## Bernoulli

• Bernoulli
• Volume 20, Number 2 (2014), 457-485.

### A central limit theorem for adaptive and interacting Markov chains

#### Abstract

Adaptive and interacting Markov Chains Monte Carlo (MCMC) algorithms are a novel class of non-Markovian algorithms aimed at improving the simulation efficiency for complicated target distributions. In this paper, we study a general (non-Markovian) simulation framework covering both the adaptive and interacting MCMC algorithms. We establish a central limit theorem for additive functionals of unbounded functions under a set of verifiable conditions, and identify the asymptotic variance. Our result extends all the results reported so far. An application to the interacting tempering algorithm (a simplified version of the equi-energy sampler) is presented to support our claims.

#### Article information

Source
Bernoulli, Volume 20, Number 2 (2014), 457-485.

Dates
First available in Project Euclid: 28 February 2014

https://projecteuclid.org/euclid.bj/1393593994

Digital Object Identifier
doi:10.3150/12-BEJ493

Mathematical Reviews number (MathSciNet)
MR3178506

Zentralblatt MATH identifier
1303.60020

#### Citation

Fort, G.; Moulines, E.; Priouret, P.; Vandekerkhove, P. A central limit theorem for adaptive and interacting Markov chains. Bernoulli 20 (2014), no. 2, 457--485. doi:10.3150/12-BEJ493. https://projecteuclid.org/euclid.bj/1393593994

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#### Supplemental materials

• Supplementary material: Supplement to “A central limit theorem for adaptive and interacting Markov chains”. We detail in this supplement: (1) the gap in the proof of Atchade’s [9] theorem, (2) the proofs of technical Lemmas 4.1, 4.3, A.1–A.3, (3) some additional proofs of [18], Section 3.1, (4) results on the variance of completely degenerated V-statistics of asymptotically stationary Markov chains, and (5) the weak law of large number for adaptive and interacting Markov chains.