  This article discusses MLMC estimators with and without weights, applied to
nested expectations of the form E [f (E [F (Y, Z)|Y ])]. More precisely, we are
interested on the assumptions needed to comply with the MLMC framework,
depending on whether the payoff function f is smooth or not. A new result to
our knowledge is given when f is not smooth in the development of the weak
error at an order higher than 1, which is needed for a successful use of MLMC
estimators with weights.
