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Random Processes & Random Fields

Basics of Probability

Let $\mathbb T$ be an index set, usually a subset of $\mathbb R^N$ for some $N$. Then a collection of random variables $\{X_t\}_{t \in \mathbb T}$ on the same probability space $(\Omega, \mathcal H, \mathbb P)$ is a random process. From another perspective, for each $\omega \in \Omega$ we have $t \rightarrow X_t(\omega)$ is a random function, or trajectory of the process. We will talk about how we may evaluate probabilities for trajectories.

When $\mathbb T$ is a subset of $\mathbb R$, usually $\mathbb R$, $[0,\infty)$ or $\mathbb Z$ or $\mathbb N$, then we often think of $\mathbb T$ as a set of times. When we do this, we may call $(X_t)$ a temporal process.

On the other hand, when $\mathbb T \subseteq \mathbb R^N$ (for $N > 1$) then we often think of $\mathbb T$ as a set of spatial positions. In this situation $(X_t)$ is called a spatial process or a random field. We will talk about a special type of spatial process, point processes, where the $t$ represent the positions of particles in a subsequent section.

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