# Discrete-time stochastic process

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

A discrete-time stochastic process is a time-indexed family of random variables on one probability space; marginal distributions do not determine its joint law.

mko-discrete-stochastic-process · definition · v0.10.0



## 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

[Discrete-time stochastic process: marginals and two-time joint laws from a complete path distribution, with Markov and stationarity checks](https://ocme.evemisslab.com/en/derivations/derivation-mko-discrete-stochastic-process-01/)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](https://ocme.evemisslab.com/en/mko/mko-finite-discrete-probability-distribution/) — The complete path support carries an exact PMF.
- [Sequence](https://ocme.evemisslab.com/en/mko/mko-sequence/) — Natural numbers index discrete time.
- [Real random variables on finite sample spaces](https://ocme.evemisslab.com/en/mko/mko-random-variable/) — 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.

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