  We investigate continuous time random walks with truncated $\alpha$-stable
trapping times. We prove distributional ergodicity for a class of observables;
namely, the time-averaged observables follow the probability density function
called the Mittag--Leffler distribution. This distributional ergodic behavior
persists for a long time, and thus the convergence to the ordinary ergodicity
is considerably slower than in the case in which the trapping-time distribution
is given by common distributions. We also find a crossover from the
distributional ergodic behavior to the ordinary ergodic behavior.
