  This paper proves several Tauberian theorems for general iterations of
operators, and provides two applications to zero-sum stochastic games where the
total payoff is a weighted sum of the stage payoffs. The first application is
to provide conditions under which the existence of the asymptotic value implies
the convergence of the values of the weighted game, as players get more and
more patient. The second application concerns stochastic games with finite
state space and action sets. This paper builds a simple class of asymptotically
optimal strategies in the weighted game, that at each stage play optimally in a
discounted game with a discount factor corresponding to the relative weight of
the current stage.
