax results using α-mixing coefficients. The loss of a logarithmic factor in the convergence rate in this last paper is balanced by the generality of the context, as the class of α-mixing models is larger than the one of β-mixing models.

However, α and β-mixing coefficients are not easy to compute for some models and are useless for others; [START_REF] Andrews | Non strong mixing autoregressive processes[END_REF] [1] proved that the mixing coefficients of the stationary solution of the AR(1) model

X t = 1 2 (X t-1 + ξ t )
, where (ξ t ) t∈Z iid with a Bernouilli law of parameter 1/2, (

do not tend to zero as the gap from the past to the future of the time series goes to infinity. Mixing coefficients do not behave nicely in this case as, through a reversion of time, the Markov chain solution of (1.1) is a dynamical system, i.e. X t-1 = T (X t ) for some transformation T , namely T (x) = 2x1 1 0≤x<1/2 + (2x -1)1 1 1/2≤x≤1 . So called weak dependence coefficients have been recently developed to deal with such processes, see Dedecker et al. [START_REF] Dedecker | Weak Dependence: Models, Theory and Applications[END_REF], Maume-Deschamps [START_REF] Maume-Deschamps | Exponential inequalities and functional estimations for weak dependent data ; applications to dynamical systems[END_REF] and references therein. Introduced by [START_REF] Dedecker | New dependence coefficients: Examples and applications to statistics[END_REF] in [START_REF] Dedecker | New dependence coefficients: Examples and applications to statistics[END_REF], φ-weak dependence coefficients give sharp bounds on the covariance terms of dynamical systems, such as the stationary solution to (1.1). Using these coefficients, we prove near-minimax results of thresholded wavelet estimators for dynamical systems called expanding maps. To our knowledge, only non adaptive density estimation has been studied in this non-mixing context, see for instance Bosq et Guegan [START_REF] Bosq | Nonparametric estimation of the chaotic function and the invariant measure of a dynamical system[END_REF], Prieur [START_REF] Prieur | Applications statistiques de suites faiblement dépendantes et de systèmes dynamiques[END_REF] and Maume-Deschamps [START_REF] Maume-Deschamps | Exponential inequalities and functional estimations for weak dependent data ; applications to dynamical systems[END_REF].

The advantage of our approach is also to treat in one draw many other contexts of dependence. We prove that near-minimaxity still holds for a very large class of models using λ-weak dependence coefficients, defined by Doukhan and Wintenberger, 2007, [14]. We pay for generality by adding up conditions on the joint densities of the couples (X 0 , X r ) for all r > 0. These conditions are not restrictive as it is satisfied for many econometric models such as ARMA, GARCH, ARCH, LARCH, MA models.

The estimation scheme is based on Donoho et al.'s procedures developed in [START_REF] Donoho | Density estimation by wavelet thresholding[END_REF] for the iid case, and it is adaptive with respect to the regularity of f . Soft and hard-threshold levels (λ j ) j 0 ≤j≤j 1 are chosen equal to K j/n for some K > 0. Note that the constant K and the highest resolution level j 1 depend on the weak dependence properties of the observations. If weak dependence properties are known, estimators are near minimax: same rates as in the iid setting are achieved for mean-L p errors with 1 ≤ p < ∞. If weak dependence properties are unknown, we develop cross-validation procedures to approximate threshold levels λj and the highest resolution level j 1 . We check on simulations that corresponding estimators are adaptive with respect to the regularity of f and to weak dependence properties of (X t ) t∈Z . The order of errors of approximations in the dependence cases are very close to the one in the iid cases.

We believe that we obtain such good results as we work on simulations of processes satisfying our main Assumption (D). This assumption consists of the exponential decay of the covariance terms. We give in this paper some simulations study of dynamical systems that do not satisfy this Assumption (D). Comparing the behavior of our estimators with the kernel ones show that ours are less efficient when covariance terms do not decrease exponentially fast. We also prove that the error terms of our estimators are unbounded in some cases, due to terms of covariances that decrease too slowly. This is a restriction of our procedure, based on thresholding wavelet coefficients.

The paper is structured as follows. In Section 2, we give notation, we introduce estimation procedures and we formulate weak dependence assumptions. Main results are given in Section 3 and examples of models in Section 4. Cross-validation procedures and their accuracy on simulations are developed in Section 5. Proofs are relegated in the last Section.

Preliminaries

2.1. Notation. We restrict ourselves to the estimation of a density f which is compactly supported. We suppose that f is supported by [-B, B] for some B > 0. For all p ≥ 1, L p denotes the space of all functions f such that

f p p = |f (x)| p dx < ∞.
2.2. Density estimators. Throughout the paper, we work within an r-regular orthonormal multiresolution analysis of L 2 (endowed with the usual inner product), associated with a compactly supported scaling function φ and a compactly supported mother wavelet ψ. Without loss of generality, we suppose that the support of functions φ and ψ is included in an interval [-A, A] for some A > 0. Let us recall that φ and ψ generate orthonormal basis by dilatations and translations: for a given primary resolution level j 0 , the functions {φ j,k : x → 2 j/2 φ(2 j x -k)} k∈Z and {ψ j,k : x → 2 j/2 ψ(2 j x -k)} k∈Z are such that the family

{φ j 0 ,k , k ∈ Z, ψ j,k j ≥ j 0 , k ∈ Z}
is an orthonormal base of L 2 . Any function f ∈ L 2 can thus be decomposed as

f = k∈Z α j 0 ,k φ j 0 ,k + ∞ j=j 0 k∈Z β j,k ψ j,k , where α j,k = f (x)φ j,k (x)dx, β j,k = f (x)ψ j,k (x)dx.
The nonlinear estimator developed in [START_REF] Donoho | Density estimation by wavelet thresholding[END_REF] is defined by the equation

f n = k∈Z α j 0 ,k φ j 0 ,k + j 1 j=j 0 k∈Z γ λ j ( β j,k )ψ j,k ,
where αj,k = n -1 n i=1 φ j,k (X i ) and βj,k = n -1 n i=1 ψ j,k (X i ), and where γ λ is a threshold function of level λ. The authors consider both hard and soft thresholding functions, corresponding respectively to γ λ (β) = β1 1 |β|>λ and γ λ (β) = (|β| -λ) + sign(β). If no distinction is done in the sequel, both hard and soft thresholding estimators are concerned.

Let s, π, r be three positive real numbers satisfying s + 1/2 -1/π > 0. We assume that f belongs to the Besov ball B s π,r (M 1 ) on the real line, i.e. f s,π,r ≤ M 1 where

f s,π,r = |α 0,0 | +   j∈N 2 j(sπ+π/2-1) k∈Z |β j,k | π r/π   1/r .
Approximation errors of an estimator f n are expressed as E f n -f p p for p ≥ 1. Associated minimax rates are the best convergence decrease α of the worst approximation error we may achieve over all estimators f n :

inf fn sup f ∈B s π,r (C) E f n -f p p = O n -pα .
The minimax rate α is determined in [START_REF] Donoho | Density estimation by wavelet thresholding[END_REF] as:

α = α + = (s/(1 + 2s) if ǫ ≥ 0, α -= (s -1/π + 1/p)/(1 + 2s -2/π) if ǫ ≤ 0, where ǫ = sπ -(p -π)/2. (2.1)
2.3. Weakly dependent assumption. Throughout the paper, the symbol δ denotes with no distinction φ or ψ and δ j,k (x) = δ j,k (x) -E δ j,k (X 0 ) for all integers j ≥ 0 and k. Define for all positive integers u, v the quantities

