CONVEX REARRANGEMENTS OF RANDOM-WALKS

The asymptotic behavior of random zonotopes (Minkowski sums of random segments [0, xi(i)] where xi(i) are i.i.d. d-dimensional random variab les) and the properties of their limits are considered. When the dimen sion d = 2 one can also use the convex rearrangements of the ordinary random walks. If the law P of xi(i) has a first order moment, the stro ng law of large numbers is applicable and in this case the limiting co nvex set is not random. Under the additional hypothesis of existence o f a second order moment some variants of CLT are found. We discuss som e applications of these results to the convexification of stochastic p rocesses. The main result is related to the case when P belongs to the domain of attraction of an alpha-stable law, alpha < 1. The weak conv ergence to a non-degenerate limiting distribution is proved. We establ ish the remarkable properties of this law: it is concentrated on the f amily of convex sets with Canter type boundary structure; one can comp arer this structure whith that of a generic set in category sense. (C) Elsevier, Paris.