Asymptotically invariant sampling and averaging from stationary-like processes

Given a process X on or , we may form a random sequence [xi]1,[xi]2,... by sampling from X at some independent points [tau]1,[tau]2,... . If X is stationary up to shifts (which holds for broad classes of Markov and Palm processes) and the distribution of ([tau]n) is asymptotically invariant (as in the case of Poisson or Bernoulli sampling, respectively) then ([xi]n) is asymptotically exchangeable, and the associated empirical distribution converges to the corresponding product random measure.