Manon J Pepels

Johanna H M J Vestjens

Maaike Boer

Marjolein Smidt

Paul J Diest

George F Borm

M.D Vivianne C G Tjan-Heijnen
email: vcg.tjan.heijnen@mumc.nl

Maaike De Boer

Paul J Van Diest

Safety of avoiding routine use of axillary dissection in early stage breast cancer: a systematic review

Keywords:

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

Introduction

Axillary lymph node dissection (ALND) has long been considered the golden standard in the work-up of early stage breast cancer patients. It offered as such the best opportunity for staging and prognostication to select patients requiring additional treatment. Whether ALND also offers a therapeutic advantage is questioned [START_REF] Fisher | Reanalysis and results after 12 years of follow-up in a randomized clinical trial comparing total mastectomy with lumpectomy with or without irradiation in the treatment of breast cancer[END_REF].

Nowadays, approximately 60 to 70 percent of breast cancer patients have nodenegative disease, due to an earlier diagnosis of breast cancer [START_REF] Siesling | Trends in incidence of and mortality from cancer in The Netherlands in the period 1989-1998[END_REF]. Obviously, if the axillary lymph nodes are truly negative, there can be no possible benefit from performing an ALND. In the late nineties, the sentinel lymph node (SN) procedure was introduced [START_REF] Giuliano | Long-term follow-up confirms the oncologic safety of sentinel node biopsy without axillary dissection in node-negative breast cancer patients[END_REF][START_REF] Ho | Implementation of sentinel node biopsy in breast cancer patients in the Netherlands[END_REF]. The intrinsic hypothesis is that SN biopsy will identify patients with a negative axilla on pathology. The false-negative rate of the SN procedure is on average 8.4%, ranging from 0 to 29% [START_REF] Lyman | American Society of Clinical Oncology guideline recommendations for sentinel lymph node biopsy in early-stage breast cancer[END_REF]. Based on the low false-negative rate, the SN procedure was rapidly implemented in routine daily practice, even before the first randomized trial on the safety of the SN procedure was finalized [START_REF] Veronesi | A randomized comparison of sentinelnode biopsy with routine axillary dissection in breast cancer[END_REF][START_REF] Veronesi | Sentinel lymph node biopsy in breast cancer: ten-year results of a randomized controlled study[END_REF].

In The Netherlands at least 70% of breast cancer patients undergo an SN procedure. Of these patients, in 65% an SN only is carried out, implying, that in daily practice in nearly 50% of newly diagnosed breast cancer patients ALND is avoided [START_REF] Veronesi | Sentinel lymph node biopsy in breast cancer: ten-year results of a randomized controlled study[END_REF]. (Completion) ALND is still the standard of care for patients with axillary metastases being identified by SN biopsy, including micrometastases [START_REF] Lyman | American Society of Clinical Oncology guideline recommendations for sentinel lymph node biopsy in early-stage breast cancer[END_REF].

To this end, many series have reported on axillary recurrence rates (ARRs). A few randomized trials evaluated the safety of the SN procedure. Recently, attention was drawn to the fact that there is a trend towards omitting ALND in patients with micrometastatic involvement of the SN, in contrast to the ASCO recommendations and despite lack of data on the safety of this strategy [START_REF] Wasif | Survey of ASCO members on management of sentinel node micrometastases in breast cancer: variation in treatment recommendations according to specialty[END_REF].

The aim of our study was to provide a systematic review on the safety of withholding ALND to clinically-node negative breast cancer patients. For this purpose we reviewed ARRs in the pre-SN era of patients who did or did not undergo an ALND, and we reviewed ARRs of patients with a negative or positive SN in observational series and randomized trials that did or did not undergo a completion ALND. As secondary endpoint we reviewed survival rates in relation to axillary surgery type.

Methods

Literature Search Strategy

We performed a systematic review of the literature. Pubmed and the Cochrane Library were searched for (1 st of January 1985 through 30 th September 2009) with the use of the Medical Subject Heading terms "breast neoplasm", "sentinel node", "axillary lymph node dissection", "recurrence" and "survival". We only included studies which were published in English language. Studies were included irrespective of number of patients included, except for case reports.

Study Inclusion Criteria

The following criteria were applied to the papers that were identified by the literature search. Studies were included when they studied ALND and/or axillary staging by the SN procedure if follow-up information was provided. Endpoints had to be available in terms of ARR and/or overall survival (OS) rates. Studies that reported on detection methods using molecular biology approaches like reverse transcriptase polymerase chain reaction, and studies in which neoadjuvant chemotherapy was administered were excluded. If duplicate or updated studies were identified, only the most recent study was included. Only full papers published in peer-reviewed journals and based on original data were included. Retrospective large database studies analyzing OS were only included if more than 1,000 patients were included. To prevent overlap of data from studies that described subpopulations besides a total population, only the total population was taken into account, and the most recent paper. For cohort studies and series based on overlapping selections, only the largest study was taken into account.

Data Extraction

Two independent investigators (MP and JV) extracted data to rule out potential bias or errors. Discrepancies were resolved by consensus, if necessary through arbitrage by a third investigator (VTH). The following data were extracted from the included papers: number of patients studied, tumor size or stage, follow-up time, false-negative rate, ARR, OS, type of surgery, number of nodes excised, use of adjuvant systemic therapy, use of axillary radiotherapy. Complete data were not always obtainable for every characteristic evaluated in each study. For some studies the range for time of follow-up had to be estimated based on date of inclusion versus date of analysis and/or date of manuscript submission. If not provided in the text or tables in the original paper, data with respect to ARRs were extracted from the recurrence-free survival curves. In some series, patients with in situ carcinoma were included; these patients were omitted in our tables, with recalculation of the ARR for invasive cancers only.

Definitions

The false-negative rate was obtained by dividing the number of patients who were SN-negative but non-SN positive by the number of patients who had a positive SN or a positive non-SN. This shows the reliability of the SN procedure, i.e., the sensitivity of the SN procedure is 1 minus the false-negative rate. This is essentially different from the percentage of patients with a negative SN, having a positive non-SN (1 minus negative predictive value). This percentage was collected or calculated and shown in the table regarding the validation and randomized studies. If a different definition was used in a paper, or if in situ carcinoma was included in the denominator, the rate was recalculated. Of note, if in a study axillary recurrence was reported as either isolated or in combination with local or distant relapse, the total number of axillary recurrences was counted. In most papers median (or mean) tumor size and T-stage were only described for the total population, both with a negative SN and positive SN. This overall number was then used in the tables.

Statistical Analyses

Most studies were very small and only reported summary statistics for the exposure and the recurrence, such as median exposure time and number of recurrences. As no estimates of efficacy were reported, for example a 5 or 10 years recurrence rate, a routine meta-analysis was impossible. Hence, we used a more informal approach and calculated the overall recurrence by dividing the total number of recurrences by the total number of patients in all studies. In addition, we estimated the median exposure by the weighted median of the median exposure times in the studies, with study size as weight.

For the single group studies, we were able to calculate Kaplan Meier estimates of the pooled 5 and 8 years ARRs. As we did not have the individual patient data, we had to estimate the follow-up duration for each patient. For patients who did not have a recurrence, we used three different scenarios. Middle scenario: almost all studies reported minimal, median and maximal duration of follow-up. In this scenario, we assumed that the follow-up durations were uniformly distributed, with half of the durations between the minimum and median follow-up and half of the durations between the median and maximum follow-up. For three studies [START_REF] Rosing | Axillary regional recurrence after sentinel lymph node biopsy for breast cancer[END_REF][START_REF] Canavese | Sentinel node biopsy compared with complete axillary dissection for staging early breast cancer with clinically negative lymph nodes: results of randomized trial[END_REF][START_REF] Veronesi | Axillary metastases in breast cancer patients with negative sentinel nodes: a follow-up of 3548 cases[END_REF] , no minimal follow up was reported, and we assumed it to be 0. When no maximal follow up was reported (one study) [START_REF] Schulze | Long-term morbidity of patients with early breast cancer after sentinel lymph node biopsy compared to axillary lymph node dissection[END_REF], we assumed it to be twice the upper quartile. Worst case scenario: we assumed that 40% of the follow-up durations were equal to the minimum follow-up. The others were uniformly distributed, as in the middle scenario.

In the worst case scenario, the follow-up duration is low, so the estimate of the recurrence rate will be high. Best case scenario: we assumed that 40% of the exposures were equal to the maximum follow-up duration. The others were uniformly distributed, as in the middle scenario.

In most studies, for patients that had a recurrence, the time of the recurrence was reported, so no assumptions about the follow-up duration were required. For the four studies, that only reported median, minimal and maximal duration till recurrence, we followed a similar approach as described above, using a middle, a worst and a best case scenario. For two studies with one event, the duration till recurrence was not reported and we set it to the median follow-up [START_REF] Zavagno | A Randomized clinical trial on sentinel lymph node biopsy versus axillary lymph node dissection in breast cancer: results of the Sentinella/GIVOM trial[END_REF].

The worst and best case scenarios are extremely conservative, because they assume worst and best case scenarios in every individual study. In practice, different studies probably had different follow up patterns that more or less cancelled each other out. We therefore think that the middle scenario is the most realistic.

In addition to the approach above we investigated the results when the exposure data were not uniformly distributed, but followed a Weibull distribution.

Further, we calculated pooled survival rates for the three randomized SN studies.

For the large database pre-SN studies we were not able to calculate pooled survival rates, because of large heterogeneity between studies.

All reported p-values are two-sided, and confidence intervals (CI) are at 95% level. All analyses were performed in SAS (version 8.2).

Results

This search strategy resulted in 61 abstracts. The reference lists of selected papers were searched for additional papers, leading to the identification of in total 68 eligible studies which included follow up of different axillary treatment strategies.

We decided to categorize the included studies according to type of axillary surgical approach. In this way, we aimed to show more clearly the differences and similarities between studies on the issue of ARR and OS. The first category consisted of studies of patients with clinically node-negative disease who did or did not undergo a conventional ALND, in the pre-SN era, with a long-term follow-up. The second category consisted of observational studies of patients who had a negative SN and who did not undergo a completion ALND. The third category consisted of comparative studies with a non-ALND and an ALND group. All patients had a negative SN and in the learning phase, one group of patients underwent a completion ALND, whereas the more recent patients did not undergo a completion ALND anymore (i.e., validation and application phase studies). We also included randomized studies assessing the safety of the SN procedure in this category. The fourth category included observational studies that reported follow-up data on patients with a positive SN who did not undergo a completion ALND. The last category concerned studies that reported on at least 5-years follow-up data on survival rates in either pre-SN or SN era.

ARRs with versus without conventional ALND: Pre-SN studies

To evaluate the role of ALND in patients with clinically node-negative disease in the pre-SN era, we included 3 randomized trials comparing ALND versus no axillary treatment [START_REF] Hayward | The significance of local control in the primary treatment of breast cancer. Lucy Wortham James clinical research award[END_REF][START_REF] Fisher | Twenty-five-year follow-up of a randomized trial comparing radical mastectomy, total mastectomy, and total mastectomy followed by irradiation[END_REF] (Table 1).

In the NSABP B-04 trial, after a follow-up of ten years the ARR was 18.4 % in the total mastectomy group as compared to 1.4% in the radical mastectomy group, including ALND, even though it was noted that a small number of lymph nodes was excised in one third of the patients in the "no axillary treatment arm" [START_REF] Fisher | Twenty-five-year follow-up of a randomized trial comparing radical mastectomy, total mastectomy, and total mastectomy followed by irradiation[END_REF]. Site of recurrence was only reported for the first event, which may have resulted in an underreported ARR.

