Mathieu Gomes

Does CSR influence M&A target choices?

Keywords: Corporate social responsibility, Mergers and acquisitions, Matched-pair analysis, Logistic regression JEL Classification: G30, G34, M14

We examine the impact of corporate social responsibility (CSR) on mergers and acquisitions (M&A) target choices. We offer evidence that CSR performance of firms matter for M&A acquirers. Indeed, our results based on 608 deals between 2003 and 2014 reveal that target firms have on average higher CSR scores than similar non-target firms. We also show directly that a firm's CSR is positively associated with its propensity to become a M&A target. These results hold for all CSR dimensions (environment, social, and governance). Overall, our results suggest that CSR matters in M&A decisions.

Introduction

Corporate social responsibility (CSR, hereafter) has been discussed in academic studies for decades. Existing studies have mainly focused on the impact of CSR on market valuation [START_REF] Ferrell | Socially responsible firms[END_REF][START_REF] Buchanan | Corporate social responsibility, firm value, and influential institutional ownership[END_REF], firm risk [START_REF] Godfrey | The relationship between corporate social responsibility and shareholder value: An empirical test of the risk management hypothesis[END_REF], financial performance [START_REF] Flammer | Does corporate social responsibility lead to superior financial performance? A regression discontinuity approach[END_REF] and asset allocation choices [START_REF] Cumming | Socially responsible institutional investment in private equity[END_REF]. A few authors have tried to study CSR within the framework of mergers and acquisitions (M&A, hereafter) transactions. [START_REF] Aktas | Do financial markets care about SRI? Evidence from mergers and acquisitions[END_REF] study how M&A announcement returns are affected by the CSR engagement of target firms and report a positive association between target CSR and M&A wealth creation. They explain this result arguing that acquirers learn from targets' CSR practices, and that more synergistic deals occur with targets that exhibit better CSR performance. More recently, [START_REF] Gomes | Does CSR impact premiums in M&A transactions?[END_REF] find that the CSR performance of target firms is positively related to M&A bid premiums, implying acquirers positively value the CSR engagement of targets. This is in line with the resource-based view (RBV, hereafter) of the firm which suggests a firm's competitive advantage results from its ability to generate capabilities that are valuable and inimitable [START_REF] Barney | Firm resources and sustained competitive advantage[END_REF][START_REF] Wernerfelt | A resource-based view of the firm[END_REF]. Because good relationships with stakeholders (proxied by CSR ratings) are hard to develop, high CSR performance can potentially provide a basis for the type of resource that serves as a source of competitive advantage under the RBV of the firm. Indeed, according to some authors, CSR activities can help firms develop valuable intangible assets such as know-how, corporate culture, and reputation (Aragon-Correa and Sharma, 2003), which can in turn provide many benefits such as increased customer loyalty [START_REF] Fombrun | Opportunity platforms and safety nets: Corporate citizenship and reputational risk[END_REF] or the ability to attract and retain valuable employees [START_REF] Turban | Corporate social performance and organizational attractiveness to prospective employees[END_REF]. To the extent that CSR activities can be the source of intangible assets and impact firms' characteristics, it follows that they should have an impact on their appeal to potential acquirers.

Our study differs from the existing literature on the impact of targets' CSR in M&A in the following ways. First of all, while M&A announcement returns convey information about how the market perceives shareholder value generation/destruction resulting from a deal, the choice of a target reflects how the acquirer's management team value the acquisition of a particular target. Second, the relationship between bid premiums and target CSR can partly convey an impact of CSR on the price paid for the target but not on target choice. Indeed, it could be possible that CSR be used as a lever in negotiating the terms of a deal once the target is picked, rather than actually driving target choice. Focusing on target choice rather than price paid allows us to determine whether CSR influences the strategic decision of picking a target.

To the best of our knowledge, no study has attempted to determine whether CSR performance impacts the choice of target firms in M&A deals. In this paper, we address this question by studying 608 M&A deals over the 2003-2014 period. We use pair-matching techniques (propensity score matching) in order to compare target firms with similar nontarget firms and find that target firms feature on average higher CSR scores than comparable non-target firms. We also make use of logistic regression analysis and find a positive association between a firm's CSR and its propensity to become a M&A target. These results hold for all dimensions of CSR (environment, social, and governance).

Our paper proceeds as follows. Section 2 presents the data. Section 3 then introduces our tests and empirical findings. Finally, section 4 concludes.

Data

Measuring CSR

To measure CSR, we use data provided by Thomson Reuters ASSET4. The ASSET4 database includes 5,000 global publicly listed companies and provides history up to fiscal year 2002 for close to 1,000 companies. Specifically, in year t, a firm is assigned a z-score for each of the CSR dimensions (environment and social), benchmarking its performance against the rest of the firms based on all the information available in fiscal year t-1. The resulting percentage is therefore a relative measure of performance, z-scored and normalized to be comprised between 0 and 100%. In this study, for each deal we use the last available ASSET4 scores before the announcement date. Following [START_REF] Cheng | Corporate social responsibility and access to finance[END_REF], we compute a firm's overall CSR score (CSR) by averaging the scores assigned to the environmental, social, and governance dimensions 1 .

Sample selection

We derive our sample in multiple steps. We download a list of international deals over the 2003-2014 period from Thomson Reuters Financial SDC Platinum database. In line with most studies, we select all offers where the bidder initially owns less than 50% of the target firm and seeks to acquire more than 50% of the target firm. We then merge this list of deals with the ASSET4 database and remove the deals for which we do not have CSR scores for the target. We then remove deals occurring in non-developed countries because of the small number of potential control firms. Finally, we merge this sample with the Thomson Reuters Datastream database to get the necessary control variables. We follow standard practice and exclude deals involving financial firms. Our final sample contains 608 deals 2 . Table 1 shows

Empirical analysis

1 Appendix A contains a description of the environment, social, and governance factors, as outlined in the ASSET4 documentation.

2 Sample size is constrained by the availability of ASSET4 ratings. It is certainly true that the firms covered by ASSET4 are not a random sample of all firms, and we acknowledge this potential limitation. However, the ASSET4 universe includes most of the largest companies worldwide with a coverage including FTSE 250, S&P 500, NASDAQ 100, DJ STOXX, Russell 1000, S&P ASX 200 and MSCI World indices. It means our results do have implications for understanding how such large companies operate and make acquisition decisions (as large firms are taken over by other large firms). 3 To address potential heterogeneity concerns, we have also conducted our study focusing on US firms only (242 deals). Results are similar to those presented in this paper and are not reported due to limited space. They are available upon request.

Propensity score matching

We now turn to the problem of building a control (non-target) sample, and in particular to the problem of determining characteristics on which control and target samples should be matched. The procedure used to create the control group varies and is often based on a single variable such as size [START_REF] Hasbrouck | The characteristics of takeover targets and other measures[END_REF]. Traditional dimension-by-dimension matching may potentially not yield good ex-ante matches because of a multi-dimensional matching problem, which makes it difficult to match simultaneously on multiple dimensions. To overcome potential selection biases resulting from the multi-dimensional matching problem, this paper employs a treatment-effect methodology. Specifically, we use a propensity score matching (PSM, hereafter) approach. PSM combines all the covariate information into estimated treatment probabilities, known as propensity scores, and use this single continuous covariate as the matching variable (Rosenbum and Rubin, 1985). We employ PSM using three different matching techniques: (1) nearest-neighbor (NN), (2) kernel4 , and (3) local linear regressions (LLR). These matching procedures ensure that control firms (i.e., non-targets) are as similar as possible to the treated firms (i.e., targets) ex ante.

Based on prior literature on takeover likelihood [START_REF] Palepu | Predicting takeover targets: A methodological and empirical analysis[END_REF][START_REF] Powell | Modelling takeover likelihood[END_REF], we use the following matching variables: firm size (log of total assets), leverage (total debt/total assets), liquidity (current assets/current liabilities), valuation (market-to-book ratio), tangibility (tangible assets/total assets), profitability (return on equity), growth (sales growth), and free cash-flow (operating cash flow/total assets). The variables definitions are detailed in Table 2. In addition, we also control for year, country, and industry5 .

Preliminary analysis

Descriptive statistics of target and control firms are presented in Table 3. The average firm size is $3.3 billion (=e 14.995 ). A typical sample firm has a total debt to asset ratio of 25.4%, a marketto-book ratio of 2.77 and a return on equity of 10.1%. The rest of the descriptive statistics are shown in column 1 of Table 3. Descriptive statistics regarding the control groups are presented in columns 2, 4, and 6 of Table 3. As expected, variables mean values for these control groups of non-targets are not statistically different from those of the target group (as evidenced by the small values of t-statistics for the difference in estimated mean values between control and target groups).

Results

Table 4 displays the results of propensity score matching analysis, i.e., the difference in CSR performance between target firms and non-target firms that had the closest ex-ante propensities to be targeted according to the set of matching variables. We see that target firms have higher CSR scores than similar non-target firms as evidenced by the positive and statistically significant (at the 1% level) difference in estimated means. This result holds regardless of the matching approach that is used. It shows that target firms have a CSR score that is on average 2.7 to 4.3 points higher than similar non-target firms. We also focus on individual dimensions of CSR, namely environment, social, and governance. Again, we find that target firms features higher scores than similar non-target firms. All results are statistically significant at conventional significance levels.

