Elvira Ruiz
email: eruizm@msps.es

Ma Isabel Santillana

Ma Teresa Nieto

Ma Eugenia Cirugeda

Jose J Sánchez

Furan determination in jarred baby food purchased from the Spanish market by headspace gas chromatography-mass spectrometry

Keywords: furan, baby-food, headspace-gas chromatography-mass spectrometry

Furan determination in jarred baby food purchased from

the Spanish market by headspace gas chromatography-mass spectrometry.

Introduction

Furan is a lipophilic contaminant which is formed during the heating processes used for the manufacture of foods. It is an aromatic heterocyclic compound, highly volatile; which was already detected, although at low concentration levels in the 1960s in a great variety of heat-processed foods (canned meat, bread, coffee, hydrolyzed proteins…) [START_REF] Stoffelsma | Volatile components or roasted coffee[END_REF][START_REF] Persson | Aroma of canned beef: Gas chromatographic and mass spectrometric analysis of the volatiles[END_REF].

Experimental models multiple precursors and alternative routes for the furan formation have been proposed: thermal oxidation of ascorbic acid, non-enzimatic browning involving reducing sugars only (Strecker degradation) or together with amino acids [Maillard reactions [START_REF] Maga | Furan in foods[END_REF], polyunsaturated fat acids oxidation [START_REF] Locas | Origin and mechanistic pathways of formation of the parent furan: A toxicant[END_REF]Becalski et al., 2005;[START_REF] Märk | Quantitation of furan and methylfuran formed in different precursor systems by proton transfer reaction mass spectrometry[END_REF]].

In 2004 US Food and Drug Administration (US FDA 2004b) reported the presence of this compound in a great variety of thermally treated food, such as meat, vegetables, and fruits, canned sauces in tins or jars especially including baby-food. The presence of furan, classified as possibly carcinogenic to human (Group 2B) (IARC 1995) in food has been confirmed by different laboratories [START_REF] Reinhard | Furan in foods on the Swiss markmethod and results[END_REF]Becalski et al., 2005;[START_REF] Heppner | Data requirements for risk assessment of furan in food[END_REF] The scientific panel on contaminants in the food chain from the European Food Safety Authority (EFSA 2004) considered furan in foodstuff as an urgent matter and made a scientific report about its presence that was presented in Brussels. The report concluded that it was necessary to generate reliable data of the furan content in products commercialized within the European Community to create a good database and study the risk of furan toxicity. In 2007, the European Commission recommended to the members states to follow the presence of furan in thermally treated food throughout 2007-2008(EC 2007)).

As an answer to the Commission and EFSA recommendation, our laboratory taking into account the lack of data available in Spain, decided to carry out a market surveillance targeted on foods intended to be consumed by infants, since they are the most susceptible consumer group [START_REF] Suk | Environmental hazards to children's health in the modern Word[END_REF] and because relatively high amounts of furan were previously detected in commercial baby-food [START_REF] Crews | A review of the occurrence, formation and analysis of furan in heat-processed foods[END_REF][START_REF] Morehouse | Survey of furan in heat processed oods by headspace gas chromatography/mass spectrometry and estimate adult exposure[END_REF].

The determination of furan is very challenging due to its high volatility, low molecular weight, and its presence in low concentrations in foodstuff. It is very important to have a suitable method to analyze this compound. The most used analytical technique is headspace extraction combined with gas chromatography-mass spectrometric determination [START_REF] Nyman | Singlelaboratory validation of a method for the determination of furan in foods by using static headspace sampling and gas chromatography/mass spectrometry[END_REF][START_REF] Nyman | Single laboratory validation of a method for the determination of furan in foods by headspace gas chromatograph/ mass spectrometry, Part 2: Low-moisture snack foods[END_REF][START_REF] Bianchi | Development and validation of a solid phase micro-extraction-gas chromatography-mass spectrometry method for the determination of furan in baby-food[END_REF]Zoller et al., 2007;Vranová et al., 2007). Our laboratory used the method proposed by FDA (US FDA 2006) as reference, with slight modifications.

The sample (purification was not necessary) was weighted in a headspace vial and placed in the static headspace autosampler, heated at 80º C for 30 min and determined by CG-MS in SIM mode, using furan-d4 as an internal standard.

Experimental

Reagents and apparatus

Furan (purity ≥ 99%) and d 4 -Furan (isotopic purity ≥ 98%) were provided by Sigma-Aldrich. Methanol (HPLC grade) from Merck. Ultrapure water was prepared using a Direct-Q system (Millipore).

The following equipment was used: Gas chromatograph (6890 Agilent) equipped with a mass spectrometer (5973 Agilent) and a Headspace injector (7694 Agilent). The chromatographic capillary column was a HP-PLOT Q by J&W Scientific (15 m x 0, 32 mm x 20 µm). 20 ml headspace vials with PTFE septa (Agilent Technologies) were used. To prevent the presence of contaminants in the vials, they were heated in a muffle oven (Heraeus) at 400º C for 1 hour before being used.

The stock solutions were prepared and transferred using microsyringes (Hamilton Gastight®).

Standards preparation

Stock furan and d 4 -furan solutions were prepared in methanol with a minimum concentration of 3000 mg.l -1 and stored in a freezer for no longer than one week.

From the stock solutions, intermediate solutions of furan and d 4 -furan were prepared (50 µl to 50 ml methanol) and from these, working solutions in water in the in the range 4 -400 µg.l -1 for furan and at 10 µg.l -1 for d 4 -furan, approximately. The precise concentrations were calculated from the stock solutions. All these processes were performed in an ice bath to avoid losses due to the volatilization of furan.

Commercial samples for the study

Thirty nine jarred baby food samples, corresponding to three different commercial brands, and including a wide range of different matrices were purchased in retail food stores of the Spanish market. The samples were stored unopened and refrigerated, until they were analyzed. The main ingredients of the samples analyzed are shown in Table 2.

Sample preparation

A 5 g aliquot of the homogenised sample (homogenisation was performed under cold conditions) was weighted in a headspace vial, diluted with 5 ml of purified water and the internal standard was added at the same concentration as used in the calibration curve. All these processes were performed in an ice bath. The vials were hermetically sealed as fast as possible to avoid analyte losses and were shaken for 1 min until a homogeneous aspect was obtained. All the samples were twice analyzed, following the procedure already described. Additionally, a triplicate for each sample was prepared spiked with furan at a concentration of 100 µg.l -1 in order to be used as a quality control sample.

HS-GC-MS analysis

Gas chromatographic conditions as well as (split ratio of 1:10) MS conditions were the same as those described in the US FDA 2006 method. The headspace operating conditions were as follows: oven temperature at 80º C with an equilibrium time of 30 min (stirring), loop at 95º C, transfer line at 110º C and an injection time of 1 min.

Selected ion monitoring (SIM), ions m/z 68, 39 (furan), 72 (furan d-4). Chromatograms for furan and d 4 -furan standards and for a baby food sample are presented in figure 1.

(Figure 1)

Validation

In-house validation was carried out following the standard protocol from our laboratory, which is based on ISO 17025 (2005). The following parameters were assesed: specificity, linearity, detection and quantification limits, precision and accuracy (expressed as recovery).

Specificity was tested verifying the lack of response at the retention time of the analyte by analysing in triplicate a blank sample. A response was considered to be present if it was ≥1/5 the value of the quantification limit.

Linear regression calibration was performed by plotting the relative response of furan (m/z 68) and d 4 -furan (m/z 72) against the furan concentration in µg.l -1 . An average calibration curve was calculated from three curves obtained in different days, each one at six concentration levels in the range 4 to 400 µg.l -1 . Linearity was established by Precision and accuracy were assessed spiking a blank sample with furan at three different levels of concentration (Table 1) Each level was analyzed three times on three different days (n = 9). Precision was expressed in terms of relative standard deviation (RSD% ≤ 20) and the accuracy as percentage of recovery for each concentration level (80 -120 %).

Results and discussion

Previous to method validation, some experiments were performed using different headspace incubation temperatures, in order to evaluate the influence on the furan response. These temperatures were: 60º C, as recommended in the FDA method, and 80ºC which is the temperature we have previously used in our laboratory for other volatile compounds, and also used by other authors [START_REF] Nyman | Singlelaboratory validation of a method for the determination of furan in foods by using static headspace sampling and gas chromatography/mass spectrometry[END_REF]Zoller et al., 2007;[START_REF] Morehouse | Survey of furan in heat processed oods by headspace gas chromatography/mass spectrometry and estimate adult exposure[END_REF].