C j,k u,v (r) = sup max s i+1 -s i =s u+1 -su=r {|Cov( δ j,k (X s 1 ) • • • δ j,k (X su ), δ j,k (X s u+1 ) • • • δ j,k (X s u+v ))|}. (2.2)
Functions φ and ψ play a symmetric role through δ in this setting. As stressed in [START_REF] Doukhan | A new weak dependence condition and applications to moment inequalities[END_REF], bounds on covariance terms C j,k u,v (r) are useful to extend asymptotic results from the iid case. Now we can state the main assumption of this paper:

(D): There exists a sequence ρ(r) such that for all r ≥ 0, all indexes j, k, u, v, we have

C j,k u,v (r) ≤ (u+v+uv)/2 (2 j/2 M 2 ) u+v-2 ρ(r) where M 2 is a constant satisfying δ ∞ ≤ M 2 . (D1)
Moreover, there exist real numbers a, b, C 0 > 0 depending only on δ, f and on the dependence properties of (X t ) t∈Z such that ρ(r) ≤ C 0 exp(-ar b ) for all r > 0. (D2)

In Section 4, we give explicit conditions on the stationary process (X t ) t∈Z in order that it satisfies Assumption (D). When it is possible, the values of the constants a, b, C 0 and M 2 are given.

Main results

Let (X t ) t∈Z be a stationary real valued time series.

3.1. A useful lemma. Under similar conditions than Assumption (D), moments inequalities of even orders and Bernstein's type inequalities for the sums n i=1 δ j,k (X i ) are respectively given in [START_REF] Doukhan | A new weak dependence condition and applications to moment inequalities[END_REF] and [START_REF] Doukhan | A Bernstein type inequality for times series[END_REF]. The following Lemma recall these inequalities applied on the quantities β j,k . They remain valid for α j,k as φ and ψ play the same role in (D). Lemma 3.1. If (D) holds then for all even integer q ≥ 2, for all λ ≥ 0 and for j, k such that 0 ≤ j ≤ log n and 0 ≤ k ≤ 2 j -1:

E| β j,k -β j,k | q ≤ C 1 n -q/2 , (3.1)
P | β j,k -β j,k | ≥ λ ≤ 2 exp - nλ 2 C 2 + C 3 (2 j /n) b/(2(1+2b)) ( √ nλ) (2+3b)/(1+2b) , (3.2)
where C 1 , C 2 and C 3 are constants depending on q and on the constants of Assumption (D): a, b, C 0 and M 2 .

The proof of this Lemma is given in Section 6.1. Notice that C 3 depends deeply on the dependence context through C 0 , see Propositions 4.1 and 4.2 for more details.

3.2.

Near-minimax results of thresholded wavelet estimators. Following results are extensions to weak dependence settings of Theorem 5 of [START_REF] Donoho | Density estimation by wavelet thresholding[END_REF].

Theorem 3.1. Suppose that f ∈ B s π,r (M 1 ) with 1/π < s < N/2 where N is the regularity of the function ψ. If (D) holds, then for each 1 ≤ p < ∞ there exists a constant C(N, p, a, b, C 0 , M 1 , M 2 , A, B) such that E[ f n -f p p ] ≤ C        log n n pα if ǫ = 0 log n n pα (log n) (p/2-(1∧π)/r) + if ǫ = 0
where the minimax rate α and the parameter ǫ are given in (2.1). Here j 0 is chosen as the smallest integer larger than log(n)(1 + N ) -1 , j 1 is the largest integer smaller than log(n log -2/b-3 (n)) and λ j = K j/n for a sufficiently large constant K > 0.

A sketch of the proof of this Theorem is given in Section 6.2.

We refer to [START_REF] Meyer | Wavelets and Operators[END_REF] for the definition of the parameter N , the regularity of the wavelet function ψ. The condition s < N/2 ensures the sparsity of the wavelet coefficients of the density f . This condition is not restrictive as N can be chosen sufficiently large in practice.

The estimators f n are the same than in the iid case given in [START_REF] Donoho | Density estimation by wavelet thresholding[END_REF], except for the highest resolution level j 1 which is smaller here. This restriction is needed in the weak dependence context due to the Bernstein's type inequality (3.2) which is not as sharp as the one in the iid case. But this restriction does not perturb the rate of convergence which is the same as the one obtained in the iid case by [START_REF] Donoho | Density estimation by wavelet thresholding[END_REF].

The constant K > 0 plays a key role in the asymptotic behavior of f n . From (3.2), we infer that the constant K depends on the parameters a, b, C 0 , M 2 through C 1 , C 2 , C 3 in an intricate way. Then how sufficiently large the constant K must be deeply depends on the dependence structure of observations (X 1 , . . . , X n ). Then contrarily to the iid case, we are not able to develop direct procedures based on the observations (X 1 , . . . , X n ) that chose a convenient parameter K like in [START_REF] Juditsky | On minimax density estimation on R[END_REF]. In Section 5, we propose cross validation procedures to determine the threshold levels λ j when the weak dependence properties of the process (X t ) t∈Z are unknown.

Examples

In this Section, we give examples of models that satisfy Assumption (D) using φ and λ-weak dependence coefficients introduced respectively in [START_REF] Dedecker | New dependence coefficients: Examples and applications to statistics[END_REF] and [START_REF] Doukhan | Invariance principle for new weakly dependent stationary models[END_REF]. For each of them, we proceed in two steps: Firstly we give sufficient conditions on these coefficients for ensuring Assumption (D) in Propositions 4.1 and 4.2 and secondly we give in Subsections 4.2 and 4.4 examples of models satisfying such conditions.

A Lipschitz function h : R u → R for some u ∈ N * is a function such that Lip (h) < ∞ with Lip (h) = sup (a 1 ,...,au) =(b 1 ,...,bu) |h(a 1 , . . . , a u ) -h(b 1 , . . . , b u )| |a 1 -b 1 | + • • • + |a u -b u | .
As Lipschitz functions play an important role in weak dependence contexts we restrict ourselves to the cases where N > 4. This assumption on the regularity of the wavelet functions implies that φ and ψ can be chosen as Lipschitz functions, as established in [START_REF] Daubechies | Ten Lectures on Wavelets[END_REF]. Note also that f ∞ < ∞ as f ∈ B s π,r (M 1 ), see equation ( 15) in [START_REF] Donoho | Density estimation by wavelet thresholding[END_REF]. For convenience, we denote with no distinction ψ j,k (x)-β j,k and φ j,k (x) -α j,k as δ j,k (x) for any integers j ≥ 0 and k.

Weak dependence coefficients is to generalize mixing ones. Let us recall that α-mixing coefficients can be defined in a similar way by two equations:

α(r) = sup 1 ≤ ℓ i + r ≤ j1 ≤ • • • ≤ j ℓ g ∞ ≤ 1 E|E (g(X i 1 , . . . , X iu )|σ({X j , j ≤ i})) -E (g(X i 1 , . . . , X iu )) |, α(r) = sup (u, v) ∈ N * × N * i1 ≤ • • • ≤ iu ≤ iu + r ≤ iu+1 ≤ • • • ≤ iu+v f ∞, g ∞ ≤ 1 |Cov(f (X i 1 , . . . , X iu ), g(X i u+1 , . . . , X i u+v ))|.
These coefficients measure the dependence between the past and the future values of the process (X t ) t∈Z as the gap r between past and future goes to infinity. As these coefficients are often too restrictive, the authors of [START_REF] Dedecker | An empirical central limit theorem for dependent sequences[END_REF] release them by considering a supremum taken on functions with bounded variations or on Lipschitz bounded functions rather than uniformly bounded functions. These different choices of functions sets lead to different coefficients of weak dependence, namely φ and λ respectively. We give hereafter the precise definition of φ and λ-weak dependence coefficients.

