◄ WORLD V · SONNY 5DART 199 · a helldive at the net

THE STATIONARY DISTRIBUTION the eigenvector every start flows into

The stationary distribution π is the one the chain stops changing: π = πP. For a connected, aperiodic chain it is unique, and every start converges to it — it is the Perron–Frobenius eigenvector of the transition matrix, the one with eigenvalue exactly 1. You find it not by algebra but by pushing any distribution through P over and over until it stops moving.

THE TECHNIQUE π = πP, the dominant eigenvector; found by power iteration

Iterate an arbitrary start through P and watch the residual |πP − π| collapse to zero — the fixed point found by pure repetition, no matrix inverse: live demo







  
HISTORY & CREDIT Perron 1907 · Frobenius 1912

  
“You solve πP = π with linear algebra.” — you can, but the honest engine is iteration. A stochastic matrix has dominant eigenvalue exactly 1 (Perron–Frobenius), so repeatedly applying P drives any start onto π geometrically. No pivoting, no inverse — just multiply and wait. cited

  
1907–12 · Oskar Perron (1907) and Georg Frobenius (1912) — a non-negative matrix has a real, largest eigenvalue with a non-negative eigenvector; for a stochastic matrix that eigenvalue is 1.
1906 · Markov's limit theorem — an irreducible, aperiodic finite chain converges to a unique π from any start (his own 1906 result); Kolmogorov (1936) extends it to countably-infinite state spaces.
now · the power method is how π is actually computed at scale — the same loop PageRank runs.

  
The stationary distribution is computed, not stored: nowhere is π written down: it is the place iteration lands. Push any distribution through P enough times and it becomes π. Perron–Frobenius






  
RECOMMEND FOR I-13 power iteration to the fixed point, computed

  
On the canonical compiler, iterating the weather chain from a pure sunny start lands exactly on the algebraic answer π = (5/6, 1/6), with probability mass conserved:

  
$ i13 run stationary.i13   # power-iterate P 60 times from [1,0]
pi0 = 0.8333333333333345    -- 5/6
pi1 = 0.1666666666666669    -- 1/6
masssum = 1.0000000000000013   -- probability conserved to a few ulp

  
Recommend: stationary distribution is LIT for I-13 — verified the power method converges to (0.83333, 0.16667), the exact (5/6, 1/6), with total mass 1.0000000000000013 (that last drift is the honest Kahan remainder). A stochastic matrix, a start vector, and one loop — no linear solver required.