Finite-state Markov chain

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

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 — A Markov chain is a process with a conditional-independence structure.
• Finite matrices: dimensions, indexing, and transpose — A finite transition kernel is a matrix in fixed state order.
• Conditional probability in finite models — 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.