ALND was compared to axillary radiotherapy in five randomized studies [START_REF] Hayward | The significance of local control in the primary treatment of breast cancer. Lucy Wortham James clinical research award[END_REF][START_REF] Louis-Silvestre | Axillary treatment in conservative management of operable breast cancer: dissection or radiotherapy? Results of a randomized study with 15 years of follow-up[END_REF][START_REF] Hamilton | The treatment of operable cancer of the breast: a clinical trial in the South-East region of Scotland[END_REF][START_REF] Johansen | Simple mastectomy with postoperative irradiation versus extended radical mastectomy in breast cancer. A twenty-five-year follow-up of a randomized trial[END_REF]. In two of these, the Guy I and Guy II trials, the radiotherapy dosage in the no-ALND arms was considered inadequate to modern standards for axillary radiotherapy [START_REF] Hayward | The significance of local control in the primary treatment of breast cancer. Lucy Wortham James clinical research award[END_REF], and for that reason these two latter studies were considered of interest for our systematic review. The ARRs were 18.8% and 12.5% without ALND compared to 0.9% and 1.4% with ALND, in the Guy I and Guy II trials respectively after 5 to 10 years of follow-up, in agreement with findings in the B-04 trial. The difference in outcome between the Guy I and Guy II trial could be explained by the fact that there were significantly more patients with T1 tumors in the Guy II trial.

ARRs for a negative SN, without ALND: observational studies

Of 50 observational SN studies [START_REF] Rosing | Axillary regional recurrence after sentinel lymph node biopsy for breast cancer[END_REF][START_REF] Canavese | Sentinel node biopsy compared with complete axillary dissection for staging early breast cancer with clinically negative lymph nodes: results of randomized trial[END_REF][START_REF] Veronesi | Axillary metastases in breast cancer patients with negative sentinel nodes: a follow-up of 3548 cases[END_REF][START_REF] Schulze | Long-term morbidity of patients with early breast cancer after sentinel lymph node biopsy compared to axillary lymph node dissection[END_REF][START_REF] Zavagno | A Randomized clinical trial on sentinel lymph node biopsy versus axillary lymph node dissection in breast cancer: results of the Sentinella/GIVOM trial[END_REF], information on ARRs was available of patients who had a negative SN and who had not undergone a completion ALND (Table 2, Figure 1). There was a large difference in tumor size and administered systemic therapy between studies.

Overall, follow-up of more than 26,000 patients was reported. The median and maximum follow-up duration of all studies pooled was 36 and 144 months, respectively. With a median follow-up duration of 36 months, the ARR was 0.6% (95%CI 0.6-0.8).

Fig. 1 shows the Kaplan-Meier curves for the middle, best and worst case scenarios. According to the middle scenario, the 5-and 8-years ARRs were 1% (95%CI 0.8-1.1) and 1.4% (95%CI 1.1-1.6), respectively. In the worst case scenario, the recurrence rates were approximately 0.1% worse.

ARRs for a negative SN, with and without ALND: comparative studies

The third category consisted of studies in which patients with a negative SN were randomized between a completion ALND or not, and of studies in which patients with a negative SN in the learning / validation phase routinely underwent a completion ALND, whereas the next patients with a negative SN did not undergo a completion ALND anymore (SN application phase) (Table 3).

In total, 12 studies [START_REF] Veronesi | Sentinel lymph node biopsy in breast cancer: ten-year results of a randomized controlled study[END_REF][START_REF] Canavese | Sentinel node biopsy compared with complete axillary dissection for staging early breast cancer with clinically negative lymph nodes: results of randomized trial[END_REF][START_REF] Schulze | Long-term morbidity of patients with early breast cancer after sentinel lymph node biopsy compared to axillary lymph node dissection[END_REF][START_REF] Zavagno | A Randomized clinical trial on sentinel lymph node biopsy versus axillary lymph node dissection in breast cancer: results of the Sentinella/GIVOM trial[END_REF]24,33,35,[START_REF] Veronesi | Sentinel-lymph-node biopsy as a staging procedure in breast cancer: update of a randomised controlled study[END_REF][START_REF] Fan | The effect of sentinel node tumor burden on non-sentinel node status and recurrence rates in breast cancer[END_REF][START_REF] Jeruss | Axillary recurrence after sentinel node biopsy[END_REF]52,54,[START_REF] Shivers | Final results of the Department of Defense multicenter breast lymphatic mapping trial[END_REF] were available with follow-up information, of which three [START_REF] Veronesi | A randomized comparison of sentinelnode biopsy with routine axillary dissection in breast cancer[END_REF][START_REF] Veronesi | Sentinel lymph node biopsy in breast cancer: ten-year results of a randomized controlled study[END_REF][START_REF] Canavese | Sentinel node biopsy compared with complete axillary dissection for staging early breast cancer with clinically negative lymph nodes: results of randomized trial[END_REF][START_REF] Zavagno | A Randomized clinical trial on sentinel lymph node biopsy versus axillary lymph node dissection in breast cancer: results of the Sentinella/GIVOM trial[END_REF][START_REF] Veronesi | Sentinel-lymph-node biopsy as a staging procedure in breast cancer: update of a randomised controlled study[END_REF] were randomized. Overall, follow-up of 8,542 patients was reported. The ARR was 0% in all but three "SN and ALND" study arms, leading to an overall recurrence rate of 0.3% (95%CI 0.1-0.6). As in one of these it was actually unclear whether the only patient with an axillary recurrence had a negative or positive SN, the recurrence rate may even be lower [START_REF] Canavese | Sentinel node biopsy compared with complete axillary dissection for staging early breast cancer with clinically negative lymph nodes: results of randomized trial[END_REF]. The median follow up duration in these studies was 47 months, with a maximum of 102 months.

On the other hand, the pooled recurrence rate for SN negative patients who did not undergo a completion ALND was 0.4% (95%CI 0.2-0.6), which is still low but slightly higher compared with the SN-negative ALND-treated patients, especially when considering the shorter median follow up duration of only 31 months, with a maximum of 98 months.

The false-negative rate ranged from 4.1 to 22.2%. Conversely, the percentage of patients with a negative SN having a positive non-SN varied from 2.0-9.2%. This latter rate reflects the potential long term risk of axillary recurrence if no systemic treatment is offered.

ARRs for a positive SN, without ALND: observational studies

One population-based study [START_REF] Bilimoria | Comparison of sentinel lymph node biopsy alone and completion axillary lymph node dissection for node-positive breast cancer[END_REF] and fifteen single centre studies [START_REF] Schulze | Long-term morbidity of patients with early breast cancer after sentinel lymph node biopsy compared to axillary lymph node dissection[END_REF]23,35,[START_REF] Fan | The effect of sentinel node tumor burden on non-sentinel node status and recurrence rates in breast cancer[END_REF][START_REF] Jeruss | Axillary recurrence after sentinel node biopsy[END_REF][START_REF] Langer | Axillary recurrence rate in breast cancer patients with negative sentinel lymph node (SLN) or SLN micrometastases: prospective analysis of 150 patients after SLN biopsy[END_REF]50,55,[START_REF] Hwang | Low locoregional failure rates in selected breast cancer patients with tumor-positive sentinel lymph nodes who do not undergo completion axillary dissection[END_REF][START_REF] Park | A declining rate of completion axillary dissection in sentinel lymph node-positive breast cancer patients is associated with the use of a multivariate nomogram[END_REF][START_REF] Pejavar | Regional nodal recurrence in breast cancer patients treated with conservative surgery and radiation therapy (BCS+RT)[END_REF][START_REF] Haid | Medium-term follow-up data after sentinel node biopsy alone for breast cancer[END_REF][START_REF] Chagpar | Clinical outcome of patients with lymph node-negative breast carcinoma who have sentinel lymph node micrometastases detected by immunohistochemistry[END_REF][START_REF] Fant | Preliminary outcome analysis in patients with breast cancer and a positive sentinel lymph node who declined axillary dissection[END_REF][START_REF] Guenther | Axillary dissection is not required for all patients with breast cancer and positive sentinel nodes[END_REF] reported on patients with a positive SN who did not undergo a completion ALND (table 4).

From the National Cancer Data Base of the United States of America, nearly 100,000 patients with clinically node-negative breast cancer who had a positive SN were identified [START_REF] Bilimoria | Comparison of sentinel lymph node biopsy alone and completion axillary lymph node dissection for node-positive breast cancer[END_REF]. Of these, 21% underwent SN only (n=1,988 patients). In this population based study, 1% (95%CI 0.6-1.5) of the patients had a recurrence, while the median duration of follow up was 64 months, with a maximum of 72 months. The authors themselves noted already that recurrences may have been underreported to cancer registries. Patients with micrometastases who did not undergo ALND probably included isolated tumor cells according to the 6 th edition of AJCC classification [START_REF] Green | American Joint Committee on Cancer Staging Manual 6th ed[END_REF], implying that a substantial number of patients actually may have had "node-negative disease" according to current definition.

The pooled single centre studies resulted in 1.7% (95%CI 1.0-2.7) recurrences, with a median and maximum follow-up duration of 30 months and 98 months, respectively. Overall, follow-up was reported from 962 patients.

In the largest single centre study from the MSKCC (n=287), patients with SN positive / no ALND had compared with the SN positive / ALND patients more favorable tumors with a lower median predicted risk of non-SN metastases (9% versus 37%, P<0.001) [START_REF] Park | A declining rate of completion axillary dissection in sentinel lymph node-positive breast cancer patients is associated with the use of a multivariate nomogram[END_REF]. The SN metastasis size was not reported, but SN positivity was detected by serial sectioning and/or immunohistochemistry in 39% of patients, indicating small volume disease in many patients. Of note, in the "no ALND" group, still a substantial number of nodes was excised, that is, 4 or more nodes (SN and non-NSs) in 14% of patients. And, fifteen percent of patients received radiotherapy to the axilla and/or supraclavicular nodes. In their series, axillary recurrences developed in 2.0% of SN positive / no ALND patients at a median followup of 23 months versus in 0.4% of SN positive /ALND patients at 30 months (P=.004). Of importance, the highest ARR was seen in the subgroup of SN positive / no ALND patients whom SNs were positive on routine H&E. In these patients the ARR was 5% after 23 months of follow-up.

In the second largest study, from the MD Anderson Cancer Center, 196 patients with a positive SN did not undergo completion ALND [START_REF] Hwang | Low locoregional failure rates in selected breast cancer patients with tumor-positive sentinel lymph nodes who do not undergo completion axillary dissection[END_REF]. Twenty-three patients had non-SNs removed, and relatively many SNs (median 4, range 1-14) were excised.

Radiotherapy covering part of the axilla was used in 64% of patients. The calculated risk of positive non-SNs based on the MSKCC nomogram was 9.8%, in line with the low risk profile according to the MDACC risk model. At a median follow-up of 30 months, no axillary recurrences were observed.

OS in randomized and large database studies: pre-SN and SN studies

Before the SN era, 6 randomized trials have assessed the role of ALND with respect to survival in clinically node-negative breast cancer patients (Table 5). Three of six studies, among others the NSABP-B04, did not show a statistically significant survival benefit, but likely due to lack of power. Although 3 studies used axillary radiotherapy in patients who did not undergo an ALND, these studies were yet included in the analysis [START_REF] Louis-Silvestre | Axillary treatment in conservative management of operable breast cancer: dissection or radiotherapy? Results of a randomized study with 15 years of follow-up[END_REF][START_REF] Hamilton | The treatment of operable cancer of the breast: a clinical trial in the South-East region of Scotland[END_REF][START_REF] Johansen | Simple mastectomy with postoperative irradiation versus extended radical mastectomy in breast cancer. A twenty-five-year follow-up of a randomized trial[END_REF] . In a Bayesian meta-analysis, the combination of trials including almost 3,000 patients showed a 5.4% (95%CI 2.7-8.0) survival benefit in favor of ALND [START_REF] Orr | The impact of prophylactic axillary node dissection on breast cancer survival--a Bayesian meta-analysis[END_REF]. Of note, essentially none of the patients were treated with systemic therapy.

