---
tag: 0QH5ENKZM1
kind: concept
name: Stochastic process
aliases: [random-process]
msc: [60G05]
uses: [0DKRR22AEQ, 0ZYBZ34RT2]
sources:
  - url: https://en.wikipedia.org/wiki/Stochastic_process
    locator: Stochastic process
same_as:
  wikidata: Q176737
  wikipedia: Stochastic process
---

## Summary

A stochastic process is a family $(X_t)_{t \in T}$ of [[random variables|0DKRR22AEQ]] on one [[probability space|0ZYBZ34RT2]] $(\Omega, \mathcal{F}, P)$, all with values in the same measurable space $(S, \mathcal{S})$.
The set $T$ is the index set, often a set of times such as $\{0, 1, 2, \dots\}$ or $[0, \infty)$. The set $S$ is the state space.
For each outcome $\omega \in \Omega$, the function $t \mapsto X_t(\omega)$ is a sample path of the process.

## Remarks

The process is in discrete time if $T$ is countable, such as $\{0, 1, 2, \dots\}$, and in continuous time if $T$ is an interval.
The finite-dimensional distributions are the laws of the vectors $(X_{t_1}, \dots, X_{t_n})$ for finite sets $\{t_1, \dots, t_n\} \subseteq T$. The Kolmogorov extension theorem builds a process from a consistent family of such laws.
Important examples are the Bernoulli process, random walks, Markov chains, the Poisson process and the Wiener process.

## Explanations

### L1 human

A stochastic process is a random quantity that changes with time.
Examples are the price of a share day by day and the number of customers in a queue.
At each time it is a random variable, and one run of the whole process gives a random path.