4.1. φ-weak dependence. Bounded variations functions are defined as follows: let BV and BV 1 denote the sets of functions g supported on [-A, A] satisfying respectively g BV < +∞ and g BV ≤ 1 where

g BV = |g(-A)| + sup n∈N sup a 0 =-A<a 1 <•••<an=A n i=1 |g(a i ) -g(a i-1 )|.
Let (Ω, A, P) be a probability space, M a σ-algebra of A and let X = (X 1 , . . . , X v ), for v ≥ 1, be a collection of real valued random variables X i defined on A. We define the coefficient φ as it was introduced in [START_REF] Dedecker | An empirical central limit theorem for dependent sequences[END_REF] by the equation:

φ(M, X 1 , . . . , X v ) = sup g 1 ,...,gv∈BV 1 v i=1 g i (x i )P X|M (dx) - v i=1 g i (x i )P X (dx) ∞ ,
where dx = (dx 1 , . . . , dx v ). The coefficients φ(r) are now defined by the equation

φ(r) = max 1≤ℓ 1 ℓ sup i+r≤j 1 ≤•••≤j ℓ φ(σ({X j ; j ≤ i}), X j 1 , . . . , X j ℓ ) .
These coefficients are multivariate extensions of φ 1 (r) defined in [START_REF] Dedecker | New dependence coefficients: Examples and applications to statistics[END_REF], see also [START_REF] Maume-Deschamps | Exponential inequalities and functional estimations for weak dependent data ; applications to dynamical systems[END_REF]. Instead of mixing coefficients, they efficiently treat the dependence structure of dynamical systems and associated Markov chains. A process (X t ) t∈Z is said to be φ-weakly dependent if the series φ(r) goes to 0 when r goes to infinity, i.e. when the gap between observations from the future and observations from the past goes to infinity. Introducing φ-weakly dependent processes is useful in the present framework due to the following links with the Assumption (D):

Proposition 4.1. Assume that (X t ) t∈Z is a process such that there exist a, b, c > 0 satisfying

φ(r) ≤ c v exp(-ar b ), (4.1)
then Assumption (D) holds with

M 2 = 2( δ ∞ + ALip δ) and C 0 = 4c( δ ∞ + ALip δ) f ∞ δ 1 .
The proof of this Proposition is given in Subsection 6.3.

4.2.

Examples of φ-weakly dependent processes. Following the work of [START_REF] Dedecker | An empirical central limit theorem for dependent sequences[END_REF], we give a general class of models where (4.1) is satisfied.

Lemma 4.1. Assume that (X t ) t∈Z is a process satisfying the Markov property, taking values in [0, 1] and such that there exist constants a, b, c > 0 satisfying, for any functions g, k with g ∈ BV 1 and E|k(X 0 )| < ∞, and for all r ≥ 0,

|Cov(k(X 0 ), g(X r ))| ≤ E|k(X 0 )| exp(ar -b ), (4.2)
E(g(X r |X 0 = •) BV ≤ c, (4.3
)

then φ v (r) ≤ c v exp(-ar b
) and the conclusions of Proposition 4.1 follow.

The proof of this Lemma is given in Subsection 6.3. Various examples of processes satisfying conditions of Lemma 4.1 are given in [START_REF] Dedecker | An empirical central limit theorem for dependent sequences[END_REF]. We recall here the case of Markov chains obtained by time reversing expanding maps, as they are extensions of Andrew's example (1.1).

Let us define stationary Markov chains (X t ) t∈Z associated with dynamical systems through a reversion of the time as non degenerate stationary solutions of the recurrent equation

X t = T i (X t-i ), ∀t ∈ Z, i ∈ N (4.4)
where T : [0, 1] → [0, 1] is a deterministic function.

Remark 1. For such Markov chains, mixing coefficients are useless. Future values write simply as functions of past values via (4.4) and then it is easy to check that α(r) are constant from their definitions. Thus α(r) does not tend to 0 and α-mixing coefficients do not evaluate the dependence of such processes.

A Markov chain (X t ) t∈Z is associated with an expanding map through a reversion of the time if T satisfies • (Regularity) The function T is differentiable, with a continuous derivate T ′ and there exists a grid 0 =

a 0 ≤ a 1 • • • ≤ a k = 1 such that |T ′ (x)| > 0 on ]a i-1 , a i [ for each i = 1, . . . , k. • (Expansivity)
For any integer i, let I i be the set on which the first derivate of T i , (T i ) ′ , is defined. There exists a > 0 and s > 1 such that inf

x∈I i {|(T i ) ′ (x)|} > as i . • (Topological mixing) For any nonempty open sets U , V , there exists i 0 ≥ 1 such that T -i (U ) ∩ V = ∅ for all i ≥ i 0 .
Under these three conditions a non degenerate stationary solution (X t ) t∈N to (4.4) exists and has remarkable properties, see [START_REF] Viana | Stochastic dynamics of deterministic systems[END_REF] for a nice survey. For instance, the process (X t ) t∈Z satisfies the conditions of Proposition 4.1 for some a, c > 0 and b = 1, see [START_REF] Dedecker | New dependence coefficients: Examples and applications to statistics[END_REF]. Moreover, the marginal distribution is absolutely regular and its distribution belongs to BV . Noticing that B 1 1,1 ⊂ BV ⊂ B 1 1,∞ , see e.g. [START_REF] Donoho | Density estimation by wavelet thresholding[END_REF], Theorem 3.1 provides adaptive estimators of the marginal density of (X t ) t∈Z in that context and extends Theorem 2.2 of [START_REF] Vanharen | Estimation par ondelettes dans les systèmes dynamiques[END_REF] where rates of the MISE for non adaptive estimators are given in such context. 4.3. λ-weak dependence. The stationary process (X t ) t∈Z is λ-weakly dependent, as defined in [START_REF] Doukhan | Invariance principle for new weakly dependent stationary models[END_REF], if there exists a sequence of non-negative real numbers λ(r) satisfying λ(r) → 0 as r → ∞ and such that:

Cov h (X i 1 , . . . X iu ) , k X i u+1 , . . . , X i u+v ≤ (u k ∞ Lip(h) + v h ∞ Lip(k) + uvLip (h)Lip (k)) λ(r) for all p-tuples, (i 1 , . . . , i p ) with i 1 ≤ • • • ≤ i u ≤ i u + r ≤ i u+1 ≤ • • • ≤ i p , and for all h ∈ Λ u and h ∈ Λ v where Λ u = {h : R u → R, Lip (h) < ∞, h ∞ < ∞} , for any u ≤ 1.
The λ-weak dependence provides simple bounds of the covariance terms:

Cov δ j,k (X i 1 ) • • • δ j,k (X iu ), δ j,k (X i u+1 ) • • • δ j,k (X i u+v ) ≤ (u + v + uv) δ j,k u+v-2 ∞ (Lip δ j,k ) 2 λ(r).
The right hand side term is bounded by (2 j/2 2 δ ∞ ) u+v-2 Lip δ 2 2 3j λ(r). At this point, we do not achieve Assumption (D1) as this bound depends on j via an extra term 2 3j . An additional assumption is needed to ensure Assumption (D):

(J): The joint densities f r of (X 0 , X r ) exist and are bounded for r > 0.

