Suppose I am collecting a set of samples from a finite-support 1D distribution (e.g. a top hat function of some finite width around 0). The distribution is continuous. If the sample size is finite, say n, I expect the largest sample for different repetitions of my measurement to fluctuate according to some distribution. This largest-sample-distribution will have some finite variance V(n).
Now suppose that I increase the sample size n. This should produce a decrease in V(n), as I now have a higher probability for the largest sample to fall within some arbitrary dx from the distribution's upper bound.
First question: how does V(n) fall with increasing n? I would expect it to tend to zero in the n-to-infinity limit. However, not in the same way it would do so for a discrete distribution, due to the different 'orders of infinity' (cardinalities) characterising sample size and sample-space size.
Second question: how does the above compare to the case of an infinite-support distribution, say a 1D Gaussian around 0? Does V(n) still tend to zero in the n-to-infinity limit? In terms of cardinality of the sample-space size there should be no difference, but is it so?