In addition, six large pre-SN database-studies, including in total 160,459 patients, showed a survival benefit in direct proportion to the number of lymph nodes removed (Table 5) [START_REF] White | Therapeutic options and results for the management of minimally invasive carcinoma of the breast: influence of axillary dissection for treatment of T1a and T1b lesions[END_REF][START_REF] Bland | Axillary Dissection in Breast-Conserving Surgery for Stage I and II Breast Cancer: A National Cancer Data Base Study of Patterns of Omission and Implications for Survival[END_REF][START_REF] Weir | Prognostic Significance of the Number of Axillary Lymph Nodes Removed in Patients With Node-Negative Breast Cancer[END_REF][START_REF] Krag | Breast Cancer Survival According to Number of Nodes Removed[END_REF][START_REF] Polednak | Survival of Lymph Node-Negative Breast Cancer Patients in relation to number of lymph nodes examined[END_REF][START_REF] Axelsson | Impact on regional recurrence and survival of axillary surgery in women with node-negative primary breast cancer[END_REF]. Although it was noted that treatment choices may have been influenced by age, presence of co-morbidity and likelihood of nodal involvement, these were not always taken into account in multivariate analyses. Only one study corrected for use of systemic therapy and observed that the risk of death by omission of ALND was diminished when systemic therapy was used [START_REF] Weir | Prognostic Significance of the Number of Axillary Lymph Nodes Removed in Patients With Node-Negative Breast Cancer[END_REF]. Another interesting observation was that even when all regional lymph nodes were pathologically negative, the number of nodes removed was associated with survival [START_REF] Krag | Breast Cancer Survival According to Number of Nodes Removed[END_REF][START_REF] Polednak | Survival of Lymph Node-Negative Breast Cancer Patients in relation to number of lymph nodes examined[END_REF].

The three published studies randomizing patients to ALND or to SN followed by ALND only if the SN was involved did not show a survival benefit of ALND, and concluded that completion ALND can be avoided in patients with negative SNs [START_REF] Canavese | Sentinel node biopsy compared with complete axillary dissection for staging early breast cancer with clinically negative lymph nodes: results of randomized trial[END_REF][START_REF] Zavagno | A Randomized clinical trial on sentinel lymph node biopsy versus axillary lymph node dissection in breast cancer: results of the Sentinella/GIVOM trial[END_REF][START_REF] Veronesi | Sentinel-lymph-node biopsy as a staging procedure in breast cancer: update of a randomised controlled study[END_REF]. Pooled analysis of the three randomized trials also showed no significant differences in 5-year OS, with a hazard ratio of -0.2 (95% CI -2.4%, 2.2%).

Discussion

In patients with clinically node-negative breast cancer the necessity of axillary surgery has been a matter of debate for many years. Leaving out ALND in all clinically-node negative patients is potentially harmful [START_REF] Orr | The impact of prophylactic axillary node dissection on breast cancer survival--a Bayesian meta-analysis[END_REF]. With the introduction of the SN procedure a strategy to identify the patients who might not need axillary surgery became available. The aim of our study was to provide a systematic review to estimate ARRs in patients with clinically node-negative breast cancer, who did or did not undergo ALND. In this systematic review, we found that patients with a negative SN had a pooled ARR of only 0.6% at 36 months. In contrast, in SN-positive studies the ARR was about 3 times higher, up to 1.7% (95%CI 1.0-2.7) at 30 months.

For patients with a positive SN by H&E examination these rates were even higher.

The SN procedure is based on the premise that it improves identification of patients with pathologically node-negative disease. Indeed, series including patients with a negative SN who did not undergo completion ALND show low ARRs of 1% after 5 years of follow-up (Fig. 1). The results of the comparative studies, i.e. summary statistics of the follow-up duration and overall recurrence, are more difficult to interpret (Tables 2 and3). The observational study results may illustrate this. The overall recurrence rate in these studies was 0.6%, the median follow-up was 36 months and the maximal follow-up was 12 years. One may be tempted to assume that 0.6% is the recurrence rate at median follow-up, or at least for some time point between the median and the maximum follow up duration. However, the Kaplan Meier graph in Fig. 1 shows that the recurrence rate at median follow up is already 0.8% and it is steadily increasing to more than 1.5% after 9 years. So, in reported studies the ARR continued to increase over time, which was different from what we had expected. Based on the comparative studies, we conclude that omitting ALND in SN negative patients is associated with a small increase in risk of axillary recurrence.

However, based on the single group studies, we conclude that this risk seems to be acceptable as long as techniques are validated and selection criteria do not change dramatically from the criteria in validated studies. New prospective randomized survival and regional recurrence data from the NSABP B-32 were very recently presented at ASCO [START_REF] Krag | Primary outcome results of NSABP B-32, a randomized phase III clinical trial to compare sentinel node resection (SNR) to conventional axillary dissection (AD) in clinically node-negative breast cancer patients[END_REF]. In B-32, including 2,011 pathologically node-negative patients undergoing SN alone and 1,975 randomized to SN plus confirmatory axillary dissection, the SN biopsy false-negative rate was 9.8% with a sample size large enough to detect a 2% difference in survival between the control and experimental arms [START_REF] Krag | Primary outcome results of NSABP B-32, a randomized phase III clinical trial to compare sentinel node resection (SNR) to conventional axillary dissection (AD) in clinically node-negative breast cancer patients[END_REF][START_REF] Giuliano | A randomized trial of axillary node dissection in women with clinical T1-2 N0 M0 breast cancer who have a positive sentinel node[END_REF]. Median follow-up was 95 months. ARRs were 0.6% and 0.4%, 5-year OS rates were 95.0% and 96.4%, respectively, both statistically not different. This recent study supports the view that in SN negative breast cancer patients with a T1 or (small) T2 tumor, omission of ALND can be regarded as safe.

In daily practice there is a shift toward omitting completion ALND in SN positive patients [START_REF] Wasif | Survey of ASCO members on management of sentinel node micrometastases in breast cancer: variation in treatment recommendations according to specialty[END_REF][START_REF] Bilimoria | Comparison of sentinel lymph node biopsy alone and completion axillary lymph node dissection for node-positive breast cancer[END_REF][START_REF] Rescigno | Patterns of axillary surgical care for breast cancer in the era of sentinel lymph node biopsy[END_REF]. At first glance, ARRs in series and a population-based registry seem not worse as opposed to series in SN negative patients (Table 4 versus Tables 2 and3). However, these low recurrence rates in SN positive patients may be explained by more favorable tumor characteristics, the presence of isolated tumor cells (even if they were classified as "micrometastases"), a considerable number of patients being treated by axillary radiotherapy, and finally, the number of nodes removed during the SN procedure being higher than normally expected. Of note, the MSKCC reported that in patients identified with a positive SN by H&E examination, the ARR was 5% at a median follow-up of 23 months, even though these were highly selected patients [START_REF] Park | A declining rate of completion axillary dissection in sentinel lymph node-positive breast cancer patients is associated with the use of a multivariate nomogram[END_REF]. In a cohort study from The Netherlands (the MIRROR study), patients with SN micrometastases (0.2-2.0 mm) who did not undergo a completion ALND were also shown to have an increased risk of axillary recurrence of 5% at 5 years follow-up [START_REF] Tjan-Heijnen | Impact of omission of completion axillary lymph node dissection (cALND) or axillary radiotherapy (ax RT) in breast cancer patients with micrometastases (pN1mi) or isolated tumor cells (pN0[i+]) in the sentinel lymph node (SN): Results from the MIRROR study[END_REF].

The results of the observational SN negative studies may help to interpret the results for SN positive patients. On the one hand, the single group SN negative studies had a median follow-up duration of 36 months and an overall recurrence of 0.6%. The 5 and 8 year recurrence rates were 1% and 1.4% respectively. For the studies in SN positive patients, the overall recurrence was 1.7%, i.e. 1.7/0.6=2.8 times higher than for the single group studies. Therefore, the 5 and 8 year recurrence rates in SN positive patients may be (1.7/0.6)*1= 2.8% and (1.7/0.6)* 1.4=4%, respectively. If we also take the median follow-up of 30 months for the SN positive studies into account, the rates will be (36/30)*2.8=3.4% and (36/30)*4= 4.8%, respectively. In a similar way, we find that for patients identified with a positive SN by H&E examination in the MSKCC study, the ARR after 5 and 8 years may be as high as (5/0.6)*(36/23)*1= 13% and (5/0.6)*(36/23)*1.4= 18%. Although these calculations are rather speculative, they are not implausible. They suggest that in many patients with a positive SN withholding ALND may not be safe. Concerns regarding safety of general omission of ALND in clinically node-negative but SN positive patients is based on the survival rates in conventional ALND versus no-ALND studies [START_REF] Orr | The impact of prophylactic axillary node dissection on breast cancer survival--a Bayesian meta-analysis[END_REF]. However, in these studies most of the patients did not receive adjuvant systemic therapy. Recently, we showed that isolated tumor cells and micrometastases were associated with a reduced 5-year rate of disease-free survival among women with favorable early-stage breast cancer who underwent an SN procedure and who did not receive adjuvant systemic therapy [85]. Among patients who received systemic therapy, the 5-year disease-free survival rate was significantly improved with an absolute benefit of nearly 10 percent, including a reduction in locoregional recurrence rate. Therefore, with more patients undergoing systemic therapy, the need for axillary treatment may change. In the studies reported so far, the rate of systemic therapy was highly varying and, therefore, impossible to analyse. This is a limitation of a systematic review on published data instead of a meta-analysis on individual data.

Considering the number of trials and low number of patients included, an individually based analysis was considered not possible.

Survival outcomes for the node-positive ACoSOG Z0011 trial were presented at ASCO [START_REF] Giuliano | A randomized trial of axillary node dissection in women with clinical T1-2 N0 M0 breast cancer who have a positive sentinel node[END_REF]. Patients with one or two H&E positive SNs were randomized to observation or ALND. The trial unfortunately failed to reach its targeted accrual. This otherwise would have been the most robust possible test of the null hypothesis that regional lymph node dissection does not improve breast cancer outcomes. That noted, among the 856 patients randomized and analyzed in an intent-to-treat analysis, at a median follow-up of 6.2 years there were no statistically significant differences in ipsilateral breast tumor or axillary recurrence. Five-year regional recurrence rate was 0.9% for SN only compared to 0.5% for ALND (p=0.11), with 5year OS rates of 92.5% and 91.9%, respectively (p=0.24). Of note, 82% of patients had received adjuvant systemic therapy. In 50% of patients the SN contained micrometastatic disease.

How should we now interpret these new findings together with existing data? On the one hand, conventional pre-SN studies showed reduced survival rates from omission of ALND in clinically node-negative breast cancer patients. In SN positive series the overall ARR was 2.8 times higher than for the single group studies in SN negative patients, with estimated ARR after 5 and 8 years as high as 13% and 18%.

On the other hand, in the one randomized phase III study omission of ALND did not result in high ARRs or reduced survival. One explanation could be the increasing use of systemic therapy over the last decade that also reduces the risk of locoregional recurrences. Another explanation could be the increased use of pre operative screening of the axilla by ultrasound, causing stage migration of clinically nodenegative disease. Apparently, the challenge will be to better define the patients that still need to undergo a completion ALND versus those who do not. The risk of non-SN metastases, if the axilla is left untreated, is on average 8% for SN negative disease, 12% for SN isolated tumor cells, 29% for micrometastases and 38% for macrometastases [86,87]. The risk of non-SN involvement is not only associated with SN status, but also with primary tumor size and presence or absence of lymph vessel invasion [86]. In the ACoSOG Z0011 study, 27% of patients in the ALND group had positive non-SNs. Further research on predictive nomograms and on impact of modern adjuvant systemic therapy is needed to improve selection of patients who will not benefit from further axillary surgery. In the decision process the risk of arm morbidity should be taken into account, because even though morbidity from axillary surgery in the NSABP-B32 was lower than expected, shoulder deficit, arm volume difference, arm numbness and tingling occurred at least twice as frequently in the ALND arm as compared to the SN alone arm.

We conclude that the SN procedure can be regarded as nearly optimal to identify patients who do not need an ALND, that is patients who have a negative SN. But, additional eligibility criteria and the role of systemic therapy need further to be elucidated to determine a potential role for avoiding ALND in selected SN-positive patients. 2). As the curves cannot be determined with certainty, the bold line shows the recurrence for the middle scenario, whereas the other lines indicate the recurrence for extreme best case and worst case scenarios, respectively.

Bulte CS, van der

The numbers at risk correspond to the middle scenario.

