Definition (Stochastic Realization)

A stochastic realization or sample function is a single outcome of a Stochastic Process, so it is formed by taking a single possible value of each Random Variable of the stochastic process. More precisely, if $\{ X(t,\omega):t\in \mathbb{T} \}$ is a stochastic process, then for any point $\omega \in\Omega$, the mapping $$X(\cdot,\omega):\mathbb{T}\to \mathbb{R}$$is called a realization or (particularly when $\mathbb{T}$ is interpreted as time) a sample path of the process.