Proposition 4.2. Assume that (X t ) t∈Z is a λ-weakly dependent process satisfying (J) and such that there exist a ′ , b ′ , c ′ > 0 and a ′′ , b ′′ , c ′′ > 0 with:

λ(r) ≤ c ′ exp(-a ′ r b ′ ) and f r ∞ ≤ c ′′ exp(a ′′ r b ′′ ) for all r > 0. If b ′′ < b ′ then (D) holds for some C 0 > 0 with M 2 = 2 δ ∞ , a = a ′ /4 and b = b ′ .
The proof of this Proposition is given in Subsection 6.3.

4.4.

Examples of λ-weakly dependent processes. Firstly we give in this Section two generic λ-weakly dependent models, Bernoulli shifts and random processes with infinite connections. Secondly we detail the conditions of Proposition 4.2 in three more specific models.

Let H : R Z → [0, 1] be a measurable function. A Bernoulli shift with innovations ξ t is defined as

X t = H ((ξ t-i ) i∈Z ) , t ∈ Z.
According to [START_REF] Doukhan | Invariance principle for new weakly dependent stationary models[END_REF], such Bernoulli shifts are λ-weakly dependent with λ(r

) ≤ 2v([r/2]). If (ξ t ) t∈Z is iid, (v(r)) r>0 is a non-increasing sequence satisfying E H (ξ j , j ∈ Z) -H ξ ′ j , j ∈ Z ≤ v(r)
for all r > 0, where the iid sequence (ξ ′ j ) j∈Z is such that ξ ′ j = ξ j for |j| ≤ r and ξ ′ j independent of ξ j otherwise. If the weak dependence coefficients λ ξ (r) refer to (ξ t ) t∈Z , we can compute these of (X t ) t∈Z . More precisely, the result in [START_REF] Doukhan | Invariance principle for new weakly dependent stationary models[END_REF] states that if there exists ℓ ≥ 0 such that

|H(x) -H(y)| ≤ b s ( z ℓ ∨ 1)|x s -y s |,
for some sequence b j ≥ 0 satisfying j |j|b j < ∞ and where x, y ∈ R Z coincide except for the index s ∈ Z and x = sup

i∈Z |x i |, if E|ξ 0 | m ′ < ∞ for some m ′ ≥ ℓ + 1 then (X t ) t∈Z is λ-weakly dependent with λ(k) ≤ c inf r≤[k/2]   |j|≥r |j|b j + (2r + 1) 2 λ ξ (k -2r) m ′ -1-ℓ m ′ -1+ℓ   , for some c > 0.
Different values of b in Assumption (D2) may arise naturally when realizing the minimum of the equation above, see below for some classical examples.

Another approach is the one of random processes with infinite connections considered in [START_REF] Doukhan | A fixed point approach to model random fields[END_REF]. Let F : [0, 1] Z/{0} × R → [0, 1] be measurable. Under suitable conditions on F , the stationary solution of the equation X t = F ((X j , j = t), ξ t ), a.s., exists and is λ-weakly dependent. We refer the reader to [START_REF] Doukhan | A fixed point approach to model random fields[END_REF] for more details and we end the Section with some specific λ-weakly dependent models.

4.4.1. Infinite moving average. A Bernoulli shift is an infinite moving average process if 4.4.2. LARCH(∞) model. Let (ξ t ) t∈Z be an iid centered real valued sequence and a, a j , j ∈ N * be real numbers. LARCH(∞) models are solutions of the recurrence equation

X t = i∈Z α i ξ t-i . ( 4
X t = ξ t   a + j =0 a j X t-j   . (4.6)
The stationary solution of (4.6) satisfies the condition on λ in Proposition 4.2 with b ′ = 1/2 if there exists K, α > 0 and α < 1 such that a j ≤ Kα |j| for all j = 0. Then Assumption (D2) is ensured with b = 1/2. See [START_REF] Doukhan | Vector valued ARCH infinity processes[END_REF] for applications of this model in econometrics.

4.4.3. Affine model. Let us consider the stationary solution (X t ) t∈Z of the equation

X t = M (X t-1 , X t-2 , . . .)ξ t + f (X t-1 , X t-2 , . . .),
where M and f are both Lipschitz functions. This model contains various time series processes such as ARCH, GARCH, ARMA, ARMA-GARCH, etc. If the ξ t are iid random variables with a bounded marginal density, then (J) holds and the joint densities are uniformly bounded, as stated in the Appendix of [START_REF] Doukhan | Weakly dependent chains with infinite memory[END_REF]. Moreover if the functions M and f have exponentially decreasing Lipschitz coefficients, then conditions of Proposition 4.2 hold with b ′ = 1/2, b ′′ = 0 and Assumption (D2) follows with b = 1/2.

Cross-validation procedures and simulations

The aim of this Section is to evaluate the applicability and the quality of the procedure on simulated data. Even if the estimators are adaptive with respect to the regularity of the density function, a constant appears in the threshold levels that we cannot calibrate if the dependence properties of the observations are unknown. Then we develop a cross-validation scheme in order to apply concretely the estimator on simulated data. We investigate several examples of dependence for the simulated observations that satisfy the convergence result of this paper. We also give counter-examples where the estimators fail to converge near-minimaxly.

5.1. Cross-validation procedures. According to Theorem 3.1, let us fix j 0 as the smallest integer larger than log(n)(1 + N ) -1 and define f HT CV n and f ST CV n respectively as the hard and soft-thresholding estimators associated with threshold levels (( λ j ) j 0 ≤j≤j * ). Here j * = log 2 n and (( λ j ) j 0 ≤j≤j * ) are determined by cross-validation procedures: λ j = Argmin λ CV j (λ) where cross validation criterion are respectively defined by the equations HTCV:

CV j (λ) = k∈Z 1 1 {| b β j,k |≥λ}   β 2 j,k - 2 n(n -1) 1≤i =h≤n ψ j,k (X i )ψ j,k (X h )   , STCV: CV j (λ) = k∈Z 1 1 {| b β j,k |≥λ}   β 2 j,k - 2 n(n -1) 1≤i =h≤n ψ j,k (X i )ψ j,k (X h ) + λ 2   .
These criterion are obtained by approximating the coefficients β j,k by βj,k and the products

β j,k β j,k ′ by h =i ψ j,k (X i )ψ j,k (X h )/(n(n -1)) in the Integrated Square Error.
The estimator j 1 is defined as the smallest integer such that CV j ( λ j ) = 0 for all j 1 ≤ j ≤ j * . Notice that cross-validation procedures may consider larger resolution levels than the estimators f n as j * is larger than j 1 given in Theorem 3.1.

Different dependent samplings satisfying (D).

To illustrate the behavior of this crossvalidation scheme, we simulate three different weak-dependence cases with the same marginal absolutely continuous distribution F . The simulations were carried out as follows:

Case 1: Independent observations are given by X i = F -1 (U i ) where the U i are simulations of independent variables, uniform on [0,1].

