Independent sampling of a stochastic process

We investigate the question of when sampling a stochastic process X={X(t): t⩾0} at the times of an independent point process ψ leads to the same empirical distribution as the time-average limiting distribution of X. Two main cases are considered. The first is when X is asymptotically stationary and ergodic, and ψ satisfies a mixing condition. In this case, the pathwise limiting distributions in function space are shown to be the same. The second main case is when X is only assumed to have a constant finite time average and ψ is assumed a positive recurrent renewal processes with a spread-out cycle length distribution. In this latter case, the averages are shown to be the same when some further conditions are placed on X and ψ.