Logistic regression analysis

In order to better assess the impact of CSR on M&A likelihood, we employ a logistic regression (LOGIT) analysis using the entire ASSET4 universe. The dependent variable is a binary variable taking the value 1 if a firm becomes a target in a given year, and 0 otherwise. Our analysis reveals a positive and statistically significant effect of CSR on the likelihood of becoming a M&A target. Results are presented in Table 5. The odds ratios (computed by exponentiating the LOGIT coefficients) are 1.0127 for the overall CSR score, 1.0075 for the environment score, 1.0093 for the social score, and 1.0069 for the governance score. Taking the overall CSR score as an illustration, the interpretation is that for each unit-increase in overall CSR score, the estimated odds of being subjected to a M&A attempt increases by 1.27%. All results are significant at the 1% level.

Discussion

Overall, our results support the idea that CSR attributes of target firms matter for acquiring firms and that acquiring firms look for targets that feature good CSR capabilities. Reasons underlying the avoidance of low-CSR targets could include the cost and difficulty of bringing a target company up to the acquirer's standards with regards to managing CSR factors 6 . The propensity of acquirers to target high-CSR companies could also be explained by cost savings 6 This was confirmed in a 2012 PRI-PWC survey in which a number of companies stated that their willingness to do a deal would be seriously impacted if it appeared to be too difficult or expensive to bring the target company up to their own internal standards on ESG factors. Source: The Integration of Environmental, Social and Governance Issues in Mergers and Acquisitions Transactions, December 2012, PwC/PRI. linked to energy efficiency, revenue growth from sales of more sustainable products or increased reputation and customer loyalty [START_REF] Fombrun | Opportunity platforms and safety nets: Corporate citizenship and reputational risk[END_REF].

The importance of target CSR for acquirers could also be explained by the risk-reduction features of CSR activities7 . Increased transparency associated with high governance standards, e.g., increased earnings quality and lower level of earnings management [START_REF] Xie | Earnings management and corporate governance: The role of the board and audit committee[END_REF], leads to reduced information asymmetry and reduced risk, potentially making high-quality governance firms more appealing to prospective buyers. Strong social attributes reduce the probability of future labor unrest, which could make potential buyers more comfortable regarding post-acquisition outcomes. Finally, strong environmental attributes decrease the probability of negative outcomes such as pollution-related hazards, thereby reducing potential future claims, litigation costs, and reputation damages.

Conclusion

While prior literature on CSR is extensive, its impact on M&A decisions has been largely neglected. In this article, we investigate the issue of whether acquirers account for targets' CSR performance when making M&A decisions. Studying a sample of 608 deals over the 2003-2014 period and making use of propensity score matching analysis, we show conclusive evidence that targets' CSR performance matters for acquirers. Specifically, we show that target firms feature on average higher CSR scores than similar non-target firms. In addition, we use logistic regression analysis and show that CSR is positively related to the likelihood of being the subject of a M&A offer. Our results hold for overall CSR performance as well as for individual CSR dimensions (environment, social, and governance).

Our findings contribute to the literature by enhancing our understanding of the determinants of target choice. Future work could investigate the impact of CSR on target choice more deeply by further decomposing CSR performance in order to determine whether some key performance indicators (audit committee independence, board membership limits, diversity, employee relations…) matter more for acquirers than others. Integration/Vision and Strategy: The integration/vision and strategy category measures a company's management commitment and effectiveness towards the creation of an overarching vision and strategy integrating financial and extra-financial aspects. It reflects a company's capacity to convincingly show and communicate that it integrates the economic (financial), social and environmental dimensions into its day-to-day decision-making processes.

  the description of our sample of deals. Panel A reports sample distribution across years, Panel B reports sample distribution across countries and Panel C reports sample distribution across industries. Our sample includes deals occurring in 15 developed countries 3 over the 2003-2014 period.

Table 1

 1 This table presents our sample distribution across years, countries, and industries.

	Sample description
		Deals	Proportion (%)
	Panel A. Distribution across years
	2003	5	0.82
	2004	6	0.99
	2005	14	2.30
	2006	53	8.72
	2007	59	9.70
	2008	56	9.21
	2009	40	6.58
	2010	69	11.35
	2011	82	13.49
	2012	81	13.32
	2013	62	10.20
	2014	81	13.32
	Panel B. Distribution across countries
	Australia	94	15.46
	Canada	59	9.70
	Finland	6	0.99
	France	8	1.32
	Germany	17	2.80
	Hong Kong	5	0.82
	Japan	24	3.95
	Netherlands	15	2.47
	Norway	5	0.82
	Singapore	5	0.82
	Spain	11	1.81
	Sweden	9	1.48
	Switzerland	6	0.99
	United Kingdom	102	16.78
	United States	242	39.80
	Panel C. Distribution across industries (2-digit SIC codes)
	Mineral industries and construction (10-17)	146	24.01
	Manufacturing (20-39)	209	34.38

Table 3

 3 This table reports the results of pair-matched analysis for CSR measures using propensity score matching. Matching is done using nearest-neighbor, kernel, and local linear regression approaches. T-statistics are in parentheses. ***, ** and * denote significance at the 1%, 5% and 10% levels, respectively.

	Financial data for target and control groups
		Target Control group (NN)	Control group	Control group (LLR)
					(Kernel)
	Variable	Mean	Mean	t-stat	Mean	t-stat	Mean	t-stat
	Size	14.995 14.961	0.42	15.253	-1.16	14.961	0.42
	Leverage	0.254	0.257	-0.34	0.239	1.50	0.257	-0.34
	Liquidity	1.529	1.676	-1.15	1.423	0.95	1.676	-1.15
	MTB	2.777	2.879	-0.65	2.893	-0.73	2.879	-0.65
	Tangibility	0.325	0.309	1.05	0.322	0.16	0.309	1.05
	ROE	0.101	0.100	0.06	0.120	-1.59	0.100	0.06
	∆Sales	0.129	0.124	0.38	0.121	0.64	0.124	0.38
	FCF	0.313	0.285	0.17	0.134	1.24	0.285	0.17
	The table presents mean values for variables used in the matching process. Data is reported for
	target and control groups. T-statistics are for the difference in estimated mean values between
	control and target groups.
	Table 4
	CSR difference between target and non-target firms
				Propensity score matching
			Nearest-neighbor	Kernel		Local linear
							regressions
	CSR	4.302***		2.665***	3.488***
		(3.34)		(2.85)	(2.71)
	Environment	5.040***		1.089**	3.998**
		(2.82)		(1.97)	(2.25)
	Social	5.100***		1.590**	4.413***
		(3.01)		(2.03)	(2.61)
	Governance	2.763**		6.331***	2.061*
		(2.02)		(6.47)	(1.90)

  This table reports the results of logistic regression analysis. The dependent variable is a binary variable taking the value one if a firm becomes a target during a particular year, and zero otherwise. T-statistics are in parentheses. ***, ** and * denote significance at the 1%, 5% and 10% levels, respectively.Environmental componentEmission reduction: The emission reduction category measures a company's management commitment and effectiveness towards reducing environmental emission in the production and operational processes. It reflects a company's capacity to reduce air emissions, waste, hazardous waste, water discharges, spills or its impacts on biodiversity and to partner with environmental organizations to reduce the environmental impact of the company in the local or broader community. Product innovation: The product innovation category measures a company's management commitment and effectiveness towards supporting the research and development of ecoefficient products or services. It reflects a company's capacity to reduce the environmental costs and burdens for its customers, and thereby creating new market opportunities through new environmental technologies and processes or eco-designed, dematerialized products with extended durability. Resource reduction: The resource reduction category measures a company's management commitment and effectiveness towards achieving an efficient use of natural resources in the production process. It reflects a company's capacity to reduce the use of materials, energy or water, and to find more eco-efficient solutions by improving supply chain management. The workforce/employment quality category measures a company's management commitment and effectiveness towards providing high-quality employment benefits and job conditions. It reflects a company's capacity to increase its workforce loyalty and productivity by distributing rewarding and fair employment benefits, and by focusing on long-term employment growth and stability by promoting from within, avoiding lay-offs and maintaining relations with trade unions. Workforce/Health and Safety: The workforce/health and safety category measures a company's management commitment and effectiveness towards providing a healthy and safe workplace. It reflects a company's capacity to increase its workforce loyalty and productivity by integrating into its day-to-day operations a concern for the physical and mental health, well-being and stress level of all employees. Workforce /Training and Development: The workforce/training and development category measures a company's management commitment and effectiveness towards providing training and development (education) for its workforce. It reflects a company's capacity to increase its intellectual capital, workforce loyalty and productivity by developing the workforce's skills, competences, employability and careers in an entrepreneurial environment. component Board of Directors/Board Functions: The board of directors/board functions category measures a company's management commitment and effectiveness towards following best practice corporate governance principles related to board activities and functions. It reflects a company's capacity to have an effective board by setting up the essential board committees with allocated tasks and responsibilities.Board of Directors/Board Structure: The board of directors/board structure category measures a company's management commitment and effectiveness towards following best practice corporate governance principles related to a well-balanced membership of the board. It reflects a company's capacity to ensure a critical exchange of ideas and an independent decisionmaking process through an experienced, diverse and independent board. Board of Directors/Compensation Policy: The board of directors/compensation policy category measures a company's management commitment and effectiveness towards following best practice corporate governance principles related to competitive and proportionate management compensation. It reflects a company's capacity to attract and retain executives and board members with the necessary skills by linking their compensation to individual or company-wide financial or extra-financial targets.