Case 2: A φ-weakly dependent process is obtained by the equation

X i = F -1 (G(Y i )) for i = 1, . . . , n with G(x) = 2 x(1 -x)/π and (Y i ) i=1,.
..,n given by

Y 1 = G -1 (U 1 ) and, recursively, Y i = T i-1 (Y 1 ) for 2 ≤ i ≤ n with T (x) = 4x(1 -x).
Note that for all 1 ≤ i ≤ n the Y i admits the repartition function G the invariant distribution of T . Moreover the sequence (Y 1 , . . . , Y n ) satisfies the assumptions of Proposition 4.1, see [START_REF] Prieur | Applications statistiques de suites faiblement dépendantes et de systèmes dynamiques[END_REF] for details.

Case 3: A λ-weakly dependent process resulting from the transform

X i = F -1 (G(Y i )) of variables (Y t ) t∈Z which are solution of Y t = 2(Y t-1 + Y t+1 )/5 + 5ξ t /21
where (ξ t ) t∈Z is an iid sequence of Bernoulli variables with parameter 1/2. The stationary solution of this equation admits the representation

Y t = j∈Z a j ξ t-j ,
where a j = 1/3(1/2) |j| . This solution belongs to [0, 1] and its marginal distribution G is the one of (U +U ′ +ξ 0 )/3 where U and U ′ are independent variables following U([0, 1]). For 1 ≤ i ≤ n, the solution Y i is approximated by Y (N ) i for 1 ≤ j ≤ N and j-N ≤ i ≤ n+N -j where Y (j) i is generated according to the convergent algorithm given in [START_REF] Doukhan | A fixed point approach to model random fields[END_REF]: the initial values Y (0) i are fixed equal to 0 and, given simulated variables (ξ i ) -N ≤i≤n+N , we define recursively Y

(j) i = 2(Y (j-1) i-1 + Y (j-1) i+1 )/5 + 5ξ i /21 for 1 ≤ i ≤ N and i -N ≤ t ≤ n + N -i.
Error of approximation are negligible as decreasing exponentially fast with the parameter N that we fix N = n, see Lemma 6 of [START_REF] Doukhan | A fixed point approach to model random fields[END_REF] for more details. Moreover, Proposition 2.1 of [START_REF] Dedecker | Weak Dependence: Models, Theory and Applications[END_REF] ensures that there exists a, C > 0 such that λ(r) ≤ C exp(-ar) for the process (Y t ) t∈Z and consequently for (X t ) t∈Z .

Two different density functions are considered. The first one is a mixture of a sinus function and a uniform distribution, presenting a discontinuitie, and the second one is a mixture of two gaussian distributions. The calculations were carried out on MATLAB on a Unix environment. We considered n = 2 10 observations repeated M = 500 times for each of the three weak dependence cases. Once the data simulated we applied cross-validation procedures. The usual DWT algorithm proposed by [START_REF] Mallat | A theory for multiresolution signal decomposition: the wavelet representation[END_REF] and implemented in the Wavelab toolbox by Donoho and his collaborators (available on [START_REF] Wavelab | [END_REF]) only gives values of wavelet functions on an equidistant grid. As one needs to compute these values at given data points, we consider an equidistant grid of I points with the number of points I huge with respect to the number of observations n. Then we approximate the values ψ j,k (X i ) by ψ j,k ([X i I]/I) where [x] denotes the closest integer from any real number x. The wavelets used for the decomposition are Daubechies Symmlets with N = 8 zero-moments. Notice that another possible scheme is the algorithm of [START_REF] Vidakovic | Pollen bases and Daubechies-Lagarias algorithm in MATLAB[END_REF] that gives directly the values ψ j,k (X i ). But as it needs much more calculus time it has not been used here for convenience.

In Figures 1 and2 are represented the estimators f HT CV n and f ST CV n and the true density function f in different weakly dependent cases. The quality of the estimators is visually good. According to Figures 1 and2 the weak dependence properties of the simulated data do not seem to affect both procedures of estimation. Density estimators presented in Figures 1 and2 do not detect the discontinuity in the density. Actually, for any finite data set, not enough simulated values are concentrated around the discontinuity to allow estimators to detect it. In Figure 3 threshold levels are represented with respect to resolution levels. Their behaviors are similar in all cases: the threshold levels increase with respect to resolution levels. For small resolutions, both HTCV and STCV procedures are close as λ 2 j is negligible in CV j (λ). For high resolution it is also the case as λ 2 j is big enough to kill almost all β j,k . Moreover these figures tend to confirm that threshold levels do not depend on weak dependence. Finally remark that the curves do not behave in square root of j as theoretical threshold levels given in Theorem 3.1.

After looking at the threshold levels values, we give in Figure 4 the frequencies of the β j,k that are less than λ j with respect to j. As these frequencies are not discretized in two values 0 and 1, we can infer that both f HT CV n and f ST CV n are not equivalent to linear estimators. It is encouraging for both methods as the results in [START_REF] Donoho | Density estimation by wavelet thresholding[END_REF] state that linear estimators are not nearminimax. One can also see in these figures that frequencies of effective thresholds are the same among the weak dependence cases. Finally, means of higher resolution levels are given in last Table 2. According to Theorem 3.1, the values of this parameter do depend on the cases of weak dependence. But no significant differences appear on simulations. Table 2. Means of j 1 on 500 simulations of n = 2 10 observations.

Mean of j 1

5.4.

Comparison with kernel estimators. As results computed in the last Subsection are systematically better for the STCV estimators than for the HTCV ones, we only present in the sequel results for the STCV estimators. Let us now consider the case of a density function that is a mixture of normal distributions. Like in [START_REF] Hardle | Wavelets Approximation and Statistical Applications[END_REF], we compare the quality of the wavelet estimator f ST CV with linear kernel estimators. The kernel used is Epanechnikov's one and we computed two choices for its width parameter: firstly we consider the width given by the rule of thumb of Matlab, (more precisely, the width is equal to (q 3 -q 1 )/(2 * 0.6745) * (4/(3 * n)) 1/5 , where q 1 and q 3 denote respectively the first and the third quartile of the empirical distribution) and secondly we consider the width obtained by cross validation on the mean integrated squared error risk.

In Figure 5 are represented the means of wavelet and kernel estimators in the different cases. Like in the last Subsection, there is no visual difference between the different cases of dependence. In this Figure, it appears that the mean of the kernel estimators fails to detect the two modes of the density when the width is choosing according to the rule of thumb. The bandwidth is overestimated in this case. The quality of the wavelet estimators f ST CV n and the kernel estimators with the width from a cross validation procedure are visually equivalent. To analyse more precisely approximations realized by f ST CV n and by kernel estimators, we represent in Figure 6 the evolution of the mean L p risk with respect to p for the three estimators in each case of dependence, i.e. E( g -f p p ) 1/p with g equals to one of the three estimators. Even if these risks are close to each others, kernel estimator with cross validation bandwidth has the smallest risk for small values p ≤ 4. Yet, approximations of this kernel estimator clearly get worse with higher values of p, while risks of f ST CV n seem relatively stable for different value of p. Concerning kernel estimator with the width parameter taken according to the rule of thumb, the L p risk is worse for small values of p but comparable with the one of wavelet estimator for higher values, even if the modes of the density are not detected.

These graphs show that an advantage of f ST CV n is that its mean L p risk seems stable for high values of p. Nevertheless, the mean of the L 2 risk is larger than the one of kernel estimators with cross-validation width and the computation time higher. One possible way to improve the quality of approximation for the cross validation procedure may be to consider different levels of thresholding at each resolution level. We do not investigate this axis of research as then the time of computation exploded.

Different dependent samplings that do not satisfy (D).

