# Variance largest sample from finite-support vs infinite-support distribution

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?

For a uniform distribution on $[0,1]$, the maximum of a sample size $n$ has a Beta distribution with parameters $\alpha=n$ and $\beta=1$, an expectation of $\dfrac{n}{n+1}$ and a variance of $\dfrac{n}{(n+1)^2(n+2)}$ which as a function of $n$ decreases towards zero as $n$ increases, in line with what you seem to expect. If the range of the uniform distribution is $k$ rather than $1$ then this multiplies the variance by $k^2$ but you get the same reducing effect as the sample size increases.
For a standard Gaussian distribution, the variance of the sample maximum seems to tend towards zero as sample size increases, but much more slowly than with a uniform distribution (roughly proportionate to $\frac{1}{\log n}$ rather than to $\frac{1}{n^2}$). For an exponential distribution with rate parameter $1$, the variance of the sample maximum seems to increase as sample size increases, to a finite limit of $\frac{\pi^2}{6}$ as suggested by the Gumbel distribution. For a Pareto power law distribution (with parameter greater than $2$) the variance of the sample maximum seems to increase without limit as sample size increases.