Definition:Realization of Stochastic Process

Definition
Let $S$ be a stochastic process.

Let $T$ be a time series of observations of $S$ which has been acquired as $S$ evolves, according to the underlying probability distribution of $S$.

Then $T$ is referred to as a realization of $S$.

Thus we can regard the observation $z_t$ at some timestamp $t$, for example $t = 25$, as the realization of a random variable with probability mass function $\map p {z_t}$.

Similarly the observations $z_{t_1}$ and $z_{t_2}$ at times $t_1$ and $t_2$ can be regarded as the realizations of two random variables with joint probability mass function $\map p {z_{t_1} }$ and $\map p {z_{t_2} }$.

Similarly, the observations making an equispaced time series can be described by an $N$-dimensional random variable $\tuple {z_1, z_2, \dotsc, z_N}$ with associated probability mass function $\map p {z_1, z_2, \dotsc, z_N}$.