Let be a random variable indexed by . Think of 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 , which is why more assumptions are helpful.

Special Cases

  • If 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 is discrete and the value of only depends on then the process is a discrete time Markov chain.