	Appendix A. Description of ASSET4 categories (from ASSET4 documents)
	Table 5 CSR and the probability of becoming a target CSR Environment (1) (2) Constant -3.017** -3.198** (-2.22) (-2.36) CSR 0.013*** 0.007*** (4.59) (3.90) Size -0.296*** -0.265*** (-5.93) (-5.53) Leverage 1.947*** 1.881*** (6.09) (5.93) Liquidity 0.005 0.001 (0.871) (0.05) MTB -0.076*** -0.074*** (-3.66) (-3.59) Tangibility -0.553*** -0.545*** (-2.62) (-2.59) ROE -0.421* -0.403 (-1.69) (-1.62) ∆Sales 0.258 0.222 (1.07) (0.92) FCF 1.929 1.842 (1.12) (1.08) Year FE Yes Yes Industry FE Yes Yes Country FE Yes Yes Observations 20,348 20,348 Pseudo-R² 0.065 0.064 Social component Customer/Product Responsibility: The customer/product responsibility category measures a Social Governance (3) (4) -2.898** -4.153*** (-2.13) (-3.12) 0.009*** 0.007** (4.50) (2.49) -0.287*** -0.209*** (-5.85) (-4.73) 1.940*** 1.786*** (6.09) (5.68) 0.004 0.001 (0.13) (0.05) -0.077*** -0.070*** (-3.70) (-3.41) -0.540** -0.535** (-2.57) (-2.55) -0.429* -0.399 (-1.73) (-1.61) 0.237 0.142 (0.98) (0.59) 1.863 1.909 (1.09) (1.11) Yes Yes Yes Yes Yes Yes 20,348 20,348 0.065 0.063 company's management commitment and effectiveness towards creating value-added products and services upholding the customer's security. It reflects a company's capacity to maintain its license to operate by producing quality goods and services integrating the customer's health and safety, and preserving its integrity and privacy also through accurate product information and labelling. Society/Community: The society/community category measures a company's management commitment and effectiveness towards maintaining the company's reputation within the general community (local, national and global). It reflects a company's capacity to maintain its license to operate by being a good citizen (donations of cash, goods or staff time, etc.), protecting public health (avoidance of industrial accidents, etc.) and respecting business ethics (avoiding bribery and corruption, etc.). Society/Human Rights: The society/human rights category measures a company's management commitment and effectiveness towards respecting the fundamental human rights conventions. It reflects a company's capacity to maintain its license to operate by guaranteeing the freedom of association and excluding child, forced or compulsory labor. Workforce/Diversity and Opportunity: The workforce/diversity and opportunity category measures a company's management commitment and effectiveness towards maintaining diversity and equal opportunities in its workforce. It reflects a company's capacity to increase its workforce loyalty and productivity by promoting an effective life-work balance, a family friendly environment and equal opportunities regardless of gender, age, ethnicity, religion or sexual orientation. Workforce/Employment Quality: Governance

Specifically, we use an Epanechnikov kernel.

We use 2-digit SIC codes.

This point is particularly important for M&A acquirers given their undiversified nature. Indeed, M&A acquirers are forced to assume a large amount of specific risk because of investment concentration and the high costs associated with the divestiture of acquired businesses.

Acknowledgment

We gratefully acknowledge the financial support of the "Value & CSR" chair. The funding source had no involvement in this research. We are grateful to Brian Lucey (the editor), and an anonymous referee for their constructive comments. We have also benefited from the comments of Mohamed Arouri, Sylvain Marsat, and the participants at the 2017 "Conférence Internationale de Gouvernance". The ideas, methodology, and findings expressed in this paper remain our sole responsibility.


Namig J Guliyev

SPECTRAL IDENTITIES FOR SCHR ÖDINGER OPERATORS

Keywords: 2010 Mathematics Subject Classification. 34B07, 34L40, 34L15, 34A55, 15B05 spectral identities, one-dimensional Schrödinger equation, Sturm-Liouville operator, boundary conditions dependent on the eigenvalue parameter

published or not. The documents may come

Spectral identities for Schrödinger operators

Introduction and main result

In our recent papers, we solved various direct and inverse spectral problems for boundary value problems generated by the one-dimensional Schrödinger equation -y (x) + q(x)y(x) = λy(x)

and the boundary conditions y (0)

y(0) = -f (λ), y (π) y(π) = F (λ), (2)
where f and F are rational Herglotz-Nevanlinna functions. As the papers [START_REF] Guliyev | Essentially isospectral transformations and their applications[END_REF] and [START_REF] Guliyev | Schrödinger operators with distributional potentials and boundary conditions dependent on the eigenvalue parameter[END_REF] show, for summable and distributional potentials respectively, their spectral theory is mostly analogous to the classical case when the coefficients f and F in (2) are constants. There also appear some new phenomena. For example, two spectra may no longer be sufficient to determine the coefficients of (1)-( 2) uniquely [START_REF] Guliyev | On two-spectra inverse problems[END_REF] and it is quite possible for eigenfunctions not to be a basis for L 2 (0, π), even after removing some of them [START_REF] Guliyev | A Riesz basis criterion for Schrödinger operators with boundary conditions dependent on the eigenvalue parameter[END_REF]. However, it is still possible to determine the coefficients of ( 1)-(2) from the knowledge of its spectral data consisting of the eigenvalues and the so-called norming constants (see [START_REF] Gwak | Massless and massive higher spins from anti-de Sitter space waveguide[END_REF] below). The purpose of this short paper is to establish some simple identities relating the boundary coefficients (i.e., f and F ) and the spectral data of ( 1)-( 2) with a real-valued potential q ∈ L 1 (0, π), and which, in particular, allow one to find the boundary coefficients in a much more direct way. Remarkably, some of the objects that we use in this paper, namely the polynomial ω f and the matrices of (4), are very similar to, but not quite identical with, those encountered in other areas of mathematics [START_REF] Holtz | Structured matrices, continued fractions, and root localization of polynomials[END_REF].

Apart from being interesting in their own right, eigenvalue problems with boundary conditions dependent on the eigenvalue parameter have found numerous applications in various fields of science. For example, they have recently been considered in the literature in connection with such diverse areas as heat transfer [START_REF] Belinskiy | Piecewise uniform optimal design of a bar with an attached mass[END_REF], string theory [START_REF] Gwak | Massless and massive higher spins from anti-de Sitter space waveguide[END_REF], fluid dynamics [START_REF] Kaoullas | Start-up and cessation Newtonian Poiseuille and Couette flows with dynamic wall slip[END_REF], biology [START_REF] Kawano | A uniqueness result on detecting a prey in a spider orb-web[END_REF], mathematical finance [START_REF] Nie | Sticky reflecting Ornstein-Uhlenbeck diffusions and the Vasicek interest rate model with the sticky zero lower bound, Stoch. Models[END_REF], and quantum computing [START_REF] Parra-Rodriguez | Quantum networks in divergence-free circuit QED[END_REF]. Interestingly, the last-mentioned paper also discusses some identities of the kind considered in Remark 1 below.

We now recall some necessary notation from [START_REF] Guliyev | Essentially isospectral transformations and their applications[END_REF] related to rational Herglotz-Nevanlinna functions. Each such function can be written as

f (λ) = h 0 λ + h + d k=1 δ k h k -λ (3)
with h 0 ≥ 0, h ∈ R, δ k > 0, and h 1 < . . . < h d . To every function f of this form we assign two polynomials f ↑ and f ↓ by writing this function as

f (λ) = f ↑ (λ) f ↓ (λ) ,
where

f ↓ (λ) := h 0 d k=1 (h k -λ), h 0 := 1/h 0 , h 0 > 0, 1, h 0 = 0.
We also associate to f its index

ind f := 2d + 1, h 0 > 0, 2d, h 0 = 0,
which counts each finite pole of this function twice and its pole at infinity (if any) once. To each rational Herglotz-Nevanlinna function f we also assigned in [START_REF] Guliyev | Essentially isospectral transformations and their applications[END_REF] a monic polynomial

ω f (λ) := (-1) ind f 2 λf ↓ λ 2 -(-1) ind f 2 f ↑ λ 2 ,
where • and • are the usual floor and ceiling functions. Now we are ready to introduce the main players of this paper. Denote by λ n the eigenvalues of the boundary value problem (1)-(2) and by γ n its norming constants defined by [START_REF] Gwak | Massless and massive higher spins from anti-de Sitter space waveguide[END_REF] below. Consider the sums of the series

σ k := ∞ n=0 λ k n γ n , k = 0, 1, . . . , ind f -1
and

σ ind f := n<L λ ind f n γ n + n=L λ ind f n γ n - 1 π + n>L λ ind f n γ n - 2 π with L := ind f + ind F 2 ,
whose convergence is an immediate consequence of the asymptotics of the eigenvalues and the norming constants [3, Theorem 4.2]. Alternatively, their convergence also follows from the proof below (see Section 2). On the other hand, the above polynomial ω f can be written as

ω f (λ) = λ ind f +1 + ω 1 λ ind f + . . . + ω ind f +1 .
It turns out that these two (ind f + 1)-tuples σ 0 , . . ., σ ind f and ω 1 , . . ., ω ind f +1 are related in a rather simple way.

Theorem. The following identities hold:

(-1) k ω 1-k + (ind f +1-k)/2 i=-k/2 σ ind f -i-k ω 2i+k = 0, k = 0, 1, . . . , ind f (4)
with the convention that ω 0 = 1 and ω m = 0 if m < 0. Moreover, this system is uniquely solvable, both when either ω 1 , . . ., ω ind f +1 or σ 0 , . . ., σ ind f are treated as variables.

