Osȩkowski, Adam - In: Statistics & Probability Letters 82 (2012) 1, pp. 186-190
Suppose that f is a martingale taking values in a Banach space B and g is its transform by a deterministic sequence of numbers in {−1,1}, such that supn‖gn‖≥1 almost surely. We show that a certain family of Φ-estimates for f holds true if and only B is a Hilbert space.