Coupling and coalescence bounds

A coupling of μ and ν is any joint pair U,V whose individual laws are μ and ν. Their dependence is free to choose. For every event A, its two membership indicators can differ only when U and V differ, giving total variation at most the mismatch probability. On finite or countable spaces, a maximal coupling achieves equality by matching the common mass at each point. That static result does not automatically produce an efficient time-consistent coupling of entire Markov paths, so a path construction still needs to be specified.

A Markovian coupling chooses a transition law for the pair of states such that each coordinate’s conditional next-state marginal is the original kernel from its own present state. Sharing random choices is one way to arrange this, but using identical numerical noise without checking the resulting marginals can be wrong. The second chain may be started from π, making its marginal stationary at every time. A bound uniform over the first starting state then gives a worst-case mixing guarantee. Independence between the copies is neither required nor usually desirable.

Define T as the first time the copies agree. To use its tail directly, construct the pair so that once agreement occurs, both copies take the same subsequent random transition. Then disagreement at time t is exactly the event T exceeds t. Merely running independent copies allows them to meet and later separate, so their first-meeting tail need not bound later mismatch. A standard repair is to use any valid pair update while states differ and switch to an identical draw from their common row whenever they coincide.

For a finite irreducible aperiodic chain, suitable couplings can establish convergence, but their quantitative efficiency can vary greatly. A poor coupling may wait much longer than the true mixing time. The coupling inequality is an upper bound, and a large upper bound does not prove slow mixing. Lower bounds instead need a distinguishable event, a slow eigenfunction or another obstruction. When the coupling starts one coordinate in equilibrium, random initial agreement can also improve the tail beyond a simpler bound that assumes the two copies initially differ.

Consider a refresh kernel that holds its present state with probability one minus α and otherwise draws a new state from a fixed probability π, where α is positive and at most one. Couple two copies using the same refresh indicator and, at a refresh, the same new π-draw. Both marginals are correct and the first refresh makes them identical forever. The probability of no refresh in t transitions is (1−α)ᵗ, furnishing a uniform upper bound. The draw is a complete refresh of the state, not one coordinate of a larger configuration.

This special kernel also has an exact formula: before any refresh it remembers the initial point, while after a refresh its law is π. Its TV distance from a point x is therefore (1−π(x))(1−α)ᵗ. The missing prefactor explains why the no-refresh bound can be loose. More elaborate couplings use monotonicity, path metrics or contraction of expected distance, but their conclusions still rest on the same requirements of correct coordinate laws and a demonstrated route from the coupling quantity to probability of disagreement.