Abstract. In this paper we consider conditionally independent processes with respect to some dynamic factor. More precisely, we assume that for all i ∈ N, the random variables X1, . . . , Xi are conditionally independent with respect to vector V i = (V1, . . . , Vi). We study the mixing properties of the Xi's when conditioning is given with respect to unbounded memory of the factor. Our work is motivated by some real examples related to risk theory.Acknowledgments.

In risk management, the distribution of underlying random variables is not always entirely known. Sometimes, only the mean value and some shape information on the distribution (decreasingness, convexity . . .) are available. The present paper provides discrete convex extrema for several situations of this type. The starting point is the class of discrete distributions whose probability mass functions are nonincreasing on a support D n ≡ {0, 1, . . . , n}. Convex extrema in that class of distributions are well-known. Our purpose is to point out how additional shape constraints of convexity type modify these extrema. Two cases are considered: the p.m.f. is globally convex on N, or it is convex only from a given point m ≥ 1. The corresponding convex extrema are derived by using simple crossing properties between two distributions. Several applications to insurance problems are briefly presented. The results provide a complement to recent works by Lefèvre and Loisel [9,10].

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