  Heiden-van der Loo M, Hennipman A (2009): Axillary recurrence rate after tumour negative and micrometastatic positive sentinel node procedures in breast cancer patients, a population based multicenter study. Eur J Surg Oncol 35:25-31 24. Kim HJ, Son BH, Park EW, et al (2009): Axillary recurrence after negative sentinel lymph node biopsy. Breast Cancer Res Treat 114:301-305 25. van der Ploeg IM, Kroon BB, Antonini N, et al (2008): Axillary and extra-axillary lymph node recurrences after a tumor-negative sentinel node biopsy for breast cancer using intralesional tracer administration. Ann Surg Oncol 15:1025-1031 26. Christiansen P, Friis E, Balslev E, et al (2008): Sentinel node biopsy in breast cancer: five years experience from Denmark. Acta Oncol 47:561-568 27. Poletti P, Fenaroli P, Milesi A, et al (2008): Axillary recurrence in sentinel lymph nodenegative breast cancer patients. Ann Oncol 19:1842-1846 28. Bergkvist L, de Boniface J, Jonsson PE, et al (2008): Axillary recurrence rate after negative sentinel node biopsy in breast cancer: three-year follow-up of the Swedish Multicenter Cohort Study. Ann Surg 247:150-156 29. Kuijt GP, Roumen RM (2008): Second thoughts on sentinel lymph node biopsy in node-negative breast cancer. Br J Surg 95:310-311 30. van Wely BJ, Smidt ML, de Kievit IM, et al (2008): False-negative sentinel lymph node biopsy. Br J Surg 95:1352-1355 31. Heuts EM, van der Ent FW, Hulsewe KW, et al (2008): Incidence of axillary recurrence in 344 sentinel node negative breast cancer patients after intermediate follow-up. A prospective study into the accuracy of sentinel node biopsy in breast cancer patients. Acta Chir Belg 108:203-207 32. Domenech A, Benitez A, Bajen MT, et al (2007): Patients with breast cancer and negative sentinel lymph node biopsy without additional axillary lymph node dissection: a follow-up study of up to 5 years. Oncology 72:27-32 33. Konstantiniuk P, Schrenk P, Reitsamer R, et al (2007): A nonrandomized follow-up comparison between standard axillary node dissection and sentinel node biopsy in breast cancer. Breast 16:520-526 34. Susini T, Nori J, Vanzi E, et al (2007): Axillary ultrasound scanning in the follow-up of breast cancer patients undergoing sentinel node biopsy. Breast 16:190-196 35. Takei H, Suemasu K, Kurosumi M, et al (2007): Recurrence after sentinel lymph node biopsy with or without axillary lymph node dissection in patients with breast cancer. Breast Cancer 14:16-24 48. Soni NK, Spillane AJ (2005): Experience of sentinel node biopsy alone in early breast cancer without further axillary dissection in patients with negative sentinel node. ANZ J Surg 75:292-299 49. Snoj M, Bracko M, Zagar I (2005): Axillary recurrence rate in breast cancer patients with negative sentinel lymph node. Croat Med J 46:377-381 50. Swenson KK, Mahipal A, Nissen MJ, et al (2005): Axillary disease recurrence after sentinel lymph node dissection for breast carcinoma. Cancer 104:1834-1839 51. Zavagno G, Carcoforo P, Franchini Z, et al (2005): Axillary recurrence after negative sentinel lymph node biopsy without axillary dissection: a study on 479 breast cancer patients. Eur J Surg Oncol 31:715-720 52. Imoto S, Wada N, Murakami K, et al (2004): Prognosis of breast cancer patients treated with sentinel node biopsy in Japan. Jpn J Clin Oncol 34:452-456 53. Torrenga H, Fabry H, van der Sijp JR, et al (2004): Omitting axillary lymph node dissection in sentinel node negative breast cancer patients is safe: a long term follow-up analysis. J Surg Oncol 88:4-7 54. Naik AM, Fey J, Gemignani M, et al (2004): The risk of axillary relapse after sentinel lymph node biopsy for breast cancer is comparable with that of axillary lymph node dissection: a follow-up study of 4008 procedures. Ann Surg 240:462-8 55. van der Vegt B, Doting MH, Jager PL, et al (2004): Axillary recurrence after sentinel lymph node biopsy. Eur J Surg Oncol 30:715-720 56. van Wessem KJ, Meijer WS (2004): Sentinel lymph node biopsy in breast cancer: results of intradermal periareolar tracer injection and follow-up of sentinel lymph nodenegative patients. Breast 13:290-296 57. Badgwell BD, Povoski SP, Abdessalam SF, et al (2003): Patterns of recurrence after sentinel lymph node biopsy for breast cancer. Ann Surg Oncol 10:376-380 58. Blanchard DK, Donohue JH, Reynolds C, et al (2003): Relapse and morbidity in patients undergoing sentinel lymph node biopsy alone or with axillary dissection for breast cancer. Arch Surg 138:482-487 59. Estourgie SH, Nieweg OE, Valdes Olmos RA, et al (2003): Eight false negative sentinel node procedures in breast cancer: what went wrong? Eur J Surg Oncol 29:336-340 60. Ponzone R, Biglia N, Maggiorotto F, et al (2003): Sentinel node dissection as definitive treatment for node negative breast cancer patients. Eur J Surg Oncol 29:703-706 85. de Boer M, van Deurzen CH, van Dijck JA, et al (2009): Micrometastases or isolated tumor cells and the outcome of breast cancer. N Engl J Med 361:653-663 86. Bolster MJ, Peer PG, Bult P, et al (2007): Risk factors for non-sentinel lymph node metastases in patients with breast cancer. The outcome of a multi-institutional study. Ann Surg Oncol 14:181-189 87. van Deurzen CH, de Boer M, Monninkhof EM, et al (2008): Non-sentinel lymph node metastases associated with isolated breast cancer cells in the sentinel node. J Natl Cancer Inst. 22:1574-1580

Figure 1 .

 1 Figure 1. Kaplan Meier curves for the percentage axillary recurrence among patients

Table 1 . Axillary recurrence rates with versus without axillary lymph node dissection (ALND) in patients with clinically node-negative breast cancer

 1

	Study	Patients (N)	pT1 (%)	Node positive with ALND (%)	Follow-up (months)	Axillary recurrence with ALND (%)	Axillary recurrence no ALND (%)
	Guy I 14	232	17	24	60-120	0.9	18.8
	Guy II 14	258	38	-	60-120	1.4	12.5
	NSABP-B04 15	727	-	39	120	1.4	18.4

Table 3 .

 3 Studies in which breast cancer patients with a negative SN were randomized between a completion axillary lymph node dissection (ALND)-or-not, and of studies in which patients with a negative SN in the learning / validation phase routinely underwent a completion ALND, whereas the next patients with a negative SN did not undergo a completion ALND anymore (SN application phase)

	Source $	Only Sn negative pts are included	Pts (No)	%T1 (or median, range in mm)	Follow-up, mo (median, range)	Axillary recurrence (%)	Pos. non-SNs (%)	False negative rate (%)
	Randomized trials
	2009 Canavese 10	SN and ALND	88	74*	mean 67 (+/-16)	1.1 @ / NR	9.2	22.2
		SN only	79	87*	mean 67 (+/-16)	0
	2008 Zavagno 13	SN and ALND	233	82*	56 (IQ 42-63)	0	7.7	16.7
		SN only	218	81*	56 (IQ 42-63)	0.5
	2006 Veronesi 7,41	SN and ALND	174	100	79 (5-97)	0	4.6	8.8
		SN only	167	100	79 (5-97)	0.6
	Studies reporting
	on Validation and
	next Application
	phase
	2009 Kim 24	SN and ALND	174	73*	40 (24-49)#	0
		SN only	293	73*	40 (24-49)#	1.0
	2007 Konstantiniuk 33	SN and ALND	355	69*	47 (0-102)	0.8	3.7	6.5
		SN only 1394	70*	31 (0-87)	0.4
	2007 Takei 35	SN and ALND	56	30	34 (2-83) #	0
		SN only 1062	34	34 (2-83) #	0.4
	2006 Schulze 12	SN and ALND	56	100	66 +/-22	0	7.1	12.1
		SN only	25	100	47 +/-15	0
	2005 Fan 42	SN and ALND	39	13 (1-60)*	31 (0.3-70)#	0	5.1
		SN only	237	13 (1-60)*	31 (0.3-70)#	0.8
	2004 Jeruss 43	SN and ALND	30	71*	27 (1-98) #	3.3	2.0
		SN only	557	71*	27 (1-98) #	0.2
	2004 Imoto 52	SN and ALND	97	31	62 (53-71)	0
		SN only	112	56	44 (36-52)	4.5
	2004 Naik 54	SN and ALND	326	66	32 (1-74)	0
		SN only 2340	89	31 (1-75)	0.1
	2002 Shivers 64	SN and ALND	250	96%T1,2*	16 (12-42) #	0	2.0	4.1
		SN only	180	96%T1,2*	16 (12-42) #	0

table 3 :

 3 $Only studies that included consecutively all patients with a negative SN irrespective of non-SN status and with follow-up information with and without ALND were included (so, different from the main results of randomized trials, in which intent-to-treat analyses was performed irrespective of SN status).Only pts with negative SN are included in this table * Tsize also included of patients from other categories (eg. Positive SN) # Follow-up time in validation studies: only median provided for total study population, whereas in reality follow-up of "SN and ALND" group was somewhat longer than of the more recent "SN only" group. In some studies, the range of follow-time was estimated from date of inclusion versus date of analysis @ unclear whether this one patient with an axillary recurrence had a positive or a negative SN Positive non-SN rate and False negative rate were calculated only for those studies that provided full information on the validation phase including patients with a positive SN

	NR = not reported
	IQ: interquartile range

Table 4 .

 4 Axillary recurrence rate in studies on selected SN-positive breast cancer patients who did not undergo completion axillary lymph node dissection (ALND)

	Source	No pts	% T1	AST % chemo / hormonal	RT axilla %	SN status	Median FU (months)	Axillary recurrence %
	2009 Bulte 23	20	71*	21 / 23 *	NR	20"micro"	46 (11-64)	0
	2009 Bilimoria 65	1,988	63	71/ 41	NR	530"micro"; 1,458macro	64 (60-72)	0.6 / 1.2
	2007 Takei 35	120	30	92	54	Not specified	34 (2-83)*	0
	2007 Hwang 66	196	72	56 / 27	64	67itc; 90micro; 39macro	30 (1-62)	0
	2007 Park 67	287	78	NR	15	Not specified	23 (6-87)	2.1 (5.0)^
	2006 Schulze 12	6	100*	3 / 68*	-	1itc; 4micro; 1macro	49 +/-17*	0
	2006 Pejavar 68	16	80*	30/34*	100	Not specified	24-60*	0
	2006 Haid 69	10	77*	32 / 93*	-	2itc; 6micro; 2macro	47 (7-90)	0
	2005 Fan 42	38	71	NR	63	27micro; 11macro	29 (6-76)	2.6
	2005Jeruss 43	73	57*	85 / 70*	-	73 "micro"	27 (1-98)	0
	2005 Langer 46	27	72*	20 / 76@	-	27 "micro"	42 (12-64)	0
	2005 Swenson 50	67	82*	42/58*	-	32 itc; 31micro; 4 macro	33 (2-73)	1.5
	2005 Chagpar 70	15	89*	33	-	2itc; 12micro; 1macro	40 (1-54)	0
	2004 Vegt 55	10	85*	NR	100	4micro; 6macro	35 (17-59)	0
	2003 Fant 71	31	81	100	3	27"micro"; 4macro	28 (21-48)	0
	2003 Guenther 72	46	67	100	2	23itc; 16"micro"; 7macro	32 ( 4-61)	0
	Legend

table 4 :

 4 * total group, not specified separately for SN-positive patients "micro": before 2002 there was no distinction between itc/micro, and these were, therefore, referred to as "micro" @ for all SN only including both SN negatives as positives ^ axillary relapse rate if positive SN was detected by H&E NR, patients received systemic therapy, but exact percentage was not reported

Acknowledgement

The authors thank Wim A.J.G. Lemmens for his assistance with statistical analyses.

The authors thank the reviewer for his/her helpful comments regarding an earlier version of this article.


