Notes on: Rosenthal, J. S. (1995): Minorization conditions and convergence rates for markov chain monte carlo

Table of Contents

Overview

  • Provides a method for proving rigorous, a priori bounds on the number of iterations required until satisfactory convergence has taken place

Notation

  • rosenthal1995_e121a8b2e3bfed20acbb01573e58ba21593ad9f7.png is the MC transition kernel on a state space rosenthal1995_76879b948635123ecd29d7cd65a1060145ba040e.png

Definitions

We say the transition kernel rosenthal1995_e121a8b2e3bfed20acbb01573e58ba21593ad9f7.png on the state space rosenthal1995_76879b948635123ecd29d7cd65a1060145ba040e.png satisfy a minorization condition or split on a subsets rosenthal1995_8d44794d2de4fe914bbd6953a677be3248881801.png if there is a probability measure rosenthal1995_bee237ab85f41f1d678f7537a5cf1cd84c4713e2.png on rosenthal1995_76879b948635123ecd29d7cd65a1060145ba040e.png, a positive integer rosenthal1995_6aca397638a8f980a1673c1172a5f075508b95ef.png, and rosenthal1995_95253ea0c2082f08ac36eed736be50b9577033e8.png, such that

rosenthal1995_51796300403ecf8304c0af48e8143d5c6cb9b43b.png

for all measureable subsets rosenthal1995_7ca4de3a4584cf5fcc7e2a982f51d46e27615028.png.