  The widespread use of Markov Chain Monte Carlo (MCMC) methods for
high-dimensional applications has motivated research into the scalability of
these algorithms with respect to the dimension of the problem. Despite this,
numerous problems concerning output analysis in high-dimensional settings have
remained unaddressed. We present novel quantitative Gaussian approximation
results for a broad range of MCMC algorithms. Notably, we analyse the
dependency of the obtained approximation errors on the dimension of both the
target distribution and the feature space. We demonstrate how these Gaussian
approximations can be applied in output analysis. This includes determining the
simulation effort required to guarantee Markov chain central limit theorems and
consistent variance estimation in high-dimensional settings. We give
quantitative convergence bounds for termination criteria and show that the
termination time of a wide class of MCMC algorithms scales polynomially in
dimension while ensuring a desired level of precision. Our results offer
guidance to practitioners for obtaining appropriate standard errors and
deciding the minimum simulation effort of MCMC algorithms in both multivariate
and high-dimensional settings.
