• Markov chain Monte Carlo methods;
  • Metropolis–Hastings algorithm;
  • Transient phase;
  • Weak convergence

Summary.  The paper considers high dimensional Metropolis and Langevin algorithms in their initial transient phase. In stationarity, these algorithms are well understood and it is now well known how to scale their proposal distribution variances. For the random-walk Metropolis algorithm, convergence during the transient phase is extremely regular—to the extent that the algo-rithm's sample path actually resembles a deterministic trajectory. In contrast, the Langevin algorithm with variance scaled to be optimal for stationarity performs rather erratically. We give weak convergence results which explain both of these types of behaviour and practical guidance on implementation based on our theory.