# Finite-state Markov chain

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

A finite time-homogeneous Markov chain uses a row-stochastic transition kernel and, when needed, an initial law to specify finite path probabilities.

mko-finite-markov-chain · definition · v0.10.0



## Mathematical statement

Fix the state order and write P_ij=Pr(X_(n+1)=j | X_n=i). Every row is nonnegative and sums exactly to 1. This slice uses a single finite time-homogeneous transition matrix; time marginals and path probabilities additionally require an initial distribution.





## Representative formula (exact TeX)

P(X)





## Assumptions


- All probabilities are exact rationals, with rows=current state and columns=next state.
- Only finite-step outcomes are computed from the kernel and initial law; stationarity, irreducibility, ergodicity and long-term convergence are not asserted.





## Symbol scope


- P — transition kernel
- X — finite Markov chain





## Intuition, definition, and misconceptions

Row-stochastic qualification precedes every stepping computation.

The one-step law uses the row-vector convention mu*P.

A finite path probability multiplies the initial-state mass by each successive transition factor.

Do not substitute column sums for row sums or claim time distributions and path probabilities without an initial law.





## Adopted derivation guide

[Finite-state Markov chain: checking a row-stochastic matrix and computing two-step time laws and path probabilities](https://ocme.evemisslab.com/en/derivations/derivation-mko-finite-markov-chain-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Discrete-time stochastic process](https://ocme.evemisslab.com/en/mko/mko-discrete-stochastic-process/) — A Markov chain is a process with a conditional-independence structure.
- [Finite matrices: dimensions, indexing, and transpose](https://ocme.evemisslab.com/en/mko/mko-matrix/) — A finite transition kernel is a matrix in fixed state order.
- [Conditional probability in finite models](https://ocme.evemisslab.com/en/mko/mko-conditional-probability/) — The Markov property is stated through conditional distributions given the present state.





## Verification boundary

A single finite-kernel model proves neither stationarity, ergodicity nor convergence.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add state classification, stationary distributions and first-hitting times.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/finite-markov-chain.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-finite-markov-chain/index.md)