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5. 2/; : : :g is exchangeable. For all N 1, k 1, 0 Ä rk < rk 1 < < r1 < r0 D s, a 2 f0; 1gN , p1 ; p2 ; : : : ; pk such that 0 Ä Np1 ; Np2 ; : : : ; Npk Ä N are integers, 2 SN ; P ÁŒN s D DP k \ a; fXrNi i D1 ÁŒN s D D ! a/; fXrNi i D1 D 1 Ä Ng k \ ! i pi g; fZrN;r i i D1 1 Ä Ng : Proof. We prove this in the case k D 2, the case k > 2 is similar. For all a1 2 f0; 1gN , we denote by Pa1 the set of partitions P of f1; 2; : : : ; N g 1 which are such that i 'P j ) ai1 D aj1 . For any i D 1; 2, let ÁN ri 1 D a , where 1 N i a 2 f0; 1g .
Acknowledgements This work was partially supported by the ANR MANEGE (contract ANR09-BLANC-0215-03). The authors wish to thank an anonymous referee, whose excellent and very detailed report permitted us to correct two errors and some imprecisions in an earlier version of this paper. 28 B. Bah et al. References 1. : Exchangeability and related topics. In: Ecole d’´et´e St Flour 1983. , vol. 1117, pp. 1–198 (1985) 2. : Convergence of Probability Measures, 2nd edn. Wiley, New York (1999) 3. : A countable representation of the Fleming Viot measure-valued diffusion.
1] a comprehensive presentation of these problems is given. We refer also to the references of this book for the related literature [2, 3]. In this work, the authors consider that the demand comes from an underlying state of demand, which is modeled as a Markov chain with a finite number of states. The fact that the number of states is finite simplifies mathematical arguments. We will here discuss the A. edu L. Decreusefond and J. 1007/978-3-642-29982-7 2, © Springer-Verlag Berlin Heidelberg 2012 29 30 A.