# Ergodic processes and the scientific method

April 22, 2021

## Data-generating processes:

Given a *data-generating process* \(X(t)\), a scientist may only collect a finite number of samples from
this process at any point in time so without loss of generality we may consider the discretised process
as a binary sequence:

\begin{equation} X_N = \{x_n\}_{n=1}^N \end{equation}

\begin{equation} \forall n \in \mathbb{N}, x_n \in \{0,1\}^* \end{equation}

where it is implicitly assumed that the state-space \(X\) is finite, i.e. \(|X| < \infty\), so each state has a binary encoding.

## Discrete-time Markov chains:

A *discrete-time Markov chain* is necessary and sufficient to describe a sequence of observables \(Z_1, Z_2, Z_3,...\)
with state-space \(Z\) that satisfy the Markov-property:

\begin{equation} P(Z_{n+1}=z|Z_1=z_1,…,Z_n=z_n) = P(Z_{n+1}=z|Z_n=z_n) \end{equation}

whenever \(P(Z_1=z_1,...,Z_n=z_n) > 0\).

## Ergodic processes:

A Markov chain with state-space \(X\) is *ergodic* if \(\forall x,y \in X\):

\begin{equation} \sum_{n \in \mathbb{N}} P(Z_{n+1} = x | Z_n = y) = 1 \end{equation}

\begin{equation} \sum_{n \in \mathbb{N}} n \cdot P(Z_{n+1} = x | Z_n = y) < \infty \end{equation}

so all states are recurrent and have finite recurrence time.

It is worth noting that the scientific method is only applicable to ergodic processes as the ergodic assumption is equivalent to the premise that all scientific experiments are repeatable in the natural sciences.

## References:

\(1.\) Steven L. Brunton. Notes on Koopman operator theory. 2019.

\(2.\) Nicolas Privault. Understanding Markov Chains: Examples and Applications. Springer Undergraduate Mathematics Series. 2018.