In this Subsection, we discuss the necessity of Assumption (D). For this, we study the convergence of the density estimators on some dynamical systems that do not satisfy this assumption. More precisely, we focus on Liverani-Saussol-Vaienti maps, see [START_REF] Liverani | A probabilistic approach to intermittency[END_REF], defined as the solution of X t = T i (X t-i ) with

T (x) = x(1 + 2 α ′ x α ′ ), 0 ≤ x ≤ 1/2 for some 0 < α ′ < 1 2x -1, 1/2 < x ≤ 1.
The process (X t ) t∈Z is stationary and such that the covariance terms Cov(f (X 0 ), g(X r )) are of order r 1-1/α ′ , see [START_REF] Young | Recurrence times and rates of mixing[END_REF] and refinement in [START_REF] Gouëzel | Central limit theorem and stable laws for intermittent maps[END_REF]. Thus the Assumption (D) is not satisfied in this case and we have the non-minimaxity of any thresholded wavelets estimators:

Proposition 5.1. Suppose that the father wavelet φ is such that φ > 0 and that the assumptions of Theorem 3.1 are satisfied. If 1 > α ′ ≥ 1/(2α + 1) with α defined by (2.1), then for the thresholded estimators of the marginal density of the Liverani-Saussol-Vaienti map of index α there exists some C > 0 such that:

n 2α E[ f n -f 2 2 ] ≥ C, for n sufficiently large.
The same result also holds for the cross validation thresholded estimator f ST CV n .

The proof of this Proposition is given in Section 6.

To simulate these dynamical systems, we simulate Z 0 according to the Lebesgue measure on [0, 1], then we apply recursively T to determine Z i and finally we set (X 1 , . . . , X n ) = (Z n+1 , . . . , Z 2n ). This approximation of the stationary solution does not affect the study of the convergence rates as the dynamical system Z is ergodic in mean with rate O(n 1-1/α ′ ), see Theorem 5 of [START_REF] Young | Recurrence times and rates of mixing[END_REF].

The analytic expression of the density f is unknown but it is proved to be continuous, locally Lipschitz and to behave like x -α ′ as x → 0, see [START_REF] Liverani | A probabilistic approach to intermittency[END_REF] and [START_REF] Young | Recurrence times and rates of mixing[END_REF]. In particular, f is unbounded on [0, 1]. Then we restrict our study on [0.01, 1] where f is bounded. As the true density f is unknown, we compare here the estimators f ST CV n with other estimators. In Figure 7 Visually, means of both estimators are closed to each other. To detect some difference between the estimators behavior, we decide to compute the moments of order k = 1, . . . , 20 of the estimators integrated on [0, 1]: For small values α ′ ≤ 0.02 and k ≤ 4, the moments of both estimators have similar values. But as α ′ growths, all the moments of f ST CV n explode more rapidly than the ones of kernel estimators as k increases. Previous simulations studies show no behavior difference for f ST CV n between independent cases and dependent cases satisfying (D). In dependent cases that do not satisfy (D), the behavior of f ST CV n depends on the decrease rates of covariance terms. On the contrary, the behavior of the kernel estimators with rule of thumb width is more stable when (D) is not satisfied.

1 0.01 E g k (t)

Proofs

In this Section are collected all the proofs of this paper. This proof is essentially based on Lemmas 6.1 and 6.2. The first one, Lemma 6.1, is simply the Theorem 2 of [9] that we recall here for completeness: Lemma 6.1. (Doukhan & Louhichi (1999)) Let (Z i ) 1≤i≤n be centeres variables. Let q be an even integer and n ≥ 2. Suppose that for all p = 2, . . . , q and for all 1 ≤ s 1 ≤ s p ≤ n satisfying max s i+1 -s i = s u+1 -s u = r, there exists V p,n such that:

n n-1 r=0 (r + 1) p-2 Cov(Z s 1 • • • Z su , Z s u+1 • • • Z sp ) ≤ V p,n .
Then, we have

E| n i=1 Z i | q ≤ (2q -2)! (q -1)! V q/2 2,n ∨ V q,n , (6.1)
We refer the reader to [START_REF] Doukhan | A new weak dependence condition and applications to moment inequalities[END_REF] for the proof of this result. We apply this result on Z i = ψ j,k (X i )-β j,k , when (D) holds. We first determine the bounds V p,n as under Assumption (D) we have:

n-1 r=0 (r + 1) p-2 Cov(Z s 1 • • • Z su , Z s u+1 • • • Z sp ) ≤ n-1 r=0 (r + 1) p-2 ρ(r)p 2 (2 j/2 M 2 ) p-2 .
Here we need the following analytic Lemma to bound the quantity ∞ r=0 (r + 1) p ρ(r): Lemma 6.2. If (D2) is satisfied, i.e. if ρ(r) ≤ C 0 e -ar b , then for all integer p we have

∞ r=0 (r + 1) p ρ(r) ≤ C 1 C p 2 (p!) 1/b , (6.2)
with some constants C 1 and C 2 that are depending on a and b.

Applying this result, whose proof is given at the end of this subsection, we directly obtain the new bound:

n-1 r=0 (r + 1) p-2 Cov(Z s 1 • • • ≤ C 1 p 2 (2 j/2 M 2 C 2 ) p-2 ((p -2)!) 1/b We set V p,n = C 1 p 2 (2 j/2 M 2 C 2 ) p-2 ((p -2)!) 1/b n
, and, applying Lemma 6.1, we obtain that:

E| n i=1 (Z i -EZ 0 )| q ≤ (2q -2)! (q -1)! (C 1 n) q/2 ∨ C 1 (2 j/2 M 2 C 2 ) q-2 ((q -2)!) 1/b .
Dividing by n -q and noticing that 2 j/2 ≤ n for 0 ≤ j ≤ log n, we derive that:

E| β j,k -β j,k )| q ≤ n q/2 (2q -2)! (q -1)! (C 1 ) q/2 ∨ C 1 (M 2 C 2 ) q-2 ((q -2)!) 1/b ,
which corresponds to the inequality (3.1). In particular, for p = 2 we have

E| n i=1 (Z i -EZ 0 )| 2 ≤ 4C 1 n.
Now the inequality (3.2) of Lemma 3.1 is a direct application of Theorem 1 in [START_REF] Doukhan | A Bernstein type inequality for times series[END_REF] with

S n = n i=1 (Z i -EZ 0 ), t = nλ, ν = 0, µ = 1/b, A n = 4C 1 n and B n = 2M 2 C 2 2 (2+b)/b 2 j/2
. We refer the reader to [START_REF] Doukhan | A Bernstein type inequality for times series[END_REF] for the definition of the parameters t, ν, µ, A n and B n .

