Dr Laetitia Rollin
email: laetitia.rollin@chu-rouen.fr

Géraldine De Blasi

Line Boucher

Evelyne Bouteyre Verdier

Jean-François Gehanno

Rollin Laétitia

De Blasi

Boucher Line

J-F Gehanno

Return-to-work support in cancer patients: Which methodology?

Keywords:

Introduction : le retour au travail après un cancer est un déterminant de la qualité de vie des patients. Les objectifs étaient de décrire et évaluer les interventions réalisées dans un centre hospitalier universitaire français pour à aider les patients atteints de cancer dans leur retour au travail Méthode : Un questionnaire a été adressé à 153 patients qui avaient bénéficié de l'accompagnement d'une consultation multidisciplinaire d'aide au retour au travail d'un centre hospitalier universitaire français.

Résultats : 121 patients ont complété le questionnaire. Le taux de retour au travail était de 50% 2 ans après le diagnostic de cancer. Il était significativement plus élevé pour les patients qui avaient accepté un suivi psychologique individuel orienté sur le retour au travail (p=0.04) et plus élevé mais non significativement pour les patients ayant effectué une visite de préreprise avec le médecin du travail durant l'arrêt de travail (p=0.08). La reconnaissance du statut de travailleur handicapé n'était pas associée à un taux de reprise plus élevé. La participation à un groupe de paroles concernait peu de patients et n'était pas non plus associée à un taux de retour plus élevé.

Discussion: le suivi psychologique individuel orienté sur le retour au travail et la consultation du médecin du travail durant l'arrêt de travail doivent être recommandés systématiquement aux patients atteints de cancer en vue de leur retour au travail. La reconnaissance du statut de travailleur handicapé est à conseiller au cas par cas, de même que la participation à un groupe de paroles.

Mots-clés:

Retour au travail -Cancer -Consultation multidisciplinaire -Facteurs prédictifs.

Abstract Introduction

Early screening and medical treatment have increased the survival of cancer patients [START_REF]Recommandation de bonne pratique. Santé et maintien en emploi : prévention de la désinsertion professionnelle des travailleurs[END_REF]. Nevertheless, these patients are often physically and psychologically scarred from the disease and the treatments [START_REF]Recommandation de bonne pratique. Santé et maintien en emploi : prévention de la désinsertion professionnelle des travailleurs[END_REF][START_REF] Rollin | Maintien en emploi des patients atteints de cancer[END_REF][START_REF] Johnsson | Work status and life changes in the first year after breast cancer diagnosis[END_REF]. These developments have led to considering cancer pathologies no longer as fatal diseases but as chronic illnesses [START_REF] Smith | The rationale, design, and implementation of the American Cancer Society's studies of cancer survivors[END_REF]. Consequently, support has been developed with the aim of increasing the quality of life of cancer patients. Among the determinants of quality of life, return to work is an important element. Indeed, for cancer patients it means not only a source of income but also a victory over the illness and a return to a "normal" life [START_REF] Peteet | Cancer and the meaning of work[END_REF][START_REF] Kennedy | Returning to work following cancer[END_REF][START_REF] Blasi | Consultation pluridisciplinaire d'aide à la reprise du travail après un cancer : psychopathologie de la rémission et retour à l'emploi[END_REF][START_REF] Grunfeld | The only way I know how to live is to work": a qualitative study of work following treatment for prostate cancer[END_REF]. Various studies have evidenced the difficulties encountered by cancer survivors when returning to work. For example, De Boer's meta-analysis showed that on average 34% of cancer patients were unemployed against 15% in the control group [START_REF]Recommandation de bonne pratique. Santé et maintien en emploi : prévention de la désinsertion professionnelle des travailleurs[END_REF][START_REF] Boer | Cancer survivors and unemployment -A meta-analysis and meta-regression[END_REF]. This observation has led to actions in favor of the return to work of cancer patients as one of the priorities of public health care in France [START_REF]Recommandation de bonne pratique. Santé et maintien en emploi : prévention de la désinsertion professionnelle des travailleurs[END_REF][START_REF] Moatti | The "Plan Cancer" in France: an economists' point of view[END_REF]. Therefore, work-directed interventions for cancer patients have been developed. However, although the factors associated with return to work have been frequently studied (sociodemographic and professional factors or disease related factors etc), far less literature is available on support methodologies [START_REF]Recommandation de bonne pratique. Santé et maintien en emploi : prévention de la désinsertion professionnelle des travailleurs[END_REF][START_REF] Islam | MyBCC study group. Factors associated with return to work of breast cancer survivors: a systematic review[END_REF][START_REF] Van Muijen | Predictors of return to work and employment in cancer survivors: a systematic review[END_REF][START_REF] Mehnert | Employment and work-related issues in cancer survivors[END_REF][START_REF] Peugniez | Factors related to return to work by women with breast cancer in Northern France[END_REF][START_REF] Tamminga | Return-to-work interventions integrated into cancer care: a systematic review[END_REF]. In fact, several studies underline the lack of consensus on the information to be given to patients concerning the return to work [START_REF]Recommandation de bonne pratique. Santé et maintien en emploi : prévention de la désinsertion professionnelle des travailleurs[END_REF][START_REF] Bains | Helping cancer survivors return to work: what providers tell us about the challenges in assisting cancer patients with work questions[END_REF]. Moreover, previous studies concluded that it it is important to maintain research on return to work in a range of countries since social security systems, contractual sick pay schemes, employer incentivization to provide vocational rehabilitation and societal attitudes to return to work are different among different countries. [START_REF] Rollin | Maintien en emploi des patients atteints de cancer[END_REF] Hence, the objective of this study was to explore return-to-work interventions applied in a French University Hospital and to evaluate its effectiveness in achieving a successful return to work.

Materials and Methods

This observational study included patients using the return-to-work platform of a French University Hospital between January 2010 and August 2013.

Multidisciplinary platform for return-to-work support

In 2007 and 2009, a multi-disciplinary return-to-work platform was tested at Rouen University Hospital, France. At the end of the 1-year trial period and after adjustments had been made to the proposed methodology the return-to-work platform was set up on a permanent basis. The objective of the platform is to help patients with cancer to return to work. The multi-disciplinary return-to-work platform is composed of an occupational physician, a social worker and a psychologist and is accessible to all persons with cancer who wish to prepare their return to work, no matter where they are being treated. The objective of the return-to-work platform is to provide patients with information and to make recommendations for their return to work.

Patients were informed of the possibility to use the return-to-work platform by flyers and posters in the waiting rooms of cancer treatment centers and by oncologists. The methodology of the platform was initially defined on the basis of scientific data available in 2009 on the return to work of patients with chronic diseases, most of the literature dealing with low back pain and return to work

Description of patient support

The strong complementarity and synergy between the occupational physician and the social worker justified their seeing the patient in tandem for the first consultation.

The recommendations made by the occupational physician were based on the possibilities of social support and, conversely, the social measures were based on the employee's working capacities. The proposed framework included assessment of the patient's emotional status and potential appropriate solutions or orientations for each case. The aim was to allow each patient to describe his/her occupational tasks, his/her pathology and to assist each patient in the elaboration of a return-to-work project. The occupational physician evaluated the patient's work ability and, based on occupational tasks, provided the patient with recommendations on optimal work accommodations, such as time-table and task-related accommodations. The occupational physician explained to the patient that the company physician would adjust these recommendations according to the context of the company. The patient was then informed and counseled on the steps to be taken. Approximate dates for performing the steps were determined by the occupational physician according to the patient's health evaluation. The recommended steps were the following:

1) Consultation during the sick-leave period with the company physician to prepare the patient's return to work, ; the company physician has detailed knowledge of the patient's occupational tasks and validates or amends the return-to-work project and communicates his/her recommendations for adjustment to the employer.

2) Request for the recognition of disability status. In France, this recognition is given by a special committee of physicians representing the National Health System. It entitles the employee and the employer to specific help with regard to accommodations in the work environment, funding possibilities for these accommodations and professional reconversion training. Of note, companies and organizations with more than 20 employees have a general obligation to employ 6% of disabled people or are liable to fines.

3) Pre-and post-return-to-work psychological support: First of all, the role of the psychologist is explained to the patient at the beginning of the consultation.

Then, the clinical interview is conducted to promote the verbalization of the patient while ensuring the confidentiality of exchange with the psychologist.

Since personal life and professional issues are intertwined, illness being an intimate experience, it seemed necessary to provide a framework allowing for the expression of problems not only linked to the return to work but also to the "post-cancer" period in general. At the end of the consultation, the psychologist and the patient planned regular follow-up after having defined the work objectives and the frequency of the interviews. When the patient declined the offer of support, the psychologist informed him/her that it was possible to receive support later if he/she felt it necessary. These sessions allowed patients to share their experiences of their disease and of their return to work, to fight a feeling of isolation and to initiate impulses of identification within the group [START_REF] Blasi | Multidisciplinary department of "Return to Work After a Cancer": a French experience of support groups for vocational rehabilitation[END_REF]. There was no obligation for the patient and the recommendations were aimed at helping him/her to make his/her own free choices so as to allow him/her to be the actor of his/her professional development.

In addition, some practical information was systematically given to the patient, "based on the patient's recurrent questions and observations from our support experience:

-Information on the notion of fatigue and lack of concentration often present when returning to work but which fade gradually; -Information on the necessity of anticipating what to say to colleagues, to the employer and possible clients about one's illness, upon returning to work; -Information on possible changes in relations with colleagues, employer and possible clients when returning to work, since they often do not know how to behave toward the employee whom they tend to consider as a sick person or a survivor rather than a worker; -Information on the post-therapeutic void [START_REF] Derzelle | Les enjeux d'une « consultation rémission[END_REF][START_REF] Stanton | Promoting adjustment after treatment for cancer[END_REF]:

-Information on the role of the company physician and the medical confidentiality by which he/she is bound.

A summary document with the plan of action and the different pieces of information was given to the patient at the end of the consultation. In addition, with the patient's agreement, a consultation report was sent to the treating physician: general practitioner, oncologist or company physician. The consultation was no substitute for any of the actors or existing schemes for job retention or return to work. It was intended to create links, support, guide and direct, as early as possible in the return to work process.

Evaluation of the results of patient support

The following data were collected during the first medical consultation: sociodemographic data (age, gender), professional data (educational level and Then early 2013, a return-to-work questionnaire was sent to all patients who had used the return-to-work platform between 2010 and 2013, along with an information letter about the study and its non-binding nature, as well as a stamped and addressed reply envelope.

The questionnaire was built by the multidisciplinary team of the return-to-work platform and tested among the first patients during the trial period in 2007-2009. The main questions of the questionnaire were asked if the patient had returned to work (yes/no), the date of return to work if this was the case, and the implementation of the measures recommended during the first consultation (consultation with the company physician: yes/no, request for recognition of disability status: yes/no). In December 2013, the questionnaire was sent once more to those patients who had not answered or returned it.

