Asymptotically invariant sampling and averaging from stationary-like processes

Given a process X on Rd or Zd, we may form a random sequence ξ1,ξ2,… by sampling from X at some independent points τ1,τ2,… . If X is stationary up to shifts (which holds for broad classes of Markov and Palm processes) and the distribution of (τn) is asymptotically invariant (as in the case of Poisson or Bernoulli sampling, respectively) then (ξn) is asymptotically exchangeable, and the associated empirical distribution converges to the corresponding product random measure.