Modern Theory of Markov Chains

1 (Couplings and monotonicity). A coupling of two or more random variables is nothing but a simultaneous reconstruction of (copies of) the variables in the same probability model. For example, let X and Y are Bernoulli random variables each taking values 0 and 1 with probability 1/2. We can construct a coupling (X̃, Ỹ ) of X and Y by generating a uniform random variable Z over the set {00, 01, 10, 11} and defining X̃ as the first bit of Z and Ỹ as the second bit of Z. Another equally valid coupling is obtained similarly if Z is chosen nonuniformly according to the distribution p with p(00) = p(11) = 1/8 and p(01) = p(10) = 3/8.