  Stochastic two-player games model systems with an environment that is both
adversarial and stochastic. In this paper, we study the expected value of the
window mean-payoff measure in stochastic games. The window mean-payoff measure
strengthens the classical mean-payoff measure by measuring the mean-payoff over
a window of bounded length that slides along an infinite path. Two variants
have been considered: in one variant, the maximum window length is fixed and
given, while in the other, it is not fixed but is required to be bounded. For
both variants, we show that the decision problem to check if the expected value
is at least a given threshold is in NP $\cap$ coNP. The result follows from
guessing the expected values of the vertices, partitioning them into so-called
value classes, and proving that a short certificate for the expected values
exists. Finally, we also show that the memory required by the players to play
optimally is no more than that in non-stochastic two-player games with the
corresponding window objectives.
