Concrete representation of martingales
Abstract
Let (fn) be a mean zero vector valued martingale sequence. Then there exist vector valued functions (dn) from [0,1]n such that int01 dn(x1,...,xn) dxn = 0 for almost all x1,...,xn-1, and such that the law of (fn) is the same as the law of (sumk=1n dk(x1,...,xk)) . Similar results for tangent sequences and sequences satisfying condition (C.I.) are presented. We also present a weaker version of a result of McConnell that provides a Skorohod like representation for vector valued martingales. This paper may be found at http://math.missouri.edu/~stephen/preprints
Turn this paper into a full lesson
ArcXiv compiles a staged curriculum from this paper: 8-12 lessons across beginner → advanced, synthesised section guides, visuals, flashcards, a quiz, exercises, and on-demand deep dives per section. Grounded in the abstract, never invented.