Statistical analyses were carried out using the 11.0 version of the STATA software program (StataCorp, College Station, TX USA). First, bivariate analysis with the χ² test or the Fisher test, according to the numbers, was used to examine the factors associated with the return to work. Then, we used multivariate analysis by logistic regression in order to check the independent effect of each variable. Factors studied were socio-demographic variables, occupational status at time of diagnosis (working or not, permanent work contract at time of diagnosis or not), localization of cancer (breast/other), treatment (surgery or not, chemotherapy or not, and radiotherapy or not), presence of symptoms such as tiredness, pain (yes/no), being in remission (yes/no), consultation with the multidisciplinary team of the return-to-work platform during the first year after diagnosis (yes/no), number of consultations with the occupational physician and the social worker of the platform (only one/ at least 2), psychological support (yes/no), participation in support group (yes/no), request for recognition of disability status (yes/no), consultation with the company physician during the sick-leave period (yes/no).

This study was part of a doctoral research project which was approved by the local ethics committee. (number 15-07-01-008).

Results

Overall, 153 questionnaires were sent, 121 of which were returned and analyzed (79%). The socio-demographic, medical and occupational characteristics of the patients who responded are presented in table 1. The characteristics were not significantly different from the initial test group. Median delay between cancer diagnosis and first consultation with the multidisciplinary team of the platform was 16 months. Thirty-four, 43 and 23% of patients consulted during the first year of diagnosis, the second year and after, respectively.

The rate of return to work was 66% [CI95% = 57-75]. The rate increased according to the length of time after diagnosis: from 7% at 6 months, to 13% at 12 months, 38% at 18 months and 50% at 24 months.

Most patients followed the recommendations made during the first consultation and few accepted participation in support groups (table 2). Recommendations concerning therapeutic part-time work are not detailed in table 2 since this measure differs from the others in that it could not be applied in cases where the patient had not returned to work. Therapeutic part-time work was advised to all salaried patients who had still a contract with an employer (n=60). Among these patients, 38 returned to work and for 30 patients this return to work was organized with a therapeutic part-time.

In the bivariate analysis, the factors significantly associated with return to work were: 3).

In the multivariate analysis, the factors significantly associated with return to work were: having a permanent work contract at the time of the diagnosis (OR=4.7, CI95%= [1.4-16], p=0.01), being in remission (OR=5, CI95%=[1.2-25], p=0.03), and having received individual psychological support (OR=2.7, CI95%=[1.0-6.9], p=0.04).

Having a consultation with the company physician was close to the level of significance (p=0.08), OR=2.9, CI95% [0.9-9.2] (table 3).

Discussion

We found a 50% rate of return to work two years after the diagnosis of cancer.

Patients who had individual psychological support for returning to work and a consultation with the company physician during their sick-leave period were more likely to return to work than patients who did not. However, patients who performed the required actions for the recognition of handicapped worker status and patients who participated in a support group were not more likely to return to work than patients who did not.

The rates of return to work were: 7% at 6 months, 13% at 12 months, 38% at 18 months and 50% at 24 months. In comparison, Mehnert, in a review of the literature observed higher rates of return to work of: 40% at 6 months, 62% at 12 months, 73% at 18 months and 89% at 24 months [START_REF] Mehnert | Employment and work-related issues in cancer survivors[END_REF]. These rates varied depending on the countries. It is interesting to examine the average French data [START_REF] Rollin | Research on return to work in European Union countries[END_REF]. The DREES study, conducted in France in 2006, showed that 67% of the patients who had a job at the time of the diagnosis returned to work two years after [START_REF] Marino | Sex differences in the return-to-work process of cancer survivors 2 years after diagnosis: results from a large French population-based sample[END_REF].

The rate of return to work of our sample is lower than the data in the literature.

However, comparison is difficult. Our sample cannot be compared with the general population as it included more women, more so-called "execution" job categories, more patients with sequelae and more patients with progressive cancer. These factors are traditionally associated with lower rates of return to work [START_REF] Blasi | Consultation pluridisciplinaire d'aide à la reprise du travail après un cancer : psychopathologie de la rémission et retour à l'emploi[END_REF][START_REF] Marino | Sex differences in the return-to-work process of cancer survivors 2 years after diagnosis: results from a large French population-based sample[END_REF]. Moreover, 66% of our patients did not use the return-to-work platform until one year after the diagnosis. Those patients who contacted the platform were probably individuals for whom returning to work was more complex. As in the literature, we evaluated the findings from the consultation in relation to the rate of return to work [START_REF] Tamminga | Return-to-work interventions integrated into cancer care: a systematic review[END_REF].

Nevertheless, during the study it seemed to us that this criterion should be viewed with caution. Indeed, in certain cases, the return to work may happen only for financial reasons leading the patient to a "forced" return potentially jeopardizing his/her health. Furthermore, 12% of patients had progressive cancer. Even if the return to work could not be effective because of temporary aggravation of the disease, the implementation of job adjustments, temporary support and hope of "returning to a normal life" were important for those patients. These elements can not be documented by merely studying the return-to-work rate. It would have been interesting to examine the findings from the consultation using other indicators, such as the quality of life, for example. The rate of return to work was significantly higher in patients who had had individual psychological support. Individual support takes into account a patient's specific needs that can vary from one patient to another. As Wells explains, the meaning of work, the role it plays in the person's identity and its financial impact are all patient-specific [START_REF] Wells | Supporting 'work-related goals' rather than 'return to work' after cancer? A systematic review and metasynthesis of 25 qualitative studies[END_REF]. "A person-centred approach supporting the achievement of survivors' work-related goals rather than return to work" is therefore recommended [START_REF] Wells | Supporting 'work-related goals' rather than 'return to work' after cancer? A systematic review and metasynthesis of 25 qualitative studies[END_REF]. Moreover, just like the announcement of the diagnosis, the news that the patient is in remission is a difficult moment: the patient does not immediately become a healthy person with the announcement of remission. The disease and the sequelae need to be "digested" [START_REF]La vie deux ans après un diagnostic de cancer -De l'annonce à l'après cancer[END_REF]. Patients often return to work during remission; psychological support often deals with this aspect. These considerations certainly explain why psychological support was identified as a factor improving return to work.

Participating in support groups was not significantly associated with returning to work.

As a matter of fact, few patients accepted participation in support groups. Yet, these groups allow patients to fight a sense of isolation and to create identification within the group [START_REF] Blasi | Multidisciplinary department of "Return to Work After a Cancer": a French experience of support groups for vocational rehabilitation[END_REF]. Consequently, a more detailed assessment of the patients for whom these groups could be beneficial is recommended but this measure should not be implemented systematically for all patients. The rate of return to work was higher in patients having seen the company physician during their sick-leave period. The company physician has access to the company and can make recommendations to the employer for adjustments in working conditions. The earlier these recommendations are made, the more time the employer has to think them over and to find solutions. Obtaining the recognition of handicapped worker status in order to fund adjustments of workstations does not seem to influence the return to work, whereas it is a specific measure which would promote return to work. The limiting factor for return to work is probably the discussion about job adjustments rather than their actual funding. What is more, the term "recognition of handicapped worker status" has a psychological impact on the patient: even if the main objective is to obtain funding for job adjustments, the patient acquires a different status to his/her colleagues and is recognized as "ill". All in all, meeting the occupational physician during the sick-leave period is largely recommended whereas the request for recognition of handicapped worker status should be proposed on a case-by-case basis. Of note, only 44% of patients consulted the company physician. This measure must be reinforced with a particular explanation of the role of the company physician and the medical secrecy by which he/she is bound.

Our study has several strengths. First, the number of patients included and the level of participation were significant (153 patients, return rate of the questionnaire 79%) which allowed for analysis of statistical data. This high level of response might be explained by the fact that the patients themselves requested a consultation with the multidisciplinary team of the return-to-work platform and by the relationship of trust established during the interviews. Second, the one-year test period before the beginning of the study allowed us to adjust the methodology and to improve the type and content of the information to be delivered to patients. To our knowledge, there is no current framework for the delivery of such information to patients. Furthermore, it has been shown that the information delivered to the patient by health professionals is often imprecise and heterogeneous [START_REF] Tamminga | Return-to-work interventions integrated into cancer care: a systematic review[END_REF].

Nethertheless, our study also has several limitations. Regarding internal validity, patients included were patients who decided to use the return-to-work platform and thus were voluntary patients, which could bias our sample. The majority of patients (66%) did not use the platform until one year after the diagnosis, whereas, according to Menhert, 62% of patients had usually returned to work or had continued to work during the year after cancer diagnosis. This confirms that the patients included in our study probably had specific characteristics and difficulty returning to work. This may explain the low rates of return to work that we found in our study. However, ourreturnto-work platform is probably interesting mostly for these specific patients. Therefore it is interesting to have the results of this specific population. Moreover, as these types of consultations should not be mandatory, the sample studied seems to be representative of the potential population of such a platform. Therefore, only return to work rate was studied. Other criteria should be developed to better describe the return-to-work process and the positive impact for the patient. It would also have been interesting to follow patients over a longer period of time to assess the rate of return to work after 3 or 5 years but this was not the objective of the present study.

Regarding external validity, this study was conducted between 2008 and 2013, at a time when there was no rehabilitation program for patients with cancer in Normandy.

It would be interesting in the future to evaluate our return-to-work platform in a new context of cancer rehabilitation programs that are currently being set up. have difficulty returning to work. Individual psychological support during the returnto-work process and meeting the company physician during the sick-leave period should be systematically recommended. The request for recognition of handicapped worker status should be proposed on a case-by-case basis. Participating in support groups need future studies to assess impact on return to work. Future studies are needed for the long-term assessment of return-to-work platforms for the benefit of cancer patients.

4 )

 4 Participation in a support group run jointly by the psychologist and the occupational physician during 4 sessions of 1h30 each over a 4-month period.

5 )

 5 Applying for therapeutic part-time working: this measure allows the patient to return to work gradually, over several weeks, starting by working half-time, without loss of income. Working time is then increased regularly until reaching the initial time worked before the disease. This requires a prescription by the general practitioner and the agreement of the employer and the company physician.

  qualifications, last occupation according to the Occupations and Socioprofessional Categories (PCS) 2003 classification, occupational status at the time of the diagnosis: yes/no, permanent work contract at the time of the consultation: yes/no and medical data on the disease (localization, evolution, treatment, sequellae and date of cancer diagnosis).

  having a job at the time of the diagnosis (OR=3.3, CI95%=[1.0-11.3]), having a permanent work contract at the time of the diagnosis (OR=5.0, CI95%=[2.0-12.6]), being in remission (OR=5, CI95%=[1.2-20]), and having a consultation with the company physician during the sick-leave period (OR=4.6, CI95%=[1.5-16.3]) (table

Table 1 :

 1 Socio-demographic. medical and occupational characteristics of patients category (cf: PCS-2003) farmers artisans. shop keepers. business leaders -

Acknowledgments

We are grateful to Nikki Sabourin-Gibbs, Rouen University Hospital, for her help in editing the manuscript.