To check the effect of the temperature, we obtained calibration curves prepared on different matrices: water, fruit juice and vegetable purée, at both temperatures. The slopes of the regression lines, including all the matrices as well as both temperatures were compared. No significant difference was found between the mean slope obtained at 60ºC (n=3) and at 80ºC (n=4), (t test, α=0.05). Therefore, we concluded that setting the headspace temperature at 80ºC would not lead to a significant increase of the furan concentration in the samples within the scope of our study.

The in-house validation was performed as it has been described using a fruit based baby-food sample, where the absence of furan had been previously checked (<1/5 LOQ), which was taken as the blank sample.

The mean calibration curve (of three curves obtained on three days) with a furan concentration range 4 to 400 µg.l -1 (equivalent to µg.kg -1 sample) showed a good linearity (Cm% = 0.25) and the correlation coefficient was 0.9999. Detection and quantification limits were calculated from the intercept value: 0.05 µg.kg -1 and 0.17 µg.kg -1 , respectively. The lowest validated concentration level was 4 µg.kg -1 , and this was considered as the quantification limit for the report of sample results.

Precision (RSD %) and accuracy (% recovery) results are presented in Table 1. Both parameters were satisfactory according to our internal requirements.

(Table 1) Therefore, method performance parameters assesed in the method validation showed that the analytical procedure was well applicable for the reliable determination of furan in baby food with a limit of quantification suitable for the purpose of our study. Furthermore, the validation results fulfil EFSA requirements (EFSA 2007): quantification limit less than 5 µg.kg -1 , precision (as RSD %) less than 20% and a minimum recovery of 80%. Also, the suitability of the method was confirmed by participating in an inter-laboratory trial (FAPAS®) on a vegetable based baby-food. The z-score obtained was 0.4, which was in the range of ≤ ±2, considered as satisfactory.

The method was applied to the analysis of the jarred baby food samples described in When the baby-food data were processed, precision data of duplicate analysis satisfied the Thompson modified Horwitz equation [START_REF] Thompson | Recent trends in interlaboratory precision at ppb and sub-ppb concentration in relation to fitness for purpose criteria in proficiency testing[END_REF]. This is equivalent to a HorRat r ≤1.

The average recovery value was 101.5% with a relative standard deviation RSD% = 5.6, which is in complete agreement with the results obtained in the method validation.

The thirty-nine baby-food samples analyzed by the method previously described were classified depending on their composition (most common ingredients) in five different groups. Table 3 summarizes the mean value per group as well as the maximum and minimum found values.

(Table 3) From Table 3 it is evident that fruit-based baby food showed the lowest furan concentrations (5 µg.kg -1 ) and the highest values were found in the meat (chicken, turkey and seabass) and vegetables group (62 to 65 µg kg -1 ). Due to the limited number of analyzed samples based only on vegetables and dairy products (n=4), the results for these groups might not be representative and have to be carefully considered.

Furan concentrations in baby food samples from the Spanish market included in our study are in agreement with the levels determined in other studies (Bekalsi et al., 2005;[START_REF] Bianchi | Development and validation of a solid phase micro-extraction-gas chromatography-mass spectrometry method for the determination of furan in baby-food[END_REF][START_REF] Morehouse | Survey of furan in heat processed oods by headspace gas chromatography/mass spectrometry and estimate adult exposure[END_REF]Zoller et al., 2007); neither are they in disagreement with those included in the EFSA database where concentration levels ranged from not detectable to 112 µg kg -1 (EFSA 2004). In our study the levels found were from <LOQ up to 65 µg kg -1 .

All samples analysed were commercial baby foods prepared by autoclaving in glass jars, technology which has been suggested to favor furan formation over other procedures used for commercial baby food preparation and domestic practices [START_REF] Roberts | Effect of consumer cooking on furan in convenience foods[END_REF]US FDA, 2004a;Zoller et al., 2007).

It must be noticed that fruit based products with added ascorbic acid contain also organic acids. It has been reported that the formation of furan from sugars or ascorbic acid is less likely at low pH (Becalski et al., 2005;[START_REF] Limacher | Formation of furan and methylfuran from ascorbic acid in model systems and food[END_REF]. Apparently, the preparation of autoclaved baby foods does not include processing conditions of low water activity at high temperatures typical for non-enzymatic browning reactions to occur; thus furan is more likely formed through other mechanisms (Maillard reaction). It is possible to speculate that most potential precursors for furan in such baby food are polyunsaturated lipids such as polyunsaturated fatty acids and carotenoids (Becalski et al., 2005;[START_REF] Locas | Origin and mechanistic pathways of formation of the parent furan: A toxicant[END_REF].However; this was not the purpose of our study

The composition of baby food seems to be a key factor to be taken into account in furan formation (presence of ascorbic acid, polyunsaturated fatty acids, aminoacids and carotenoids), as has already been described in previous studies regarding this analyte [START_REF] Jestoi | Furan in the baby-food simples purchased from the Finnish markets. Determination with SPME-GC-MS[END_REF][START_REF] Lachenmeier | Risk assessment of furan in commercially jarred baby foods, including insights into its occurrence and formation in freshly homecooked foods for infants and young children[END_REF].

F

Where

  estimating the relative standard deviation of the mean slope, according to the following equation: Sm is the standard deviation of the mean slope and m is the slope of the mean calibration curve. Values of Cm ≤ 5 % are considered satisfactory. The correlation coefficient of the linear regresión was also calculated (≥ 0.996).Limits of quantification (LOQ) and detection (LOD) were calculated from the mean calibration curve using the following equations: is the intercept standard deviation, m is the slope of the mean calibration curve and n is the number of calibration curves.

Table 2 .

 2 All the samples were analyzed twice and a parallel recovery analysis was done for all of them. The results are shown in Table2.

	(Table 2)

Table 1 :

 1 US Food and Drug Administration (FDA). 2004b. Determination of furan in foods.7 May 2004(updated 27 October 2006). Available from: http//www.cfsan.fda.gov/dms/furan.html Vranová J, Bednáriková A, Ciesarová Z. 2007. In-house validation of a simple headspace gas chromatography-mass spectrometry method for determination of furan levels in food. Mean recoveries and relative standard deviations of furan determinations at three different spiking levels used. Each spiking level is represented by 9 replicates.

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Table 2 :

 2 The different jarred baby food samples analysed with the ingredients reported by the manufactures and furan mean concentration found.

				Food Additives and Contaminants	Page 14 of 17
	Number Brand	Product	Ingredients	Furan µ µ µ µg.kg -1
	29	C		Cream of béchamel and vegetables with chicken	Milk, water, chicken, potato, carrot, corn starch, butter, leak and green beans.	43.9
	30	C		Rice and veal	Water, potato, carrot, veil, rice, onion, tomato, olive oil, leak and corn starch.	37.2
	31	C		Banana, apple, orange and cereals.	Banana, apple, flour, honey, orange, lemon and vitamin C.	NQ*
	32	C		Béchamel sauce and hake	Water, milk, hake, rice flour, corn starch, butter, salt and nutmeg.	34.2
	33	C		Chicken with vegetables	Potato, water, carrot, chicken meat, leak, peas, onion, olive oil, corn starch and salt.	42.8
	34	C		Gluten and additive free varied fruit.	Orange, apple, pear, apricot, water, sugar, rice flour and vitamin C.	7.4
	35 36 37 38 39	C C C C C	Gluten and additive free Six fruit dessert. F Banana, orange and biscuit. o Gluten and additives free chicken and rice. r Vegetable puree and turkey. Dinner for little ones. P e Vegetables and seabass.	Apple, peach, apricot, banana, orange, pineapple, sugar, water, rice flour and vitamin C. Banana, orange, water, sugar, gluten free biscuits, rice flour, corn, lemon and vitamin C. Water, milk, chicken, potato, carrot, rice, corn starch, olive oil and salt. Water, carrots, potatoes, milk, turkey, onion, green beans, olive oil, corn starch and salt. Water, carrots, potato, seabass, milk, turkey, starch and salt. onion, peas, green beans, olive oil, leak, corn	7.8 9.0 64.6 62.8 62.2
	r e * NQ = Not quantifying, LOQ = 4 µg.kg -1
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Table 3 :

 3 Mean and maximum-minimum values per group depending on their composition. To calculate the mean value we have taken value 2 (LOQ/2) for the Non Quantifying samples.

