A Note on Some Rates of Convergence in First-Passage Percolation

A variation is given of the van den Berg-Kesten inequality on the probability of disjoint occurrence of events enabling it to apply to random variables, rather than just to events, associated with various subsets of an index set. This is used to establish superadditivity of a certain family of generating functions associated with first-passage percolation. This leads to improved estimates for the rates of convergence of the expected values of certain passage times.