## On nonlinear Markov chain Monte Carlo.(English)Zbl 1241.60037

Markov chain Monte Carlo (MCMC) algorithms are developed for simulating from complicated distributions, for example, when the target distribution has multiple modes and/or possesses strong dependencies between subcomponents of the state space. In the article, nonlinear kernels of the form $K_{\mu}(x,dy)=(1-\varepsilon)K(x,dy)+\varepsilon\Phi(\mu)(dy)$ are introduced, where $$K(x,dy)$$ is the original kernel with complicated target distribution and $$\epsilon\Phi(\mu)(dy)$$ is added in order to improve algorithmic performance.
Such nonlinear kernels cannot be simulated exactly, so approximations of the nonlinear kernels are constructed using auxiliary or potentially self-interacting chains. Several nonlinear kernels are presented, and it is demonstrated that, under some conditions, the associated approximations exhibit a strong law of large numbers; the proof technique uses the Poisson equation and Foster-Lyapunov conditions. The performance of the approximations is investigated with some simulations.

### MSC:

 60J22 Computational methods in Markov chains 65C05 Monte Carlo methods 65C40 Numerical analysis or methods applied to Markov chains
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