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,