In this paper we develop an in-depth analysis of non-reversible Markov chains on denumerable state space from a similarity orbit perspective. In particular, we study the class of Markov chains whose transition kernel is in the similarity orbit of a normal transition kernel, such as that of birth–death chains or reversible Markov chains. We start by identifying a set of sufficient conditions for a Markov chain to belong to the similarity orbit of a birth–death chain. As by-products, we obtain a spectral representation in terms of non-self-adjoint resolutions of identity in the sense of Dunford [21] and offer a detailed analysis on the convergence rate, separation cutoff and L2-cutoff of this class of non-reversible Markov chains. We also look into the problem of estimating the integral functionals from discrete observations for this class. In the last part of this paper we investigate a particular similarity orbit of reversible Markov kernels, which we call the pure birth orbit, and analyse various possibly non-reversible variants of classical birth–death processes in this orbit.

中文翻译：

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通过相似轨道分析不可逆马尔可夫链

在本文中，我们从相似轨道的角度深入分析了可数状态空间上的不可逆马尔可夫链。特别是，我们研究了转移核在正常转移核的相似轨道上的一类马尔可夫链，例如生死链或可逆马尔可夫链。我们首先确定马尔可夫链属于生死链相似轨道的一组充分条件。作为副产品，我们获得了 Dunford [21] 意义上的非自伴随身份分辨率的光谱表示，并提供了对收敛速度、分离截止和 L 的详细分析2-这类不可逆马尔可夫链的截止。我们还研究了从此类的离散观察中估计积分泛函的问题。在本文的最后一部分，我们研究了可逆马尔可夫核的一个特殊相似轨道，我们称之为纯出生轨道，并分析了该轨道中经典生死过程的各种可能不可逆的变体。