Abstract
Coupling methods originated as a probabilistic tool for the analysis of a given process by (possibly dependently) linking its sample path behavior to that of a “target process” whose long term properties may be better understood or known. This chapter illustrates the reach of coupling, extending well beyond Doeblin’s original ideas, with a sample of applications to random fields.
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Notes
- 1.
Although not covered in the present book, it is worthy of mention that the power of Doeblin’s coupling was fully realized in the constructions of optimal couplings for continuous parameter Markov chains, as well as for a class of diffusions on manifolds, by Chen and Wang (1995), Chen (1996) and his student. These couplings involve sharp estimates of the L 2-spectral gap of the infinitesimal generator and are remarkable for the contribution to mathematics outside of probability theory. See Chen (1997) for an insightful overview.
- 2.
The Choquet–Deny theorem is valid more generally for irreducible general random walks on locally compact Abelian groups; Choquet and Deny (1960).
- 3.
See Appendix D for more background from functional analysis.
- 4.
See BCPT, Theorem 7.10, p. 144.
- 5.
See BCPT, Theorem 7.4, p. 140.
- 6.
See BCPT, Proposition 7.6, p. 142.
- 7.
Holley (1974).
- 8.
Fortuin et al. (1971).
- 9.
- 10.
- 11.
Path coupling was introduced by Bubley and Dyer (1997).
- 12.
Further extensions of path coupling to aggregate path coupling were developed by Kovchegov and Otto (2018).
- 13.
A more general version of this result for probabilities on Polish spaces is given in the monograph Lindvall (2002). This provides a proof of the maximality of the coupling used for the Poisson approximation.
- 14.
BCPT, p. 136.
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Bhattacharya, R., Waymire, E. (2022). Special Topic: More on Coupling Methods and Applications. In: Stationary Processes and Discrete Parameter Markov Processes. Graduate Texts in Mathematics, vol 293. Springer, Cham. https://doi.org/10.1007/978-3-031-00943-3_24
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