E. Belsley, Rates of convergence of Markov chains related to Association schemes, PhD thesis, Dept. Math., Harvard University, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....an important topic. Diaconis [10] Sinclair and Jerrum [39] Jerrum and Sinclair [22] Diaconis and Stroock [13] Sinclair [38] and Diaconis and Salo Coste [11] proved general results on nite state spaces. Hanlon [18] Frieze et al. 15] Frigessi et al. 16] Ingrassia [20] and Belsley [5] proved results speci cally for Markov chain Monte Carlo. On general state spaces, not many results have been found yet. For partial results, see Amit and Grenander [2] Amit [1] Hwang et al. 19] Lawler and Sokal [24] Meyn and Tweedie [25] Rosenthal [33, 34, 35, 36] Baxter and Rosenthal [4] ....

E.D. Belsley, Rates of convergence of Markov chains related to association schemes, Ph.D. dissertation, Department of Mathematics, Harvard University (1993). 90 BIBLIOGRAPHY 91

.... spaces, see Diaconis [9] Sinclair and Jerrum [35] Jerrum and Sinclair [20] Diaconis and Stroock [12] Sinclair [34] and Diaconis and Salo Coste [10] For results speci cally for Markov chain Monte Carlo, see Hanlon [16] Frieze et al. 13] Frigessi et al. 14] Ingrassia [18] and Belsley [4]. For results on discrete time 1 Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3. Internet: yuen math.toronto.edu 1 general state spaces, see Amit and Grenander [2] Amit [1] Hwang et al. 17] Lawler and Sokal [22] Meyn and Tweedie [23] Rosenthal [29, 30, ....

E.D. Belsley, Rates of convergence of Markov chains related to association schemes, Ph.D. dissertation, Department of Mathematics, Harvard University (1993).

....resulting Metropolis chain are given by the coecients of Jack s symmetric functions (expanded in terms of the power sum symmetric functions) and they used the corresponding eigenvalues to give a complete analysis of the running time. Similar analyses were carried out in the Ph.D. theses of Belsley [Be2] and Silver [Si] They worked in abelian groups with proportional to (y) where is the length function with respect to a natural set of generators. In several cases they found that the eigenfunctions were natural deformations of classical orthogonal polynomials. Ross and Xu [RX] studied ....

E. Belsley, Rates of convergence of Markov chains related to association schemes, Ph.D. thesis, Harvard, 1993. 34 persi diaconis and arun ram

....process (X k ; Y k ) visits the subset R Theta R, it has probability fi of coupling. Using drift conditions , it may be possible to bound the number of such returns to R Theta R, and then use coupling as in the uniform case; see Rosenthal (1993b) A related approach is presented in Meyn and Tweedie (1993), who use minorizations, drift conditions, splittings, and careful bounding to obtain bounds on k k Gamma k directly, without introducing a second, coupled chain. Instead of trying to bound k k Gamma k directly, or use coupling, another approach is as follows. Consider a single Markov chain ....

E.D. Belsley (1993), Rates of convergence of Markov chains related to association schemes.

....process (X k ; Y k ) visits the subset R Theta R, it has probability fi of coupling. Using drift conditions , it may be possible to bound the number of such returns to R Theta R, and then use coupling as in the uniform case; see Rosenthal (1993b) A related approach is presented in Meyn and Tweedie (1993), who use minorizations, drift conditions, splittings, and careful bounding to obtain bounds on k k Gamma k directly, without introducing a second, coupled chain. Instead of trying to bound k k Gamma k directly, or use coupling, another approach is as follows. Consider a single Markov ....

E.D. Belsley (1993), Rates of convergence of Markov chains related to association schemes.

....use in guiding a simulation. Finally, in section 6, we present (Theorem 12) a simplified version of our main result, which involves verifying a simpler drift condition than does Theorem 5. Remark. Since originally completing this manuscript, we have learned of recent similar work by Meyn and Tweedie (1993b) Using minorization conditions and a simple drift condition on the chain, they obtain computable bounds on the distance to stationarity under certain conditions. Their methods require slightly less information than do ours, however their bounds appear to be weaker in specific examples. I am ....

....Q(A) for all x 2 R ; for all measurable subsets A X . Minorization conditions are closely related to the notion of Harris Recurrence. They were introduced in Athreya and Ney (1978) see also Athreya, McDonald and Ney (1978) Nummelin (1984) Asmussen (1989) Lindvall (1992) and Meyn and Tweedie (1993a) They have been used to analyze MCMC in Roberts and Polson (1990) Tierney (1991) Rosenthal (1993, 1991) and Mykland et al. 1992) Most of the present paper is based on the following theorem. Special cases of the theorem were used in Rosenthal (1993, 1991) for similar purposes. Theorem 1. ....

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E.D. Belsley (1993), Rates of convergence of Markov chains related to association schemes.

.... Frigessi, Hwang, Sheu, and Di Stefano, 1993; Ingrassia, 1994; Liu, 1992; Belsley, 1993) On infinite state spaces, however, progress is much more limited (though for partial results see Lawler and Sokal, 1988; Amit and Grenander, 1991; Amit, 1991, 1993; Hwang, Hwang Ma and Sheu, 1993; Meyn and Tweedie, 1993; Rosenthal, 1995a, 1995b, 1994; Baxter and Rosenthal, 1995; Roberts and Rosenthal, 1994) In this paper we consider the extent to which previous bounds for finite chains (especially those involving choices of paths) can be extended to bounds for infinite chains. Our results fall into two ....

....inf d 1 kP d fi fi W k L 2 (1= d ) fi, then k k Gamma k var 1 2 k 0 Gamma k L 2 (1= fi k : This corollary says that we can bound the distance to stationarity on the countably infinite chain X by any uniform bound on the sequence of finite chains fX d g. A similar idea is used in Belsley, 1993, Theorem VI 4 2. To make use of this fact, we make the following definition. A set of paths ffl xy g on X is unfolding if there exists a sequence of finite subsets X d of X with X 1 X 2 : and X = d X d , such that for any x; y 2 X d , the path fl xy connecting x to y lies entirely ....

E.D. Belsley (1993), Rates of convergence of Markov chains related to association schemes. Ph.D. dissertation, Dept. of Mathematics, Harvard University.

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E. Belsley, Rates of convergence of Markov chains related to Association schemes, PhD thesis, Dept. Math., Harvard University, 1993.

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E.D. Belsley, Rates of convergence of Markov chains related to association schemes, Ph.D. dissertation, Department of Mathematics, Harvard University (1993). 89

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E.D. Belsley, Rates of convergence of Markov chains related to association schemes, Ph.D. dissertation, Department of Mathematics, Harvard University (1993).

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E.D. Belsley, Rates of convergence of Markov chains related to association schemes, Ph.D. dissertation, Department of Mathematics, Harvard University (1993).

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E.D. Belsley, Rates of convergence of Markov chains related to association schemes, Ph.D. dissertation, Department of Mathematics, Harvard University (1993).

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