7. Consider a random variable Z with values on a countable set X and p.m.f P(Z=i)i for all iX. Let (Xn) be an ergodic Markov chain with state space X and transition matrix P=(pij). Define the matrix Qij={pijmin{1,ipijjpji}1j=iQiji=ji=j Show that Q is the transition matrix of an ergodic Markov chain that is time reversible in equilibrium and whose stationary distribution is the distribution of random variable Z. Comment on this method. In the above formula, it appears the function g1(z)=min{1,z} with z=ipijjpji If you feel adventurous, verify that g1(z) is not the only possible function g:[0,)[0,1] of z that can be used (with the same z defined above) so that Q is the transition matrix for a time reversible chain with stationary distribution equal to the pmf of Z. What constraints should such functions satisfy? 7. Consider a random variable Z with values on a countable set X and p.m.f P(Z=i)i for all iX. Let (Xn) be an ergodic Markov chain with state space X and transition matrix P=(pij). Define the matrix Qij={pijmin{1,ipijjpji}1j=iQiji=ji=j Show that Q is the transition matrix of an ergodic Markov chain that is time reversible in equilibrium and whose stationary distribution is the distribution of random variable Z. Comment on this method. In the above formula, it appears the function g1(z)=min{1,z} with z=ipijjpji If you feel adventurous, verify that g1(z) is not the only possible function g:[0,)[0,1] of z that can be used (with the same z defined above) so that Q is the transition matrix for a time reversible chain with stationary distribution equal to the pmf of Z. What constraints should such functions satisfy