	Baby food based on: Fruits Vegetables Meat and vegetables Fish and vegetables Dairy products (1) Page 15 of 17	n 10 2 20 6 2 Food Additives and Contaminants Mean value (µg.kg -1 ) (1) 5.0 37.8 25.2 33.8 30.5	Maximum value-Minimum value (µg.kg -1 ) NQ -9.0 22.2 -53.4 NQ -64.6 13.9 -62.2 21.2 -39.8

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http://mc.manuscriptcentral.com/tfac Email: fac@tandf.co.ukFood Additives and Contaminants

Acknowledgements

The authors would like to thank Juana Bustos and Marta Tello for technical assistance and critical reading of the manuscript.


Paolo Peterlongo

Irene Catucci

Graziella Pasquini

Paolo Verderio

Bernard Peissel

Monica Barile

Liliana Varesco

Mirko Riboni

Stefano Fortuzzi

Siranoush Manoukian

Paolo Radice
email: paolo.radice@istitutotumori.mi.it.

germline mutations in familial breast cancer cases with personal and family history of pancreatic cancer

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

To the Editor, PALB2 (partner and localizer of BRCA2) has been recently described as a breast cancer predisposing gene [START_REF] Rahman | PALB2, which encodes a BRCA2-interacting protein, is a breast cancer susceptibility gene[END_REF][START_REF] Stratton | The emerging landscape of breast cancer susceptibility[END_REF]. In the first report, PALB2 truncating mutations were identified in 10/923 (1.1%) English familial breast cancer cases [START_REF] Rahman | PALB2, which encodes a BRCA2-interacting protein, is a breast cancer susceptibility gene[END_REF]. Subsequent studies in Spanish, Chinese and Italian familial breast cancer found truncating mutations with frequencies ranging between 0.8 and 1.1% [START_REF] García | Analysis of FANCB and FANCN/PALB2 fanconi anemia genes in BRCA1/2-negative Spanish breast cancer families[END_REF][START_REF] Cao | The prevalence of PALB2 germline mutations in BRCA1/BRCA2 negative Chinese women with early onset breast cancer or affected relatives[END_REF][START_REF] Papi | A PALB2 germline mutation associated with hereditary breast cancer in Italy[END_REF][START_REF] Balia | PALB2: a novel inactivating mutation in a Italian breast cancer family[END_REF]. Also, PALB2 founder mutations were observed in 2.7% Finnish [START_REF] Erkko | A recurrent mutation in PALB2 in Finnish cancer families[END_REF] and 0.6% Polish [START_REF] Dansonka-Mieszkowska | A novel germline PALB2 deletion in Polish breast and ovarian cancer patients[END_REF] familial breast cancer cases, and in 0.7% French-Canadian breast cancer cases with early onset disease [START_REF] Ghadirian | The contribution of founder mutations to early-onset breast cancer in French-Canadian women[END_REF].

Recently, exomic sequencing revealed a germline truncating mutation of PALB2 in a familial pancreatic cancer case [START_REF] Jones | Exomic sequencing identifies PALB2 as a pancreatic cancer susceptibility gene[END_REF]. In the same study, the sequencing of PALB2 in 96 additional familial pancreatic cancer patients, ascertained in the US, identified truncating mutations in three cases. A subsequent larger survey analyzed 254 sporadic and familial pancreatic cancer cases ascertained in Toronto and Montreal, including nine individuals who were also diagnosed with breast cancer, one of which was found to carry a 6.7-kb deletion involving exons 12 and 13 [START_REF] Tischkowitz | Analysis of the gene coding for the BRCA2-interacting protein PALB2 in familial and sporadic pancreatic cancer[END_REF]. Interestingly, of the five PALB2 mutated pancreatic cancer cases reported in the above studies, two also developed breast cancer, including one patient with three additional breast cancer cases in the family. Moreover, two of the three remaining cases had ≥2 relatives affected with breast cancer. The last case belonged to a family negative for breast cancer, but the reported pedigree was small and included only one female individual [START_REF] Jones | Exomic sequencing identifies PALB2 as a pancreatic cancer susceptibility gene[END_REF] In this contest, we sought to investigate the frequency of PALB2 germline pathogenic mutations in index cases from breast cancer families with cases of pancreatic carcinoma. We took advantage of the sequencing of the entire coding region of PALB2 that we performed in 575 Italian female familial breast cancer cases. These were ascertained at the Medical Genetics Units of the Fondazione IRCCS Istituto Nazionale Tumori and the Istituto Oncologico Europeo in Milan, from March 2003 to August 2008. All subjects, therein referred to as BRCAX cases, were probands of families fulfilling previously reported diagnostic criteria for hereditary breast/ovarian cancer, based on family history and age at disease onset [START_REF] Manoukian | Germline mutations of TP53 and BRCA2 genes in breast cancer/sarcoma families[END_REF], and were ascertained not to carry deleterious mutations or unclassified variants in BRCA1 or BRCA2 genes, following direct sequencing or denaturing highperformance liquid chromatography (DHPLC) analysis of all coding exons and adjacent splice sites, as previously described [START_REF] Manoukian | Germline mutations of TP53 and BRCA2 genes in breast cancer/sarcoma families[END_REF]. In the 575 BRCAX cases we found 12 carriers of truncating mutations for a frequency of 2.1%. Overall, a total of nine different alterations were detected, of which five were frameshift, three were nonsense and one was a splicing mutation (manuscript in preparation).

Mutations of PALB2 have been described as having moderate penetrance [START_REF] Rahman | PALB2, which encodes a BRCA2-interacting protein, is a breast cancer susceptibility gene[END_REF][START_REF] Stratton | The emerging landscape of breast cancer susceptibility[END_REF], although it was recently suggested that the risk conferred by these mutations may be high in women with strong family histories [START_REF] Byrnes | Are the so-called low penetrance breast cancer genes, ATM, BRIP1, PALB2 and CHEK2, high risk for women with strong family histories?[END_REF]. While this aspect requires further investigation, we conservatively treated PALB2 mutations as moderate-penetrance factors and, consequently, as expected to show incomplete segregation with the disease [START_REF] Stratton | The emerging landscape of breast cancer susceptibility[END_REF]. Thus, through pedigrees analysis of the 575 BRCAX families we selected those in which at least one pancreatic carcinoma case was reported by the index case in 1 stor 2 nd -degree relatives, irrespectively of the breast cancer family branch. Whenever possible, the diagnosis of pancreatic carcinoma was confirmed by reviewing medical and pathological reports.

Thirty-nine breast/pancreatic cancer pedigrees were identified, including three families (M1475, M1203 and M1504) carrying PALB2 truncating mutations, namely the c.3497delG (p.Gly1166fs), c.72delG (p.Leu24fs) and c.1027C>T (p.Gln343X) (Figure 1). This frequency of 7.7% (3/39) suggested that, among breast cancer families, those with cases of pancreatic carcinoma might be enriched in PALB2 mutations. To verify this hypothesis, we examined a second group of 23 probands from breast/pancreatic families selected among additional BRCAX cases, using the same criteria described above, (i.e. female breast cancer cases from families fulfilling diagnostic criteria for hereditary breast/ovarian cancer, who tested negative for BRCA gene mutations and reported ≥1 1 st or 2 nd degree relatives affected with pancreatic carcinoma). One of these 23 BRCAX cases was affected also with pancreatic cancer. No PALB2 truncating mutations were detected in any of these individuals, leading to an overall frequency in the two combined groups of breast/pancreatic cancer families of 3/62 (4.8%; exact 95% confidence Interval = 0.99-13.29).

In addition, we tested one individual from a family (B276) with multiple cases of cancer to several organs, but not complying with the above inclusion criteria, who developed carcinomas of breast, pancreas, small intestine and colorectum, (Figure 1). In this individual the c.1314delA (p. Lys438fs) truncating mutation was identified.

