  We provide general conditions ensuring that the value functions of some
nonlinear stopping problems with finite horizon converge to the value functions
of the corresponding problems with infinite horizon. Our result can be
formulated as result on stability, with respect to time horizon, of nonlinear
$f$-expectations. We also study the rate of convergence. Many examples are
given to illustrate our results. They include the analysis of time asymptotics
of the fair prices of American options in a multidimensional exponential L\'evy
model.
