Okay, actually, this is something where my TA said "and with this example, we can see that probability surpasses analysis"
Let's consider a sequence of independent random variables uniformly distributed over
For any continuous function , and hence bounded (since continuous over a compact set), we have :
where 1 is the indicator function.
This formula is what we call "the transfer formula" (it may be derived from the Radon-Nikodym theorem - okay, it's an analysis theorem xD)
This gives :
And we know from the weak of large numbers (which can be applied here) that :
in probability, and thus in distribution.
But
Convergence in distribution implies that for any bounded & continuous function f, we have :
And this completes the proof.
Sidenotes :
- it's long because I tried to explain it as much as possible
- if you don't like probability, you surely wouldn't like this proof
- it is possible to prove, similarly, that :
where
and f is a continuous function
where
and f is a real valued continuous and bounded function defined over