Conflit d'intérêt : aucun


K Huang

H Le Jeannic

J Ruaudel

O Morin

J Laurat

Microcontroller-based locking in optics experiments

Optics experiments critically require the stable and accurate locking of relative phases between light beams or the stabilization of Fabry-Perot cavity lengths. Here, we present a simple and inexpensive technique based on a stand-alone microcontroller unit to perform such tasks. Easily programmed in C language, this reconfigurable digital locking system also enables automatic relocking and sequential functioning. Different algorithms are detailed and applied to fringe locking and to low-and high-finesse optical cavity stabilization, without the need of external modulations or error signals. This technique can readily replace a number of analog locking systems advantageously in a variety of optical experiments.

I. INTRODUCTION

Optical setup usually relies on various locking systems for actively stabilizing interferometer paths or cavity lengths. Traditionally, a technique, such as, among others, the Pound-Drever-Hall 1 (PDH), the dither-and-lock 2 or the tilt-locking 3 processes, provides an error signal, which is then fed into an analog proportional-integralderivative (PID) controller. In our quantum optics lab, for example, cavities containing non-linear crystals, i.e. optical parametric oscillators [4][5][6] , or empty cavities for spatial-mode cleaning and noise filtering are commonly used [START_REF] Bachor | A guide to experiments in quantum optics[END_REF][START_REF] Keller | [END_REF] . Also, fringe locking is very often required when various beams are mixed or when light is injected into a parametric amplifier as the gain depends on the relative phase between the pump and this seed 9 . Beyond the stability of the lock, other features can be very useful in these implementations: automatic searching of the locking point (especially for cavity peak locking), automatic relocking when the lock is lost, sequential locking by a sample-and-hold behavior. These practical features, which can enable for instance long-term data acquisition or just simpler functioning on a daily-basis, can be more easily implemented using digital systems.

Digital locking in optics experiments has indeed attracted more and more attention since the end of the nineties. It has been used for the first time in laser frequency stabilization 10,11 and various implementations have been subsequently developed [12][13][14][15] . Very recently, the fast development of high-speed and low-cost field programmable gate arrays (FPGA) and microcontroller units (MCU) opened a wealth of possibilities. For instance, in Ref. 16, a FPGA programmed with National instruments LabView software is used to generate the modulation signal applied to an electro-optic modulator in a PDH scenario. The FPGA enables automatic relocking and includes some inbuilt locking analysis tools. Similarly, in Ref. 17, a microcontroller unit is also used in such a scenario to lock a fiber laser to an optical cavity.

a) Electronic mail: julien.laurat@upmc.fr In these implementations, the logic unit is usually combined with traditional locking processes. It enables the digitalization and processing of the signal, replacing the usual PID control and lock-in-amplifier, and adds novel capabilities and further scalability.

Here, we present a microcontroller-based locking system that does not require any modulation or external error signal. The core process is based on a maximum (or minimum) -searching algorithm, making the way to lock simple and intuitive. Additionally, the cost is extremely reduced as no other components, either optical or electrical, are required. We will detail the implemented algorithms and show the performances obtained for the locking of a Mach-Zehnder interferometer and of optical cavities with a moderate finesse (100) and a high finesse (1000). The performances are comparable to what can be obtained with typical analog lockings.

Among the large variety of microcontroller units now available, we chose the ADuC7020 from Analog Devices. Advantageously, this chip already contains five 12-bit ADCs (with sampling rate of 1MHz) and four 12-bit DACs (with voltage output settling time of 10 µs) [START_REF]ADuC70xx Series: Precision Analog Microcontroller[END_REF] . Furthermore, the commercially-available evaluation board (EVAL-ADUC7020QSZ, around 50 euros) simplifies its use as no additional wiring is required.

There are ongoing discussions between choosing a FPGA or a conventional MCU for locking purpose. Generally speaking, FPGA can be much faster. However, for many lockings, fast digital controller is not necessary since the frequency bandwidth of the system is usually limited by the sensors, actuators and/or transducers. For instance, the bandwidth of typical piezoelectric transducers (PZT) used for controlling the optical path length is only tens of kHz, which makes FPGAs largely exceeding the needs. Last but not least, microcontrollers can be programmed in C language, which makes them easily accessible by relatively inexperienced users.

II. LOCKING A MACH-ZEHNDER INTERFEROMETER

As a first implementation, we consider the case of a Mach-Zehnder interferometer with 1.5 meter-long arms, as depicted in Fig. 1(a). One photodiode is used to measure the output power and the signal is fed into the ADC input of the microcontroller unit. The MCU output is then sent to a piezoelectric transducer (PI P-016.00H) to control the optical length of one arm. The algorithm used here is a minimum-searching approach with an additional automatic PZT recentering, as detailed now.

A. Searching algorithm

A flowchart of the program execution is given in Fig. 1(b). The voltage on the PZT is increased with a constant step unless the signal value (Y1) is smaller than the previous one, which would lead to a sign-flip of the sweep. As a result, the locked signal can always stay at its maximal value. To avoid disturbances induced by high frequency noise of the detector (P D B ) the signal is averaged over a user-defined numbers of ADC sampling (typically 50).

Importantly, in such long-arm interferometers, fluctuations can lead to a change of the paths by many wavelengths, larger than the limited range of typical actuators. In order to keep the locking, one has thus to shift back the position of the PZT. Usual analog PID controllers do not have this kind of feature, and would need elaborated developments. In contrast, this can be easily achieved by a programmable device. The recentering can indeed be trivially taken into account with simple codes in the locking program. Specifically, when the detected peak position is near the ends of the DAC output range, the output is shifted back to the center (not shown on the flowchart). As a result, the phase can be automatically relocked around the center of the DAC range. After a certain time (usually a few milliseconds), a stable locking is recovered. Such automatic locking enables a very long-term stability, as shown in Fig. 1(c) for a 7-hour recording. Typically, recentering occurred once per hour.

The inset in Fig. 1(c) corresponds to a one-second zoom. The standard deviation (with a sampling rate of 250 kHz) is 0.50±0.05 mV, which can be translated into a phase fluctuation of 1.7 • ±0.1 • (1.5 • if corrected from detector noise). Over 1 hour, the standard deviation raises to 1.7 ± 0.3 mV, corresponding to 3.1 • ± 0.2 • . The MCUbased locking enables thus a very good phase stability on the short as well as on the long time scales, compatible with most applications, including for instance the use of high squeezing levels 6,[START_REF] Takeno | [END_REF]20 in demanding quantum optics and quantum information experiments.

B. Simple model and set parameters

A simple model can be used to estimate the locking performances based on this algorithm. The phase stability is indeed limited by the ADC resolution of the microcontroller. In our case, with a number of bits N =12 and a maximum reference voltage V 0 = 2.5 V, the resolu- tion is given by = V 0 /2 12 0.61mV. Only the voltage fluctuations larger than can be detected, leading to a minimum detectable phase change ∆θ 0 .

To minimize this value, the best strategy is to use the full dynamic range, and thus having a fringe with close to unity visibility, i.e. a detected voltage V = V 0 sin 2 (ϕ/2). In this case, the minimum detectable phase change is given by:

∆θ 0 2 V 0 = 2 √ 2 -N 1.8 • . (1)
For step sizes in the algorithm leading to phase changes smaller than this value, the locking noise, i.e. the measured standard deviation, is expected to be constant, with a value slightly smaller than . In the general case of a non-unit visibility, V 0 has to be replaced by the fringe amplitude (2.2 V in our case). One way to increase the locking performance would be to combine two 12-bit ADCs to obtain a 24-bit system 17 . Another possibility is to use a logarithmic amplifier in order to increase the resolution around the set point.

We now consider step sizes leading to phase changes larger than the previous one. The resulting phase increment depends on the number of bits N of the DAC, the number n of fringe periods for a full scan and an integer M defined by the user:

∆θ = M. 2πn 2 N . (2)
When the phase step ∆θ is larger than ∆θ 0 , the phase fluctuation scales with ∆θ. The noise is driven by the steps. For proper functioning, this regime should be avoided.

For a full characterization of the locking system, and to illustrate this simple model, we studied in a systematic way the behavior obtained for various parameters in the algorithm. Figure 2(a) shows the standard deviation for one second as a function of the step size. As predicted, the measured voltage noise exhibits first a plateau and then scales as the square of the step. If the step size is increased further, a stable locking cannot be obtained anymore.

One parameter more difficult to include in this simple model is the temporal delay between the output (DAC) and read (ADC) blocks. With a fixed step size, the delay plays an important role in the locking behavior as shown in Fig. 2(b). Specifically, when the delay is too long, the bandwidth is too much reduced and cannot compensate for the fluctuations. When the delay is too short, there is not enough time for the PZT to respond. Because the locking algorithm relies on the comparison of the two sequential signal values, the improper response of PZT will produce some disturbance for each step, finally altering the locking. A 1 ms-delay is typically added in the program. A bandwidth around a few kHz is achieved here, limited by the PZT.

Finally, Figure 2(c) provides the noise spectra for different step sizes. The plateau above 3 kHz is due the limited vertical resolution (8 bits) of the oscilloscope used for the data acquisition.

III. LOCKING AN OPTICAL CAVITY

The microcontroller-based locking is now applied to a more complicated task, i.e. the locking of optical cavities. Two cavities with moderate and high finesse, respectively around 100 and 1000, will be considered, requiring two different execution programs.

A. Low-finesse cavity

The locking of a cavity specifically requires a scanning mode to first identify the peak heights and positions of the cavity resonance. This function can be easily realized with simple programming for the generation of triangle signal. The flowchart of the program execution is given in Fig. 3(a). The microcontroller first sweeps the cavity length and subsequently defines a high and low threshold (Y th1 and Y th2 ). It then sweeps again the length to reach an initial start point above the high threshold. If the locked signal becomes smaller than the low threshold, the microcontroller will get out of the locking mode and go back to the scanning mode.

To test this method, we used an optical parametric oscillator, i.e. a cavity with a non-linear crystal inserted into it 6 . The finesse for the pump is around 100 and the 4 cm-long cavity was previously locked by the PDH technique, with a phase-modulation at 12 MHz. Figure 3(b) confirms the long-term stability and the inset shows a one-second zoom. The standard deviation normalized to the peak height is equal to 7.0 × 10 -4 , similar to what is obtained here with the PDH locking (6.6 × 10 -4 ). For 15 minutes, the normalized standard deviation raises to 4 × 10 -3 .

To obtain this result, the locking algorithm has been slightly modified. Indeed, with the maximum-searching algorithm presented previously in Sec. II, the standard deviation is a bit larger (8.0 × 10 -4 ). This simple approach is however sufficient for many applications. Here, the control has been improved by implementing some proportional feedback using more sophisticated programming. For instance, one can numerically calculate the corresponding derivative of the signal, or simply use the signal difference between the sequential steps, as the gain of the feedback signal. We call this method the PI-like algorithm. The measured noise spectra are finally given in Fig. 3(c) for the three different techniques.

B. High-finesse cavity

Compared to a low-finesse cavity, locking a high-finesse cavity is more challenging due to the limited resolution of the DAC. For instance, for a finesse of 1000, the number of sampling points to cover the whole peak is around 4096/1000 4 which is obviously insufficient to lock the cavity. To overcome this problem, one can combine two 12-bit DACs in order to obtain a 24-bit ones as mentioned previously. Another way to address the problem is to use two different scanning modes, namely a long scan and a short scan, as explained now.