As already mentioned, previous reports have shown that the majority of familial pancreatic cancer cases with PALB2 germline mutations have a personal and/or family history of breast cancer [START_REF] Jones | Exomic sequencing identifies PALB2 as a pancreatic cancer susceptibility gene[END_REF][START_REF] Tischkowitz | Analysis of the gene coding for the BRCA2-interacting protein PALB2 in familial and sporadic pancreatic cancer[END_REF]. This was recently confirmed by a study, published while this manuscript was in preparation, that analyzed 81 European familial pancreatic cancer cases and identified three carriers of PALB2 truncating mutations, all of whom had ≥1 relative affected with breast cancer [START_REF] Slater | PALB2 mutations in European familial pancreatic cancer families[END_REF]. These observations suggested that PALB2 mutations are preferentially associated with families with history of both breast and pancreatic cancer. However, by taking into consideration the pertinent uncertainty (i.e. exact 95% confidence Interval), in our study the overall frequency of PALB2 mutations detected among breast cancer families with 1 case of pancreatic cancer appears to be comparable to the frequency detected in the overall group of BRCAX families (2.1%) and to those previously reported [START_REF] Rahman | PALB2, which encodes a BRCA2-interacting protein, is a breast cancer susceptibility gene[END_REF][START_REF] García | Analysis of FANCB and FANCN/PALB2 fanconi anemia genes in BRCA1/2-negative Spanish breast cancer families[END_REF][START_REF] Cao | The prevalence of PALB2 germline mutations in BRCA1/BRCA2 negative Chinese women with early onset breast cancer or affected relatives[END_REF][START_REF] Papi | A PALB2 germline mutation associated with hereditary breast cancer in Italy[END_REF][START_REF] Balia | PALB2: a novel inactivating mutation in a Italian breast cancer family[END_REF]. Thus, if an excess of PALB2 mutations is present in families with occurrences of both breast and pancreatic cancer, this is likely to be modest and detectable only by larger surveys.

Interestingly, we found a PALB2 truncating mutations in one out of two examined individuals who developed both breast and pancreatic cancer. When merged with the data reported by Tischkowitz et al. on patients with the same phenotype [START_REF] Tischkowitz | Analysis of the gene coding for the BRCA2-interacting protein PALB2 in familial and sporadic pancreatic cancer[END_REF], this leads to an overall frequency of 2/11 (18.2%). This suggests that pathogenic mutations of PALB2 might be relatively frequent in individuals affected with both breast and pancreatic cancer. Further analyses are needed to verify this association.

Finally, in the four identified PALB2-mutated families, a variety of different cancer types were reported (Figure 1). In particular, the proband of family B276, in addition to carcinomas of the small intestine, breast and pancreas, developed multiple colorectal cancers (CRCs), which were diagnosed also in two of her relatives, while a third relative was reported with a gastrointestinal cancer of unknown site. Of note, in this family immunohistochemistry and molecular analyses, performed on the proband"s intestinal cancers, showed normal expression of DNA mismatch repair (MMR) proteins and no evidence of microsatellite instability (MSI), thus ruling out Lynch syndrome. The proband also tested negative at mutation screening for MUTYH-associated polyposis, while other known syndromic conditions associated with CRC susceptibility were excluded due to the lack of related phenotypes.

Different gastrointestinal cancers, in addition to pancreatic carcinomas, were also reported in the other PALB2 mutation positive families, including two CRCs. Although none of 288 sporadic and 188 familial CRCs from Finland was found to carry the c.1592delT founder mutation [START_REF] Erkko | A recurrent mutation in PALB2 in Finnish cancer families[END_REF], an association between PALB2 and CRC was previously suggested by a study reporting a large mutation positive breast cancer family with multiple CRCs [START_REF] Foulkes | Identification of a novel truncating PALB2 mutation and analysis of its contribution to early-onset breast cancer in French-Canadian women[END_REF]. Taken together, these observations support the interest in evaluating the role of PALB2 in the susceptibility to CRC and, possibly, other gastrointestinal cancers not yet investigated.

In conclusion, our data failed to demonstrate that PALB2 mutations are preferentially associated with breast cancer families with cases of pancreatic cancer, although this might be due to the relative small number of examined individuals. However, screening for PALB2 germline mutation might be recommended in cases affected with both breast and pancreatic cancer. Moreover, our findings, together with previous observations, indicate that PALB2 mutations might increase the risk of CRC. Searching for PALB2 germline mutations in families with cases of breast cancer and CRC should be considered.

Figure 1 .

 1 Figure 1. Pedigrees of the four families in which index cases carried a PALB2 truncating mutation. No additional relatives in these families were available for PALB2 mutation testing. Index cases are indicated by arrow and PALB2 mutations are described. Cancer type and age at diagnosis are reported, when known. Individuals affected with documented cancer are indicated by asterisk. Bn, brain cancer; Br, breast cancer; Br bil, bilateral breast cancer; CR, colorectal cancer; GC, granulosa cell cancer; GI, gastrointestinal cancer (site unknown); La, larynx cancer; Le, leukemia; Li, liver cancer; Lu, lung cancer; Pa, pancreatic cancer; Pr, prostate cancer; SI, small intestine cancer; St, stomach cancer; Ut, uterine cancer; VC, vocal cords cancer.

Acknowledgements

We thank all patients and families who participated to this study. This study was funded by grants from Fondazione Italiana per la Ricerca sul Cancro (Special Project "Hereditary tumors"), Associazione Italiana per la Ricerca sul Cancro (4017), Ministero della Salute (RFPS-2006-3-340203; Extraordinary National Cancer Program 2006, "Alleanza contro il Cancro"; "Progetto Tumori Femminili), Ministero dell"Universita" e Ricerca (RBLAO3-BETH), Istituto Superiore di Sanita` (526D/41), Lega Italiana per la Lotta contro i Tumori (progetto 02/12/R/47) and by funds from Italian citizens who allocated the 5x1000 share of their tax payment in support of the Fondazione IRCCS Istituto Nazionale Tumori, according to Italian laws (INT-Institutional strategic projects "5x1000").


Arthur Charpentier
email: arthur.charpentier@ensae.fr

Johan Segers
email: segers@stat.ucl.ac.

Convergence of Archimedean copulas

Keywords: 62E20, 62H20 JEL: C14, C16 Archimedean copula, generator, Kendall distribution function

Convergence of a sequence of bivariate Archimedean copulas to another Archimedean copula or to the comonotone copula is shown to be equivalent with convergence of the corresponding sequence of Kendall distribution functions. No extra differentiability conditions on the generators are needed.

Introduction

Let C n be a sequence of bivariate Archimedean copulas with generators ψ n and Kendall distribution functions K n . In this note, we establish necessary and sufficient conditions in terms of ψ n or K n for convergence of the sequence of copulas C n to a limiting copula C which is either Archimedean or comonotone. In particular, we extend results in Genest and MacKay (1986, Proposition 4.2 A c c e p t e d m a n u s c r i p t and 4.3) and Nelsen (1999, Theorems 4.4.7 and 4.4.8) to generators that are possibly not everywhere differentiable. Moreover, we show that convergence of the sequence of copulas is equivalent to convergence of the corresponding sequence of Kendall distribution functions. These extensions are used in [START_REF] Charpentier | Lower tail dependence for Archimedean copulas: characterizations and pitfalls[END_REF] to study lower tail dependence for Archimedean copulas, extending and partially correcting [START_REF] Juri | Copula convergence theorems for tail events[END_REF].

The link between Archimedean copulas and their Kendall distribution functions has already been exploited in the context of statistical inference on Archimedean copulas [START_REF] Genest | Statistical Inference Procedures for Bivariate Archimedean Copulas[END_REF][START_REF] Barbe | On Kendall's process[END_REF][START_REF] Wang | Model Selection and Semiparametric Inference for Bivariate Failure-Time Data[END_REF][START_REF] Genest | Goodness-of-fit procedures for copula models based on the probability integral transformation[END_REF]. Our findings add perspective to these papers by showing that closeness of Kendall distribution functions indeed implies closeness of the corresponding Archimedean copulas.

The structure of the paper is as follows. We start with some preliminaries in Section 2. In Section 3, we extend to the case of general Archimedean copulas C the expression in Genest and MacKay (1986, Proposition 3.3) for the joint distribution function of the pair of random variables (X, C(X, Y )), where (X, Y ) is itself a random pair with distribution function C. The main results in this paper involve characterizations of the convergence of a sequence of Archimedean copulas to another Archimedean copula or to the comonotone copula (Sections 4 and 5). Extensions to higher dimensions are treated in Section 6. We conclude in Section 7 with a counterexample showing that not every limit copula of a sequence of Archimedean copulas is necessarily Archimedean or comonotone.

