This is a stars and bars problem (or, equivalently, an “indistinguishable balls in distinct buckets” problem). For example, the configuration
★|★★★||★ corresponds to sampling once, sampling three times, sampling zero times, and sampling once. In general, there are stars and bars, and hence the total number of configurations is .
First, note that
Therefore, by the tower property, . Next, note that
The above can also be expressed as where is the unbiased sample variance of . Next, note that
where and . Now, recall that for any random variable ,
Therefore, by the tower property,
Applying this to our setting,
As such, we can conclude that
where the asymptotic is in the limit of large .
The distribution of is given in the solution of Question 2 of Chapter 6.
Let be a bootstrap resample. Then,