The setting of this Chapter is a finite-state irreducible Markov chain $(X_{t})$, either in discrete time $(t=0,1,2,\ldots)$ or in continuous time $(0\leq t<\infty)$. Highlights of the elementary theory of general (i.e. not-necessarily-reversible) Markov chains are readily available in several dedicated textbooks and in chapters of numerous texts on introductory probability or stochastic processes (see the Notes), so we just give a rapid review in sections 2.1 and 2.1.2. Subsequent sections emphasize several specific topics which are useful for our purposes but not easy to find in any one textbook: using the fundamental matrix in mean hitting times and the central limit theorem, metrics on distributions and submultiplicativity, Matthews’ method for cover times, and martingale methods.

- 2.1 Notation and reminders of fundamental results
- 2.2 Identities for mean hitting times and occupation times
- 2.3 Variances of sums
- 2.4 Two metrics on distributions
- 2.5 Distributional identities
- 2.6 Matthews’ method for cover times
- 2.7 New chains from old
- 2.8 Miscellaneous methods
- 2.9 Notes on Chapter 2.
- 2.10 Move to other chapters

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