Mátyás Barczy
email: barczy.matyas@inf.unideb.hu

Jean Bertoin
email: jean.bertoin@upmc.fr

Functional limit theorems for Lévy processes satisfying Cramér's condition

Keywords: 2000 Mathematics Subject Classification: 60G51, 60G18, 60B10 Lévy process, Cramér's condition, self-similar Markov process

We consider a Lévy process that starts from x < 0 and conditioned on having a positive maximum. When Cramér's condition holds, we provide two weak limit theorems as

x → -∞ for the law of the (two-sided) path shifted at the first instant when it enters (0, ∞), respectively shifted at the instant when its overall maximum is reached. The comparison of these two asymptotic results yields some interesting identities related to time-reversal, insurance risk, and self-similar Markov processes.

Introduction

This work should be viewed as a continuation of [START_REF] Bertoin | Some applications of duality for Lévy processes in a half-line[END_REF] which dealt with certain weak limit theorems for real-valued Lévy processes having integrable ladder heights. Recall that this includes the case when the Lévy process has a finite and positive first moment and rules out the case when the Lévy process drifts to -∞. The main result of [START_REF] Bertoin | Some applications of duality for Lévy processes in a half-line[END_REF] is that the distribution of such a Lévy process started from x and shifted at the time of its first entrance in [0, ∞), has a weak limit as x → -∞. The latter is the law of a process indexed by the entire real line, which remains negative for negative times, crosses 0 at the instant 0 and then evolves as the original Lévy process. This two-sided process fulfills a remarkable property of spatial stationarity and yields in particular a construction of self-similar Markov processes in (0, ∞) starting from the entrance boundary point 0 via a Lamperti-type transformation, see Corollaries 3 and 4 in [START_REF] Bertoin | Some applications of duality for Lévy processes in a half-line[END_REF].

In the present paper, we consider a Lévy process ξ which fulfills Cramér's condition, namely we suppose there exists θ > 0 such that E(exp(θξ 1 )) = 1 .

Recall that a fortiori ξ drifts to -∞ and hence the ladder heights are now defective. It is well-known that if we introduce the exponentially tilted probability measure (also known as Esscher transform) P (Λ) = E(exp

(θξ t )1 Λ ) , Λ ∈ F t ,
where (F t ) stands for the natural filtration of ξ, then ξ remains a Lévy process under P and has a finite and positive first moment whenever

E(|ξ 1 | exp(θξ 1 )) < ∞ .
Applying classical techniques of changes of probability which have been developed by Asmussen [START_REF] Asmussen | Conditioned limit theorems relating a random walk to its associate, with applications to risk reserve processes and the GI/G/1 queue[END_REF] in the setting of random walks (see also [START_REF] Asmussen | Ruin Probabilities[END_REF] and [START_REF] Kyprianou | Introductory lectures on fluctuations of Lévy processes with applications[END_REF] for Lévy processes), it is then easy to deduce from [START_REF] Bertoin | Some applications of duality for Lévy processes in a half-line[END_REF] that the initial Lévy process started from x, conditioned to visit [0, ∞) and shifted in time to its first entrance in [0, ∞), has a weak limit as x → -∞ which can be expressed in terms of the spatially stationary version of the exponentially tilted Lévy process.

The starting point of this work lies in the observation that one can also investigate the asymptotics of the preceding conditional laws using a path decomposition at the instant when the process reaches its maximum, instead of the first entrance time in [0, ∞). We thus have two equivalent descriptions (see Theorems 1 and 2 below) of the same two-sided process and the comparison yields interesting identities in distribution. One of the main applications concerns again self-similar Markov processes and more precisely their recurrent extensions in [0, ∞). We will derive a path-decomposition à la Williams [START_REF] Williams | Path decomposition and continuity of local time for one-dimensional diffusions[END_REF] (i.e. at the instant when the maximum is attained) under Itō's measure of their excursions away from 0. The latter involves through a Lamperti's type transformation two independent versions of the underlying Lévy process conditioned to stay positive and to stay negative, respectively. In a different direction, we also derive an application to risk theory. Namely, we determine the asymptotic distribution of the total time when an insurance is indebted for large initial reserve and conditionally on the rare event that ruin occurs.

Very recently, Griffin and Maller [START_REF] Griffin | Convolution equivalent Lévy processes and first passage times[END_REF][START_REF] Griffin | Path decomposition of ruinous behaviour for a general Lévy insurance risk process[END_REF] have investigated a closely related topic. Basically, they deal with Lévy processes whose Lévy measure belongs to the convolution equivalent class S (α) and also consider asymptotic path-decompositions conditionally on the rare event that the process crosses a large level. There are however several fundamental differences, besides the simple fact that the classes of Lévy processes which are considered in [START_REF] Griffin | Convolution equivalent Lévy processes and first passage times[END_REF][START_REF] Griffin | Path decomposition of ruinous behaviour for a general Lévy insurance risk process[END_REF] and here are not the same. First, our results involve a single limit (we let the starting point of the Lévy process tend to -∞) whereas Griffin and Maller have a two-step procedure (they consider the first passage time above u -x given that the process exceeds level u, and let first u → ∞ and then x → ∞; cf. Theorem 3.1 in [START_REF] Griffin | Path decomposition of ruinous behaviour for a general Lévy insurance risk process[END_REF] or Theorem 6.2 in [START_REF] Griffin | Convolution equivalent Lévy processes and first passage times[END_REF]). A second key difference is that Griffin and Maller describe processes forward in time, while our description is two-sided and uses time reversal for one part.

The plan of the rest of this paper is as follows. Section 2 is devoted to notations and preliminaries on conditioning and Cramér's hypothesis. The two main limit theorems are established in Section 3, and Section 4 is dedicated to some applications.

Preliminaries

We introduce notations that may look at first sight more complicated than needed, but will turn out to be useful for our purposes.

Canonical notation

We work with generic càdlàg paths, possibly with finite lifetime, which can be either one-sided ω : [0, ∞) → R ∪ {-∞} or two-sided ω : R → R ∪ {-∞}, where -∞ is viewed as an isolated extra point added to R. We shall frequently view a two-sided path ω as a pair of one-sided paths (ω(t), t ≥ 0) and (-ω(-t-), t ≥ 0) put back-to-back. We write respectively Ω and Ω for the spaces of such paths and endow both with the Skorokhod-Lindvall topology.

It will sometimes be convenient to extend a one-sided path ω ∈ Ω to R by declaring that ω(t) = -∞ for all t < 0. More generally, by a slight abuse of notation, we shall often consider a real valued path ω ′ : [a, b) → R defined on some interval [a, b) with -∞ < a < b ≤ ∞ as an element of Ω (provided that a ≥ 0) or of Ω by agreeing that ω ′ (s) = -∞ for s ∈ [a, b). We then refer to [a, b) as the life-interval of ω ′ . In this direction, we will use the notation sup ω ′ = sup a≤t<b ω ′ (t) and inf ω ′ = inf a≤t<b ω ′ (t) respectively for the supremum and the infimum over the life-interval of ω ′ . We denote by ξ = (ξ t ) the canonical process on Ω or Ω, viz. ξ t (ω) = ω(t), and by (F t ) the canonical filtration, that is F t is the smallest sigma-field on Ω or Ω such that ξ s is measurable for all s ≤ t. We consider a probability measure P on Ω under which ξ is a Lévy process. By this, we mean that under P , the life-interval is [0, ∞) a.s. (i.e. ξ t ∈ R for all t ≥ 0, P -a.s.), and for every s, t ≥ 0, the increment ξ t+s -ξ t is independent of F t and has the same law as ξ s . For every x ∈ R, we write P x for the law of x + ξ under P ; in other words P x is the law of the Lévy process started from x at time 0.

Conditioning to stay negative

We shall assume throughout this work that 0 is regular for both half-lines. This means that if we introduce the entrance times

T = inf{t > 0 : ξ t < 0} and τ = inf{t > 0 : ξ t > 0} , then T = τ = 0 , P -a.s. ( 1
)
The hypothesis ( 1) is merely taken for granted for the sake of convenience, and it should not be difficult for an interested reader to adapt the arguments of this work and deal with the situation where 0 is irregular for (-∞, 0) or for (0, ∞). Note that (1) in particular rules out the compound Poisson case. Further essential assumptions will be made in due time.

We also suppose that the Lévy process drifts to -∞, in the sense that lim t→∞ ξ t = -∞ P -a.s.

Since for x < 0, there is a positive probability that the Lévy process started from x always remains negative, we can introduce the conditional law

P ↓ x := P x (• | sup ξ < 0) .
According to Theorem 2 of Chaumont and Doney [START_REF] Chaumont | On Lévy processes conditioned to stay positive[END_REF], (1) ensures that P ↓ x has a weak limit as x → 0-which we denote by P ↓ , or sometimes also by P ↓ 0 when it is more convenient. This is the law of a Markov process started from 0 and which is negative at positive times; its semigroup is that of the Lévy process conditioned to stay negative.

Cramér's condition

We also assume that Cramér's condition is fulfilled, that is

∃θ > 0 such that E(exp(θξ t )) = 1 (2)
and furthermore that

E(|ξ t | exp(θξ t )) < ∞ , (3)
for some (and then all) t > 0. Recall that this automatically implies that the Lévy process drifts to -∞. Cramér's condition (2) entails that the exponential Lévy process exp(θξ t ) is a P -martingale, and this enables us to introduce the tilted probability measure

P (Λ) = E(exp(θξ t )1 Λ ) , Λ ∈ F t ,
where t ≥ 0 is arbitrary. It is well-known that under P , the canonical process ξ is still a Lévy process which now drifts to ∞, i.e. P (lim t→∞ ξ t = ∞) = 1; one should think of P as the law of the Lévy process conditioned to drift to ∞.

We now state a slight extension of Cramér's classical estimate (cf. [START_REF] Bertoin | Cramér's estimate for Lévy processes[END_REF]) which will be useful later on. Denote by σ = inf{t ≥ 0 : ξ t ∨ ξ t-= sup ξ} the (first) instant when the canonical process reaches its overall supremum. It is well-known that when (1) holds, then ξ hits its maximum continuously and precisely at time σ, i.e.

P (ξ σ = ξ σ-= sup ξ and ξ t ∨ ξ t-< sup ξ for all t = σ) = 1 ; see for instance Corollary 3.11 in [START_REF] Kyprianou | Introductory lectures on fluctuations of Lévy processes with applications[END_REF].

Lemma 1 Assume that (1), ( 2) and (3) hold. There is a constant C > 0 such that for every t ≥ 0, we have lim

x→-∞ e -θx P x (sup ξ ≥ 0, σ ≥ t) = C .
We stress that the constant C in Lemma 1 can be represented in several different ways, see [START_REF] Bertoin | Cramér's estimate for Lévy processes[END_REF].

Proof: For t = 0, this is just Cramér's estimate, so assume that t > 0. Recall that τ denotes the first entrance time in (0, ∞), so for all x ∈ R e -θx P x (sup ξ ≥ 0, σ < t) ≤ e -θx P x (τ < t) .

We then use the exponential tilting and the optional sampling theorem to express the right-hand side as Ẽx (e -θξτ , τ < t) ≤ Px (τ < t) ,

where Px denotes the law of x + ξ under P . Plainly the preceding quantity tends to 0 as x → -∞, which yields our claim.

Next, just as in the preceding section, we can introduce for x > 0 the conditional law

P ↑ x = Px (• | inf ξ > 0)
together with its weak limit

P ↑ = P ↑ 0 = lim x↓0 P ↑ x .
Recall also that τ denotes the first instant when ξ enters (0, ∞) and that due to (3), under the tilted law P , the Lévy process ξ has a finite positive expectation (and hence its ladder heights are integrable). In this situation, the distribution under Pz of the pair (-ξ τ -, ξ τ ) formed by the under-shoot and over-shoot, has a weak limit as z → -∞,

lim z→-∞ Pz (-ξ τ -∈ dx, ξ τ ∈ dy) = ρ(dx, dy) , x, y ∈ R + , (4)
where the limiting probability measure ρ is described by Equation ( 5) in [START_REF] Bertoin | Some applications of duality for Lévy processes in a half-line[END_REF]; see for instance Lemma 3 there.

