Suppose for , is a uniformly distributed independent random variable.
Am I right in thinking that for any continuous function :
where is the expectation operator? (That is if these integrals are actually defined?) Is there a reference for this?
I imagine it's a straight forward application of the MCT and the LLN, but I'm just a little bit worried by the fact that Brownian motion is often referred to as an integral of white noise, though perhaps more strictly it's an integral with respect to white noise, which is why that's different?
Thanks in advance,
Tom
There are a continuum of independent uniform random variables indexed by . For each such , is the realisation of that random variable, that is to say, a sample from it. (So a value in .)
(If you want to be precise the state space is the set of all functions from to , and each random variable is a functional on this space defined by . If is the realised state then .)
I perhaps should have been clearer in my original message. It's traditional in the field I work in to blur the distinction between random variables and samples from them, for notational convenience.
Ahh I misunderstood your original point. But yeah, standard uniforms are fine (i.e. on ), though it's trivial to see that if my claim holds for any univariate probability distribution admitting a density (not least any uniform), then it holds for all univariate probability distributions which admit a density.
Which brings us back to the original question, is my claim that:
true?
Thanks,
Tom