Preliminaries

A function C : [0, 1] 2 → [0, 1] is called a (bivariate) copula if it is the distribution function of a (bivariate) random vector with uniform-(0, 1) margins, or more precisely, the restriction to

[0, 1] 2 thereof. A function ψ : [0, 1] → [0, ∞]
is called a generator if it is convex, decreasing and ψ(1) = 0. The generalized inverse of ψ is denoted by

ψ ← (t) = inf{u ∈ [0, 1] | ψ(u) ≤ t}, t ∈ [0, ∞].
A copula C is called Archimedean if there exists a generator ψ such that [START_REF] Schweizer | Probabilistic Metric Spaces[END_REF][START_REF] Genest | Copules archimédiennes et familles de lois bidimensionnelles dont les marges sont données[END_REF]. The conditions imposed on ψ are necessary and sufficient for the expression in the previous display to define a copula. The copula C determines the generator ψ uniquely The class of Archimedean copulas encompasses many well known bivariate parametric distributions, such as the Frank, Clayton or Gumbel copulas (Nelsen, 1999, Table 4.1). Furthermore, if the inverse of the generator, ψ ← , is the Laplace transform of a nonnegative random variable, then the corresponding Archimedean copula C reduces to the proportional frailty model in [START_REF] Marshall | Families of multivariate distributions[END_REF] and [START_REF] Oakes | Bivariate survival models induced by frailties[END_REF].

C(u, v) = ψ ← {ψ(u) + ψ(v)}, (u, v) ∈ [0, 1] 2 (
If the generator is a natural way to identify the Archimedean copula, other functions can be considered as well. The Kendall distribution function K of a copula C is defined as the distribution function of the random variable C(X, Y ), where (X, Y ) is a random pair with distribution function C, so

K(t) = Pr[C(X, Y ) ≤ t], t ∈ [0, 1].
If the copula C is Archimedean with generator ψ, then K(t) = t -λ(t) with λ(t) = ψ(t)/ψ (t) and ψ is the right-hand derivative of ψ on [0, 1) (Genest and Rivest, 1993, Proposition 1.1). Conversely, from K or λ it is possible to reconstruct ψ up to a multiplicative constant via

ψ(u) = ψ(u 0 ) exp u u 0 1 λ(t) dt , 0 < u 0 < 1; 0 ≤ u ≤ 1.

Auxiliary result

The following result is a useful device to deduce properties of the generator ψ of an Archimedean copula C from the copula itself. For twice continuously differentiable generators, the result can already be found in Genest and MacKay (1986, Proposition 3.3).

Proposition 1 Let (X, Y ) be a bivariate random vector with joint distribution function C, a bivariate Archimedean copula with generator ψ. Let ψ be the right-hand derivative of ψ on [0, 1).

Put Z = C(X, Y ). For (z, x) ∈ [0, 1] 2 , Pr[X ≤ x, Z ≤ z] =                            x if x ≤ z ≤ 1, z + ψ(x) ψ (z) - ψ(z) ψ (z) if 0 < z < x ≤ 1, ψ(x) -ψ(0) ψ (0) if z = 0 < x and ψ (0) > -∞, 0 if z = 0 < x and ψ (0) = -∞.

A c c e p t e d m a n u s c r i p t

Proof

. Since Z = C(X, Y ) ≤ X, we have Pr[X ≤ x, Z ≤ z] = Pr[X ≤ x] = x for x ≤ z ≤ 1.
Hence we can restrict attention to z < x.

The case z = 0 < x follows from the case 0 < z < x by the fact that ψ (0) = lim z↓0 ψ (z) and the fact that lim z↓0 ψ(z)/ψ (z) = 0 if ψ (z) = -∞, the latter property following from convexity.

Hence we can restrict attention to the case 0 < z < x. Since both ψ and the function z → Pr[X ≤ x, Z ≤ z] are right-continuous, it suffices to prove the stated equality for z such that ψ is continuous in z.

We have Pr

[X ≤ x, Z ≤ z] = z + Pr[z < X ≤ x, Z ≤ z].
We can focus on the last term on the right-hand side. Let n be a positive integer, and let

z = u 0 < u 1 < • • • < u n = x be such that ψ(u i ) = 1 - i n ψ(z) + i n ψ(x), i = 0, 1, . . . , n.
We have

Pr[z < X ≤ x, Z ≤ z] = n i=1 Pr[u i-1 < X ≤ u i , Z ≤ z]. If u i-1 < X ≤ u i , then C(u i-1 , Y ) ≤ Z ≤ C(u i , Y ). Hence n i=1 Pr[u i-1 < X ≤ u i , C(u i , Y ) ≤ z] ≤ Pr[z < X ≤ x, Z ≤ z] ≤ n i=1 Pr[u i-1 < X ≤ u i , C(u i-1 , Y ) ≤ z]. Further, for z ≤ u ≤ 1, since ψ and ψ ← are decreasing, C(u, Y ) ≤ z is equivalent to Y ≤ ψ ← {ψ(z) -ψ(u)}. We find that Pr[z < X ≤ x, Z ≤ z] ≤ n i=1 Pr[u i-1 < X ≤ u i , Y ≤ ψ ← {ψ(z) -ψ(u i-1 )}] = n i=1 (C(u i , ψ ← {ψ(z) -ψ(u i-1 )}) -C(u i-1 , ψ ← {ψ(z) -ψ(u i-1 )})) = n i=1 (ψ ← {ψ(u i ) + ψ(z) -ψ(u i-1 )} -ψ ← {ψ(z)}) .

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Our choice of the grid {u i } is such that

ψ(u i ) -ψ(u i-1 ) = -{ψ(z) -ψ(x)}/n, i = 1, . . . , n. Hence Pr[z < X ≤ x, Z ≤ z] ≤ n ψ ← [ψ(z) -{ψ(z) -ψ(x)}/n] -ψ ← {ψ(z)} Since ψ ← is convex with nondecreasing derivative 1/(ψ • ψ ← ), ψ ← (a) -ψ(b) ≤ (a -b) 1 ψ {ψ ← (a)} , 0 < a < b < ψ(0).
Combine the two previous displays to find

Pr[z < X ≤ x, Z ≤ z] ≤ - ψ(z) -ψ(x) ψ (ψ ← [ψ(z) -{ψ(z) -ψ(x)}/n])
Let n tend to infinity and use the fact that z is a continuity point of ψ to find

Pr[z < X ≤ x, Z ≤ z] ≤ - ψ(z) -ψ(x) ψ (z) .
The inequality in the other direction follows in a completely similar fashion, so the details are omitted.

4 Convergence to an Archimedean copula

In this section, we investigate necessary and sufficient conditions on a sequence of Archimedean copulas C n with generators ψ n to converge to an Archimedean copula C with generator ψ. Let ψ n and ψ be the right-hand derivatives of ψ n and ψ, respectively, and denote

λ n = ψ n /ψ n , λ = ψ/ψ , K n (t) = t -λ n (t), and
K(t) = t -λ(t).
For twice continuously differentiable generators, the equivalence of (i) and (ii) in Proposition 2 below was already established in Genest and MacKay (1986, Proposition 4.2). The claim that characterizations (iii) and (v) are sufficient for copula convergence seems to be new. Note that condition (ii) requires convergence of ψ n (x)/ψ n (y) for all x and y while condition (iii) only requires convergence of ψ n (x)/ψ n (x) for all x. Condition (v) implies that the map from an Archimedean copula to its Kendall distribution function is a homeomorphism with respect to the topology of weak convergence.

Proposition 2

The following five conditions are equivalent:

(i) lim n→∞ C n (x, y) = C(x, y) for all (x, y) ∈ [0, 1] 2

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(ii) lim n→∞ ψ n (x)/ψ n (y) = ψ(x)/ψ (y) for every x ∈ (0, 1] and y ∈ (0, 1) such that ψ is continuous in y. (iii) lim n→∞ λ n (x) = λ(x) for every x ∈ (0, 1) such that λ is continuous in x.

(iv) There exist positive constants κ n such that lim n→∞ κ n ψ n (x) = ψ(x) for all x ∈ [0, 1]. (v) lim n→∞ K n (x) = K(x) for every x ∈ (0, 1) such that K is continuous in x.

