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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?

kjetil b halvorsen
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Alec
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    You would be interested in a related thread about this question. – whuber Jan 16 '12 at 20:14
  • Indeed, a random variable maps events from a sample space into either a density or a probability which are even more constrained than just "real numbers". Can we relax some of the conventional assumptions about probability and densities, such that they are non-negative and that they "sum up" to 1, that they're real valued (as opposed to complex valued)? Sure, that's all of functional analysis, but instead of a probability space it's a measure space, and instead of "random variable" it's a function. – AdamO Jun 15 '23 at 19:47
  • Of course, you when you lose that, you lose essential probabilistic theorems, like De Finetti, Markov Inequality, etc. The assumptions are there because they're really useful for solving probability problems. – AdamO Jun 15 '23 at 19:50

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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.

Elvis
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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.

Daniel Johnson
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