The flowchart of the program execution is given in Fig.

4(a)

. One output of the DAC (DAC1) is used for the long scan, which spans over more than a free spectral range in order to identify the rough peak position. Another output (DAC0) is used for the short scan around the peak position identified by the long scan mode and also used for the locking. The two outputs are summed with different gains. Such uneven allocation of sampling over the free spectral range enables the full use of the DAC resolution. Since there are four DACs available in the microcontroller development board, this method can be easily implemented without the need of any additional electronic building. Figure 4 gives an example of the process for locking a 0.5 mm-long cavity 4 with a finesse of 1000. Note that in order to make the re-lock process faster when the cavity is unlocked, the program goes firstly to the short-scan mode instead of directly using the long-scan one. If the short scan still cannot find the peak after 10 sweeps then the program jumps to the long-scan mode.

The performances are displayed in Fig. 5. Figure 5(a) provides the temporal stability using the PI-like algorithm for the locking, while Fig. 5(b) gives the spectra for PI-like, maximum-searching and analog dither-andlock (DTH) techniques. For one second, the standard deviation normalized to the peak height is found to be 4.8×10 -3 for the PI-like, which is slightly better than the 5.6×10 -3 given by DTH. The direct maximum-searching provides a standard deviation of 6 × 10 -3 . For 15 minutes, the normalized standard deviation reaches 8×10 -3 . These results show the suitability and efficiency of the microcontroller-based locking with high-finesse cavities.

As an additional illustration of functions easily implemented by programming, figure 6 finally shows the temporal trace obtained for a sequential locking. The cavity is locked for 50 ms and the light is then blocked. The unit keeps the last output value for a specified time, until the light is switched on again. This sample-and-hold functioning is central to many experiments: for instance, it enables to lock an experiment and then to proceed to acquisition in the photon-counting regime where locking lights have to be off 9 .

IV. CONCLUSION

In conclusion, we have shown that the currently available microcontroller units, such as the 12-bit ADuC7020, can be used for various optical lockings, including highfinesse cavities. It can advantageously replace traditional analog systems, offering similar performances in the presented examples but also additional features that are easy to tailor by C programming. In complex experiments where a great number of lockings are involved, this practical digital control also enables for instance delayed locking when systems are cascaded or very long dataacquisition. Besides their use in laser-based researches, these microcontroller systems can also be easily implemented in undergraduate educational physics laboratory as a good introduction to feedback and control systems.

FIG. 1 .

 1 FIG. 1. (Color online) (a) Experimental setup: locking a 1.5-meter-arm Mach-Zehnder interferometer with a microcontroller unit. HR stands for high-reflective mirror, BS for beam-splitter, PZT for piezoelectric transducer and PD for photodiode. (b) Flowchart of the program execution for the maximum-searching algorithm. (c) Long-term stability. The inset gives a one-second zoom and the associated standard deviation. (d) Noise spectrum at low frequencies with and without locking.

FIG. 2 .

 2 FIG. 2. (Color online) (a) Standard deviation as a function of the step size given by the integer M . The solid line is a guide for the eyes, following the model given in the text. (b) Standard deviation as a function of the delay between the output (DAC) and the acquisition (ADC). (c) Noise spectrum for different step sizes. The delay is set to 800 µs.

FIG. 3 .

 3 FIG. 3. (Color online) (a) Flowchart of the program execution to lock a low-finesse cavity, including peak searching and automatic relocking. (b) Long-term stability. The inset gives a one-second zoom. The standard deviation is normalized to the cavity peak height. (c) Noise spectra for maximumsearching locking, PI-like locking and for locking with the standard analog Pound-Drever-Hall technique.

FIG. 4 .

 4 FIG. 4. (Color online) (a) Flowchart of the program execution to lock a high-finesse cavity. This scheme includes two different scanning modes controlled by two outputs in order to use the full DAC resolution. (b) Experimental locking process of a cavity with a finesse equal to 1000. The light is measured in reflexion. (I) Cavity is locked. (II) Light is blocked to disrupt the lock and the program goes thus to the short-scan mode.(III) As the short-scan did not enable the relocking in this specific example, the long-scan is started in order to learn the rough peak position. (IV) The output is set at a rough start point determined in the previous step. (V) A short-scan is started to find the precise peak position. (VI) The output is set to the precise start point. (VI) Locking is on.

FIG. 5 .FIG. 6 .

 56 FIG. 5. (Color online) (a) Long-term stability for the highfinesse cavity. The inset gives a one-second zoom. The standard deviation is normalized to the cavity peak height. (b) Noise spectrum for maximum-searching locking, PI-like locking and the traditional analog dither-and-lock technique.

ACKNOWLEDGMENTS

This work is supported by the ERA-Net CHIST-ERA (QScale), the Emergence program from Ville de Paris (NanoQIP) and the European Research Council Starting grant HybridNet. K. Huang acknowledges the support from the Foundation for the Author of National Excellent Doctoral Dissertation of China (PY2012004) and the China Scholarship Council. J. Laurat is a member of the Institut Universitaire de France.


L Galtchouk

V Konev

On asymptotic normality of sequential LS-estimates of unstable autoregressive processes. *

Keywords: AMS 1991 Subject Classification : 62L10, 62L12 Autoregressive process, least squares estimate, sequential estimation, uniform asymptotic normality

For estimating the unknown parameters in an unstable autoregressive AR(p), the paper proposes sequential least squares estimates with a special stopping time defined by the trace of the observed Fisher information matrix. The limiting distribution of the sequential LSE is shown to be normal for the parameter vector lying both inside the stability region and on some part of its boundary in contrast to the ordinary LSE. The asymptotic normality of the sequential LSE is provided by a new property of the observed Fisher information matrix which holds both inside the stability region of AR(p) process and on the part of its boundary. The asymptotic distribution of the stopping time is derived.

Introduction

Consider the autoregressive AR(p) model

x n = θ 1 x n-1 + . . . + θ p x n-p + ε n , n = 1, 2, . . . , (1.1) where (x n ) is the observation, (ε n ) is the noise which is a sequence of independent identically distributed (i.i.d.) random variables with Eε 1 = 0 and 0 < Eε 2 1 = σ 2 < ∞, σ 2 is known, x 0 = x -1 = . . . = x 1-p = 0; parameters of the model θ 1 , . . . , θ p are unknown. This model can be expressed in vector form as

X n = AX n-1 + ξ n , (1.2)
where X n = (x n , x n-1 , . . . , x n-p+1 ) ′ , ξ = (ε n , 0, . . . , 0) ′ ,

A = θ 1 . . . θ p I p-1 0 , (1.3)
the prime denotes the transposition.

A commonly used estimate of the parameter vector θ = (θ 1 , . . . , θ p ) ′ is the least squares estimate (LSE)

θ(n) = M -1 n n k=1 X k-1 x k , M n = n k=1 X k-1 X ′ k-1 , (1.4)
where M -1 n denotes the inverse of matrix M n if det M n > 0 and M -1 n = 0 otherwise. Let P(z) = z pθ 1 z p-1 -. . .θ p (1.5) denote the characteristic polynomial of the autoregressive model (1.1 ). The process (1.1 ) is said to be stable (asymptotically stationary) if all roots z i = z i (θ) of the characteristic polynomial (1.5 ) lie inside the unit circle, that is the parameter vector θ = (θ 1 , . . . , θ p ) ′ belongs to the parametric stability region Λ p defined as

Λ p = {θ ∈ R p : |z i (θ)| < 1, i = 1, . . . , p} . (1.6)
The process (1.1 ) is called unstable if the roots of P(z) lie on or inside the unit circle, that is, θ ∈ [Λ p ]; [Λ p ] denotes the closure of the stability region Λ p . It is well known (see,e.g. [START_REF] Anderson | The statistical analysis of time series[END_REF], Th.5.5.7) that the LSE θ(n) is asymptotically normal for all θ ∈ Λ p , that is

√ n(θ(n) -θ) L =⇒ N (0, F ), as n → ∞,
where F = F (θ) is a positive definite matrix, L =⇒ indicates convergence in law. It should be noted that the asymptotic normality of θ(n) is provided by the following asymptotic property of the observed Fisher information matrix lim n→∞ M n /n = F a.s.

(1.7) for all θ ∈ Λ p . On the boundary ∂Λ p of the stability region Λ p , this property does not hold and the distribution of θ(n) is no longer asymptotically normal.

The investigation of the asymptotic distribution of LSE θ(n) when x n is unstable goes back to the late fifties with the paper of [START_REF] White | The limiting distribution of the serial correlation coefficient in the explosive case[END_REF] (see also [START_REF] Ahtola | Distribution of least squares estimators of autoregressive parameters for a process with complex roots on the unit circle[END_REF], [START_REF] Dickey | Distribution of the estimators for autoregressive time series with a unit root[END_REF], [START_REF] Rao | Asymptotic distribution of an estimator of the boundary parameter of an unstable process[END_REF], [START_REF] Sriram | Sequential estimation of the mean of a first order stationary autoregressive process[END_REF],( 1988)) who considered the AR(1) model with i.i.d. N (0, σ 2 ) random errors ε n and θ 1 = 1 and established that

n(θ(n) -1) L =⇒ (W 2 (1) -1)/ 1 0 W 2 (t)dt ,
where W (t) is a standard brownian motion. Subsequently the research of the limiting distribution of θ(n) for unstable AR(p) processes has been receiving considerable attention due to important applications in time series analysis, in modeling economic and financial data and in system identification and control. We can not go into the detail here and refer the reader to the paper by [START_REF] Chan | Limiting distribution of least squares estimates of unstable autoregressive processes[END_REF] who derived the limiting distribution of LSE θ(n) for the general unstable AR(p) model. By making use of the functional central limit theorem approach, Chan and Wei expressed the limiting distribution of LSE θ(n) in terms of functionals of standard brownian motions. However, the closed forms of the distribution functions of these functionals are not known and that may cause difficulties in practice (see section 4 in Chan and Wei).

For the unstable AR(1) model with i.i.d. random errors and -1 ≤ θ 1 ≤ 1, [START_REF] Lai | Fixed-accuracy estimation of an autoregressive parameter[END_REF] proposed to use the sequential least squares estimate for θ 1 which is obtained from the LSE

θ 1 (n) = n k=1 x k-1 x k / n k=1 x 2 k-1
by replacing n with a special stopping time τ based on the observed Fisher information. They proved that, in contrast with the ordinary LSE θ 1 (n), the sequential LSE is asymptotically normal uniformly in θ ∈ [-1, 1]. For the unstable AR(2) model, [START_REF] Galtchouk | Sequential estimation of the parameters in unstable AR(2)[END_REF] applied the sequential LSE with a particular stopping time and established that it is asymptotically normal not only inside the stability rigion Λ but also for its boundary points θ corresponding to a pair of conjugate complex roots z 1 = e iφ , z 2 = e -iφ of the polynomial (1.5 ).