Proof. (i) implies (ii). Let (X, Y ) and (X n , Y n ) be pairs of random variables with joint distribution functions C and C n , respectively. Also, put Z = Y ) and

Z n = C n (X n , Y n ). By (i), (X n , Y n ) converges in distribution to (X, Y ) as n → ∞. Moreover, since C is a continuous distribution function, the convergence of C n to C is necessarily uniform in (x, y) ∈ [0, 1] 2 . Hence (X n , Z n ) converges in distribution to (X, Z) as n → ∞. By Proposition 1, we have lim n→∞ ψ n (x) -ψ n (y) ψ n (y) = ψ(x) -ψ(y) ψ (y) for all 0 < y < x ≤ 1 such that ψ is continuous in y. Choose x = 1 to find lim n→∞ ψ n (y) ψ n (y) = ψ(y) ψ (y) .
Combine the two previous displays to get

lim n→∞ ψ n (x) ψ n (y) = ψ(x) ψ (y)
for every 0 < y ≤ x ≤ 1 such that y < 1 and ψ is continuous in y. Let 0 < x i < 1 for i = 1, 2 and apply the above display to (x 1 , y) and (x 2 , y) for some 0 < y < min(x 1 , x 2 ) in which ψ is continuous to arrive at

lim n→∞ ψ n (x 1 ) ψ n (x 2 ) = ψ(x 1 ) ψ(x 2 ) .
Combine the last two displays to arrive at (ii).

(ii) implies (iii). Trivial.

(iii) implies (iv). For 0 < x < y < 1, we have

log ψ n (y) -log ψ n (x) = y x ψ n (z) ψ n (z) dz = y x 1 λ n (z)
dz.

Suppose that we can show that the limit of the integral of the right-hand side of the previous display is equal to the integral of the (almost everywhere) limit of the integrand. Then we have lim n→∞ {log ψ n (y) -log ψ n (x)} = log ψ(y) -log ψ(x).

A c c e p t e d m a n u s c r i p t

This, in turn, obviously implies (iv).

In order to justify interchanging limit and integral in the previous paragraph, we will show that (iii) implies lim sup

n→∞ sup z∈[x,y] ψ n (z) ψ n (z) < ∞. Let 0 < ε < x be such that |ψ (x -ε)| ≤ ψ(y)/(4ε).
By (iii), we have

lim n→∞ 1 |ψ n (z)| ψ n (z) > 2 |ψ (z)| ψ(z) = 0
for almost every z ∈ [x -ε, y]. Since the above indicator variables are bounded and converge pointwise to zero, there exists a positive integer n ε such that

y x-ε 1 |ψ n (z)| ψ n (z) > 2 |ψ (z)| ψ(z) dz < ε for all integer n ≥ n ε . Hence, for z ∈ [x, y] and integer n ≥ n ε , there exist z -ε < u < z such that |ψ n (u)| ψ n (u) ≤ 2 |ψ (u)| ψ(u) ≤ 2 |ψ (x -ε)| ψ(y) ≤ 1 2ε
But then, since ψ n and |ψ n | are both nonincreasing,

ψ n (z) |ψ n (z)| ≥ ψ n (u) -(z -u)|ψ n (u)| |ψ n (u)| ≥ 2ε -ε = ε,
as required.

(iv) implies (i). Let φ n = κ n ψ n . Then φ n is a generator of C n . Since each φ n is monotone and since ψ is monotone and continuous, we have lim

n→∞ φ n (x n ) = ψ(x) whenever lim n→∞ x n = x in [0, 1]. Hence also lim n→∞ φ ← n (t n ) = ψ ← (t) whenever lim n→∞ t n = t in [0, ∞]. Hence, for every (x, y) ∈ [0, 1] 2 , C n (x, y) = φ ← n {φ n (x) + φ n (y)} → ψ ← {ψ(x) + ψ(y)} = C(x, y),
as n → ∞.

(v) implies (iii) and conversely. Trivial.

2

Example 3 In Prange and Scherer (2006, section 2.4), in the context of basket type credit derivatives, the dependence between asset value and latent variable is modeled by an Archimedean copula with a piecewise linear generator. That

A c c e p t e d m a n u s c r i p t

is, on a set of nodes 0 = x 0 < x 1 < . . . < x n = 1, values 1 = y 0 > y 1 > . . . > y n = 0 are specified and the generator ψ is defined to be a piecewise linear convex function with ψ(x i ) = y i . The points y i , i = 1, . . . , n -1 must be chosen to satisfy the convexity of ψ.

By Proposition 2, the corresponding Archimedean copula is a continuous function of the abscissa points x i and the ordinates y i . Moreover, the class of Archimedean copulas with piecewise linear generators is dense in the set of Archimedean copulas: For a given Archimedean copula C with generator ψ we can find a sequence of piecewise linear generators ψ n such that lim n→∞ ψ n (x) = ψ(x) for every x ∈ [0, 1]; for instance, let ψ n be the linear interpolation of ψ on the set {i/n; i = 1, . . . , n}, with ψ n (0) chosen in such a way that ψ n is linear on [0, 2/n]. The corresponding sequence of Archimedean copulas C n then converges pointwise to C.

Example 4 The countermonotone copula C(x, y) = max(x + y -1, 0) is Archimedean with generator ψ(t) = 1 -t for t ∈ [0, 1]. By Proposition 2, a sequence of Archimedean copulas C n with Kendall distribution functions K n converges to the countermonotone copula if and only if lim n→∞ K n (x) = 0 for every x ∈ (0, 1).

Convergence to the comonotone copula

The comonotone copula is itself not an Archimedean copula, so that Proposition 2 is not suitable for deciding whether a sequence of copulas converges to the comonotone copula. The following resulting, extending Nelsen (1999, Theorem 4.4.8) to arbitrary generators, gives such a criterion. Let C n , ψ n , λ n , and K n be as in the beginning of Section 4.

Proposition 5

The following four conditions are equivalent:

(i) lim n→∞ C n (x, y) = min(x, y) for all (x, y) ∈ [0, 1] 2 (ii) lim n→∞ λ n (x) = 0 for every x ∈ (0, 1). (iii) lim n→∞ ψ n (y)/ψ n (x) = 0 for every 0 ≤ x < y ≤ 1. (iv) lim n→∞ K n (x) =
x for every x ∈ (0, 1).

Proof. (i) implies (ii). Let (X n , Y n ) be a pair of random variables with distribution function C n . Since the limit of C n is the comonotone copula, (X n , Y n ) converges in distribution to (X, X), where X is a uniform random variable on (0, 1). But since the convergence in (i) is necessarily uniform, we find that Z n = C n (X n , Y n ) converges in distribution to min(X, X) = X, whence

Pr[Z n ≤ z] = z + ψ n (z) ψ n (z) , 0 < z < 1.
Hence we arrive at (ii).

(ii) implies (iii). Let 0 < x < y < 1 (the cases x = 0 or y = 1 follow by monotonicity of ψ n ). We have

ψ n (x) ψ n (y) -1 = ψ n (x) -ψ n (y) ψ n (y) ≥ (y -x)|ψ n (y)| ψ n (y)
.

By (ii), the right-hand side diverges to infinity as n → ∞.

(iii) implies (i). Since each C n is a symmetric copula, it suffices to consider 0 < x ≤ y < 1. Take 0 < w < x. By (ii), we have ψ n (w) ≥ 2ψ n (x) ≥ ψ n (x) + ψ n (y) for all sufficiently large integer n, whence

w ≤ ψ ← n {ψ n (x) + ψ n (y)} = C n (x, y) ≤ x.
Let first n → ∞ and then w ↑ x to find that lim n→∞ C n (x, y) = x.

6 Extension to higher dimensions

Propositions 2 and 5 can be readily extended to the general multivariate case. Let d be an integer at least two. A d-variate copula C is the distribution function of a d-variate random vector (X 1 , . . . , X d ),

C(x 1 , . . . , x d ) = Pr[X 1 ≤ x 1 , . . . , X d ≤ x d ]
the components of which are uniformly distributed on the interval [0, 1]. A d-variate copula C is called Archimedean if there exists a generator ψ such that

C(x 1 , . . . , x d ) = ψ ← {ψ(x 1 ) + • • • + ψ(x d )} for all (x 1 , . . . , x d ) ∈ [0, 1] d .
In general, extra conditions on the generator ψ are required to ensure that the expression in the above display defines a genuine copula. A sufficient condition is for instance that ψ ← is d-times differentiable and (-D) j ψ ← ≥ 0 for every j = 1, . . . , d; see for instance Kimberling (1974, Theorems 1 and 2), [START_REF] Schweizer | Probabilistic Metric Spaces[END_REF], Barbe et al. (1996, Example 3), andNelsen (1999, Section 4.6). A special case arises if ψ ← is completely monotonic, that is, if ψ ← is infinitely times differentiable and if (-D) j ψ ← ≥ 0 for every positive integer j. In this case, ψ ← is the Laplace transform of a nonnegative random variable and C reduces to the proportional frailty model [START_REF] Marshall | Families of multivariate distributions[END_REF][START_REF] Oakes | Bivariate survival models induced by frailties[END_REF].

