Convergence theorems for adapted sequences of random sets

Abstract

We consider random sets with values in a separable Banach space. We study set-valued amarts, L¹-amarts, uniform amarts and submartingales. For all these classes of random sets, we prove convergence theorems in all main modes of set convergence (weak, Wijsman, Mosco, and Hausdorff). We also prove new convergence theorems for vector-valued subpramarts and pramarts.

Keywords:

• Random sets
• set convergence
• amarts
• submartingales
• support function
• distance function

2010 AMS Subject Classification:

• 28B20
• 46B22
• 60G42
• 60G49

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research was supported by the National Science Center of Poland under Project No. 2015/19/B/ST1/01169.