It has been pointed out in [START_REF] Bertoin | Some applications of duality for Lévy processes in a half-line[END_REF] that the weak convergence (4) of the under-shoot and overshoot can be extended to paths. In this direction, we first introduce a probability measure P on Ω by P(dω) = P(dω, dω ′ ) =

R + ×R + ρ(dx, dy) P ↑ x (dω) Py (dω ′ ),
where a two-sided path ω is identified with a pair of one-sided paths (ω, ω ′ ) as it has been explained in the preliminaries. In other words, under P the distribution of (-ξ 0-, ξ 0 ) is ρ, and conditionally on (-ξ 0-, ξ 0 ) = (x, y), the processes (-ξ -t-, t ≥ 0) and (ξ t , t ≥ 0) are independent with laws P ↑ x and Py , respectively. According to Theorem 3 in [START_REF] Bertoin | Some applications of duality for Lévy processes in a half-line[END_REF], we have

L(ξ τ +• , Px ) =⇒ L(ξ, P) as x → -∞ , (5)
where the notation L(η, Q) refers to the law of the process η under a probability measure Q and ⇒ to weak convergence in the sense of Skorokhod-Lindvall, and by a slight abuse of notation, we view the process shifted in time at its first entrance in (0, ∞), ξ τ +• : t → ξ τ +t , as a two-sided process by agreeing that ξ τ +t = -∞ for t < -τ . For being precise, we note that Theorem 3 in [START_REF] Bertoin | Some applications of duality for Lévy processes in a half-line[END_REF] states that the weak convergence (5) holds on D([b, ∞)) for any fixed b ∈ R. However, it is equivalent that this holds on D(R), see for instance [START_REF] Whitt | Some useful functions for functional limit theorems[END_REF].

Limits under conditioning to have a large maximum

Our main goal is to investigate the asymptotic behavior of the initial Lévy process in the neighborhood of the instant when it reaches its maximum, conditionally on the event that its maximum is abnormally large. We shall first tackle this question via exponential tilting, relying on (5), and then independently using classical techniques of path decomposition at the instant when the maximum is reached.

Shifting at the first entrance time in [0, ∞)

We start by recalling that for every x ∈ R, the process (exp(-θξ t ), t ≥ 0) is a Px -martingale, and that for every (F t )-stopping time S which is finite Px -a.s., there is the identity

P x (Λ, S < ∞) = e θx Ẽx (exp(-θξ S )1 Λ ) , Λ ∈ F S .
This enables us to introduce a probability measure P on Ω by converse exponential tilting of the limit law P. Specifically we set

P(Λ, S < ∞) = c(θ) Ẽ(exp(-θξ S )1 Λ ) , Λ ∈ F S , (6)
where S ≥ 0 is an (F t )-stopping time which is P-a.s. finite, Ẽ refers to the expectation under P, and c(θ) > 0 is the normalizing constant, i.e.

1 c(θ) = R + ×R + e -θy ρ(dx, dy) = Ẽ(exp(-θξ 0 )) .
Recalling the construction of P in the preceding section, we obtain an equivalent description of P. First, P(-ξ 0-∈ dx, ξ 0 ∈ dy) = c(θ)e -θy ρ(dx, dy) := ρ(dx, dy) ,

x, y ∈ R + ,

and then, under the conditional law P(• | -ξ 0-= x, ξ 0 = y), the processes (-ξ -t-, t ≥ 0) and (ξ t , t ≥ 0) are independent with respective laws P ↑ x and P y . Note that lim t→±∞ ξ t = -∞, P-a.s Recall from Corollary 3 in [START_REF] Bertoin | Some applications of duality for Lévy processes in a half-line[END_REF] that P fulfills the spatial stationarity property: for all y ∈ R L(ξ τy+• , P) = L(y + ξ, P) ,

where τ y = inf{t : ξ t > y} denotes the first passage time above y. This yields the spatial quasi-stationarity property for P, namely for every y > 0, there is the identity

L(ξ τy+• , P(• | sup ξ > y)) = L(y + ξ, P) . (9)
Indeed, specifying [START_REF] Bertoin | Some applications of duality for Lévy processes in a half-line[END_REF] for S = τ y , we find in particular using (8) that P(sup ξ > y) = e -θy , and then (9) follows from ( 8) and ( 6) applied to S = τ y + t ′ for t ′ ≥ 0 arbitrarily large.

Similarly, the following limit theorem derives readily from Theorem 3 in [START_REF] Bertoin | Some applications of duality for Lévy processes in a half-line[END_REF] .

Theorem 1 Assume (1), ( 2) and (3). We have that

L(ξ τ +• , P x (• | sup ξ > 0)) =⇒ P as x → -∞ .
Proof: Fix a < 0 < b, consider a bounded continuous functional Φ : Ω → R which only depends on the restriction of ω to the time-interval [a, b], and write η = Φ((ξ τ +t ) a≤t≤b ). We have for all x ∈ R that

E x (η | sup ξ > 0) = E x (η, τ < ∞) P x (sup ξ > 0) = e θx P x (sup ξ > 0)
Ẽx (η exp(-θξ τ +b )) .

On the one hand, [START_REF] Bertoin | Cramér's estimate for Lévy processes[END_REF] shows that

lim x→-∞ Ẽx (η exp(-θξ τ +b )) = Ẽ (Φ((ξ t ) a≤t≤b ) exp(-θξ b )) .
On the other hand, we know from Cramér's estimate that

P x (sup ξ > 0) ∼ Ce θx as x → -∞ ,
where C is the expected value of exp(-θγ) with a variable γ distributed as the stationary overshoot for the tilted Lévy process (i.e. as the second marginal of ρ); see Remark (2) in [START_REF] Bertoin | Cramér's estimate for Lévy processes[END_REF].

Hence we have c(θ)C = 1, which completes the proof.

Shifting at the time of the maximum

We start by introducing another probability measure Q on Ω as follows. Consider two independent processes (η ↓ t ) t≥0 and (η ↑ t ) t≥0 with respective laws P ↓ and P ↑ , together with an independent exponential variable ǫ with parameter θ. Then define the two-sided process (η t ) t∈R by

η t = ǫ + η ↓ t if t ≥ 0 , ǫ -η↑ -t- if t ≤ 0 .
We denote the law of η on Ω by Q. Observe that η reaches its overall maximum at time 0, and that this maximum is exponentially distributed, so that by the lack of memory of the exponential law, Q also fulfills a spatial quasi-stationarity property. More precisely, for every y > 0, there is the identity

L(ξ, Q(• | sup ξ > y)) = L(y + ξ, Q) .
Note that this is slightly different from (9) as here there is no shift in time.

Recall that σ = inf{t ≥ 0 : ξ t = sup ξ} denotes the instant when the path reaches its overall maximum; we now claim a second weak limit theorem for the Lévy process conditioned to have a large maximum.

Theorem 2 Assume (1), ( 2) and (3). We have that

L(ξ σ+• , P x (• | sup ξ > 0)) =⇒ Q as x → -∞ .
It is well-known that under P , the post-supremum process (ξ σ+t -sup ξ) t≥0 has the law P ↓ and is independent of the pre-supremum process (ξ t ) 0≤t<σ ; see for instance Theorems 3.1 and 3.4 in [START_REF] Bertoin | Splitting at the infimum and excursions in half-lines for random-walks and Lévy processes[END_REF]. This reduces the proof of Theorem 2 to the following weaker statement.

Lemma 2 Assume (1), ( 2) and (3). The distribution of the process (ξ (σ-t)-) 0≤t<σ under the conditional law P x (• | sup ξ > 0) converges weakly as x → -∞ to that of (ǫ -η↑ t ) t≥0 , where η↑ has the law P ↑ and ǫ is an independent exponential variable with parameter θ.

The rest of this section is devoted to the proof of Lemma 2. In this direction, it is convenient to denote by ←ξ the reversed pre-maximum path which has life-interval [0, σ) and is given by

← - ξ t = sup ξ -ξ (σ-t)-, 0 ≤ t < σ .
To start with, we recall that the Lévy process reflected at its running infimum (ξ t -inf 0≤s≤t ξ s ) t≥0 is Markovian under P . We denote by n the Itō measure of its excursions away from 0, which is viewed as a sigma-finite measure on paths killed when entering the negative half-line (i.e. at time T ). Similarly, we write ñ for the Itō's excursion measure under the tilted law P . The connexion between the two excursion measures and the law P ↑ is that for every t > 0 and every

F t -measurable functional F : Ω → R + we have n(F (ξ)e θξt , t < T ) = ñ(F (ξ), t < T ) = Ẽ↑ (F (ξ)/ h(ξ t )) , (10)
where h(x) = Px (inf ξ ≥ 0). Indeed this follows readily from Corollary 1 in [START_REF] Chaumont | On Lévy processes conditioned to stay positive[END_REF] and Theorem 3 in [START_REF] Chaumont | Conditionings and path decompositions for Lévy processes[END_REF]. More precisely, the excursion measures n and ñ are only defined up to a constant factor that depends on the normalization of the local time at 0, and hence we implicitly assume that the normalization has been chosen such that (10) holds.

We also denote by κ the Laplace exponent of the inverse local time at 0 of the reflected Lévy process (also called the descending ladder times) which is given by

κ(q) = n(1 -e -qT ) , q ≥ 0 , (11)
since it is well-known that the regularity assumption (1) implies that the descending ladder time process has zero drift. Last, we introduce k t , the killing operator at time t on Ω, so that the path ξ • k t coincides with ξ on the time-interval [0, t) and takes the value -∞ on [t, ∞).

We are now able to express the law of the reversed pre-maximum process in terms of n as follows.

Lemma 3 Assume (1). Let F be some nonnegative functional on Ω. We have

E(F ( ← - ξ )) = 1 κ ′ (0) ∞ 0 n (F (ξ • k t ), t < T ) dt .
Proof: For r > 0, denote by P r the law of the initial Lévy process killed at rate r, i.e. the law of ξ •k ǫ(r) under P , where ǫ(r) stands for an independent exponential variable with parameter r. It is a classical consequence of the duality lemma (Lemma II.2 in [START_REF] Bertoin | Lévy processes[END_REF]) that under P r , ←ξ has the same law as the post-minimum process ξ -→ , where ξ -→ t = ξ σ+t -min s<ǫ(r) ξ s for 0 ≤ t < ǫ(r) -σ and σ is the a.s. unique instant at which the path reaches its overall minimum.

Imagine that we mark times in [0, ∞) using an independent Poisson process with intensity r. Recall that since 0 is regular for (0, ∞), the set of times t at which ξ t = inf 0≤s≤t ξ s has zero Lebesgue measure, and hence contains no mark P -a.s. In this setting the post-minimum process ξ -→ under P r can be identified as the first excursion of the Lévy process reflected at its running infimum away from 0 which contains a mark and killed at this mark. It should then be plain from excursion theory and elementary properties of marked Poisson point processes that the law of ξ -→ under P r coincides with the distribution of the excursion conditioned to have at least one mark on its life-interval, and killed at this first mark. In other words, there is the identity

E r (F ( ξ -→ )) = r κ(r) ∞ 0 n (F (ξ • k t ), t < T ) e -rt dt ,
where κ(r) = n(1 -e -rT ) is the Itō measure of the excursions containing at least one mark according to [START_REF] Fitzsimmons | On the existence of recurrent extensions of self-similar Markov processes[END_REF]. Our statement thus follows letting r → 0+; recall that κ ′ (0) < ∞ since ξ drifts to -∞ under P .

Next we introduce the potential operator for the Lévy process killed when entering the negative half-line

V g(x) = E x T 0 g(ξ t )dt , x ≥ 0 ,
where g : R → R denotes a generic measurable function.

We now state a consequence of the classical Cramér's estimate for Lévy processes. Proof: An application of the strong Markov property at the first-passage time T yields

V g x (y) = Ug x (y) -E y (Ug x (ξ T )) ,
where U denotes the potential operator of the original Lévy process, i.e.

Uf (y) = E ∞ 0 f (ξ t + y)dt .
Because Ug x (y) = Ug(x + y), we may rewrite the preceding as

V g x (y) = Ug(x + y) -E y (Ug(x + ξ T )) .
Next denote by Ũ the potential operator of the tilted Lévy process,

Ũf (y) = Ẽ ∞ 0 f (ξ t + y)dt = E ∞ 0 e θξt f (ξ t + y)dt ,
and observe the identity Uf (y) = e θy Ũ f (y) , where f (z) = e -θz f (z) .

