  We prove a Tauberian theorem for nonexpansive operators, and apply it to the
model of zero-sum stochastic game. Under mild assumptions, we prove that the
value of the lambda-discounted game v_{lambda} converges uniformly when lambda
goes to 0 if and only if the value of the n-stage game v_n converges uniformly
when n goes to infinity. This generalizes the Tauberian theorem of Lehrer and
Sorin (1992) to the two-player zero-sum case. We also provide the first example
of a stochastic game with public signals on the state and perfect observation
of actions, with finite state space, signal sets and action sets, in which for
some initial state k_1 known by both players, (v_{lambda}(k_1)) and (v_n(k_1))
converge to distinct limits.
