Random variables mapping to a state space that isn't a set of real numbers?

Question

A random variable usually has a definition like "a function that maps from the original sample space to a new sample space, typically a set of real numbers". Is it ever useful to map into a new space that isn't a set of real numbers?

Answer 1

As Xi'an says in his comment, you can define random variables taking values in function spaces: you can "draw a function at random" among a set of functions. You could also imagine drawing a functional (a transformation which is applied to functions) at random...

You can also, for example, define variables taking values in p-adic fields. A google search on "p-adic random variable" will gives you lots of answers.

Answer 2

Though they could be viewed as an entirely separate construct from the random variable, random measures map into a set of measures. However, the measures are generally parameterized by real valued random variables, which kind of make this a somewhat unsatisfying answer to your question.

Also, one might wish to pick a random colour or random word or something similar, but, in these cases, one can just parameterize the space you are mapping to by real numbers and use random variables under the standard definition.