Markov Chain Stationary Explorer

Markov chains rigidly model strictly memoryless probabilistic transitions over discrete time. As the transition matrix is repeatedly raised to infinite powers, the system relentlessly converges onto a unique stationary eigenvalue vector, obliterating all initial conditions and exposing the pure long-term stochastic equilibrium.

Each row of the transition matrix is just a set of odds: given you are in one state, how likely you jump to each other one on the next step. Multiply your current probability vector by P once per step and the numbers drift toward a fixed mix, the stationary distribution π. That mix tells you the fraction of time the system spends in each state over the long haul — try the weather sample and watch p(k) settle.

Set the two-state matrix to [[0,1],[1,0]] — a coin that always flips. It has a perfectly valid, unique stationary distribution π = (0.5, 0.5), yet start at (1, 0) and p(k) bounces 1,0 → 0,1 → 1,0 forever, never converging. Such periodic chains are exactly why the explorer can report converged: no; guaranteed convergence needs an aperiodic chain, not just a unique π.