We will prove this theorem in the next section. It is interesting to note that the system (4) resembles, in a sense, the Gelfand-Levitan integral equation. Indeed, the numbers ω k are defined in terms of the coefficient f and play the role of the Gelfand-Levitan kernel. The numbers σ k , on the other hand, are defined in terms of the eigenvalues and the norming constants, and thus play the role of the other function of two variables from the Gelfand-Levitan equation, usually denoted by

f (•, •) or F (•, •).
Remark 1. The above theorem allows one to determine f in terms of the coefficients of the polynomials f ↓ and f ↑ . It is also possible to obtain some identities involving the coefficients in the representation [START_REF] Guliyev | Essentially isospectral transformations and their applications[END_REF]. Recall that [3, Section 2.2] the boundary value problem ( 1)-( 2) can also be stated as an eigenvalue problem for a self-adjoint operator in a Hilbert space of the form L 2 ⊕ C M for a suitable M , namely

M := ind f 2 + ind F 2 .
Parseval's identity for the eigenvectors of this operator yields the identities

1 δ k = ∞ n=0 1 γ n f ↓ (λ n ) λ n -h k 2 , k = 1, . . . , d and 1 h 0 = ∞ n=0 f 2 ↓ (λ n ) γ n ,
including the possibility that both sides of this last identity are infinite. However, it seems that we still need the above theorem to determine h 1 , . . ., h d , or equivalently f ↓ . Such identities were obtained in [START_REF] Guliyev | A uniqueness theorem for Sturm-Liouville equations with a spectral parameter linearly contained in the boundary conditions[END_REF] for the case when ind f = ind F = 2.

Remark 2. Of course, all of the above identities have their counterparts for the right endpoint. One only needs to replace γ n by β 2 n γ n everywhere, where β n is defined by [START_REF] Guliyev | A Riesz basis criterion for Schrödinger operators with boundary conditions dependent on the eigenvalue parameter[END_REF] below. This can be proven by repeating the corresponding arguments for the left endpoint. A much shorter proof can also be given, based on a simple symmetry argument: for the boundary value problem of the form (1)-(2) with q(x), f , and F replaced by q(π -x), F , and f respectively, the roles of the solutions ϕ and ψ defined by ( 5) are interchanged, and hence that problem has the same eigenvalues λ n and the norming constants β 2 n γ n . In addition to being interesting for their own sake, these identities also have a multitude of applications to inverse problems. For example, as we have already pointed out, they allow one to avoid the whole machinery of [START_REF] Guliyev | Essentially isospectral transformations and their applications[END_REF] and compute the coefficients of f directly. Moreover, for constant f (i.e., when ind f = 0), the system (4) consists of only one identity and this identity is a key ingredient in Korotyaev and Chelkak's parametrization of sets of isospectral problems [START_REF] Korotyaev | The inverse Sturm-Liouville problem with mixed boundary conditions (Russian)[END_REF]. The identities (4) also play an important role in the solution of the two-spectra inverse problem [START_REF] Guliyev | On two-spectra inverse problems[END_REF] for boundary value problems of the form (1)- [START_REF] Guliyev | A uniqueness theorem for Sturm-Liouville equations with a spectral parameter linearly contained in the boundary conditions[END_REF].

The result of this paper can also be useful in the so-called incomplete inverse problems. To determine the boundary value problem (1)-( 2) completely, one needs to know all of the eigenvalues and the norming constants. But if we have some extra information about the coefficients of this boundary value problem, a part of this spectral data can be omitted (see, e.g., [3, Section 4.6], [START_REF] Wei | On the uniqueness of inverse spectral problems associated with incomplete spectral data[END_REF], and the references therein). In particular, if the boundary coefficients f and F are known and some of the eigenvalues and the norming constants are missing, with the number of missing ones not exceeding ind f + ind F + 2, then the identities (4) (and their counterparts for the right endpoint, mentioned in Remark 2) give us algebraic equations for these missing ones.

Proof

Before turning to the proof, we need some more definitions and preliminary results. Let ϕ(x, λ) and ψ(x, λ) be the solutions of (1) satisfying the initial conditions

ϕ(0, λ) = f ↓ (λ), ϕ (0, λ) = -f ↑ (λ), ψ(π, λ) = F ↓ (λ), ψ (π, λ) = F ↑ (λ). (5)
Then the eigenvalues of the boundary value problem ( 1)-( 2) coincide with the zeros of the characteristic function

χ(λ) := F ↑ (λ)ϕ(π, λ) -F ↓ (λ)ϕ (π, λ) = f ↓ (λ)ψ (0, λ) + f ↑ (λ)ψ(0, λ)
and for each eigenvalue λ n there exists a unique number

β n = 0 such that ψ(x, λ n ) = β n ϕ(x, λ n ). (6)
We define the norming constants as

γ n := π 0 ϕ 2 (x, λ n ) dx + f (λ n )ϕ 2 (0, λ n ) + F (λ n )ϕ 2 (π, λ n ), (7)
where the second (respectively, the third) summand on the right-hand side is omitted if λ n coincides with one of the poles of the function f (respectively, F ). The three sequences {λ n } n≥0 , {β n } n≥0 , and {γ n } n≥0 are related by the identity ([3, Lemma 2.1])

χ (λ n ) = β n γ n . (8)
The solution ψ and its first derivative with respect to x satisfy the asymptotic estimates

ψ(0, λ) = λ ind F/2 cos √ λ + ind F 2 π + 1 2 π 0 q(x) dx + Ω 1 λ (ind F -1)/2 sin √ λ + ind F 2 π + o λ (ind F -1)/2 e | Im √ λπ| and ψ (0, λ) = λ (ind F +1)/2 sin √ λ + ind F 2 π - 1 2 π 0 q(x) dx + Ω 1 λ ind F/2 cos √ λ + ind F 2 π + o λ ind F/2 e | Im √ λπ| ,
which in turn imply the asymptotic formula

χ(λ) = λ L+1/2 sin √ λ + L π - 1 2 π 0 q(x) dx + ω 1 + Ω 1 λ L cos √ λ + L π + o λ L e | Im √ λπ|
for the characteristic function. These estimates can be obtained by expressing ψ as a linear combination of the cosine-and sine-type solutions and using the wellknown estimates for them. Another possible way to obtain these formulas is to observe that they all are (up to a change in sign) the characteristic functions of problems of the form ( 1)-( 2) (with the Dirichlet condition at the left endpoint in the case of ψ(0, λ) and the Neumann condition in the case of ψ (0, λ)) and then apply a lemma of Marchenko and Ostrovskii (see [START_REF] Marchenko | Sturm-Liouville operators and applications[END_REF]Lemma 3.4.2]) to their infinite product representations (cf. [4, Appendix A]).

We are now going to obtain the identities (4) as a consequence of a well-known result from complex analysis: if g is a meromorphic function satisfying the estimate

sup |λ|=R N |g(λ)| = o 1 R N with R N := N -L - 1 2 2 ,
then the sum of the residues of this function, defined as 1 2πi lim N →∞ |λ|=R N g(λ) dλ, is zero [START_REF] Pöschel | Inverse spectral theory[END_REF]Lemma 3.2].

It is convenient to consider the cases of odd and even ind f separately. However, since the arguments are very similar in both cases, we give the details only for the former case. Taking g(λ) = λ j ψ(0, λ)/χ(λ) and g(λ) = λ j ψ (0, λ)/χ(λ) respectively in the above result for j = 0, 1, . . . , d -1 and using [START_REF] Holtz | Structured matrices, continued fractions, and root localization of polynomials[END_REF], we obtain

∞ n=0 λ j n f ↓ (λ n ) γ n = ∞ n=0 λ j n β n f ↓ (λ n ) χ (λ n ) = ∞ n=0 Res λ=λn λ j ψ(0, λ) χ(λ) = 0 and ∞ n=0 λ j n f ↑ (λ n ) γ n = ∞ n=0 λ j n β n f ↑ (λ n ) χ (λ n ) = - ∞ n=0
Res λ=λn λ j ψ (0, λ) χ(λ) = 0, which are exactly the identities (4) for k = 2, 3, . . . , ind f . To obtain the case k = 1 we take

g(λ) = λ d ψ(0, λ) χ(λ) - (-1) d λ .
Finally, the case k = 0 is obtained by taking

g(λ) = (-λ) d ψ (0, λ) χ(λ) + λ -1/2 cot √ λ + L π + 1 2 π 0 q(x) dx + ω 1 + Ω 1 λ -1 cot 2 √ λ + L π + 1 2 π 0 q(x) dx + Ω 1 λ -1
and using the above asymptotic formulas for ψ, ψ , and χ together with the obvious equalities

Res λ=(n-L) 2 λ -1/2 cot √ λ + L π = 2 π , n > L, 1 π , n = L and Res λ=(n-L) 2 λ -1 cot 2 √ λ + L π = - 2 π 2 (n-L) 2 , n > L, - 2
3 , n = L.

We now turn to uniqueness. When ωs are treated as variables, taking into account the fact that all the equations of the system (4) except the one with k = 0 contain only ωs with indices of the same parity, this system can be solved in two steps: one first determines ω k with k of the same parity as ind f (i.e., those coming from f ↓ ) and then finds the remaining ones. More precisely, if ind f = 2d + 1 then the subsystem of (4) consisting of the equations with odd k is a system of linear equations with respect to ω 1 , ω 3 , . . ., ω 2d+1 whose matrix is a strictly positive definite Hankel matrix, and is thus uniquely solvable (see the proof of [4, Lemma 2.1] for details). Now it only remains to observe that the matrix of the subsystem of (4) with even k is the same strictly positive definite Hankel matrix as above. Similarly, if ind f = 2d then ω 2 , ω 4 , . . ., ω 2d are uniquely determined from the subsystem of (4) consisting of the equations with nonzero even k, ω 1 is then found from the equation with k = 0, and finally the remaining ωs are determined from the equations with odd k.

