Let $X_t$ be a random variable indexed by $t \in T$. Think of $t$ as being time. Stochastic processes are sets of random variables of the form

For actual data we can only ever have one observation of each $X_t$, which is why more assumptions are helpful.

### Special Cases

• If $X_t$ is vector valued then this is called a random field.
• If a linear combination of samples from a stochastic process is jointly normal then this is called a Gaussian process, or Gaussian random field for the multivariate case.
• If $T$ is discrete and the value of $X_{t+1}$ only depends on $X_t$ then the process is a discrete time Markov chain.