The waste-recycling Monte Carlo (WR) algorithm, introduced by Frenkel,
  is a  modification of  the Metropolis-Hastings algorithm,  which makes
  use  of all  the proposals,  whereas the  standard Metropolis-Hastings
  algorithm only  uses the accepted proposals. We  prove the convergence
  of the WR  algorithm and its asymptotic normality.  We give an example
  which shows  that in  general the WR  algorithm is  not asymptotically
  better than  the Metropolis-Hastings algorithm : the  WR algorithm can
  have   an   asymptotic  variance   larger   than   the   one  of   the
  Metropolis-Hastings algorithm.  However, in the particular case of the
  Metropolis-Hastings  algorithm called  Boltzmann  algorithm, we  prove
  that   the   WR   algorithm   is  asymptotically   better   than   the
  Metropolis-Hastings algorithm.