the limiting distribution and convergence to equilibrium

Run a well-behaved chain for a long time and a beautiful thing happens: it forgets where it started. The probabilities of being in each state settle down to fixed numbers that no longer depend on the initial state. That settled long-run distribution is the limiting distribution — the answer to 'where is the chain likely to be, far in the future, regardless of today?'.

Precisely, the limiting distribution is the limit of P^(n)(i, j) as n goes to infinity, when that limit exists and does not depend on the start i. Equivalently the rows of P^n all converge to the same row, and pi_n = pi_0 times P^n converges to it from any pi_0. The fundamental limit theorem says: for a chain that is irreducible, aperiodic, and positive recurrent, this limit exists, is the same from every starting state, and equals the unique stationary distribution pi. All three conditions earn their keep — irreducibility ties the states into one system, positive recurrence guarantees an equilibrium exists, aperiodicity stops the chain from oscillating forever.

Convergence is usually geometric: the distance to equilibrium shrinks roughly like the second-largest eigenvalue magnitude raised to the n. The 'mixing time' — how long until the chain is close enough to equilibrium — is set by that eigenvalue gap, and it is the central practical concern when you simulate. The limiting distribution is why PageRank converges, why a shuffled deck becomes uniform, and why MCMC eventually samples from its target.