◄ WORLD V · SONNY 5DART 198 · a helldive at the net

THE MARKOV CHAIN the next step forgets everything but now

A Markov chain is a walk over states where the next state depends only on the current one — not the whole history before it. Encode it as a transition matrix P: row i lists the probabilities of where state i goes next. The distribution after n steps is v·Pⁿ, and for a well-connected chain every starting point drifts toward the same long-run distribution — the chain forgets where it began.

THE TECHNIQUE next state depends only on now; vₙ₊₁ = vₙ · P

Evolve a two-state weather chain from a chosen start and watch the distribution converge — the memory of the start washing out one step at a time: live demo







  
HISTORY & CREDIT Andrey Markov, 1906

  
“A process with a fixed rule must depend on its whole past.” — the opposite is the defining Markov property: given the present, the future is independent of the past. Markov built the idea precisely to prove the law of large numbers for dependent variables — against Nekrasov's claim that independence was required for it. cited

  
1906 · Andrey Markov — introduces chains of dependent trials to extend the law of large numbers beyond independence, answering Pavel Nekrasov.
1913 · Markov counts the vowels and consonants in 20,000 letters of Pushkin's Eugene Onegin — the first Markov chain fitted to real data.
1933 · Kolmogorov — puts probability, and Markov processes, on an axiomatic footing.

  
Every ergodic chain forgets its start: run it long enough and the distribution is the stationary one, whatever you began with. That washing-out is the engine under every dart in this batch. Markov 1906






  
RECOMMEND FOR I-13 n-step distribution converges, computed

  
On the canonical compiler the distribution of a 2-state chain, started fully in sunny, marches step by step toward its stationary value:

  
$ i13 run markov.i13   # weather P=[[.9,.1],[.5,.5]], start = sunny [1,0]
P(sunny) after 1 step = 0.9
P(sunny) after 2 steps = 0.8600000000000001
P(sunny) after 5 steps = 0.8350400000000002   -- drifting to 5/6 = 0.8333..., start forgotten

  
Recommend: the Markov chain is LIT and native to I-13 — a transition matrix is a flat f64 array, a step is one matrix–vector multiply, and recursion is the clock. Verified: from a pure sunny start the chain gives 0.9 → 0.86 → 0.835 over five steps, converging to the stationary 5/6. The whole batch stands on this one primitive.