Martingale type, the Gamlen-Gaudet construction and a greedy algorithm

Abstract

In the present paper we identify those filtered probability spaces (Ω,F,(Fn),P) that determine already the martingale type of a Banach space X. We isolate intrinsic conditions on the filtration (Fn) of purely atomic σ-algebras which determine that the upper ℓp estimates‖f‖Lp(Ω,X)p⩽Cp(‖E(f|F0)‖Lp(Ω,X)p+∑n=1∞‖E(f|Fn)−E(f|Fn−1)‖Lp(Ω,X)p),f∈Lp(Ω,X) imply that the Banach space X is of martingale type p. Our paper complements G. Pisier's investigation [12] and continues the work by S. Geiss and second named author in [3].