Proof of Lemma 6.2. Define g(x) = (1 + x)e -ax b for all x ≤ 0. Studying its derivative, we can easily see that it exists x a,b such that the function g decreases on [x a,b , +∞). If we denote k ≥ 1 the smallest integer greater than x a,b , we can infer from (D2) that

∞ r=0 (r + 1) p ρ(r) ≤ C 0 k-1 r=0 (r + 1) p ρ(r) + ∞ k-1 (x + 1) p exp(-ax b )dx ≤ C 0 C a,b + ∞ 0 (x + 1) p exp(-ax b )dx .
With a convex inequality on x → x p , we achieve the bound

∞ r=0 (r + 1) p ρ(r) ≤ C 0 C a,b + 2 p-1 ∞ 0 x p exp(-ax b )dx + ∞ 0 exp(-ax b )dx . Then, writing u = ax b , ∞ r=0 (r + 1) p ρ(r) ≤ C 0 C a,b + 2 p-1 b -1 a -(p+1)/b Γ p + 1 b + a -1/b Γ 1 b ,
with Γ defined by Γ(x) = ∞ 0 u x-1 exp(-u)du for all x > 0. Let ⌈x⌉ denotes the largest integer smaller than x > 0 and note Γ = sup x∈]0,1] Γ(x). Using the inequalities Γ((p + 1)/b) ≤ Γ⌈(p + 1)/b⌉! ≤ Γe p/b (p !) 1/b and the fact that this last bound is also available for Γ(1/b), we have: The Lemma comes immediately by choosing the appropriated constants. 6.2. A sketch of the proof of Theorem 3.1. The proof of Theorem 3.1 is very similar to the one in the iid case given in [START_REF] Donoho | Density estimation by wavelet thresholding[END_REF]. It is well known for the community that both inequalities (3.1) and (3.2) are sufficient to set rates of convergence. We just give here a sketch of the proof.

As f ∈ L 2 we write it in the basis of wavelets:

f = k∈S j 0 α j 0 ,k φ j 0 ,k E j 0 f + ∞ j=j 0 k∈S j β j,k ψ j,k D j 0 f , (6.3)
with S j = {k, |2 -j k| < A + T }. Note that the number of elements in S j is less or equal than 2 j (A + T ) and that every k in S j is bounded by 2 j (A + T ). We decompose the estimators f n of f in the same way:

f n = k∈S j 0 α j 0 ,k φ j 0 ,k b E j 0 f + j 1 j=j 0 k∈S j γ λ j ( β j,k )ψ j,k b D j 0 f , (6.4)
where the γ λ j denotes without distinction the soft and hard-threshold function.

Thanks to Minkowski's inequality, the risk of f n is divided in two terms:

E[ f n -f p p ] ≤ 2 p-1   E[ E j0 f -E j0 f p p ] T 1 + E[ D j 0 ,j 1 f -D j 0 f p p ] T 2    .
The following Lemmas given in [START_REF] Meyer | Wavelets and Operators[END_REF] give bounds for these terms. Here for any p ≥ 1 we denote • ℓp the ℓ p -norm defined by a p ℓp = i |a i | p for any sequence of real number (a i ) i≥0 and we denote δ the wavelet φ or ψ with no distinction. Lemma 6.3. For any 1 ≤ p ≤ ∞, there exists c 1 , c 2 > 0 such that for all j ≥ 0 and all sequence (a k ) 0≤k≤2 j -1 we have c 1 2 j(p/2-1) a p ℓp ≤ k∈S j a k δ j,k p p ≤ c 2 2 j(p/2-1) a p ℓp .

The proof of Theorem 3.1 is based on multiple applications of Lemma 6.3 and its refinements given in [START_REF] Donoho | Density estimation by wavelet thresholding[END_REF], see Eq. ( 21), with a j,k = γ λ j ( β j,k ) -β j,k . From the linearity of the expectation, bounds for the mean of the approximation in L p express as functions of

E γ λ j ( β j,k ) -β j,k p ℓp = k∈S j E|γ λ j ( β j,k ) -β j,k | p .
Then one uses the following estimate, common for hard and soft-thresholding:

|γ λ j ( β j,k ) -β j,k | p ≤ 2 p-1 | β j,k -β j,k | p + |λ j | p 1 1 {| b β j,k |>λ j } + |β j,k | p 1 1 {| b β j,k |≤λ j } . (6.5)
The indicators terms are bounded by using (3.2) and the centered moments of the coefficients are bounded using (3.1). We refer the reader to [START_REF] Donoho | Density estimation by wavelet thresholding[END_REF] for more details. Denoting with no distinction ψ j,k (x) -β j,k and φ j,k (x) -α j,k as δ j,k (x) for any j, k, we collect here some inequalities useful in this Section:

E| δ j,k (X 0 )| ≤ 2 f ∞ δ 1 2 -j/2 , E| δ j,k (X 0 )| 2 ≤ f ∞ , δ j,k ∞ ≤ 2 δ ∞ 2 j/2
, Lip δ j,k ≤ Lip δ2 3j/2 and δ j,k BV ≤ ( δ ∞ + ALip δ)2 j/2+1 for all j ≥ 1. The last assertion comes from the fact that δ j,k is a bounded Lipschitz function supported by [(-A + k)2 -j , (A + k)2 -j ].

Proof of Proposition 4.1. As for any j, k the function δ j,k has bounded variations we have Cov δ j,k (X s 1 ) • • • δ j,k (X su ), δ j,k (X s u+1 ) • • • δ j,k (X s u+v ) ≤ vE δ j,k (X s 1 ) • • • δ j,k (X su ) δ j,k v BV φ v (r).

Noticing that δ j,k ∞ ≤ δ j,k BV , it follows

C j,k u,v (r) ≤ v δ j,k u+v-2 BV δ j,k BV E| δ j,k (X 0 )| φ v (r)
≤ (u + v + uv)(2 j/2+1 c( δ ∞ + ALip δ)) u+v-1 2 f ∞ δ 1 2 -j/2 exp(-ar b ).

Then Proposition 4.1 is proved.

Proof of Lemma 4.1. The proof is very close to the one given in [START_REF] Dedecker | An empirical central limit theorem for dependent sequences[END_REF]. First notice that (4.3) for r = 0 implies that c ≥ 1 = sup g∈BV 1 g BV . From (4.2) we infer that φ(σ(X 0 ), X r ) ≤ exp(ar -b ) applying Lemma 4 of [START_REF] Dedecker | New dependence coefficients: Examples and applications to statistics[END_REF] on (4.2). From the Markov property we get φ(σ({X j , j ≤ 0}), X r ) = φ(σ(X 0 ), X r ). Now for any ℓ ≤ 1, for all r ≤ i 1 ≤ • • • ≤ i ℓ consider any g i j ∈ BV 1 for 1 ≤ j ≤ ℓ.

Collecting these facts and using that |S j 0 | is up to a constant equal to 2 j 0 , we obtain that n 2α E( f n -f 2 2 ) is larger than some positive constant and the result of Proposition 5.1 is proved.

. 5 )

 5 If ξ t are iid random variables satisfying E|ξ 0 | ≤ 1 then (X t ) t∈Z is λ-weakly dependent with λ(r) ≤ 4 |j|>[r/2] |a j |. If a j ≤ Kα |j| for j = 0, K > 0 and 0 < α < 1 then the condition on λ in Proposition 4.2 holds with b ′ = 1. Then Assumption (D2) is ensured with b = 1.

5. 3 ..

 3 Comparison of f HT CV n and f ST CV n The first density considered is a mixture between a sinus function and a uniform distribution.

Figure 1 .

 1 Figure 1. Examples of estimators f HT CV n obtained on 2 10 observations. The true distribution is represented in dashed lines. Figures from left to right correspond respectively to Case 1, Case 2 and Case 3.

  0

Figure 2 .

 2 Figure 2. Examples of estimators f ST CV n obtained on 2 10 observations. The true distribution is represented in dashed lines. Figures from left to right correspond respectively to Case 1, Case 2 and Case 3.

Figure 3 .

 3 Figure 3. Means of the proportions of the threshold levels obtained by cross-validation with respect to the resolution levels for hard-thresholding (left) and soft-thresholding (right). Case 1 corresponds to the solid line, Case 2 to the dashed line and Case 3 to the dotted line.

