Convergence of adaptive and interacting Markov chain Monte Carlo algorithms

Adaptive and interacting Markov chain Monte Carlo algorithms (MCMC) have been recently introduced in the literature. These novel simulation algorithms are designed to increase the simulation efficiency to sample complex distributions. Motivated by some recently introduced algorithms (such as the adaptive Metropolis algorithm and the interacting tempering algorithm), we develop a general methodological and theoretical framework to establish both the convergence of the marginal distribution and a strong law of large numbers. This framework weakens the conditions introduced in the pioneering paper by Roberts and Rosenthal [J. Appl. Probab. 44 (2007) 458--475]. It also covers the case when the target distribution $\pi$ is sampled by using Markov transition kernels with a stationary distribution that differs from $\pi$.