  The Deffuant-Weisbuch (DW) model is a bounded-confidence opinion dynamics
model that has attracted much recent interest. Despite its simplicity and
appeal, the DW model has proved technically hard to analyze and its most basic
convergence properties, easy to observe numerically, are only conjectures. This
paper solves the convergence problem for the heterogeneous DW model with the
weighting factor not less than $1/2$. We establish that, for any positive
confidence bounds and initial values, the opinion of each agent will converge
to a limit value almost surely, and the convergence rate is exponential in mean
square. Moreover, we show that the limiting opinions of any two agents either
are the same or have a distance larger than the confidence bounds of the two
agents. Finally, we provide some sufficient conditions for the heterogeneous DW
model to reach consensus.