Finally, when σs are treated as variables, the determinant of the system (4) is simply equal to the resultant of the polynomials f ↓ and f ↑ , up to a possible change in sign. As these two polynomials have no common roots, their resultant is nonzero.


Nicolas Juillet

A COUPLING PROOF OF CONVEX ORDERING FOR COMPOUND DISTRIBUTIONS

In this paper, we give an alternative proof of the fact that, when compounding a nonnegative probability distribution, convex ordering between the distributions of the number of summands implies convex ordering between the resulting compound distributions. Although this is a classical textbook result in risk theory, our proof exhibits a concrete coupling between the compound distributions being compared, using the representation of one-period discrete martingale laws as a mixture of the corresponding extremal measures.

 for the definition and main properties of the convex ordering of probability measures.

, the theorems are stated with respect to the increasing convex order, and involve two sequences of random variables instead of just one, so that (1) appears as a special case of these results. We refer to Section 4.3 for a discussion of how the more general case can be deduced from (1).

3 The proof given in [13] is similar, using functions of the special form f (x) = (x -t) + .

1

Introduction

Consider an i.i.d. sequence of nonnegative random variables X = (X i ) i 1 , and two integer-valued random variables M, N , independent from the sequence X . Assume that E(X 1 ) < +∞, E(M ) < +∞, E(N ) < +∞, and that a comparison between M and N holds with respect to the convex 1 ordering: M ≺ cx N . We then have the following comparison between the compound variables X 1 + • • • + X M and X 1 + • • • + X N with respect to the convex ordering:

(1)

X 1 + • • • + X M ≺ cx X 1 + • • • + X N .
This is a classical result (see e.g. Theorem 4.A.9 in [START_REF] Shaked | Stochastic orders[END_REF] or Theorem 4.3.6 in [START_REF] Müller | Comparison methods for stochastic models and risks[END_REF]) 2 , useful in the context of risk theory (see e.g. [START_REF] Kaas | Modern actuarial risk theory: using R[END_REF], chap. 7) where its interpretation is that compounding with a riskier frequency distribution leads to a riskier aggregated loss distribution. The proof given in [START_REF] Shaked | Stochastic orders[END_REF] is analytical in nature, and consists in showing that, given a non-decreasing convex function f : [0, +∞[→ R, the sequence (u n ) n 0 defined by u n = Ef (X 1 + • • • + X n ) satisfies, for all n 0, the condition u n+2 -u n+1 u n+1 -u n (provided that u n+2 , u n+1 , u n have a finite value) 3 . Here, we give an alternative proof of (1), based on a coupling between the two random variables being compared, which provides an explicit realization of Strassen's condition (see [START_REF] Strassen | The existence of probability measures with given marginals[END_REF] or e.g. [START_REF] Müller | Comparison methods for stochastic models and risks[END_REF] section 1.5.2) for convex ordering: we construct a pair of random variables (A, B) such that (2)

A d = X 1 + • • • + X M , B d = X 1 + • • • + X N ,
and A = E(B|σ(A)) a.s.

Our proof relies a representation result which we call a diatomic representation of convex ordering, stated as Theorem 1 in Section 2, where we review several approaches for proving the existence of this representation, including an explicit algorithm in the case of discrete distributions. Section 3 contains the coupling construction leading to [START_REF] Beiglböck | Shadow couplings[END_REF] and the proof of [START_REF] Beiglböck | On a problem of optimal transport under marginal martingale constraints[END_REF]. Finally, in Section 4, we discuss various extensions of these results.

Diatomic representation

Given two real numbers x < y, we define, for all z the normalized barycentric coordinates: α x,y (z) = y -z y -x and β x,y (z) = 1 -α x,y (z) = z -x y -x .

In the case x = y, we extend the above definition by setting α x,y (z) = 1 and β x,y (z) = 0. Whenever x z y, both α x,y (z) and β x,y (z) lie in the interval [0, 1], and z can be written as the convex combination of x and y : z = α x,y (z) • x + β x,y (z) • y.

Theorem 1. Given two probability distributions µ, ν on R possessing a finite expectation, the comparison µ ≺ cx ν holds if and only if there exists a triple of random variables (V -, U, V + ) defined on the same probability space and such that:

(3) U ∈ [V -, V + ] (4) Law(U ) = µ (5) ν = E α V -,V + (U ) • δ V -+ β V -,V + (U ) • δ V +
We call such a triple (V -, U, V + ) a diatomic representation of the stochastic ordering µ ≺ cx ν.

A more concrete statement of ( 5) is that Law(V ) = ν, where V is a random variable whose conditional distribution with respect to V -, U, V + is given by: V = V -with probability α V -,V + (U ) V + with probability β V -,V + (U )

Theorem 1 may not have been stated under this specific form in the mathematical literature, but its content is certainly not new. In the following subsections we review several ways of proving this result.

2.1. Proof of Theorem 1 via Choquet's and Douglas' theorems (for compactly supported measures ν). We prove here the result only in the case of measures µ ≺ cx ν concentrated on some closed interval K.

Consider the following space of measures on R 2 :

S K = π ∈ M + (K 2 ) : ∀a ∈ R, ∀b ∈ C b (R), a + b(u).(v -u)dπ(u, v) = a .
Here M + (K 2 ) is the space of finite positive Borel measures on R 2 with support contained in K 2 , and C b (R) the space of real-valued continuous and bounded functions on R. In other words, denoting by F the space of functions f a,b : (u, v) ∈ R 2 → a + b(u)(v -u), the above definition reads:

S K = {π ∈ M + (K 2 ) : ∀f a,b ∈ F, f a,b dπ = a}.
It turns out that S K is a non-empty space of probability measures, known as martingale measures in the literature since for (X 1 , X 2 ) ∼ π ∈ S K , the process (X i ) i=1,2 is a martingale on its natural filtration.

Our task amounts to proving that a measure π ∈ S K is represented as a mixture of 'triplet' measures of the form δ z ⊗ [αδ x + βδ y ], where z = αx + βy and α, β are nonnegative and satisfy α + β = 1.

We admit without proof that S K is convex and compact for the weak topology. Choquet's Theorem (see [START_REF] Choquet | Existence des représentations intégrales au moyen des points extrémaux dans les cônes convexes[END_REF] or e.g. [START_REF] Robert | Lectures on Choquet's theorem[END_REF]) then implies that every measure in S K can be represented as a mixture of extremal measures in S K . So we shall be done as soon as we can prove that the extremal measures in S K are triplet measures concentrated on S K . Let η be such an extremal measure and µ, ν its marginals. We aim at proving that µ is supported on a single point and that ν is supported on at most two points. Striving for a contradiction suppose that µ is not a Dirac measure. Hence, there exists a Borel set E such that µ(E) / ∈ {0, 1}. We consider the L 1 (η) distance between f : (u, v) → 1 E (u) and the functions f a,b ∈ F . Letting (U, V ) be distributed according to η we find

f -f a,b L 1 (η) E E(f (U, V ) -f a,b (U, V )| U ) E E(1 E (U ) -a| U ) = P(U ∈ E)|1 -a| + P(U / ∈ E)|a| 1/2 min(P(U ∈ E), P(U / ∈ E)).
Note that this lower bounds only depends on f . We have thus proved that F is not dense in L 1 (η). According to Douglas's theorem [START_REF] Douglas | On extremal measures and subspace density[END_REF] (see also [16, Chapter V]) this in contradiction with the fact that η is extremal. Thus η is of type δ u 0 ⊗ν where u 0 is the barycenter of ν, i.e v dν(v) = u 0 . Next, again striving for a contradiction suppose that there exists a partition (A i ) i∈{1,2,3} of R such that ν(A i ) > 0 holds for every i. From Douglas's Theorem again the set F of functions is dense in L 1 (η). In particular any function

g c 1 ,c 2 ,c 3 : (u, v) → 3 i=1 c i 1 A i (v) can be approximated in L 1 (η) by functions f a,b . The linear map (c 1 , c 2 , c 3 ) → ( A 1 g c 1 ,c 2 ,c 3 dη, A 2 g c 1 ,c 2 ,c 3 dη, A 3 g c 1 ,c 2 ,c 3 dη)
is clearly linear and onto. It follows that the linear map

(a, b) ∈ R 2 → ( A 1 f a,b dη, A 2 f a,b dη, A 3 f a,b dη) ∈ R 3
is onto as well, a contradiction. Therefore, extremal measures of S K are of type δ z ⊗ [αδ x + βδ y ] where z = αx + βy, as it was required. Remark. The search for martingale kernels k : u → k u is a key question in the field of martingale optimal transport.The first completely canonical method seems to be the left-curtain coupling by Beiglböck and Juillet [START_REF] Beiglböck | On a problem of optimal transport under marginal martingale constraints[END_REF] that is also of particular interest to us. Not only is k u canonical but when µ is diffuse its kernels k u are automatically diatomic (this also holds for the former coupling by Hobson and Neuberger [START_REF] Hobson | Robust bounds for forward start options[END_REF] under more general assumptions). This is not the case if µ possesses atoms. However a quantile version of the left-curtain coupling is described in a second paper by the same authors [START_REF] Beiglböck | Shadow couplings[END_REF] where the martingale measure π directly appears as a mixture over the set [0, 1] of quantile levels ω ∈ [0, 1] of diatomic kernels

δ zω ⊗ (α xω,yω (z ω )δ xω + β xω,yω (z ω )δ yω )
where z ω is the ω-quantile of µ. Note that the same can be said of the recent coupling by Jourdain and Margheriti [START_REF] Jourdain | A new family of one dimensional martingale couplings[END_REF].

Algorithmic proof of Theorem 1 (for finitely supported µ, ν).

