Hamiltonian Monte Carlo (HMC) is currently one of the most popular Markov Chain Monte Carlo algorithms to sample smooth distributions over continuous state space. This paper discusses the irreducibility and geometric ergodicity of the HMC algorithm. We consider cases where the number of steps of the StörmerVerlet integrator is either fixed or random. Under mild conditions on the potential U associated with target distribution π, we first show that the Markov kernel associated to the HMC algorithm is irreducible and positive recurrent. Under more stringent conditions, we then establish that the Markov kernel is Harris recurrent. We provide verifiable conditions on U under which the HMC sampler is geometrically ergodic. Finally, we illustrate our results on several examples.

中文翻译：

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Hamiltonian Monte Carlo 的不可约性和几何遍历性

Hamiltonian Monte Carlo (HMC) 是目前最流行的马尔可夫链蒙特卡罗算法之一，用于在连续状态空间上对平滑分布进行采样。本文讨论了HMC算法的不可约性和几何遍历性。我们考虑 StörmerVerlet 积分器的步数是固定的或随机的情况。在与目标分布 π 相关的潜在 U 的温和条件下，我们首先证明与 HMC 算法相关的马尔可夫核是不可约的和正循环的。在更严格的条件下，我们然后确定马尔可夫核是哈里斯循环的。我们在 U 上提供了可验证的条件，在这些条件下 HMC 采样器是几何遍历的。最后，我们用几个例子来说明我们的结果。