A stochastic process is a sequence of measurable functions, that is, a random variable X defined on a probability space (Ω, Pr) with values in a space of functions F. The space F in turn consists of functions I → D. Thus a stochastic process can also be regarded as an indexed collection of random variables {Xi}, where the index i ranges through an index set I, defined on the probability space (Ω, Pr) and taking values on the same codomain D (often the real numbers R). This view of a stochastic process as an indexed collection of random variables is the most common one.
A notable special case is where the index set is a discrete set I, often the nonnegative integers {0, 1, 2, 3, ...}.
In a continuous stochastic process the index set is continuous (usually space or time), resulting in an uncountably infinite number of random variables.
Each point in the sample space Ω corresponds to a particular value for each of the random variables and the resulting function (mapping a point in the index set to the value of the random variable attached to it) is known as a realisation of the stochastic process. In the case the index family is a real (finite or infinite) interval, the resulting function is called a sample path.
A particular stochastic process is determined by specifying the joint probability distributions of the various random variables.
Stochastic processes may be defined in higher dimensions by attaching a multivariate random variable to each point in the index set, which is equivalent to using a multidimensional index set. Indeed a multivariate random variable can itself be viewed as a stochastic process with index set {1, ..., n}.