We now describe an explicit algorithmic construction, inspired by [START_REF] Beiglböck | On a problem of optimal transport under marginal martingale constraints[END_REF][START_REF] Beiglböck | Shadow couplings[END_REF], leading to a diatomic decomposition in the case where both µ and ν are finitely supported probability measures. This algorithm is used to produce the simulation shown in Fig. 1.

Let us write µ = p i=1 µ(u i )δ u i and ν = q j=1 ν(v j )δ v j .

Initialization: µ * ← µ, ν * ← ν, θ * ← 1, T ← ∅ Loop: Repeat the following steps while θ * > 0 Pick a triple (v j-, u i , v j+ ) such that a. ν * (v j-) > 0, µ * (u i ) > 0, ν * (v j+ ) > 0 b. v j-u i v j+ c. ν * ((v j -, v j +)) = 0 s ← min (µ * (u i ), ν * (v j-)/α v j-,v j+ (u i ), ν * (v j+ )/β v j-,v j+ (u i ))) µ * ← µ * -sδ u i ν * ← ν * -sα v j-,v j+ (u i )δ v j--sβ v j-,v j+ (u i )δ v j+ θ * ← θ * -s T ← T ∪ {((v j-, u i , v j+ ), s)} Result: return the set T The probability distribution of (V -, U, V +) is then deduced from T as (v-,u,v+,s)∈T sδ (v-,u,v+) .
The reason why the above algorithms stops lies in the fact that the transformation performed on µ * and ν * keeps the comparison µ * ≺ cx ν * valid throughout the execution of the algorithm (see the proof of Lemma 2.8 in [START_REF] Beiglböck | On a problem of optimal transport under marginal martingale constraints[END_REF]), with µ * and ν * having equal total mass. As a consequence, as long as µ * and ν * do not have zero total mass, the comparaison µ * ≺ cx ν * ensures that a triple (v j-, u i , v j+ ) satisfying conditions a.-b.-c. exists. Finally, since at each step at least one of the three numbers ν * (v j-), µ * (u i ), ν * (v j+ ) is set to zero, the total mass of both µ * and ν * must reach zero after a finite number of steps.

Remark. If, in the loop part of the algorithm, one systematically choses the unique triple (v -, u, v + ) such that u is the leftmost point in the support of µ * (such a choice is always possible, see the proof of Lemma 2.8 in [START_REF] Beiglböck | On a problem of optimal transport under marginal martingale constraints[END_REF]), the end-result of the algorithm is the so-called left-curtain coupling.

Coupling construction

We now describe the coupling construction leading to our proof of (1). Consider a triple (N -, M, N + ) as in Theorem 1, with µ = Law(M ) and ν = Law(N ), and an i.i.d. sequence X = (X i ) i 1 independent from (N -, M, N + ). For all integer k 1, we let S k = k i=1 X i , with the convention that S 0 = 0. Finally, we let

F = σ(M, N -, N + , S N -, S N + ) and G = σ(M, N -, N + , S M , S N -, S N + ). Note that G = F ∨ σ(S M ).
We start our construction by setting:

(6) A = S M .
Next, we specify B by the requirement that, conditional upon G, the distribution of B is:

(7) α S N -,S N + (A) • δ S N -+ β S N -,S N + (A) • δ S N + .
Note that α S N -,S N + (A) and β S N -,S N + (A) do indeed lie in the interval [0, 1] thanks to the assumption that the random variables X i are nonnegative, so that S N -A = S M S N + , since N -M N + . We now proceed to checking that all three properties listed in (2) are satisfied by the random variables A and B. From (6), it is immediate that A has the required distribution. Moreover, from the definition of α and β, one has the identity

α S N -,S N + (A) • S N -+ β S N -,S N + (A) • S N + = A,
which rewrites as:

E [B|G] = A a.s.,
whence, taking the conditional expectation E(•|σ(A)) on both sides, and using the fact that σ

(A) ⊂ G since A = S M is G-measurable, E [B|σ(A)] = A a.s.,

as required by (2).

To conclude the proof, it remains to check that B d = S N . By construction, the conditional distribution of B given F can be written as:

E α S N -,S N + (A) F • δ S N -+ E β S N -,S N + (A) F • δ S N + .
Now observe that, by symmetry, given integers n - m n + such that n -< n + , we have:

E S m -S n -σ(S n -, S n + ) = m -n - n + -n - (S n + -S n -) a.s., from which we deduce that 4 E α Sn -,Sn + (S m ) σ(S n -, S n + ) = α n -,n + (m) a.s.
and

E β Sn -,Sn + (S m ) σ(S n -, S n + ) = β n -,n + (m) a.s.
As a consequence, the conditional distribution of B given F is none but:

α N -,N + (M ) • δ S N -+ β N -,N + (M ) • δ S N + .
From the fact that the sequence X = (X i ) i 1 is independent from (N -, M, N + ), and from condition (5), we deduce that

B d = S N ,
which concludes the proof. 4 In the case where n-= m = n+, these identities are still (obviously) valid.

Final remarks and extensions

4.1. Continuous time. We note that the coupling construction described in Section 3 can be extended in continuous time. For instance, let N = (N t ) t 0 be a standard Poisson process and S ≺ cx T nonnegative integrable random variables independent from N . Then one has N S ≺ cx N T , and it is straightforward to extend our approach to define a pair of random variables (A, B) such that (8)

A d = N S , B d = N T , and A = E(B|σ(A)) a.s.
The same approach still works in exactly the same way if we consider an integrable subordinator instead of a Poisson process.

4.2. Exchangeable random variables. If the sequence of random variables (X i ) i 1 is assumed to be exchangeable instead of i.i.d., the coupling construction described in Section 3 works in exactly the same way. (The classical proof found in [START_REF] Shaked | Stochastic orders[END_REF][START_REF] Müller | Comparison methods for stochastic models and risks[END_REF] also works in this case.) Note that, in the case of an infinite exchangeable sequence of random variables, one can directly deduce (1) from the i.i.d. case, using the De Finetti representation of such a sequence as a mixture of (distributions of) i.i.d. sequences, and the characterization of (1) through the inequality ( 9)

Ef (X 1 + • • • + X M ) Ef (X 1 + • • • + X N
) for all convex functions f . On the other hand, if M and N are assumed to have finite support, say {0, 1, . . . , q}, and one considers a finite exchangeable sequence of random variables X 1 , . . . , X q , the extension of De Finetti's theorem to this case (see [START_REF] Kerns | De Finetti's theorem for abstract finite exchangeable sequences[END_REF][START_REF] Janson | On a representation theorem for finitely exchangeable random vectors[END_REF]) leads in general to a signed mixture of i.i.d. sequences, so one cannot integrate the inequality (9) with respect to the mixing measure in order to directly deduce (1).

4.3.

Increasing convex ordering. Assume that a comparison between M and N holds with respect to the increasing convex ordering: M ≺ icx N . We then have the following modified version of (1):

(10) X 1 + • • • + X M ≺ icx X 1 + • • • + X N .
To deduce ( 10) from ( 1), we note that the comparison M ≺ icx N implies that there exists an integer-valued 5 random variable N 0 such that M ≺ st N 0 and N 0 ≺ cx N , where ≺ st denotes the usual stochastic ordering. Given an 5 The existence of a random variable N0 such that M ≺ st N0 and N0 ≺cx N is a classical and easily proved decomposition result for the increasing convex order. That N0 can, in addition, be chosen to be integer-valued is less standard. One possible proof of this fact is that, when µ ≺ icx ν, there exists a kernel k : R → P(R) such that µ almost surely k u = k(u) has mean u and it holds µ • k = ν. In turn, k u can be written as a mixture

k u = αuk u 1 + (1 -αu)k u 2 , where µ almost surely αu ∈ [0, 1], k u 1 has mean u, and the support of k u 2 is contained in [u, +∞[. For k (u) = αuδu + (1 -αu)k u 2 we have µ ≺ st (µ • k ) ≺cx ν
and a coupling of the corresponding random variables can also be easily deduced. Alternatively for another proof one can consider Kellerer's kernels defined in [11, §2.1] for connecting µ ≺ icx ν. i.i.d. sequence X = (X i ) i 1 independent from M, N 0 , N , the fact that the X i s are nonnegative random variables, combined with M ≺ st N 0 , yields the comparison

X 1 + • • • + X M ≺ st X 1 + • • • + X N 0 . Then, using (1), we deduce that X 1 + • • • + X N 0 ≺ cx X 1 + • • • + X N , so that (10) is proved. Now consider two sequences X = (X i ) i 1 and Y = (Y i ) i 1 of i.i.d.
nonnegative random variables, and, in addition to M ≺ icx N , assume that X i ≺ icx Y i . We then have the following extension of ( 10):

(11) X 1 + • • • + X M ≺ icx Y 1 + • • • + Y N ,
which is a straightforward consequence of ( 10) and of the fact that the increasing convex ordering is preserved by convolution.

