Geometric L 2 and L 1 convergence are equivalent for reversible Markov chains

By Gareth O. Roberts, Richard L. Tweedie. Journal of Applied Probability, 2001

93 citations recorded by the source corpus.

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

Markov chain decomposition for convergence rate analysis (2002, background): “We note that convergence in the ||·||2 metric implies convergence at the same rate in the “total variation” norm ||µ1 − µ2||Total Variation = sup A⊂Ω |µ1(A) − µ2(A)| (Roberts and Rosenthal (1997), Roberts and Tweedie (2000)).”
To Center or Not to Center: That Is Not the Question—An Ancillarity–Sufficiency Interweaving Strategy (ASIS) for Boosting MCMC Efficiency (2011, unknown): “This is both the L1 and L2 geometric rate, because for time-reversible Markov chains such as the DA algorithm, geometric L1 convergence and L2 convergence are equivalent (see Roberts and Tweedie 2001).”
Convergence rates and asymptotic standard errors for Markov chain Monte Carlo algorithms for Bayesian probit regression (2007, background): “a reversible Markov chain (that satisfies the usual regularity conditions) is geometrically ergodic if and only if the norm of the associated Markov operator is less than 1 (Roberts and Rosenthal, 1997; Roberts and Tweedie, 2001 ).”