  Markov Chain Monte Carlo (MCMC) is a class of algorithms to sample complex
and high-dimensional probability distributions. The Metropolis-Hastings (MH)
algorithm, the workhorse of MCMC, provides a simple recipe to construct
reversible Markov kernels. Reversibility is a tractable property that implies a
less tractable but essential property here, invariance. Reversibility is
however not necessarily desirable when considering performance. This has
prompted recent interest in designing kernels breaking this property. At the
same time, an active stream of research has focused on the design of novel
versions of the MH kernel, some nonreversible, relying on the use of complex
invertible deterministic transforms. While standard implementations of the MH
kernel are well understood, the aforementioned developments have not received
the same systematic treatment to ensure their validity. This paper fills the
gap by developing general tools to ensure that a class of nonreversible Markov
kernels, possibly relying on complex transforms, has the desired invariance
property and leads to convergent algorithms. This leads to a set of simple and
practically verifiable conditions.