4.4. Counterexample to (1) when the X i are not positive random variables. We let M ≡ 1 and N ∼ 1 2 (δ 0 +δ 2 ), so that the condition M ≺ cx N is satisfied. Then let X 1 , X 2 be i.i.d. random variables, independent from N (and M ), with

X i ∼ 1 2 (δ -1 + δ 1 ). We find that X M ∼ 1 2 (δ -1 + δ 1 ), while X N ∼ 1 8 (δ -2 + 6δ 0 + δ 2 ). Hence 1 = E(|X N |) < E(|X M |) = 1 2 so that X M ≺ cx X N cannot hold.
q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 5 10

Figure 1 . 1 4 δ n and ν = 1 4 δ 1 + 1 4 δ 3 + 6 n=4 1 6

 111 Figure 1. The figure shows 10 4 simulated pairs (A, B), with µ = 5 n=2

  2.2. Proof of Theorem 1 via Strassen's theorem (general case). Another approach to proving Theorem 1 is to use Strassen's theorem instead of Choquet's and Douglas's theorems: there exists a kernel k : R → P(R) such that µ almost surely k

u = k(u) has mean u and it holds µ • k = ν. (Such kernels are known as dilations or martingale kernels in the literature.) Hence the mixture with weight µ of the measures δ u ⊗ k u defines a probability measure π on R 2 whose marginals are µ and ν. To complete the proof, it remains to check that each measure k u can be represented as a mixture of diatomic measures with mean u. This last fact is a classical step in the proof of Skorokhod's representation theorem (see e.g.

[START_REF] Durrett | Probability-theory and examples[END_REF] Theorem 8.1.1]

): every probability measure on R with mean u can be represented as a mixture of diatomic measures with mean u. See [9, §5.1] for another approach.


Jean Bérard

Nicolas Juillet

RECONCILIATION OF PROBABILITY MEASURES

published or not. The documents may come

Reconciliation of probability measures

Introduction

Consider a finite number n 2 of probability spaces all built upon the same sample space Ω, and denoted by (Ω, F 1 , P 1 ), . . . , (Ω, F n , P n ). We ask whether it is possible to reconcile these n probability spaces, meaning that there exists a probability measure P on F = σ(F 1 , . . . , F n ) such that, for all 1 i n, the restriction of P to F i coincides with P i . In such a case, we say that P provides a reconciliation of the probability measures P 1 , . . . , P n . This is a natural problem from a modeling perspective, where several probabilistic models may be available, each describing a specific aspect of the situation under study. The question is then the existence of a probabilistic model which simultaneously incorporates the previous specific models into a global one. Scenario aggregation (see e.g. [4]) and coherent belief modeling (see e.g. [1]) are two examples where related (though not equivalent) problems appear.

Note that, in general, the σ-fields F 1 , . . . , F n correspond to distinct but not completely unrelated families of events, which leads to additional constraints on a potential reconciliation P beyond the mere requirement that P |F i = P i for all i. For instance, one may have two events A i ∈ F i and A j ∈ F j for which A i ∩ A j = ∅ with i = j, so that any potential reconciliation P should satisfy P(A i ∩ A j ) = 0. As a consequence, even in the simple case where all σ-fields are assumed to be finite, the existence of a reconciliation P is neither automatic nor a trivial question.

In this paper, we first state a characterization of the existence of a reconciliation in the case of finite σ-fields. For n = 2, a very simple criterion is obtained, while the corresponding criterion in the general case n 2 looks more complicated. Through a counterexample, we show that as simple a criterion as in the case n = 2 cannot be expected to hold in general. We then discuss the extension of these results to the case of more general (infinite) σ-fields. Through a counterexample, we show that additional conditions are required for such an extension to hold, then give one example of such conditions.

Note that the positive results stated in Theorem 1.1 and Theorem 1.2 can in fact be derived as corollaries of results obtained some decades ago by various authors, as we gradually became aware while completing the present study. The reconciliation problem as formulated above is neither the main motivation nor a notable example in these works, and we believe that the short self-contained proofs provided in the present paper are still of interest. Moreover, the counterexamples we provide highlight some interesting issues that are specific to the reconciliation problem.

The rest of the paper is organized as follows. Part 1.1 is devoted to the statement of the results (positive and negative). Connections with earlier work are discussed in Part 1.2. Finally, proofs of the various results are collected in Section 2.

1.1. Statement of results.

1.1.1. The finite case. Throughout this section, the σ-fields F 1 , . . . , F n on Ω are assumed to comprise a finite number of events. Extensions to the infinite case are discussed in the next section.

The first result deals with the case of two probability spaces (n = 2), where we have the following characterization of when a reconciliation exists.

Theorem 1.1. Assume that F 1 and F 2 comprise a finite number of events. Then it is possible to reconcile (Ω, F 1 , P 1 ) and (Ω, F 2 , P 2 ) if and only if

P 1 (E 1 ) P 2 (E 2 ) for every E 1 ⊂ E 2 with E 1 ∈ F 1 and E 2 ∈ F 2 .
(1) Note that condition (1) is clearly necessary since, given a reconciliation P, one must have P 1 (E 1 ) = P(E 1 ) P(E 2 ) P 2 (E 2 ) as soon as E 1 ⊂ E 2 . The non-trivial part of the theorem lies in the fact that condition (1) is indeed sufficient to ensure the existence of a reconciliation. Also note that, taking complementary sets, condition (1) is equivalent to the symmetric condition

E 2 ⊂ E 1 ⇒ P 2 (E 2 ) P 1 (E 1 ).
The next result deals with the general case n 2. The existence of a reconciliation is characterized in terms of elementary integer-valued measurable functions. Consider integer numbers m 1 1, . . . , m n 1, and, for all 1 i n, a family of m i pairwise disjoint events

E (i) 1 ∈ F i , . . . , E (i) m i ∈ F i , and a family of m i integer numbers d (i) 1 ∈ Z, . . . , d (i) m i ∈ Z. Then set (2) f i = m i ℓ=1 d (i) ℓ • 1 E (i) ℓ and f = n i=1 f i .
Note that each f i is F i -measurable, and that one has

(3)

f i dP i = m i ℓ=1 d (i) ℓ • P i (E (i) ℓ ).
Theorem 1.2. Assume that F 1 , . . . , F n comprise a finite number of events. Then it is possible to reconcile (Ω, F 1 , P 1 ), . . . , (Ω, F n , P n ) if and only if, for all functions f and f 1 , . . . , f n of the form given by (2), one has that

(4) n i=1 f i = f 0 ⇒ n i=1 f i dP i 0.
The necessity of condition (4) above is easy to see, for, given a reconciliation P, and a non-negative function f , one must have

0 f dP = n i=1 f i dP = n i=1 f i dP i .
As a consequence, the non-trivial part of Theorem 1.2 is the fact that condition ( 4) is indeed sufficient for the existence of a reconciliation P.

Note that we insisted on giving a formulation in terms of integer-valued functions (instead of general real-valued functions) since we are interested in having a combinatorial interpretation of the criterion in terms of comparisons between probabilities of sets. Indeed, condition (4) in Theorem 1.1 has a very clear such combinatorial interpretation, and one may hope for an equally clear criterion in the general case n 2. Since, in the case n = 2, the existence of a reconciliation can be checked by looking at a very specific subset of the conditions appearing in Theorem 1.2 (namely, those involving

m 1 = 1, m 2 = 1, d (1) 1 = -1, d (2) 1
= 1), a natural question is whether, in the general case n 2, it is still possible to characterize the existence of a reconciliation through a subset of conditions that involve only "small" integer values. Unfortunately, the following result shows that the answer is negative.

Theorem 1.3. For all K > 0, there exists a triple of probability spaces (Ω, F 1 , P 1 ), (Ω, F 2 , P 2 ), (Ω, F 3 , P 3 ), where F i comprises a finite number of events for i = 1, 2, 3, such that:

• no reconciliation exists; • condition (4) is satisfied whenever |d (i) ℓ | K for all i, ℓ.
Theorem 1.3 shows that, even when n = 3, there is no upper bound on how large the integer coefficients d (i) ℓ in (2) have to be in order to check the existence of a reconciliation. Note that, with K = 1 (and Theorem 1.1), the theorem also provides an example of a triple of probability spaces among which every pair admits a reconciliation, while no overall reconciliation exists for the triple.

1.1.2. Extension to the infinite case. We now consider the case where the σ-fields F i on Ω may comprise an infinite number of events. We start with a negative result showing that one cannot extend the previous results to such a general case without additional assumptions.

Theorem 1.4. There exists a pair (Ω, F 1 , P 1 ), (Ω, F 2 , P 2 ) such that condition (1) is satisfied, but for which no reconciliation exists.

Remark 1.5. The proof of Theorem 1.1 shows that conditions (1) and ( 4) are in fact equivalent when n = 2. As a consequence, in the counterexample used to prove Theorem 1.4, condition (4) is also satisfied, showing that, in the infinite case, neither (1) nor ( 4) is sufficient to imply the existence of a reconciliation.

The next theorem states that, under a (reasonably mild and general) additional assumption on the σ-fields F 1 , . . . , F n , can indeed be extended.

Theorem 1.6. Assume that the σ-fields F 1 , . . . , F n (with n = 2 in the case of Theorem 1.1) are of the form

F i = σ(X i ), where X i is a map from Ω to R d i (equipped with the Borel σ-field), with d i 1. Moreover, assume that the set (X 1 , . . . , X n )(Ω) is a closed set in R d 1 +•••+dn
Then the conclusions of Theorems 1.1 and 1.2 hold. Namely, in the case n = 2, there exists a reconciliation between P 1 and P 2 if and only if condition (1) holds, and, in the general case n 2, there exists a reconciliation between P 1 , . . . , P n if and only if condition (4) holds.

The two following corollaries are immediate by-products of the theorem.

Corollary 1.7. The conclusions of Theorems 1.1 and 1.2 hold when the σ-fields F 1 , . . . , F n are generated by a countable number of atoms.

