On Averages Seen by Arrivals in Discrete Time

We study the limiting behavior of averages from an embedded stochastic process obtained by sampling a discrete-time stochastic process at points of an associated discrete-time stochastic point process. We determine when the limit of the averages from the embedded process coincides with the limit of the averages from the original process. In a certain stationary Markov framework, this happens if and only if the point process is a Bernoulli sequence with future points being independent of the state of the Markov process.