  The purpose of this paper is to introduce a new Markov chain Monte Carlo
method and exhibit its efficiency by simulation and high-dimensional asymptotic
theory. Key fact is that our algorithm has a reversible proposal transition
kernel, which is designed to have a heavy-tailed invariant probability
distribution. The high-dimensional asymptotic theory is studied for a class of
heavy-tailed target probability distribution. As the number of dimension of the
state space goes to infinity, we will show that our algorithm has a much better
convergence rate than that of the preconditioned Crank Nicolson (pCN) algorithm
and the random-walk Metropolis (RWM) algorithm. We also show that our algorithm
is at least as good as the pCN algorithm and better than the RWM algorithm for
light-tailed target probability distribution.
