Composition of Haar Paraproducts: The Random Case
  When is the composition of paraproducts bounded? This is an important, and difficult question, related to to a question of Sarason on composition of Hankel matrices, and the two-weight problem for the Hilbert transform. We consider randomized variants of this question, finding non-classical characterizations, for dyadic paraproducts. 
Dmitriy Bilyk, Michael Lacey, Xiaochun Li, Brett Wick