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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\u2013death 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\u2013death 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 L<jats:sup>2<\/jats:sup>-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\u2013death processes in this orbit.<\/jats:p>","DOI":"10.1017\/s0963548320000024","type":"journal-article","created":{"date-parts":[[2020,2,18]],"date-time":"2020-02-18T08:03:56Z","timestamp":1582013036000},"page":"508-536","update-policy":"http:\/\/dx.doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":1,"title":["Analysis of non-reversible Markov chains via similarity orbits"],"prefix":"10.1017","volume":"29","author":[{"given":"Michael C. H.","family":"Choi","sequence":"first","affiliation":[]},{"given":"Pierre","family":"Patie","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2020,2,18]]},"reference":[{"key":"S0963548320000024_ref49","unstructured":"[49] Patie, P. and Savov, M. 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