In this paper, for the case of unstable AR(p) process, we propose a sequential LSE for θ and find the conditions on θ (see Conditions 1-3 in the next section) ensuring its asymptotic normality. The set Λp of the points θ , satisfying these conditions includes the stability region Λ p and some part of its boundary. It is shown that the convergence of the sequential LSE to the normal distribution is uniform in θ ∈ K for any compact set K ∈ Λp (see Theorem 2.1). The extension of the property of asymptotic normality of the sequential estimate to the part of the boundary ∂Λ p is achieved by making use of a new property of observed Fisher information matrix M n , which holds in a broader subset of [Λ p ] as compared with (1.7 ) (see Lemma 3.3).

The remainder of this paper is arranged as follows. In Section 2 we introduce a sequential procedure for estimating the parameter vector θ = (θ 1 , . . . , θ p ) ′ in (1.1 ) and study its properties. Section 3 gives a new property of the observed Fisher information matrix and establishes some technical results. In Section 4 we prove Theorem 2.2 from Section 2 on asymptotic distribution of the stopping time.

2

Sequential least squares estimate. Uniform asymptotic normality.

In this section we consider the sequential least squares estimate and study its asymptotic properties. We define the sequential LSE for the parameter vector θ = (θ 1 , . . . , θ p ) ′ in model (1.1 ) as

θ(τ (h)) = M -1 τ (h) τ (h) k=1 X k-1 x k , (2.1)
where

τ (h) = inf n ≥ 1 : tr M n ≥ hσ 2 , inf{∅} = +∞, ( 2.2)
is stopping time, h is a positive number (threshold). Assume that the parameter vector θ = (θ 1 , . . . , θ p ) ′ in (1.1 ) satisfies the following Conditions.

Condition 1. Parameter θ = (θ 1 , . . . , θ p ) ′ is such that all roots z i = z i (θ) of the characteristic polynomial (1.5 ) lie inside or on the unite circle.

Condition 2. All the roots z i = z i (θ) of P(z), which are equal to one in modulus, are simple. Condition 3. The system of linear equations with respect to Y 1 , . . . , Y p-1

   Y 1 -p l=2 θ l Y l-1 = θ 1 -j-1 k=1 θ j-k + Y j -p-j k=1 θ k+j Y k = θ j , 2 ≤ j ≤ p -1, (2.3)
has a unique solution (Y 1 , . . . , Y p-1 ), Y i = κ i (θ), 1 ≤ i ≤ p -1, and the matrix

L(θ 1 , . . . , θ p ) =      1 κ 1 (θ) κ 2 (θ) . . . κ p-1 (θ) κ 1 (θ) 1 κ 1 (θ) . . . κ p-2 (θ) . . . . . . . . . . . . . . . κ p-1 (θ) κ p-2 (θ) . . . κ 1 (θ) 1      (2.4)
is positive definite. Let

• Λp denote all θ = (θ 1 , . . . , θ p ) ′ in (1.1 ) which satisfy Conditions 1,2, and Λp -all θ = (θ 1 , . . . , θ p ) ′ satisfying all Conditions 1-3.

Example 2.1. For AR(2) process, one finds

Λ 2 = {θ = (θ 1 , θ 2 ) ′ : -1 + θ 2 < θ 1 < 1 -θ 2 , |θ 2 | < 1} , [Λ 2 ] = {θ = (θ 1 , θ 2 ) ′ : -1 + θ 2 ≤ θ 1 ≤ 1 -θ 2 , |θ 2 | ≤ 1} , • Λp = [Λ 2 ] \ {(-2, -1), (2, -1)} , Λ2 = {θ = (θ 1 , θ 2 ) ′ : -1 + θ 2 < θ 1 < 1 -θ 2 , -1 ≤ θ 2 < 1} , L(θ 1 , θ 2 ) = 1 θ 1 /(1 -θ 2 ) θ 1 /(1 -θ 2 )
1 .

Example 2.2. By numerical calculation for AR(3) process, one can check that Conditions 1-3 are satisfied, for example, for the values of θ = (θ 1 , θ 2 , θ 3 ) such that z 1 (θ) = e iφ , z 2 (θ) = e -iφ with 3π/10 ≤ φ ≤ 3π/5 and -1 ≤ z 3 (θ) ≤ -0.5.

As is shown in Lemma 3.3 (Section 3), Conditions 1-3, imposed on the parameter θ = (θ 1 , . . . , θ p ) ′ in (1.1 ), provide the convergence of the ratio M n / n k=1 x 2 k-1 to the matrix L(θ 1 , . . . , θ p ) given in (2.4 ). This property can be viewed as an extension of (1.7 ) outside the stability region (1.6 ).

Remark 2.1 It will be observed that Λ p ⊂ Λp and, for all θ ∈ Λp , one has

lim n→∞ M n n k=1 x 2 k-1 = pF tr F = Λ(θ 1 , . . . , θ p ) a.s., (2.5)
where F is the same as in (1.7 ). Indeed, by making use of the identity

n k=1 x 2 k-1 = 1 p n k=1 X k-1 2 + 1 p p i=2 n l=n-i+2 x 2 l-1 , (2.6)
one obtains

M n n k=1 x 2 k-1 = M n n 1 p tr M n n (1 + ( n k=1 X k-1 2 ) -1 p i=2 n l=n-i+2 x 2 l-1 ) -1
.

Limiting n → ∞, one comes to (2.5 ), in view of (1.7 ).

Theorem 2.1 Suppose that in the AR(p) model (1.1 ), (ε n ) is a sequence of i.i.d. random variables with Eε n = 0 and Eε 2 n = σ 2 < ∞ and the parameter vector θ = (θ 1 , . . . , θ p ) ′ satisfies Conditions 1-3. Then for any compact set

K ⊂ Λp lim h→∞ sup θ∈K sup t∈R p P θ M 1/2 τ (h) (θ(τ (h)) -θ) ≤ t -Φ p ( t σ ) = 0 , (2.7)
where

Φ p (t) = Φ(t 1 ) • • • Φ(t p ), Φ is the standard normal distribution function, Λp is defined in Condition 3.
Proof. Substituting (1.1 ) in (2.1 ) yields

M 1/2 τ (h) (θ(τ (h) -θ) = M -1/2 τ (h) τ (h) k=1 X k-1 ε k = √ hM -1/2 τ (h) L 1/2 (θ 1 , . . . , θ p )Y h , (2.8)
where

Y h = 1 √ h τ (h) k=1 L -1/2 (θ 1 , . . . , θ p )X k-1 ε k (2.9) and L(θ 1 , . . . , θ p ) is given in (2.4 ). Denote G τ (h) = L -1/2 (θ 1 , . . . , θ p )M τ (h) L -1/2 (θ 1 , . . . , θ p ).
One can easily verify that

L -1/2 (θ 1 , . . . , θ p ) M 1/2 τ (h) √ h -I p 2 = 1 √ h G 1/2 τ (h) -I p 2 ≤ h -1 G τ (h) -I p 2 ≤ tr L -1 (θ 1 , . . . , θ p ) h -1 M τ (h) -L(θ 1 , . . . , θ p ) 2 .
From here, by making use of Lemma 3.4 from Section 3, one gets, for any compact set K ⊂ Λp and δ > 0,

lim h→∞ sup θ∈K P θ √ hM -1/2 τ (h) L 1/2 (θ 1 , . . . , θ p ) -I p > δ = 0 . (2.10)
Now we prove that for any compact set K ⊂ Λp and for each constant vector

v ∈ R p with v = 1 lim h→∞ sup θ∈K sup t∈R |P θ (v ′ Y h ≤ t) -Φ(t)| = 0 . (2.11)
In view of (2.9 ), one has

v ′ Y h = 1 √ h τ (h) k=1 g k-1 ε k , g k-1 = v ′ L -1/2 (θ 1 , . . . , θ p )X k-1 .
For each h > 0, we define an auxiliary stopping time as

τ 0 = τ 0 (h) = inf{n ≥ 1 : n k=1 g 2 k-1 ≥ h}, inf{∅} = +∞.
Further we make use of the representation

v ′ Y h = 1 √ h τ 0 (h) k=1 g k-1 ε k + η(h) + ∆(h) , where ∆(h) = 4 i=1 ∆ i (h), ∆ 1 (h) = h -1/2 I (τ (h)=1) g 0 ε 1 , ∆ 2 (h) = h -1/2 g τ (h)-1 ε τ (h) , ∆ 3 (h) = -h -1/2 I (τ 0 (h)=1) g 0 ε 1 , ∆ 4 (h) = h -1/2 g τ 0 (h)-1 ε τ 0 (h) , η(h) = 1 √ h τ (h)-1 k=1 g k-1 ε k - 1 √ h τ 0 (h)-1 k=1 I (τ 0 (h)>1) g k-1 ε k . Now we show that lim h→∞ sup θ∈K sup t∈R |P θ ( 1 √ h τ 0 (h) k=1 g k-1 ε k ≤ t) -Φ(t)| = 0 (2.12)
and for any δ > 0 lim

h→∞ sup θ∈K P θ (|η(h)| > δ) = 0 , (2.13) lim h→∞ sup θ∈K P θ (|∆(h)| > δ) = 0 . (2.14)
The proof of (2.12 ) is based on Proposition 3.1 from the paper by [START_REF] Lai | Fixed-accuracy estimation of an autoregressive parameter[END_REF]. Actually one needs to check only the condition A 6 , that is, for each δ > 0,

lim m→∞ sup θ∈K P θ (g 2 n ≥ δ n k=1 g 2 k-1 for some n ≥ m) = 0 . (2.

