Discrete-time stochastic process

Mathematical statement

A discrete-time stochastic process (X_n) is a family of random variables on the same probability space. This slice specifies a complete path distribution over a finite horizon to obtain the joint law and derive time marginals and two-time joint distributions. Neither the Markov property nor stationarity follows automatically.

Representative formula (exact TeX)

X(n)

Assumptions

• Exercises list complete support consisting of distinct equal-length finite paths and an exact PMF.
• A finite-horizon joint law does not extend to infinite time, a transition law, stationarity or ergodicity.

Symbol scope

• X — discrete-time stochastic process
• n — discrete time index

Intuition, definition, and misconceptions

A complete path law specifies dependence between times jointly.

A time marginal aggregates path masses at the fixed time, but different joint laws can share the same marginals.

The Markov property requires conditional-distribution obligations and cannot be inferred from marginal tables.

Do not treat a collection of time marginals as a fully specified stochastic process or automatically assert the Markov property.

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

• Finite discrete probability distribution — The complete path support carries an exact PMF.
• Sequence — Natural numbers index discrete time.
• Real random variables on finite sample spaces — Each time variable belongs to the same probability space.

Verification boundary

Only complete finite-horizon joint path laws are handled.

not_formalized

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

Next obligation

Add finite Markov chains, transition matrices and path probabilities.

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.