As a consequence there is the upper-bound sup

x,y∈R

e -θ(x+y) |V g x (y)| ≤ sup x,y∈R e -θ(x+y) U|g|(x + y) ≤ sup z∈R Ũ |g|(z) ,
where g is a continuous function with compact support. It follows readily that the right-hand side above is finite.

Recall also from the renewal theorem (see e.g. Theorem I.21 in [START_REF] Bertoin | Lévy processes[END_REF]) that for every function f which is continuous with compact support, we have lim

x→-∞ Ũf (x + y) = 1 m ∞ -∞ f (z)dz , y ∈ R ,
with m = Ẽ(ξ 1 ) ∈ (0, ∞). As a consequence, we get that for every fixed y ∈ R,

lim x→-∞ e -θx Ug(x + y) = e θy m ∞ -∞
e -θz g(z)dz .

Putting the pieces together, we finally conclude using dominated convergence that

lim x→-∞ e -θx V g x (y) = 1 m ∞ -∞
e -θz g(z)dz e θy -E y (e θξ T ) .

Note that the last factor in the right-hand side can also be expressed as e θy (1 -Py (T < ∞)) = e θy Py (T = ∞) , since the stopping time T is finite P y -a.s.

We are now able to tackle the proof of Lemma 2.

Proof of Lemma 2: Fix t > 0, consider a bounded continuous functional F : D[0, t] → R + and a continuous function g : R + → R + with compact support. It will be convenient to agree that g(y) = 0 for y < 0.

We aim at estimating

E x F ( ← - ξ s , 0 ≤ s ≤ t)g(sup ξ), σ > t = E F ( ← - ξ s , 0 ≤ s ≤ t)g(x + sup ξ), σ > t when x → -∞.
We first use Lemma 3 and the notation g x (y) = g(x + y) to express this quantity as

1 κ ′ (0) ∞ t n (F (ξ • k t )g x (ξ s ), s < T ) ds = 1 κ ′ (0) n F (ξ • k t )E ξt T 0 g x (ξ s )ds , t < T = 1 κ ′ (0) n (F (ξ • k t )V g x (ξ t ), t < T ) ,
where the first equality follows from the Markov property of the excursion measure n, and in the second, V denotes the potential operator of the Lévy process killed when it enters the negative half-line.

Next we condition on sup ξ > 0 and apply [START_REF] Chaumont | Fluctuation theory and exit systems for positive self-similar Markov processes[END_REF] to get

E x F ( ← - ξ s , 0 ≤ s ≤ t)g(sup ξ), σ > t | sup ξ > 0 = 1 κ ′ (0)P x (sup ξ > 0) n (F (ξ • k t )V g x (ξ t ), t < T ) = e θx κ ′ (0)P x (sup ξ > 0) ñ F (ξ • k t )e -θ(x+ξt) V g x (ξ t ), t < T .
Then, recall Cramér's estimate (Lemma 1), that ñ(T > t) < ∞ for all t > 0 and from the first part of Lemma 4, that sup x,y e -θ(x+y) |V g x (y)| < ∞. Lemma 4 now entails by dominated convergence that the preceding quantity converges as x → -∞ towards

c ñ F (ξ • k t ) h(ξ t ), t < T ∞ 0 e -θz g(z)dz = c Ẽ↑ (F (ξ • k t )) ∞ 0 e -θz g(z)dz ,
where c > 0 is some constant, h(y) = Py (inf ξ > 0) and the identity follows from [START_REF] Chaumont | Fluctuation theory and exit systems for positive self-similar Markov processes[END_REF].

Putting the pieces together, we have thus shown that lim

x→-∞ E x F ( ← - ξ s , 0 ≤ s ≤ t)g(sup ξ), σ > t | sup ξ > 0 = c Ẽ↑ (F (ξ • k t ))
∞ 0 e -θz g(z)dz .

Taking F ≡ 1 and invoking Lemma 1, we realize that c = θ. This completes the proof.

Some applications

We conclude this work by presenting applications of the two main limit theorems, first to timereversal identities, then to insurance risk and ruin, and finally to self-similar Markov processes.

Time-reversal identities

Roughly speaking, Theorems 1 and 2 describe the same two-sided process up-to some timeshift. More precisely, consider a two-sided path ω that has a finite positive maximum and limit -∞ at ±∞; it should be plain that the two-step transformation that consists of shifting ω first at the first instant τ when it enters (0, ∞) and then shifting again at the first instant when the new path reaches its overall maximum, is equivalent to shifting directly ω to the first instant σ when it reaches its overall maximum. We deduce that the law under P of ξ σ+• , the path shifted at the instant σ when it attains its maximum, coincides with Q, i.e.

L(ξ σ+•

, P) = Q . ( 12
)
Conversely, the law under Q of ξ τ +• , the process shifted at the first instant τ when it enters (0, ∞), coincides with P,

L(ξ τ +• , Q) = P . ( 13
)
We now derive a few consequences of these identities. The first concerns ←ξ , the reversed pre-maximum process.

Corollary 1 The distribution of ( ←ξ t = sup ξ -ξ (σ-t)-, 0 ≤ t < σ) under P is the same as that of the process (η ↑ t , 0 ≤ t < ℓ(ǫ)), where η↑ has the law P ↑ , ǫ is an independent exponential variable with parameter θ and ℓ(ǫ) = sup{t ≥ 0 : η↑ t ≤ ǫ} denotes the last passage time of η↑ below ǫ.

Proof: Indeed the law of ( ←ξ t ) 0≤t<σ under P y does not depend on y, and as a consequence, it is the same under P as under P. So the claim is immediate from [START_REF] Griffin | Convolution equivalent Lévy processes and first passage times[END_REF] and the construction of Q. Alternatively, the interested reader may wish to check directly this identity from Lemma 4 and (10).

Remark. We see from Corollary 1 that the distribution of sup ξ under P is stochastically dominated by the exponential distribution with parameter θ, a property that can also be verified directly from exponential tilting. It turns out that the law of sup ξ under P is in fact a divisor of the exponential distribution in the following sense. Let A be a variable distributed as sup ξ under P and B an independent variable distributed according to the second marginal of ρ, where ρ is the probability law on R 2 + defined in [START_REF] Bertoin | Exponential functionals of Lévy processes[END_REF]. Then A + B has the exponential law with parameter θ. Indeed, we see from the construction of P that the law of the one-sided path (ξ t ) t≥0 under P coincides with that of (B + ξ t ) t≥0 under P , assuming implicitly that B and ξ are independent. The claim is now plain from [START_REF] Griffin | Convolution equivalent Lévy processes and first passage times[END_REF] and the fact that under Q, sup ξ = ǫ has the exponential distribution with parameter θ.

We next present a second application to time-reversal. In this direction, we introduce the law P ′ of -ξ under P , and observe that P ′ also fulfills Cramér's condition with the same exponent θ, and that the exponentially tilted probability measure associated to P ′ is the law P of -ξ under P . We also require the analog of (3) to hold for P ′ , which amounts to

E(|ξ 1 |) < ∞ . (14)
Last, we denote by P ′ the law on Ω constructed from P ′ as in section 3.1, and which corresponds to weak limit of the path shifted at its first passage time above 0 under

P ′ x (• | sup ξ > 0) as
x → -∞, see Theorem 1. We point at the following connexion with P involving last-passage times.

Corollary 2 Assume (1), ( 2), ( 3) and ( 14), and introduce

ℓ = sup{t ∈ R : ξ t > 0} ,
the last passage time above 0. Then we have

L (ξ (ℓ-t)-) t∈R , P = P ′ .
Proof: Let Q ′ denote the law which is constructed from P ′ as in section 3.2; it is immediately seen from the discussion preceding the statement that Q ′ coincides with the law of the timereversed process (ξ -t-) t∈R under Q, viz.

L ((ξ -t-) t∈R , Q) = Q ′ . (15)
We can now combine ( 12) and ( 15) and deduce that

L((ξ σ-t ) t∈R , P) = L((ξ σ+t ) t∈R , P ′ ).
Hence the law of γ := sup{t ≥ 0 : ξ σ+t > 0} under P ′ coincides with that of σ under P and then, since ℓ = σ + γ, we have

L((ξ -t-) t∈R , P) = L((ξ ℓ+t ) t∈R , P ′ ),
which yields the identity of the statement.

An application to insurance risk

We shall now derive an application of the preceding section to insurance risk. The reserves of an insurance company are modeled by a process R = (R t ) t≥0 ; in the classical theory [START_REF] Asmussen | Ruin Probabilities[END_REF], R is often assumed to be a compound Poisson process with drift, where the drift coefficient represents the premium rate and the negative jumps the claims. More general models based on Lévy processes are nowadays largely used in the literature, see for instance [START_REF] Klüppelberg | Ruin probabilities and overshoots for general Lévy insurance risk processes[END_REF].

Imagine that ruin occurs (i.e. the reserves become negative at some time), but that then the insurance company is not bankrupted and rather can borrow and survive until debt is reimbursed and reserves become positive again. We are interested in the total time when the insurance company is indebted,

D = ∞ 0 1 {Rt<0} dt .
We aim at determining the asymptotic law of D when the initial reserve R 0 is large and R is given by a spectrally negative Lévy process, conditionally on the event that ruin occurs.

In this direction, it is convenient to express the reserve process as R = -ξ and work under a probability measure P x for which ξ is a spectrally positive Lévy process. In particular the initial reserve is R 0 = -x > 0 and

D = ∞ 0 1 {ξt>0} dt = ∞ 0 1 {ξ τ +t >0} dt .
Theorem 1 has the following consequence: 2), ( 3) and ( 14), and that ξ has no negative jumps P -a.s. Then the law of D under P x (• | sup ξ > 0) has a weak limit as x → -∞; more precisely

Corollary 3 Assume (1), (
lim x→-∞ P x (D ∈ dt | sup ξ > 0) = θE(ξ - t exp(θξ t )) dt t , t > 0 ,
where ξ - t = (-ξ t ) ∨ 0 denotes the negative part of ξ t .

Proof: It should be clear from Theorem 1 that the law of D under P x (• | sup ξ > 0) has a weak limit as x → -∞, and that the limiting law corresponds to the distribution under P of

J := ∞ -∞ 1 {ξt>0} dt = ℓ 0 1 {ξt>0} dt = ℓ 0 1 {ξ ℓ-t >0} dt ,
with ℓ = sup{t ∈ R : ξ t > 0} the last passage time of ξ above 0. We can then invoke Corollary 2 (recall also the notation which is used there) to deduce that J has the same law under P as under P ′ .

The point in working with P ′ instead of P is that, since ξ has no positive jumps P ′ -a.s., we have P ′ (ξ 0 = 0) = 1, and hence the law of J under P ′ is the same as under P ′ . According to a celebrated identity due to Sparre Andersen (see for instance Lemma VI.15 in [START_REF] Bertoin | Lévy processes[END_REF]), under P ′ , J has the same law as σ, the instant when the path reaches its maximum. All in all, there are the identities L(J, P) = L(J, P ′ ) = L(J, P ′ ) = L(σ, P ′ ) .

The Laplace transform of the latter distribution is well-known for spectrally negative Lévy processes; see e.g. Theorem VII.4(i) in [START_REF] Bertoin | Lévy processes[END_REF]. Specifically, if we denote by Φ : [0, ∞) → [θ, ∞) the Laplace exponent of the killed subordinator (τ x , x ≥ 0) of the first passage times under P ′ , i.e. This establishes our claim since P ′ (ξ t ∈ dy) = P (-ξ t ∈ dy) = e -θy P (-ξ t ∈ dy).

Remark. It can be checked directly that the limit distribution in Corollary 1 is a probability measure. Indeed, we have from the Ballot Theorem that

∞ 0 E(ξ - t exp(θξ t )) dt t = ∞ 0 Ẽ(ξ - t ) dt t = ∞ 0 P ′ (τ y < ∞)dy .
Since under P ′ the process (τ y ) y≥0 of the first passage times is a subordinator killed at rate θ, the right-hand side above equals 1/θ, showing that the limit distribution in Corollary 3 is indeed a probability measure.