15)

Conditions A 1 -A 5 are evidently satisfied. It will be noted that

n k=1 g 2 k-1 = v ′ L -1/2 (M n / n k=1 x 2 k-1 -L)L -1/2 v + 1 n k=1 x 2 k-1 .
Proceeding from this equality one gets the inclusion

{g 2 n ≥ δ n k=1 g 2 k-1 for some n ≥ m} ⊆ ⊆ { X n 2 ≥ δ 1 n k=1 g 2 k-1 for some n ≥ m} = { X n 2 ≥ δ 1 n k=1 x 2 k-1 [1+v ′ L -1/2 (M n / n k=1 x 2 k-1 -L)L -1/2 v] for some n ≥ m} ⊆ { X n 2 ≥ δ 1 n k=1 x 2 k-1 [1 -L 1/2 v 2 M n / n k=1 x 2 k-1 -L ] for some n ≥ m} ⊆ { X n 2 ≥ δ 1 n k=1 x 2 k-1 [1 -a * M n / n k=1 x 2 k-1 -L ] for some n ≥ m} ⊂ { M n / n k=1 x 2 k-1 -L ≥ (2a * ) -1 for some n ≥ m} ∪{ X n 2 ≥ δ 1 2 n k=1 x 2 k-1 for some n ≥ m} ,
where

δ 1 = δ/a * , a * = sup θ∈K v ′ L -1/2 2 .
This, in view of Lemmas 3.1,3.3, yields (2.15 ). It will be observed that (2.15 ) enables one to show (by the same argument as in Lemma 3.5) that, for any compact set K ⊂ Λp and δ > 0

lim h→∞ sup θ∈K P θ (g 2 τ 0 -1 / τ 0 -1 k=1 g 2 k-1 ≥ δ) = 0 . (2.16)
Now we check (2.13 ). One can easily verify that

E θ η 2 (h) = E θ u(h), u(h) = 1 h τ -1 k=1 g 2 k-1 - τ 0 -1 k=1 g 2 k-1 .
The random variable u(h) is uniformly bounded from above uniformly in

θ ∈ K because u(h) ≤ 1 h τ -1 k=1 g 2 k-1 + 1 = 1 h v ′ L -1/2 M τ (h)-1 L -1/2 v + 1 ≤ a * h τ -1 k=1 X k-1 2 + 1 ≤ a * + 1 .
Therefore, it suffices to show that for each δ > 0

lim h→∞ sup θ∈K P θ (u(h) ≥ δ) = 0 .
(2.17)

To this end, one can use the following estimate

u(h) = 1 h v ′ L -1/2 M τ (h)-1 L -1/2 v - τ 0 -1 k=1 g 2 k-1 = h -1 τ -1 k=1 x 2 k-1 v ′ L -1/2 M τ (h)-1 / τ -1 k=1 x 2 k-1 -L L -1/2 v +h -1 τ -1 k=1 x 2 k-1 -h -1 τ 0 -1 k=1 g 2 k-1 ≤ a * M τ (h)-1 / τ (h)-1 k=1 x 2 k-1 -L + x 2 τ (h)-1 / τ (h)-1 k=1 x 2 k-1 + g 2 τ 0 (h)-1 / τ 0 (h)-1 k=1 g 2 k-1 .
From here, by making use of ( Now we will study the properties of the stopping time τ (h) defined by (2.2 ). Further we need the following functionals

J 1 (x; t) = t 0 x 2 (s)ds, J 2 (x, y; t) = t 0 (x 2 (s) + y 2 (s))ds, J 3 (x, y; t) = t |z k (θ)| < 1, k = 4, p}, Γ 6 (p) = {θ ∈ ∂ Λp : z 1 (θ) = 1, z 2 (θ) = e iφ , z 3 (θ) = e -iφ , φ ∈ (0, π), |z k (θ)| < 1, k = 4, p} Γ 7 (p) = {θ ∈ ∂ Λp : z 1 (θ) = -1, z 2 (θ) = 1, z 3 (θ) = e iφ , z 4 (θ) = e -iφ , φ ∈ (0, π), |z k (θ)| < 1, k = 5, p}, (2.19
) where z k (θ) are roots of the characteristic polynomial (1.5 ).

It will be noted that all these sets will be used only for the AR(p) model (1.1 ) with p ≥ 5. In the case when p ≤ 4, it is obious which of the sets Γ i (p) are odd and how to amend the remaining subsets Γ i (p). Theorem 2.2 Suppose that in the AR(p) model (1.1 ), (ε n ) n≥1 is a sequence of i.i.d. random variables with Eε n = 0, 0 < Eε2 n = σ 2 < ∞ and the parameter vector θ = (θ 1 , . . . , θ p ) satisfies Conditions 1-3. Let τ (h) be defined by (2.2 ). Then, for each θ ∈ Λ p ,

P θ -lim h→∞ τ (h) h = σ 2 trF .
(2.20)

Moreover, for each θ ∈ ∂ Λp , as h → ∞, τ (h) b i √ h L =⇒ ν i , if θ ∈ Γ i (p), 1 ≤ i ≤ 7 , (2.21)
where Λ p is given in (1.6 ); The proof of Theorem 2.2 is given in the Appendix.

ν 1 = inf {t ≥ 0 : J 1 (W 1 ; t) ≥ 1} , ν 2 = inf {t ≥ 0 : J 1 (W 2 ; t) ≥ 1} , ν 3 = inf {t ≥ 0 : J 2 (W 1 , W 2 ; t) ≥ 1} , ν 4 = inf {t ≥ 0 : J 3 (W 1 , W 2 ; t) ≥ 1} , ν i = inf {t ≥ 0 : J 4 (W 1 , W 2 , W 3 ; t) ≥ 1} , i = 5, 6, ν 7 = inf {t ≥ 0 : J 5 (W 1 , W 2 , W 3 , W 4 ; t) ≥ 1} ; b 1 , . . . ,
3 Auxiliary propositions.

In this Section we establish some properties of the process (1.1 ) and the observed Fisher information matrix M n used in Section 2.

We need some notations. Let z 1 (θ), . . . , z q (θ) denote all the distinct roots of the characteristic polynomial (1.5 ), m i (θ) be the multiplicity of z i (θ), ( q i=1 m i (θ) = p). Let

ρ(θ) = max(m i (θ) : |z i (θ)| = 1) if max |z i (θ)| = 1, 0 if max |z i (θ)| = 1 for all i = 1, q.
Formally the set

•

Λp introduced in Condition 3 can be written as where

• Λp = Λ p ∪ {θ : max 1≤i≤q |z i (θ)| = 1, ρ(θ) = 1}. ( 3
B m (δ) = { X n 2 ≥ δ n k=1 X k-1 2 for some n ≥ m}.
Now we estimate the ratio X n 2 / n k=1 X k-1 2 from above. For each 1 ≤ s < n, we introduce the quantity

l s = min{1 ≤ i ≤ s : min 1≤j≤s X k-j 2 = X k-i 2 }
and have the inequality

n k=1 X k-1 2 ≥ s k=1 X n-k 2 ≥ s X n-ls 2 . (3.2)
On the other hand, it follows from (1.1 ) that

X n = A ls X n-ls + ls-1 i=0 A i ξ n-i
and, therefore, one gets

X n 2 ≤ 2 A ls 2 X n-ls 2 + 2 ls-1 i=0 A i ξ n-i 2 ≤ 2 A ls 2 X n-ls 2 +2 s-1 i=0 A i ξ n-i 2 ≤ 2 A ls 2 X n-ls 2 +2s s-1 i=1 A i ξ n-i 2 ≤ 2 A ls 2 X n-ls 2 + 2s s-1 i=1 A i 2 ε 2 n-i . (3.3)
Further it will be observed that, for every compact set K ⊂

• Λp , there exists a positive number κ such that sup θ∈K A n 2 ≤ κ, n ≥ 1 .

(3.4) Indeed, we express A in its Jordan normal form

A = SDS -1 , (3.5)
where D = diag(J 1 , . . . , J q ), J l is the m l × m l submatrix of the form

J l =     z l 1 0 . . . 0 0 z l 1 . . . 0 0 . . . z l 1 0 . . . z l    
if z l is a multiple root with multiplicity m l ≥ 2, and J l = z l if z l is a simple root.

By direct computation with (3.5 ) one finds

A n = SD n S -1 , D n = diag(J n 1 , . . . , J n q )
, where the powers of the matrix J l are equal to z n l for a simple root z l and consist of the elements (see, R.Varga (2000))

< J n l > ij =      0, j < i , n j -i z n-j+i l , i ≤ j ≤ min(m l , n + i) , 0, n + i < j ≤ m l ,
for the roots z l with multiplicity m l ≥ 2, n ji is the binomial coefficient. From here, in view of the definition (3.1 ), one comes to (3.4 ). By making use of (3.3 ) and (3.4 ), one obtains

X n 2 ≤ 2κ X n-ls 2 + 2sκ s-1 i=0 ε 2 n-i .
Combining this inequality and (3.2 ) yields

X n 2 n k=1 X k-1 2 ≤ 2κ s + 2sκ s-1 i=0 ε 2 n-i n k=1 X k-1 2 .

It remains to use elementary inequality

n-1 i=0 ε 2 k ≤ 2(1 + A 2 ) n k=1 X k-1 2 ,
which follows from (1.2 ), to derive the desired estimate for the ratio

X n 2 n k=1 X k-1 2 ≤ 2κ s + 2sκ 2(1 + A 2 ) s-1 i=0 ε 2 n-i n-1 k=1 ε 2 k .

This inequality implies the inclusion

B m (δ) ⊂ {2κ/s > δ/2 for some n ≥ m} ∪{4(1 + A 2 )sκ s-1 i=0 ε 2 i-1 / n-1 k=1 ε 2 k > δ 2
for some n ≥ m} .

Therefore, for sufficiently large s, one gets

sup θ∈K P θ (B m (δ) ≤ P{ν s-1 i=0 ε 2 i-1 / n-1 k=1 ε 2 k > δ 2
for some n ≥ m} , where ν = sup θ∈K 4(1 + A 2 )sκ. Limiting m → ∞ and applying the law of large numbers one comes to the assertion of Lemma 3.1.

Lemma 3.2 Let (x n ) n≥0 be an autoregressive process defined by (1.1 ). Then for any compact set K ⊂

• Λp and each l = 1, p -1 lim m→∞ sup θ∈K P θ ( n k=1 x k-l ε k ≥ δ n k=1 x 2 k-1 for some n ≥ m) = 0 , (3.6)
where

• Λp is given in (3.1 )
Proof. We will apply Lemma 2.2 from the paper by Lai and Siegmund (1983)) given in the Appendix. Let c n = n 3/4 . For the set of interest one has the following inclusions

| n k=1 x k-l ε k | > δ n k=1 x 2 k-1 for some n ≥ m ⊆ | n k=1 x k-l ε k | > δ n k=1 x 2 k-l for some n ≥ m = | n k=1 x k-l ε k | ( n k=1 x 2 k-l ) 2/3 ∨ c n • ( n k=1 x 2 k-l ) 2/3 ∨ c n n k=1 x 2 k-l > δ for some n ≥ m ⊆ | n k=1 x k-l ε k | ( n k=1 x 2 k-l ) 2/3 ∨ c n > √ δ for some n ≥ m ∪ ( n k=1 x 2 k-l ) -1/3 ∨ c n ( n k=1 x 2 k-l ) -1 > √ δ for some n ≥ m ⊂    | n k=1 x k-l ε k | ( n k=1 x 2 k-l ) 2/3 ∨ c n -1 > √ δ for some n ≥ m    ∪ ∪ ( n k=1 x 2 k-l ) -1 > δ 3/2 for some n ≥ m ∪ ∪ c n ( n k=1 x 2 k-l ) -1 > √ δ for some n ≥ m ⊂ | n k=1 x k-l ε k | > δ( n k=1 x 2 k-l ) 2/3 ∨ c n for some n ≥ m ∪ c n ( n k=1 x 2 k-l ) -1 > √ δ ∧ δ 3/2 for some n ≥ m .
From here, it follows that

