  We show that the class of $L^2$ functions for which ergodic averages of a
reversible Markov chain have finite asymptotic variance is determined by the
class of $L^2$ functions for which ergodic averages of its associated jump
chain have finite asymptotic variance. This allows us to characterize
completely which ergodic averages have finite asymptotic variance when the
Markov chain is an independence sampler. In addition, we obtain a simple
sufficient condition for all ergodic averages of $L^2$ functions of the primary
variable in a pseudo-marginal Markov chain to have finite asymptotic variance.