Applications to self-similar Markov processes

This section is devoted to applications of Theorems 1 and 2 to self-similar Markov processes (in short, ssMp); we start by briefly recalling well-known facts due to Lamperti [START_REF] Lamperti | Semi-stable Markov processes[END_REF] in this area.

We consider a Markov process X = (X t ) t≥0 with values in [0, ∞) and denote by P y its law started from y > 0. We suppose self-similarity in the sense that L((cX t/c ) t≥0 , P y ) = P cy for every c > 0 and y > 0 .

Note that for the sake of simplicity, we focus on the case where the index of self-similarity is 1; of course this induces no loss of generality as we can always replace X by X a for some adequate power a = 0. The boundary point 0 will have a special role; we denote by

ζ = inf{t ≥ 0 : X t = 0}
the first hitting time of 0 and always assume that X reaches 0 continuously whenever ζ < ∞, that is

P y (X ζ-> 0, ζ < ∞) = 0 .
Lamperti [START_REF] Lamperti | Semi-stable Markov processes[END_REF] pointed out that ssMp are related to Lévy processes by a simple transformation that we now describe. The exponential integral

I = ∞ 0 exp(ξ t )dt
of a Lévy process (ξ t ) t≥0 converges P -a.s. if ξ drifts to -∞, and diverges a.s. otherwise, see e.g. Theorem 1 in [START_REF] Bertoin | Exponential functionals of Lévy processes[END_REF]. If we define the time-substitution γ : [0, I) → R + by γ(t) 0 exp(ξ s )ds = t , 0 ≤ t < I , then under P x the process X t = exp(ξ γ(t) ) for 0 ≤ t < I is a ssMp started from y = e x and (possibly) killed at time ζ = I when it hits the boundary point 0. This transformation yields the construction of an arbitrary ssMp on (0, ∞) which reaches 0 continuously.

We are interested in the case when ζ < ∞, P y -a.s., which is equivalent to assuming that ξ drifts to -∞ under P . It is then natural to ask whether X possesses a recurrent extension that leaves 0 continuously. Roughly speaking, we know from the fundamental work of Itō that this amounts to the existence of an excursion measure n under which the process leaves 0 continuously. This question was raised by Lamperti and solved by him in the special case where X is a Brownian motion killed at 0. It was then tackled by Vuolle-Apiala [START_REF] Vuolle-Apiala | Itō excursion theory for self-similar Markov processes[END_REF] and next by Rivero [START_REF] Rivero | Recurrent extensions of self-similar Markov processes and Cramér's condition[END_REF] who obtained sufficient conditions. The problem has been completely solved by Fitzsimmons [START_REF] Fitzsimmons | On the existence of recurrent extensions of self-similar Markov processes[END_REF] and Rivero [START_REF] Rivero | Recurrent extensions of self-similar Markov processes and Cramér's condition II[END_REF] who have shown that the existence of a (unique) recurrent extension that leaves 0 continuously is equivalent to Cramér's condition (2) with θ < 1 (recall that we assume that the index of self-similarity is 1).

In the case when furthermore (3) holds, Rivero [START_REF] Rivero | Recurrent extensions of self-similar Markov processes and Cramér's condition[END_REF] has obtained some explicit descriptions of the excursion measure n, including via its entrance law (Theorem 1 in [START_REF] Rivero | Recurrent extensions of self-similar Markov processes and Cramér's condition[END_REF]) and à la Itō, i.e. conditionally on its duration (section 4 in [START_REF] Rivero | Recurrent extensions of self-similar Markov processes and Cramér's condition[END_REF]). The purpose of this section is to propose further descriptions of n from the point of view of its height

H = sup 0≤t<ζ X t .
We start with the following Lamperti-type representation for the conditional law n(• | H > 1) in terms of P.

Corollary 4 Assume (1), ( 2) and (3), and that θ < 1. The quantity

Ī = ∞ -∞ exp(ξ t )dt
is finite P-a.s. and if we define the time-change γ : (0, Ī) → (-∞, ∞) by

γ(t) -∞ exp(ξ s )ds = t , 0 < t < Ī ,
then the distribution of the process (exp(ξ γ(t) ), 0 < t < Ī) under P coincides with that of (X t , 0 < t < ζ) under the conditional law n(• | H > 1).

Remark.

The assumption that θ < 1 will play no role in the proof; it is only needed in Corollary 4 to ensure the existence of a true excursion measure. If this assumption is dropped, the statement still holds, except that now n is only a pseudo-excursion measure in the sense used by Rivero [START_REF] Rivero | Recurrent extensions of self-similar Markov processes and Cramér's condition[END_REF].

Proof: Since ξ drifts to ∞ under P , the exponential integral ∞ 0 exp(-ξ t )dt is finite Px -a.s. for every x ≥ 0, and thus also almost surely under the conditional laws P ↑

x . On the other hand, ξ drifts to -∞ under P and hence ∞ 0 exp(ξ t )dt < ∞ P y -a.s. for every y ≥ 0. By conditioning the bivariate path (ξ t ) t∈R on (-ξ 0-, ξ 0 ), we now see that Ī < ∞, P-a.s.

Next consider the ssMp which is associated by Lamperti's transformation to the tilted Lévy process with law P . Recall that (3) ensures that the expectation Ẽ(ξ 1 ) exists in (0, ∞). In turn this guaranties that 0 is an entrance boundary for the ssMp (see Chaumont et al. [START_REF] Chaumont | Fluctuation theory and exit systems for positive self-similar Markov processes[END_REF] or [START_REF] Bertoin | Some applications of duality for Lévy processes in a half-line[END_REF]), and we denote by P0 the law of the latter starting from 0. Then according to the proof of Theorem 1 of Rivero [START_REF] Rivero | Recurrent extensions of self-similar Markov processes and Cramér's condition[END_REF] (beware that our notations differ), if θ < 1, then the excursion measure n of the recurrent extension of X can be expressed as the Doob's transformation of P0 corresponding to the excessive function x → x -θ , x > 0. This means that for every stopping time S which is finite P0 -a.s., there is the identity

n(F ((X s ) 0≤s≤S ), S < ζ) = Ẽ0 (F ((X s ) 0≤s≤S ))X -θ S ) ,
where F : D[0, S] → R + is a generic functional that depends on the path only up-to time S.

We can now invoke the connexion between the laws P0 and P via a Lamperti's type transformation described in Corollary 4 of [START_REF] Bertoin | Some applications of duality for Lévy processes in a half-line[END_REF]. In the notation of the present statement, P( Ī = ∞) = 1 and the law of (X t = exp(ξ γ(t) ), t > 0) under P is P0 . It is readily checked in this setting that where the last equality follows by the local equivalence of P and P and P(γ(S 1 ) = 0) = 1.

We shall now provide a more precise description of the excursion measure n, now conditionally on the height H. In this direction we recall from Corollary 1(ii) of [START_REF] Rivero | Recurrent extensions of self-similar Markov processes and Cramér's condition[END_REF] that

n(H > z) = cz -θ , z > 0 ,
where c > 0 is some constant depending on the choice of the normalization of the excursion measure. Roughly speaking, we shall show that if we decompose the excursion of X at the instant λ when it reaches its maximum, then conditionally on the value of that maximum, the two parts can be described as Lamperti's type transforms of the underlying Lévy process conditioned to stay negative and conditioned to stay positive, respectively. More precisely, consider two independent processes η ↓ and η↑ with respective laws P ↓ and P ↑ . Introduce the exponential integrals Finally define the two-sided path with life-interval (-Ĩ↑ , I ↓ )

I ↓ = ∞ 0 exp(η ↓ t )dt
Y t = exp(-η ↑ γ↑ (-t)-) if -Ĩ↑ < t ≤ 0 , exp(η ↓ γ ↓ (t) ) if 0 ≤ t < I ↓ .
Corollary 5 Assume (1), ( 2) and (3), and that θ < 1. In the notation above, for every y > 0, the distribution of the process (yY t/y , -y Ĩ↑ < t < yI ↓ ) is a version of that of (X λ+t ) -λ<t<ζ-λ under the conditional law n(• | H = y), where λ denotes the a.s. unique instant when X reaches its maximum.

Proof: Since the laws P and Q are related by a simple time-shift [START_REF] Griffin | Convolution equivalent Lévy processes and first passage times[END_REF], and the time-shift has no effect on the Lamperti's type transformation described in Corollary 4, the latter still holds if we replace P by Q. This yields a Lamperti's type representation for the excursion of the ssMp shifted at the time when it reaches its maximum, conditionally on H > 1. Specifically, let ǫ be an exponential variable with parameter θ which is independent of η ↓ and η↑ . If we define Y (ǫ) t = exp(ǫ -η↑ γ↑ (-te -ǫ ) ) if -e ǫ Ĩ↑ < t ≤ 0 , exp(ǫ + η ↓ γ ↓ (te -ǫ ) ) if 0 ≤ t < e ǫ I ↓ , then Y (ǫ) is a version of that of (X λ+t ) -λ<t<ζ-λ under the conditional law n(• | H > 1), where λ denotes the a.s. unique instant when X reaches its maximum. By conditioning on exp(ǫ), this yields our claim when y > 1, and then thanks to the scaling property, for arbitrary y > 0.

We conclude by noting that this path decomposition à la Williams of the excursion of a ssMp enlightens a duality identity obtained by Rivero, see Proposition 6(ii) in [START_REF] Rivero | Recurrent extensions of self-similar Markov processes and Cramér's condition[END_REF].

Lemma 4

 4 Assume that (2) and (3) hold. Let g : R → R be continuous with compact support and set g x (y) = g(x + y) for every x, y ∈ R. Thensup x,y∈R e -θ(x+y) |V g x (y)| < ∞ and lim x→-∞ e -θx V g x (y) = 1 me θy h(y) ∞ -∞ e -θz g(z)dz , y ∈ R ,where m = Ẽ(ξ 1 ) = E(ξ 1 exp(θξ 1 )) and h(y) = Py (inf ξ > 0).

Ee 0 ∞ 0 e

 00 ′ (exp(-aτ x )) = exp(-xΦ(a)) , a, x ≥ 0 , then E ′ (exp(-aσ)) = θ/Φ(a) , a ≥ 0 .Observe that we can re-writeθ -at P ′ (τ y ∈ dt)dy .We then use a version of the Ballot Theorem (Corollary VII.3 in[START_REF] Bertoin | Lévy processes[END_REF]) to express the right-hand side as θ ∞ -at y t P ′ (ξ t ∈ dy)dt .

S 1 :

 1 = inf{s ≥ 0 : X s > 1} = 0 -∞e ξs ds , and hence P(γ(S 1 ) = 0) = 1. Applying this construction to any stopping time S ≥ S 1 which is finite P0 -a.s. givesn(F ((X s ) 0≤s≤S ), S < ζ) = Ẽ(F ((exp(ξ γ(s) )) 0≤s≤S )) exp(-θξ γ(S) )) .By[START_REF] Bertoin | Some applications of duality for Lévy processes in a half-line[END_REF], the right hand side can be expressed as1 c(θ) E(F ((exp(ξ γ(s) )) 0≤s≤S )), γ(S) < ∞) , which establishes our claim, since n(H > 1) = 1 c(θ) P(γ(S 1 ) < ∞) = 1 c(θ) ,

  finite a.s. Then define the time-changes γ ↓ : [0,I ↓ ) → [0, ∞) and γ↑ : [0, Ĩ↑ ) → [0, ∞) by γ ↓ (t) 0 exp(η ↓ s )ds = t for 0 ≤ t < I ↓and γ↑ (t) 0 exp(-η ↑ s )ds = t for 0 ≤ t < Ĩ↑ .

Acknowledgment : This work has been supported by ANR-08-BLAN-0220-01. It has been undertaken while M. Barczy was on a post-doctoral position at the Laboratoire de Probabilités et Modèles Aléatoires, University Pierre-et-Marie Curie, thanks to NKTH-OTKA-EU FP7 (Marie Curie action) co-funded 'MOBILITY' Grant No. MB08-A 81263. M. Barczy was also supported by the Hungarian Scientific Research Fund under Grant No. OTKA T-079128.