  We study convergence properties of pseudo-marginal Markov chain Monte Carlo
algorithms (Andrieu and Roberts [Ann. Statist. 37 (2009) 697-725]). We find
that the asymptotic variance of the pseudo-marginal algorithm is always at
least as large as that of the marginal algorithm. We show that if the marginal
chain admits a (right) spectral gap and the weights (normalised estimates of
the target density) are uniformly bounded, then the pseudo-marginal chain has a
spectral gap. In many cases, a similar result holds for the absolute spectral
gap, which is equivalent to geometric ergodicity. We consider also unbounded
weight distributions and recover polynomial convergence rates in more specific
cases, when the marginal algorithm is uniformly ergodic or an independent
Metropolis-Hastings or a random-walk Metropolis targeting a super-exponential
density with regular contours. Our results on geometric and polynomial
convergence rates imply central limit theorems. We also prove that under
general conditions, the asymptotic variance of the pseudo-marginal algorithm
converges to the asymptotic variance of the marginal algorithm if the accuracy
of the estimators is increased.