P θ | n k=1 x k-l ε k | > δ n k=1 x 2 k-l for some n ≥ m ≤ P θ   | n k=1 x k-l ε k | > δ n k=1 x 2 k-l 2/3 ∨ c n for some n ≥ m   +P θ n 3/4 ( n k=1 x 2 k-l ) -1 > √ δ ∧ δ 3/2 for some n ≥ m . (3.7)
By making use of (1.1 ) and the elementary inequalities, one obtains

n k=1 ε 2 k = n k=1 (x k -θ 1 x k-1 -. . . -θ p x k-p ) 2 ≤ (p + 1) n k=1 x 2 k + p j=1 θ 2 j n k=1 x 2 k-j ≤ (p + 1) max 1≤j≤p θ 2 j ( n k=1 x 2 k + p j=1 n k=1 x 2 k-j ) ≤ (p + 1) 2 µ k n k=1 x 2 k-l 1 + n k=n-l+1 x 2 k / n k=1 x 2 k-l
, where µ k = sup θ∈K θ 2 . Therefore the second summand in the right-hand side of (3.6 ) can be estimated as

P θ n 3/4 ( n k=1 x 2 k-l ) -1 > √ δ ∧ δ 3/2 for some n ≥ m ≤ P θ 2(p + 1) 2 µ k n 3/4 ( n k=1 ε 2 k ) -1 > √ δ ∧ δ 3/2 for some n ≥ m +P θ n k=n-l+1 x 2 k / n k=1 x 2 k-l ≥ 1 for some n ≥ m .
Combining this and (3. Proof. Each diagonal element of the matrix M n can be expressed through

n l=1 x 2 l-1 as < M n > ii = n k=i x 2 k-i = n-i+1 l=1 x 2 l-1 = n l=1 x 2 l-1 - n l=n-i+2 x 2 l-1 , 2 ≤ i ≤ p . (3.8)
Further it will be observed that each element < M n > ij , 2 ≤ i < j ≤ p, of M n standing above the principal diadonal and below the first row can be expressed through some element of the first row as

< M n > ij = n k=1 x k-i x k-j = n l=1 x l-1 x l-1+i-j - n l=n-i+2 x l-1 x l-1+i-j (3.9) =< M n > i,j-i+1 - n t=n-i+2
x t-1+i-j .

Now we derive the equations relating the elements of the first row, that is, < M n > 1s , 2 ≤ s ≤ p. Making use of the equation (1.1 ), one gets

< M n > 1s = n k=1 x k-1 x k-s = n k=1 p l=1 θ l x k-l-1 + ε k-1 x k-s = p l=1 θ l n k=1 x k-l-1 x k-s + n k=1 x k-s ε k-1 . Since n k=1 x k-l-1 x k-s =    < M n > ss , if l = s -1, < M n > s,l+1 , if l ≥ s, < M n > l+1,s , if l ≤ s -2,
this implies the following system of equations

< M n > 12 = θ 1 < M n > 22 + p l=2 θ l < M n > 2,l+1 + n k=1 x k-2 ε k-1 , < M n > 1s = s-2 l=2 θ l < M n > l+1,s +θ s-1 < M n > ss + p l=s θ l < M n > s,l+1 + n k=1 x k-s ε k-1 , 3 ≤ s ≤ p.
Taking into account (3.8 ),(3.9 ) one can represent this system as

< M n > 12 = θ 1 n k=1 x 2 k-1 + p l=2 θ l < M n > 1,l +η 1,2 (n), (3.10)
< M n > 1s = s-2 l=1 θ l < M n > 1,s-l +θ s-1 n k=1 x 2 k-1 + p l=s θ l < M n > 1,l-s+2 +η 1,s (n), where 3 ≤ s ≤ p, η 1,2 (n) = -θ 1 x 2 n-1 - p l=2 θ l x n-1 x n-l + n k=1 x k-2 ε k-1 , η 1,s (n) = - s-2 l=1 θ l n t=n-l+1 x t-1 x t+l-s -θ s-1 n l=n-s+2 x 2 l-1 - p l=s θ l n t=n-s+2 x t-1 x t+s-l-2 + n k=1 x k-s ε k-1 .
Denote

z i (n) = < M n > 1,i+1 n k=1 x 2 k-1 , η1,i+1 = η 1,i+1 (n) n k=1 x 2 k-1 , i = 1, p -1.
Then the system of equations (3.10 ) takes the form

   z 1 (n) -p l=2 θ l z l-1 (n) = θ 1 + η1,2 , -j-1 k=1 θ j-k z k (n) + z j (n) -p-j k=1 θ k+j z k (n) = θ j + η1,j+1 , j = 2, p -1 (3.11)
In virtue of Lemmas 3.1,3.2, for any compact set K ⊂ Λp and δ > 0, one has

lim m→∞ sup θ∈K P θ ( max 2≤s≤p |η 1,s (n)| > δ for some n ≥ m) = 0.
From here and the Condition 3, which holds for each vector θ ∈ K, it follows that the solution of the system (3.11 ) converges, as n → ∞, to the unique solution of system (2.3 ) uniformly in θ ∈ K, that is,

lim m→∞ sup θ∈K P θ ( max 1≤i≤p-1 |z i (n) -κ i (θ)| > δ for some n ≥ m) = 0.
This, in view of (3.9 ) and Lemma 3.1, implies the desired convergence of the remaining elements of the matrix M n . Hence Lemma 3.3. 2 Lemma 3.4 Let M n , τ (h) and L(θ) = L(θ 1 , . . . , θ p ) be given by (1.4 ), (2. 2) and (2.4 ), respectively. Then, for any compact set K ⊂

• Λp and δ > 0, lim h→∞ sup θ∈K P θ M τ (h) h -L(θ 1 , . . . , θ p ) > δ = 0, (3.12)
where

•
Λp is defined in Condition 3.

Proof. By making use of the equality

M τ (h) h -L(θ) = M τ (h) τ (h) k=1 x 2 k-1 -L(θ) + M τ (h) 1 h - 1 τ (h) k=1 x 2 k-1
one gets the estimate

M τ (h) h -L(θ) ≤ M τ (h) τ (h) k=1 x 2 k-1 -L(θ) +   M τ (h) / τ (h) k=1 x 2 k-1     τ (h) k=1 x 2 k-1 -h   /h. Therefore M τ (h) h -L(θ) > δ ⊂ M τ (h) τ (h) k=1 x 2 k-1 -L(θ) > δ/2 (3.13) ∪    M τ (h) τ (h) k=1 x 2 k-1 τ (h) k=1 x 2 k-1 -h h > δ/2    .
Further one has the inclusions

M τ (h) τ (h) k=1 x 2 k-1 -L(θ) > δ/2 ⊆ {τ (h) ≤ m} ∪ M n n k=1 x 2 k-1 -L(θ) > δ/2 for some n ≥ m ,    M τ (h) τ (h) k=1 x 2 k-1   τ (h) k=1 x 2 k-1 -h   /h > δ/2    ⊂ {τ (h) ≤ m} ∪ M n n k=1 x 2 k-1 -L(θ) > δ/2 for some n ≥ m ,    M τ (h) τ (h) k=1 x 2 k-1   τ (h) k=1 x 2 k-1 -h   /h > δ/2    ⊂ {τ (h) ≤ m} ∪ M n n k=1 x 2 k-1 • x 2 n-1 n-1 k=1 x 2 k-1 > δ/2 for some n ≥ m .
From here and (3.13 ), it follows that

P θ M τ (h) h -L(θ) > δ ≤ 2P θ {τ (h) ≤ m} (3.14) +P θ M n n k=1 x 2 k-1 -L(θ) > δ/2 for some n ≥ m +P θ M n n k=1 x 2 k-1 • x 2 n-1 n-1 k=1 x 2 k-1 > δ/2 for some n ≥ m .
By the definition of τ (h) in (2.2 )

{τ (h) < m} = m k=1 (x 2 k-1 + • • • + x 2 k-p ) > h = m k=1 (x 2 k-1 + • • • + x 2 k-p ) > h, max 1≤j≤m (x 2 j-1 + • • • + x 2 j-p ) < l + m k=1 (x 2 k-1 + • • • + x 2 k-p ) > h, max 1≤j≤m (x 2 j-1 + • • • + x 2 j-p ) ≥ l ⊂ {ml > h} ∪ ∪ m j=1 {(x 2 j-1 + • • • + x 2 j-p ) ≥ l}. (3.15)
This yields

P θ {τ (h) < m} ≤ I (ml>h) + m k=1 P θ {x 2 k-1 + • • • + x 2 k-p ≥ l}.
Consider the last term in (3.14 ). By the inequality

M n n k=1 x 2 k-1 ≤ M n n k=1 x 2 k-1 -L(θ) + L(θ)
one has

P θ M n n k=1 x 2 k-1 • x 2 n-1 n-1 k=1 x 2 k-1 > δ/2 for some n ≥ m ≤ P θ M n n k=1 x 2 k-1 -L(θ) > δ/4 for some n ≥ m (3.16) +P θ L * k x 2 n-1 n-1 k=1 x 2 k-1 > δ/4 for some n ≥ m +P θ L * k x 2 n-1 n-1 k=1 x 2 k-1 > δ/4 for some n ≥ m , where L * k = sup θ∈K L(θ) . Combining (3.14 )-(3.16 ) yields sup θ∈K P θ M τ (h) h -L(θ) > δ ≤ 2I (ml>h) + 2 m k=1 sup θ∈K P θ { X k-1 2 ≥ l} +P θ M n n k=1 x 2 k-1 -L(θ) > 1 2 (δ ∧ √ δ) for some n ≥ m +2P θ L * k x 2 n-1 n-1 k=1 x 2 k-1 > 1 2 (δ ∧ √ δ) for some n ≥ m . Limiting h → ∞, l → ∞, m → ∞
P θ   x 2 τ (h)-1 / τ (h)-1 k=1 x 2 k-1 > δ   = 0. (3.17)
Proof. In view of the inclusion

x 2 τ (h)-1 τ (h)-1 k=1 x 2 k-1 > δ ⊂ {τ (h) ≤ m} x 2 n n-1 k=1 x 2 k-1 > δ for some n ≥ m . one has sup θ∈K P θ x 2 τ (h)-1 τ (h)-1 k=1 x 2 k-1 > δ ≤ I (ml>h) + m j=1 sup θ∈K P θ X j-1 2 ≥ l + sup θ∈K P θ x 2 n n-1 k=1 x 2 k-1 > δ for some n ≥ m .
Limiting h → ∞, l → ∞, m → ∞ and applying Lemma 3.1 lead to (3.17 ). This completes the proof of Lemma 3.5. 2 4 Appendix.

In this Section we cite the probabilistic result from the paper of [START_REF] Lai | Fixed-accuracy estimation of an autoregressive parameter[END_REF] used in Section 3 and give the proof of Theorem 2.2.

