A Second Look at General Markov ChainsDavid
نویسنده
چکیده
In the spirit of Chapter 2, this is an unsystematic treatment of scattered topics which are related to topics discussed for reversible chains, but where reversibility plays no essential role. Section 1 treats constructions of stopping times with various optimality properties. Section 2 discusses random spanning trees associated with Markov chains, the probabilistic elaboration of \the matrix-tree theorem". Section 3 discusses self-verifying algorithms for sampling from a stationary distribution. Section 4 discusses \reversib-lizations" of irreversible chains. Section 5 gives an example to show that the nonasymptotic interpretation of relaxation time, so useful in the reversible setting, may fail completely in the general case. At rst sight these topics may seem entirely unrelated, but we shall see a few subtle connections. Throughout the chapter, our setting is a nite irreducible discrete-time Markov chain (X n) with transition matrix P = (p ij). 1 Minimal constructions and mixing times Chapter 4 Theorem yyy involved three mixing time parameters; 1 related to variation distance to stationarity, (1) 1 related to \separation" from station-arity, and (2) 1 related to stationary times (see below). In Chapter 4 these parameters were deened under worst-case initial distributions, and our focus was on \equivalence" of these parameters for reversible chains. Here we discuss underlying \exact" results. Fix an initial distribution. Then
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