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Obviously, if the distribution function of the random vector (X 1 , . . . , X d ) is given by the d-variate Archimedean copula C with generator ψ, then the distribution function of every bivariate subvector (X i , X j ), with i = j, is given by the bivariate Archimedean copula with the same generator. This property can be used to upgrade Propositions 2 and 5 to the general multivariate case.

Let C n be a sequence of d-variate Archimedean copulas with generators ψ n .

On the one hand, if C n converges to another d-variate Archimedean copula C with generator ψ or to the d-variate comonotone copula, then the sequence of bivariate Archimedean copulas with generators ψ n must converge to the bivariate Archimedean copula with generator ψ or to the bivariate comonotone copula, respectively. Hence, the stated conditions on the sequence of generators are certainly necessary for convergence of the sequence of copulas. On the other hand, they are also sufficient, as the proofs of the implications "(iv) implies (i)" in Proposition 2 and "(iii) implies (i)" in Proposition 5 carry over to the d-variate case with only notational changes.

Note that there is no analogon in dimension d ≥ 3 of Example 4: if d ≥ 3, then the Fréchet-Hoeffding lower bound max(x 1 + • • • + x d -1, 0) is not a copula, whereas the point-wise limit of a sequence of copulas is necessarily a copula.

Counterexample

From Propositions 2 and 5, one might get the impression that every limit copula of a sequence of Archimedean copulas is necessarily Archimedean or comonotone. This is not true, as is demonstrated by the following example.

For integer n ≥ 2, define a generator ψ n by

ψ n (x) =      n -2(n -1)x if 0 ≤ x ≤ 1/2, 2(1 -x) if 1/2 ≤ x ≤ 1.
In words, ψ n is piecewise linear with knots ψ n (0) = n, ψ n (1/2) = 1, and ψ n (1) = 0. Denoting the right-hand derivative of ψ n with ψ n , we have

K n (x) = x - ψ n (x) ψ n (x) =      n/{2(n -1)} if 0 ≤ x < 1/2, 1 if 1/2 ≤ x ≤ 1.
Let C n be the Archimedean copula with generator ψ n . For real z, denote z + = max(z, 0). By direct computation, we find that lim n→∞ C n (x, y) = C(x, y)

  multiplicative constant. If ψ(0) = ∞, then ψ ← is the ordinary inverse of ψ; in this case, the generator and the Archimedean copula are called strict.

  n→∞ Pr[Z n ≤ z] = z for all z ∈ [0, 1]. But by Proposition 1,

Acknowledgment. The authors gratefully acknowledge helpful discussions with Christian Genest as well as an extensive list of constructive comments by an anonymous referee.

be (Johan SegersSupported by a VENI grant of the Netherlands Organization for Scientific Research (NWO) and by IAP research network grant nr. P5/24 of the Belgian government (Belgian Science Policy).

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where

(1)

The copula C corresponds to the uniform distribution with respect to onedimensional Lebesgue measure on the union of the two line segments

By Genest and Rivest (1993, Proposition 1.2), the Kendall distribution function K of an Archimedean copula necessarily satisfies lim t↑v K(t) < v for every 0 < v < 1. As the function K in (2) violates this condition in v = 1/2, the copula C in (1) cannot be Archimedean.


Carla Henriques
email: carlahenriq@mat.estv.ipv.pt

Paulo Eduardo Oliveira

Dep Matemática

Large deviations for the empirical mean of associated random variables

Keywords: association, large deviations, stationarity AMS Classification: 60E15, 60F10

We find conditions under which the sequence of empirical means of associated random variables, {Xn, n ≥ 1}, satisfies the large deviation principle.

Introduction, assumptions and auxiliary results

Consider a sequence {X n , n ≥ 1} of associated and strictly stationary real valued random variables. In this paper, we prove a large deviation principle (LDP) for the sequence of partial means.

We recall that a sequence of random variables {X n , n ≥ 1} is said to be associated if for any m ∈ IN and any two real-valued coordinatewise nondecreasing

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functions f and g it holds Cov f (X 1 , . . . , X m ) , g (X 1 , . . . , X m ) ≥ 0 , whenever this covariance exists. This definition was introduced in statistics by [START_REF] Esary | Association of random variables, with applications[END_REF] who were motivated from applications in reliability theory. Independently, this notion appeared also in statistical mechanics, referred as the KFG inequalities after the work of Fortuin, Kasteleyn and Ginibre (1971). There has been an intense research on the subject matter of association. For a brief review of the relevant literature on this subject, we suggest Roussas (1999) and Dewan and Prakasa Rao (2001),which include, besides numerous references, several applications of this concept.

As mentioned before, we will find conditions under which the large deviation principle (LDP) holds for the sequence of empirical means of associated random variables. When the random variables are independent and identically distributed (i.i.d.), this is a classical result known in the literature of large deviations as the Cramér's theorem. As is well known, this result has been extended in several directions, one of which is the case of non-independence of the underlying random variables. The literature on this subject is now extensive. For an account of the relevant results on LDPs, see, for example, Dembo and Zeitouni (1998) and the references therein. We will prove a LDP assuming a hyper-geometric decrease rate on the covariances Cov(X 1 , X n ) of associated variables. This is in accordance with analogous results obtained under mixing assumptions, where deviations from independence were conveniently controlled (see, for example, [START_REF] Nummelin | Large deviations for funccionals of stationary processes[END_REF][START_REF] Bryc | On large deviations for uniformly strong mixing sequences[END_REF][START_REF] Bryc | Large deviations and strong mixing[END_REF].

In fact, as is well known, the covariance structure of a collection of associated random variables highly determines its approximate independence (cf. Newman, 1984). So, the referred condition on the decrease rate of the covariances is, for association, the counterpart of the hyper-geometric mixing rates assumed to establish the LDP under φ-mixing and α-mixing in [START_REF] Bryc | On large deviations for uniformly strong mixing sequences[END_REF] , 1996).

We now present the assumptions to be considered in this paper.

(A1) The sequence {X n , n ≥ 1} is associated and strictly stationary.

(A2) The variables of the sequence {X n , n ≥ 1} are uniformly bounded, that is,

|X n | ≤ M, n ≥ 1.
(A3) For each n ≥ 1, X n has density function bounded by a 1 (B 1 ) n , for some a 1 > 0 and some B 1 > 1.

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These assumptions are usual in associated literature, except for (A3). Concerning this last assumption, we remark that when we consider a sequence of independent and identically distributed random variables, say Y 1 , Y 2 , . . ., having a common bounded density with compact support, and take

X n = c (Y n + . . . + Y n+m )
, for some fixed m ∈ IN and c ∈ IR, then these variables verify (A3). Notice further that this procedure is a commonly used method to generate associated sequences.

In addition, we will consider an hyper-geometric decay rate of the covariances. We state this condition in terms of the sequence

u(n) = ∞ j=n+1 Cov(X 1 , X j ) . (H) u(n) ≤ a 0 exp(-n log 1+a n),
for some a 0 > 0 and a > 0.

We notice that (H) holds true if Cov(X 1 , X n ) = a 1 exp(-n log 1+a n), for some a 1 > 0 and a > 0. In fact, if we take

v K (x) = +∞ x K(t) dt, with K(t) = exp(-t log 1+a t), from the l'Hospital rule it easily follows that v K (t)/K(t) → 0, as t -→ +∞. Now just recall that u(n) ≤ a 1 v K (n).
The following two lemmas will be needed for the proof of the main result.