Corollary 1.8. Let V be a closed set of R d × R d and µ, ν two probability measures on R d . Then, using transport terminology, there exists a transport plan π from µ to ν concentrated on V , i.e, a measure π ∈ P(R d × R d ) with marginals µ and ν and π(V ) = 1, if and only if for every Borel sets

A ⊂ R d and B ⊂ R d µ(A) ν(B) as soon as (A × R d ) ∩ V ⊂ (R d × B) ∩ V.
(5) Note that Corollary 1.8 is obtained from Theorem 1.6 by setting Ω = V , with the random variables X i being given by coordinate functions. Also, note that Theorem 1.4 is based on a construction where V is not closed and the conclusion of Corollary 1.8 fails. Take Ω = V = {(x 1 , x 2 ) ∈ R 2 ; 0 < x 1 < x 2 < 1}, and let X 1 and X 2 denote the coordinate functions. Then no reconciliation exists if one asks that both X 1 and X 2 follow the uniform distribution on ]0, 1[, see the proof of Theorem 1.4.

Connections with earlier work.

Early references bearing directly on Theorem 1.1 are Fréchet [7] and Dall'Aglio [5] (where an unpublished result of Berge is also quoted). There, a general result is proved on the existence of two-dimensional discrete distributions with prescribed marginals and upper bounds, from which Theorem 1.1 can be derived. Subsequent work by Kellerer (Satz 3.2 in [9]) in a general framework (finite dimensional distributions on abstract spaces) can be used to derive Theorem 1.2. Finally, key ingredients needed to prove Theorem 1.6 can be taken from Theorem 11 in Strassen [14]. We refer to the book [6] for additional references and a more detailed historical perspective on this body of work.

Here, a step-by-step self-contained approach to the proofs is given. Indeed, in our specific framework, Theorem 1.2 stems from a rather straightforward application of Farkas' Lemma. Theorem 1.1 is then derived from Theorem 1.2 through a simple reduction argument. Finally, Theorem 1.6 is what one more or less readily obtains by passing to the limit within a sequence of discrete approximations provided by Theorems 1.1 and 1.2.

To conclude this section, we would like to emphasize the connections of Corollary 1.8 (which is in fact a variant of Theorem 11 in [14], and also related with ) with recent developments in optimal transport theory.

Although it appears a little less obvious, condition ( 5) is necessary in Corollary 1.8 for the same reason why (1) is in Theorem 1.1. Its natural interpretation is not probabilistic, but in terms of transport, as follows. We recall that π(A × B) represents the mass transported from A to B. The constraint on the capacity of the transport plan is given by V : no mass can travel from x to y if (x, y) / ∈ V . Gravel located in A with mass µ(A) can be displaced according to the capacity transport constraint encoded by V only to the set B 0 = proj 2 ((A × R) ∩ V ). The storage size of this (universally measurable) set is ν(B 0 ). In order for the transport to be manageable, it has to be larger than µ(A), which is condition (5). Corollary 1.8 states that under this condition there exists a transport plan that satisfies the capacity constraint. As for the usual Monge-Kantorovich optimal transport problem, the goal in constrained versions of the problem is to obtain information on the minimizers of π → c dπ, where c : R d × R d is a lower semicontinuous cost function. In particular, is there a Monge solution, that is in the form π = (Id ⊗ T ) # µ ?

We are aware of two works in the transport literature that correspond to this problem. In [8] the special case of a constraint displacement vector has been encoded by

V = {(x, y) ∈ R 2d : y -x ∈ -→ V } where -→ V ⊂ R d is a closed convex set
with some additional properties. The authors look at the shape of optimizers for a quadratic (constraint) cost

c(x, y) = |y -x| 2 if y -x ∈ -→ V , +∞ if y -x / ∈ -→ V .
They prove that, provided µ is absolutely continuous, any solution π * of the Monge-Kantorovich transport problem is a Monge transport plan π * = (Id ⊗ T ) # µ where T (x) ∈ x + -→ V , µ-almost surely, and hence, π * is uniquely determined. This occurs under the assumption made that a transport plan π with finite cost does exist, i.e, c dπ < ∞ for some admissible π. In this respect, condition (5) is namely required in order for the problem to have finite minimal total cost. Note that (5) reads in this case ν(A + -→ V ) µ(A). The other work is [2] where -→ V is the unit Euclidean ball B 1 (0) and c is the so-called relativistic cost

c(x, y) = 1 -1 -|y -x| 2 if |y -x| 1, +∞ if |y -x| > 1.
As c is bounded on its domain, ( 5) is a necessary and sufficient condition to have finite total cost. The authors also introduce c t := c(x/t, y/t) and the critical speed T = T µ,ν = inf{t ∈ R + : the total cost is finite for c t }. Therefore T is also the minimal t such that ν(A+B t (0)) µ(A) for every Borel set A. Let us finally mention [12,11] where another Monge-Kantorovich problem under capacity constraint is investigated: a transport plan π is admissible if it possesses a density h on X × Y ⊂ R d 1 +d 2 that satisfies 0 < h h for a given h. In this case the authors of [12] cite (p. 575) two functional criteria for the existence of an admissible transport plan. These are due to Kellerer [9] and Levin [13]. An equivalent set-based criterion due to Kellerer [10, Satz 4.2] directly transposes in the continuous settings the (already mentioned) ones obtained by Fréchet [7] and Dall 'Aglio [5] in the discrete setting.

Proofs

We start with the proof of Theorem 1.2, which consists merely in an application of Farkas' Lemma.

Proof of Theorem 1.2. As already noted after the statement of the theorem, condition (4) is clearly necessary for the existence of a reconciliation. We now assume that (4) holds, and prove the existence of a reconciliation.

Denote by H the (finite-dimensional) vector space formed by real-valued functions f on Ω of the form f = f 1 + • • • + f n , where each f i is an F i -measurable real-valued function on Ω. Note that condition (4) is assumed to hold for such functions f when each f i is integer-valued. Since we consider functions on a finite state space, condition (4) in fact holds for any f in H, as can be seen by approximating each f i by functions with rational values, which in turn can be written as integer values divided by the g.c.d. of these values. Now, for any ω ∈ Ω, denote by ϕ ω the linear form on H defined by ϕ ω (f ) = f (ω). On the other hand, denote by θ the linear form on H defined by θ(f ) = n i=1 f dP i . Condition (4) (extended to all functions in H) says that, if f ∈ H is such that ϕ ω (f ) 0 for all ω ∈ Ω, then one has θ(f ) 0. Farkas' Lemma then guarantees the existence of a family (t ω ) ω∈Ω of non-negative real numbers such that, for all f ∈ H, one has the identity ( 6)

θ(f ) = ω∈Ω t ω • f (ω).
Applying ( 6) with e.g. f 1 ≡ 1 and f 2 = • • • = f n ≡ 0, we see that the nonnegative numbers (t ω ) ω∈Ω sum up to 1, so that we can define P({ω}) = t ω . Given A i ∈ F i , we can apply (6) with f i = 1 A i and f j ≡ 0 for j = i, and we deduce that P(A i ) = P i (A i ).

We now prove Theorem 1.1 from Theorem 1.2, through a suitable reduction argument.

Proof of Theorem 1.1. As already noted after the statement of the theorem, condition (1) is clearly necessary for the existence of a reconciliation. We now assume that (1) holds, and prove the existence of a reconciliation through Theorem 1.2.

Without loss of generality, we start with a function f of the form f = f 1 + f 2 , with f 1 = i λ i 1 A i and f 2 =j µ j 1 B j , where the λ i and µ j are in Z, and where the (A i ) i (resp. (B j ) j ) is a finite family of events in F 1 (resp. F 2 ) that forms a finite partition of Ω. In the sequel, the λ i and µ j are called the coefficients of f .

Our goal is to establish condition (4) so as to apply Theorem 1.2. So we now have to prove that, if f 0, then i λ i P 1 (A i )j µ j P 2 (B j ) 0.

Step 1: We may assume that f has coefficients in N: Adding to f the function 0

= C -C = C( i 1 A i ) -C( j 1 B j )
where C = max i,j (|λ i |, |µ j |), we see that i (λ i + C)1 A ij (µ j + C)1 B j has non-negative coefficients λ ′ i = λ i + C and µ ′ j = µ j + C, while keeping λ ′ i P 1 (A i ) + µ ′ j P 2 (B j ) = λ i P 1 (A i ) + µ j P 2 (B j ), since CP 1 (A i ) -CP 2 (B j ) = 0.

Step 2: We may assume that f has coefficients in {0, 1}: Consider f such that f 0, and, given Step 1, assume that the coefficients in f satisfy λ i , µ j ∈ N. We set λ ′ i = max(λ i -1, 0) and µ ′ j = max(µ j -1, 0), and write f = g + h, with g = λ ′ i 1 A iµ ′ j 1 B j and h = (λ iλ ′ i )1 A i -(µ jµ ′ j )1 B j . Since λ i 1 A i µ j 1 B j , we must have, for every (i, j) with A i ∩ B j = ∅, the fact that λ i µ j , and therefore that λ ′ i µ ′ j and λ iλ ′ i µ jµ ′ j . As a consequence, we have g 0 and h 0, and g and h have non-negative integer coefficients. Moreover, unless they are already equal to 0 or 1, the coefficients of g and h are strictly smaller than the corresponding coefficients of f . Indeed, if λ i 2, one has λ ′ i < λ i and λ iλ ′ i < λ i , and similarly for µ j . As a consequence, after a finite number of iterations, we can write f as a sum of non-negative functions with integer coefficients in {0, 1}.

Conclusion:

For a function f 0 with coefficients in {0, 1}, the difference i λ i P 1 (A i )j µ j P 2 (B j ) is of the form P 1 (E 1 ) -P 2 (E 2 ) with E 2 ⊂ E 1 , so that condition (1) implies the fact that i λ i P 1 (A i )j µ j P 2 (B j ) 0.