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Association criteria for m-infinitely-divisible and u-infinitely-divisible random sets

Karłowska-Pik J., Schreiber T.

Probability and Mathematical Statistics

2008

Vol. 28, Fasc. 2

169--178

We study random convex compact sets infinitely divisible with respect to the Minkowski addition and establish a sufficient condition for their association as well as a necessary and sufficient condition for the so-called infinite association. Further, we show also that every union infinitely-divisible random closed set and every convex compact set infinitely divisible for convex hulls of unions are associated.

Large deviation principle for set-valued union processes

Schreiber T.

Probability and Mathematical Statistics

2000

Vol. 20, Fasc. 2

273--285

The purpose of the paper is to establish a large deviation principle for a certain class of increasing set-valued processes obeyingMarkovian dynamics. The obtained result is then applied to investigate the asymptotics of the sequence of successive convex hulls generated by uniform samples on a d-dimensional ball.