First we state a result contained in [START_REF] Newman | Normal fluctuations and the FKG inequalities[END_REF] are random variables such that E f (X 1 ) 2 < +∞ and E g(X 2 ) 2 < +∞, then

Cov (f (X 1 ), g(X 2 )) = +∞ -∞ +∞ -∞ f (x 1 )g (x 2 )H 1,2 (x 1 , x 2 ) dx 1 dx 2 ,
where

H 1,2 (x 1 , x 2 ) = Cov I (x1,+∞) (X 1 ), I (x2,+∞) (X 2 ) .
The next result follows from relation (21) in [START_REF] Newman | Normal fluctuations and the FKG inequalities[END_REF] and Corollary to Theorem 1 in Sadikova (1996) (see Lemma 2.6 in Roussas, 1995, for details).

Lemma 1.2 [START_REF] Roussas | Asymptotic normality of a smooth estimate of a random field distribution function under association[END_REF] Let X 1 and X 2 be two associated random variables having density functions bounded by B 0 . Then, for all

x 1 , x 2 ∈ IR, Cov I (-∞,x1] (X 1 ), I (-∞,x2] (X 2 ) ≤ B 1 Cov 1/3 (X 1 , X 2 ) ,
where

B 1 = 2 max(2/π 2 , 45B 0 ).

A c c e p t e d m a n u s c r i p t 2 Large deviation principle

For the proof of the LDP, we follow the methodology of proof of Theorem 6.4.4

of Dembo and Zeitouni (1998), which deals with the large deviation principle of the empirical mean, under a certain mixing assumption.

For n, m ≥ 1, define

X m n = 1 n-m n i=m+1 X i .
For the sake of simplicity, we write X n instead of X 0 n . The following two results are key tools to prove the LDP for the empirical mean X n . These results are the analogues for our framework of Lemmas 6.4.6 and 6.4.7 of Dembo and Zeitouni (1998). The proofs follow the same sort of arguments, so we will concentrate only on the technical details that are specific to associated variables. Lemma 2.1 Suppose that (A1), (A3) and (H) are satisfied. Then, for each function g : IR -→ IR concave, continuous and bounded above, the following limit exists Xn) .

Λ g = lim n→+∞ 1 n log E e n g(
Proof: Let g : IR -→ IR be concave, continuous and bounded above. Being concave and continuous, g is also Lipschitz continuous on [-M, M ], that is,

there exists L > 0 such that, for all x, y ∈ [-M, M ], |g(x) -g(y)| ≤ L |x -y|.
Without loss of generality, we assume that -∞ < -B ≤ g(x) ≤ 0, for all

x ∈ [-M, M ].
As in the proof of Lemma 6.4.6 of Dembo and Zeitouni (1998), define h(n) = log E e ng (Xn) and obtain h(n + m) ≤ 2lLMlog E e ng(Xn) e mg(X n+l n+m+l ) .

(1)

For each n ∈ IN , define f n (x) = e ng(x) , x ∈ [-M, M ],
) = [-M,M ] 2 f n (x)f m (y)Cov I (-∞,x] (X n ), I (-∞,y] (X n+l n+m+l ) dx dy ≤ nmL 2 Cov X n , X n+l n+m+l ,
remembering that all the covariances above are non-negative by association.

Using the stationarity assumption (A1), we get Cov e ng(Xn) , e mg(X n+l n+m+l )

≤ L 2 n ∞ i=l+2 Cov(X 1 , X i ) = L 2 (n + m)u(l) ,
yields

h(n + m) ≤ 2lLM + h(n) + h(m) -log (Θ(l, n + m) ∨ 0) . (2)
By (H), we have, for each κ < a and for each c ∈ IR,

lim n→+∞ n u n log 1+κ n e cn = 0 . ( 3
)
Let 0 < δ < a, where a is given in (H). From (3) it is obvious that In what follows we will use the notation

Θ n log 1+δ n , n = 1 -L 2 nu n log 1+δ n e nB -→
S δ x =]x -δ, x + δ[. Lemma 2.2 Suppose that (A1), (A3), (A4) and (H) are satisfied. If x 1 , x 2 ∈ IR are such that, for each δ > 0, lim inf n→+∞ 1 n log P X n ∈ S δ xi > -∞, i = 1, 2 , then inf δ>0 lim inf n→+∞ 1 n log P X 2n ∈ S δ x 1 +x 2 2 P X n ∈ S δ/2 x1 P X n ∈ S δ/2 x2 ≥ 0.
Proof: Fix δ > 0. From the assumptions of the lemma, there exists c 1 > 0 such that, for every sufficiently large n,

P X n ∈ S δ/2 x1 P X n ∈ S δ/2 x2 ≥ exp(-nc 1 ). (4)
By (A4), we may apply Lemma 1.2 to obtain

P X n ∈ S δ/2 x1 , X n+l 2n+l ∈ S δ/2 x2 -P X n ∈ S δ/2 x1 P X n+l 2n+l ∈ S δ/2 x2 ≤ 4B n Cov 1/3 X n , X n+l 2n+l ,
where B n = 2 max(2/π 2 , 45 a 1 B n 1 ). Therefore, by the stationarity assumption (A1),

P X n ∈ S δ/2 x1 , X n+l 2n+l ∈ S δ/2 x2 -P X n ∈ S δ/2 x1 P X n ∈ S δ/2 x2 ≤ 4B n 1 n 2 n ∞ i=l+1 Cov(X 1 , X i ) 1/3 = 4B n u(l) n
P X n ∈ S δ/2 x1 , X n+l 2n+l ∈ S δ/2 x2 P X n ∈ S δ/2 x1 P X n ∈ S δ/2 x2 ≥ 1 -4B n u(l) n 1/3 exp(c 1 n), (5)
for each l ∈ IN and n large enough, taking into account [START_REF] Dewan | Associated sequences and related inference problems[END_REF].

Following the arguments used in the proof of Lemma 6.4.7 in Dembo and Zeitouni (1998), we get, for l = δ n 2M ,

P X 2n ∈ S δ x 1 +x 2 2 ≥ P X n ∈ S δ/2 x1 , X n+l 2n+l ∈ S δ/2 x2 , so that, from (5), lim inf n→+∞ 1 n log P X 2n ∈ S δ x 1 +x 2 2 P X n ∈ S δ/2 x1 P X n ∈ S δ/2 x2 ≥ lim inf n→+∞ 1 n log      1 -4B n u δ n 2M n 1/3 exp(c 1 n)    ∨ 0   . By (3)
, which is valid under (H), the right-hand side above is equal to zero, from which the desired result follows.

We may now formulate the main result of this section. The rest of the proof proceeds exactly as in Theorem 6.4.4 of Dembo and Zeitouni (1998).

A c c e p t e d m a n u s c r i p t and then E≥ 1 -

 then1 e ng(Xn) e mg(X n+l n+m+l ) E e ng(Xn) E e mg(X n+l n+m+l ) L 2 (n + m)u(l)e (n+m)B , as g(x) ≥ -B, for all x ∈ [-M, M ]. Now, define Θ(l, n) = 1 -L 2 nu(l)e nB , l, n ∈ IN . As in the proof of Lemma 6.4.6 of Dembo and Zeitouni (1998), by means of the preceding inequality, (

  1, as n → +∞. So, we may take l = n+m log 1+δ (n+m) as in the proof of Lemma 6.4.6 of Dembo and Zeitouni (1998) and proceed in the same way to conclude the proof.

Theorem 2 . 3 A c c e p t e d m a n u s c r i p t Lemma 2 .

 232 Under (A1), (A2), (A3) and (H), the sequence {X n , n ≥ 1} satisfies the large deviation principle with rate function given byΛ * (x) = sup t∈I R {tx -Λ(t)}, x ∈ IR , which is the Fenchel-Legendre transform of Λ(t) = lim n→+∞ 1 nlog E e n t Xn .Proof: The proof goes along the same lines as that of Theorem 6.4.4 of Dembo and Zeitouni (1998), using our Lemma 2.1, together with Lemma 4.4.8 and Theorem 4.4.10 of Dembo and Zeitouni (1998), to ensure that {X n , n ≥ 1} satisfies the large deviation principle with good rate function I. The convexity of I follows in the same way as in Dembo and Zeitouni (1998), applying our 2 to obtain, for x 1 , x 2 ∈ IR such that I(x 1 ) < +∞ and I(x 2 ) < +∞,

† Author supported by Centro de Matemática da Universidade de Coimbra and Fundo Social Europeu (FSE) -PRODEP III (Action 5.3) ‡ Author supported by Centro de Matemática da Universidade de Coimbra, Fundação para a Ciência e Tecnologia (FCT) and POCTI.

A c c e p t e d m a n u s c r i p t