---------------------------- Original Message ---------------------------- Subject: Your open problem 12 From: "Christophe Leuridan" Date: Wed, November 30, 2005 8:38 am To: aldous@stat.Berkeley.EDU -------------------------------------------------------------------------- Jean Brossard and I read your recent paper 'A Survey of Max-type Recursive Distributional Equations' (in Annals of Probability). In our paper "chaines de Markov constructives indexees par Z" (submitted for publication) we proved a theorem (theorem 9) that is less general but similar to your theorem 1.1. You may be interested by our proof of implication 3 => 2 in theorem 9, corresponding to point (b) of your theorem 1.1. Our proof can be adapted to your situation and does not use any continuity condition. Thus, it gives a positive answer to your open problem 12. Here is how it works in your context. If X is a RTP with inovation xi as in theorem 1.1, consider a process Y such that X and Y are i.i.d. conditionnally to xi (enlarge the probability space if necessary). Thus (X,Y) is a bivariate version. By uniqueness property, P[X=Y]=1 and so P[X=Y|xi]=1. This means that the conditionnal distribution of X is a Dirac mass. That is endogeny. Furthermore, we believe that the results of our first part on the conditionnal law of X are still true in your situation, although we did not check all the details at this time. According to these results, for example, we can say that if the asymptotic sigma-field is trivial, then the conditionnal law of X with respect to the innovations xi is a.s. diffuse or a.s. a uniform law on a finite set with constant cardinal. Sincerly yours, Christophe Leuridan. PS: our paper can be found on http://www-fourier.ujf-grenoble.fr/ by clicking on the links 'Séminaires' and 'Index Global'. Its number is 677.