This talk will discuss the Diaconis-Holmes-Neal Markov chain sampler. This Markov chain sampler is a modification of the Metropolis algorithm based on the nearest neighbor random walk and involves a duplication of the state space. Both the Metropolis algorithm and this modification involve a desired stationary probability, and transition probabilities involve the ratio of the stationary probabilities at neighboring states. For many choices of the stationary probability, the Diaconis-Holmes-Neal Markov chain sampler converges (with appropriate choices of a parameter) to the stationary distribution faster than the corresponding Metropolis algorithm. This talk will discuss the proof of rates of convergence of the Diaconis-Holmes-Neal Markov chain sampler for certain stationary probabilities such as a V-shaped stationary probability.