Lemma 2.2 (by [START_REF] Lai | Fixed-accuracy estimation of an autoregressive parameter[END_REF]). Let (F n ) n≥0 be a filtration on a measurable space (Ω, F ), (x n ) n≥0 and (ε n ) n≥0 be sequences of random variables adapted to (F n ) n≥0 . Let (P θ , θ ∈ Θ) be a family of probability measures on (Ω, F ) such that under every P θ ε n is independent of F n-1 for each n ≥ 1. Then, for each γ > 1/2, δ > 0, and increasing sequence of positive constants c n → ∞,

sup θ∈Θ P θ | n i=1 x i-1 ε i | ≥ δ max(c n , ( n i=1 x 2 i-1 ) γ ) for some n ≥ m → 0 , as m → ∞.
Proof of Theorem 2.2. Assertion (2.20 ) easely follows from Lemma 3.12 in [START_REF] Galtchouk | On uniform asymptotic normality of sequential least squares estimators for the parameters in a stable AR(p)[END_REF]. For the points θ belonging to ∂Λ p we decompose the original time series (1.2 ) into several components depending on the number of the roots of the characteristic polynomial (1.5 ) lying on the unit circle and their values. To this end, the characteristic polynomial (1.5 ) is represented as

P(z) = (z + 1) δ 1 (z -1) δ 2 (z 2 -2z cos φ + 1) δ 3 ϕ(z),
where δ i are either zero or 1 with δ 1 + δ 2 + δ 3 ≥ 1, ϕ(z) is the polynomial of order r = p -δ 1 -δ 2 -2δ 3 which has all roots inside the unit circle. Assuming (without loss of generality) that r ≥ 1, one has

ϕ(z) = z r + β 1 z r-1 + . . . + β r .
By applying the backshift operator q -1 (i.e. q -1 x n = x n-1 ) one can write down (1.1 ) as q -δ 1 (q + 1) δ 1 q -δ 2 (q -1) δ 2 q -2δ 3 (q 2 -2q cos φ + 1) δ 3 q -r ϕ(q)x n = ε n (4.1) Let θ ∈ Γ 1 (ρ). Then this equation, in view of (2.19 ), takes the form q -1 (q + 1)q -p+1 ϕ(q)x n = ε n .

Denote

u n = q -p+1 ϕ(q)x n , v n = q -1 (q + 1)x n , that is u n = x n + β 1 x n-1 + . . . + β p-1 x n-p+1 , v n = x n + x n-1 . (4.2)
Introducing the vector V n = (v n , . . . , v n-p+1 ) ′ and the matrix

Q =        1 β 1 β 2 . . . β p-1 1 1 0 . . . 0 0 1 1 . . . 0 . . . 0 . . . 1 1        (4.3)
By a similar argument, one check (2.21 ) for θ ∈ Γ 2 (ρ) with

b 2 2 = 1/(σ 2 < (Q 2 Q ′ 2 ) -1 > 11 ) , (4.8)
where

Q 2 =        1 β 1 β 2 . . . β p-1 1 -1 0 . . . 0 0 1 -1 . . . 0 . . . 0 0 . . . 1 -1        .
Assume that θ ∈ Γ 3 (ρ). Then using the equation q -2 (q 2 -2q cos φ -1)q -p+2 ϕ(q)x n = ε n , we decompose the process (1.2 ) into two processes

U n = (u n , u n-1 ) ′ , V n = (v n , . . . , v n-p+3 ) ′ ,
which obey the equations

U n = 2 cos φ -1 1 0 U n-1 + ε n 0 , (4.9)
v n + β 1 v n-1 + . . . + β p-2 v n-p+2 = ε n .
These processes are related with X n = (x n , . . . , x n-p+1 ) ′ by the following transformation where

U n V n = Q 3 X n with Q 3 =          1 β 1 . . . β p-2 0 0 1 β 1 . . . β p
r 2 = σ 2 tr(Q 3 Q ′ 3 ) -1 1 cos φ cos φ 1 .
From here one comes to (2.21 ) with b 3 = (2 sin φ)/r . (4.10)

Now assume that θ ∈ Γ 4 (ρ). In this case the process (1.2 ) is decomposed into two vector processes U n = (u n , u n-1 ) ′ , V n = (v n , . . . , v n-p+3 ) ′ , which are defined by the formulae

u n = x n + β 1 x n-1 + . . . + β p-2 x n-p+2 , v n = x n -x n-2
and satisfy the equations

u n = u n-2 + ε n , v n + β 1 v n-1 + . . . + β p-2 v n-p+2 = ε n .
These processes are related to the original process (1.2 ) by the transformation

U n V n = Q 4 X n , (4.11)
where Further we represent U n as

Q 4 =        1
U n = T -1 y n z n , (4.12)
where processes y n and z n satisfy the equations

y n = y n-1 + ε, z n = -z n-1 + ε , (4.13)
where J 3 is defined in (2.18 ). Assume that θ ∈ Γ 5 (ρ). Then equation (4.1 ) has the form q -1 (q + 1)q -2 (q 2 -2q cos φ + 1)q r ϕ(q)x n = ε n , r = p -3.

Decompose x n into three processes u n = q -2 (q 2 -2q cos φ + 1)q -r ϕ(q)x n = x n + γ 1 x n-1 + . . . + γ p-1 x n-p+1 , v n = q -1 (q + 1)q -r ϕ(q)x n = x n + f 1 x n-1 + . . . + f p-2 x n-p+2 , w n = q -1 (q + 1)q -2 (q 2 -2q cos φ + 1)x n = x n + t 1 x n-1 + t 2 x n-2 + t 3 x n-3 , (4.18) where γ i , f j , t k are the coefficients of the corresponding polynomials. These processes satisfy the following equations

u n = -u n-1 + ε n , v n = 2v n-1 cos φ -v n-2 + ε n , w n = -β 1 w n-1 -. . . -β p-3 w n-p+3 + ε n .
Introducing vectors V n = (v n , v n-1 ) ′ , W n = (w n , w n-1 , . . . , w n-p+4 ) ′ and the matrix

Q 5 =           1 γ 1 . . . γ p
  U n V n W n   = Q 5 X n . (4.20)
From here and (1.4 )

n -2 tr M [nt] = tr(Q 5 Q ′ 5 ) -1 C [nt] /n 2 , 0 ≤ t ≤ 1, (4.21)
where

C n =   n k=1 u 2 k-1 n k=1 U k-1 V ′ k-1 n k=1 U k-1 W ′ k-1 n k=1 V k-1 u k-1 n k=1 V k-1 V ′ k-1 n k=1 V k-1 W ′ k-1 n k=1 W k-1 u k-1 n k=1 W k-1 V ′ k-1 n k=1 W k-1 W ′ k-1   .

. 1 )Lemma 3 . 1 • 1

 1311 It includes both the stability region Λ p and the points θ of its boundary for which all the roots of the polynomial (1.5 ), lying on the unit circle, are simple. Let (x n ) n≥0 be an autoregressive process defined by (1.1 ). Then for any compact set K ⊂ Λp for some n ≥ m) = 0 .Proof. Taking into account the equation (1.2 ), it suffices to show that for any compact set K ⊂ • Λp and δ > 0 lim m→∞ sup θ∈K P θ (B m (δ)) = 0 ,

2 Lemma 3 . 3 1 -

 2331 [START_REF] Galtchouk | Sequential estimation of the parameters in unstable AR(2)[END_REF] and applying Lemma 3.1, and Lemma 2.2 by Lai and Siegmund (see Section 4) yield(3.6 ).This completes the proof of Lemma 3.2. Let parameters θ 1 , . . . , θ p in the equation (1.1 ) satisfy Conditions 1-3 and p × p matrix M n be given by(1.3 ). Then, for any compact set K ⊂ Λp and each δ > 0L(θ 1 , . . . , θ p ) ≥ δ for some n ≥ m = 0, where L(θ 1 , . . . , θ p ) is defined in (2.4 ).

  b 7 are defined by (4.7 ),(4.8 ),(4.10 ),(4.17 ), (4.27 ) and (4.28 ), respectively; W 1 (t), . . . , W 4 (t) are independent standard brownian motions.

  and taking into account Lemma 3.3, one comes to (3.12 ). Hence Lemma 3.4. 2 Lemma 3.5 Let x k and τ (h) be defined by (1.1 ) and (2.2 ). Then for any compact set K ⊂

	• Λ0 and δ > 0
	lim h→∞	sup θ∈K

  -1 1 f 1 . . . f p-2 0 0 1 f 1 . . . f p-2 1 t 1 t 2 t 3 0 . . . 0 0 1 t 1 t 2 t 3 0 . . .

					
					
					
					
					 	,	(4.19)
				0	 
	. . .				 
	0 . . .	0 1 t 1	t 2	t 3
	we write down (4.18 ) as

(x 2 (s) + µ

y

(s))ds, J 4 (x, y, z; t) = J 2 (x, y; t) + µ 2 J 1 (z; t), J 5 (x, y, z, u; t) = J 2 (x, y; t) + µ

J 1 (z; t) + µ

J 1 (u; t),(2.18) where µ i , i = 1, 4 are defined by (4.17 ),(4.27 ) and (4.28 ). For the set Λp of the parameter vector θ = (θ 1 , . . . , θ p ) ′ satisfying Conditions 1-3, we introduce the following subsets belonging to its boundary ∂ ΛpΓ 1 (p) = {θ ∈ ∂ Λp : z 1 (θ) = -1, |z k (θ)| < 1, k = 2, p} Γ 2 (p) = {θ ∈ ∂ Λp : z 1 (θ) = 1, |z k (θ)| < 1, k = 2, p} Γ 3 (p) = {θ ∈ ∂ Λp : z 1 (θ) = e iφ , z 2 (θ) = e -iφ , φ ∈ (0, π), |z k (θ)| < 1, k = 3, p} Γ 4 (p) = {θ ∈ ∂ Λp : z 1 (θ) = -1, z 2 (θ) = 1, |z k (θ)| < 1, k = 3, p} Γ

(p) = {θ ∈ ∂ Λp : z 1 (θ) = -1, z 2 (θ) = e iφ , z 3 (θ) = e -iφ , φ ∈ (0, π),

* This research was carried out at the Department of Mathematics, the Strasbourg University, France. The second author is partially supported by the RFFI-DFG-Grant 02-01-04001.

1

one can rewrite equations (4.2 ) in the vector form

The processes u n and v n satisfy the equations

Substituting (4.4 ) in (1.4 ) yields

By Theorem 3.4.2. in Chan and Wei (1988)

Since the process V n is stable

as n → ∞. By making use of these limiting relations in (4.5 ), one get

where

This and (4.6 ) imply the validity of (2.21 ) for θ ∈ Γ 1 (ρ) with

By making use of (4.11 )-(4.12 ) one gets

By Theorems 3.4.1 and 2.3 in Chan and Wei (1988), one has, as n → ∞,

where 0 ≤ t ≤ 1. From here and (4.11 ), it follows that

where

It easy to check that

where

as n → ∞. Due to the stability of the process W k ,

By the Theorem 2.3 therein

) where J 1 and J 2 are defined in (2.18 ),

Limiting n → ∞ in (4.21 ) and taking into account (4.22 )-(4.24 ) one gets

where

By easy calculation in (4.26 ) one finds

where ≥ 1) = P(J 4 (W 1 , W 2 , W 3 ; t) ≥ 1) , where J 4 is defined in (2.18 ), that is, (2.21 ) holds for θ ∈ Γ 5 (ρ).

One can check that for θ ∈ Γ 6 (ρ) the limiting distribution of τ (h) coincides with that for θ ∈ Γ 5 (ρ) with b 6 = b 5 .

By a similar argument it can be shown that for θ ∈ Γ 7 (ρ)