Figure 4 .

 4 Figure 4. Means of the proportions of thresholded coefficients with respect to the resolution levels for hard-thresholding (left) and soft-thresholding (right). Case 1 corresponds to the solid line, Case 2 to the dashed line and Case 3 to the dotted line.

Figure 5 .

 5 Figure 5. Means of estimators f ST CV n obtained from 2 10 observations on 500 simulations. The mean of the wavelet estimators is represented in dashed lines while the means of the kernel estimators are represented respectively in line with dots for the rule of thumb's width (kernel estimator 1) and in dots for the cross-validation width (kernel estimator 2). Figures from left to right correspond respectively to Case 1, Case 2 and Case 3.

Figure 6 .

 6 Figure 6. Evolution of the L p risk of estimators f ST CV n and kernel estimators obtained on 2 10 observations. The wavelet estimator is represented in dashed lines while the kernel estimators are represented respectively in dots line for the rule of thumb width (kernel estimator 1) and in dots for the cross-validation width (kernel estimator 2). Figures from left to right correspond respectively to Case 1, Case 2 and Case 3.

  , we plot the mean of M = 100 estimators f ST CV n and Epanechnikov kernel estimators given by Matlab's rule of thumb for 9 different values of α ′ . for α'=0.9

Figure 7 .

 7 Figure 7. Means of the estimators f ST CV n obtained on 2 10 observations and 500 simulations. Means of the kernel estimators are represented in dashed lines.

  function g is alternatively f ST CV n or kernel estimators.

Fluctuations of the estimators α=0. 9 Figure 8 . 6 . 1 .

 9861 Figure 8. Moments of the estimators f ST CV n obtained on 2 10 observations and 500 simulations. In dashed lines are represented the moments of the kernel estimators.

  1) p ρ(r) ≤ C 0 C a,b + 2 p-1 b -1 a -(p+1)/b + a -1/b Γe p/b (p !) 1/b ≤ C 0 C a,b + 2 -1 b -1 a -1/b Γ 2e 1/b (a -1/b ∨ 1) p (p !) 1/b .

6. 3 .

 3 Proofs of results given in Sections 5 and 4. In this Section are collected the proof of Proposition 4.1, Lemma 4.1, Proposition 4.2 and Propostion 5.1.

Table 1

 1 gives approximations by Monte-Carlo method of the MISE. MISE values of the estimators have the same order whereas the weak dependent cases and f ST CV

	n	is preferable in all the

Table 1 .

 1 MISE approximated by MC on 500 simulations of samples of size n = 2 10 .

Acknowledgements. We thank Karine Tribouley and Vincent Rivoirard for their helpful advices to one of the author. We also thank referees for having pointed out lapses in the first version of this paper.

Let us prove that we can restrict ourselves to the ℓ-uplets satisfying i 1 < • • • < i ℓ . On the one hand, if i j = i j ′ = r then we have E(g r (X r )g r (X r )|X i 1 = •) BV ≤ c g 2 r BV from (4.3). As from assumption g r ∞ ≤ g r BV ≤ 1 then g 2 r BV ≤ g r BV + g r BV ≤ 2 and we achieve the bound E(g r (X r )g r (X r )|X i 1 = •) BV ≤ 2c. On the other hand, if i j < i j ′ then, from the Markov property, we get the equation

We proceed in two steps. Firstly from (4.3), we infer thet x → g i j ,i j ′ (x) = E(g i j ′ (X i j ′ )|X i j = x) has variations bounded by c. Notice also that g i j ,i j ′ ∞ ≤ g i j ′ ∞ ≤ 1 and therefore we deduce the bound g i j g i j ,i j ′ BV ≤ g i j BV + g i j ,i j ′ BV ≤ 1+c. Secondly, using (4.3) on g i j ,i j ′ / g i j ,i j ′ BV , we infer that E(g i j (X i j )g

As c ≥ 1, this bound is larger than the one in the case i j = i j ′ . Now, by straightforward recurrences in the worst cases i 1 < • • • < i ℓ , we get that

x) for all x, we have almost surely the equation

From the definition of coefficients φ we get

For all 1 ≤ ℓ ≤ v this bound holds uniformly for all g i j ∈ BV 1 , 1 ≤ j ≤ ℓ, using the definition of φ v (r) we conclude that φ v (r) ≤ c v φ(σ({X j , j ≤ 0}), X r ) ≤ c v exp(-ar b ).

Proof of Proposition 4.2. We use the direct bound given in the proof of Lemma 1 of [START_REF] Ragache | Convergence rates for density estimators of weakly dependent time series[END_REF] under (J):

Noticing that γ ∧β ≤ γ 1/4 β 3/4 for any positive numbers γ and β, we combine the two bounds on the covariance terms and we infer that (D1) is satisfied with

Assumptions of Proposition 4.2 on respective decrease and increase rates of λ(r) and f r ∞ yield the existence of C 0 > 0 such that ρ(r) ≤ C 0 exp(-a ′ r b ′ /4).

Proof of Propostion 5.1. We give the proof for f n but it also holds for f ST CV n . We begin with recalling the result of Corollary 7.1 in [START_REF] Gouëzel | Central limit theorem and stable laws for intermittent maps[END_REF]: Lemma 6.4 [START_REF] Gouëzel | Central limit theorem and stable laws for intermittent maps[END_REF]. For any Lipschitz function δ 1 , bounded measurable function δ 2 such that δ 1 , δ 2 = 0 in a neighborhood of 0, then for any 0 < α ′ < 1, there exists some constant

We use the decomposition of f ∈ L 2 in the orthogonal basis of wavelets functions, see (6.3) and (6.4), and we obtain

If we develop E( α j 0 -k -α j 0 -k ) 2 using the covariance terms and denoting δ j,k (x) = φ j,k (x) -α j,k for x ∈ [0.01, 1] and null elsewhere, it comes:

We want to apply Lemma 6.4 with δ 1 = δ 2 = δ j 0 ,k . We check easily the assumptions of this Lemma because of the definition of δ j 0 ,k , resulting from the fact that we estimate the density on [0.01, 1]. Moreover φ j 0 ,k = 2 -j 0 /2 φ with φ > 0 from assumption and then the covariance terms Cov( δ j 0 ,k (X 0 ), δ j 0 ,k (X r )) are equivalent to C2 -j 0 r 1-1/α ′ for some C > 0 as r goes to infinity.

Let n 0 be such that for all n ≥ n 0 , u r,n = n-r n 2 Cov( δ j 0 ,k (X 0 ), δ j 0 ,k (X r )) is nonnegative. For some m > m ′ > 2, we decompose the sum of covariance terms, for n sufficiently large, in four sums:

where [a] denotes the integer part of a. The first term goes to 0 with rate n. Then, by definition of n 0 , the second and the last terms are nonnegative. Concerning the sum

The minimax rate α is such that α < 1/2 and then by hypothesis α ′ ≥ 1/(2α+1) > 1/2. Consequently, when n goes to infinity, the sum

r=[n/m] u r,n is larger than a partial sum equivalent to C2 -j 0 n 1-1/α ′ for some C > 0 as n → ∞. As we assume 2α ≥ 1/α ′ -1 we obtain the existence of some C > 0 such that n 2α 2 j 0 n-1 r=1 n -r n 2 Cov( δ j 0 ,k (X 0 ), δ j 0 ,k (X r )) ≥ C, for n sufficiently large.