# Why is one of the two approximations in the bootstrap worse than the other?

My statistics text has the following diagram:

$$\mathbb{V}_F(T_n) \overbrace{\approx}^{ \text{not so small} } \mathbb{V}_{\hat{F}_n}(T_n) \overbrace{\approx}^{ \text{small} } v_{\text{boot}}$$

Where $$T_n$$ is a statistic, $$F$$ is the true CDF and $$\hat{F}_n$$ is the empirical distribution function, $$\mathbb{V}$$ denotes variance, and $$v_{\text{boot}}$$ is the variance of the statistic that we got from bootstrap replications.

Why is one "small" and the other is "not so small"?

The second approximation is obtained by montecarlo simulation. Hence you can make it as small as you wish taking $$B_{boot} \to \infty$$ where $$B_{boot}$$ is the number of bootstrap replications. The first approximation is due to estimating $$F$$ by $$F_n$$. By Glivenko Cantelli theorem this is small if $$n$$ is sufficently large. The problem is that you can not control it unless you can get as many samples as you wish i.e make $$n\to \infty$$.
Note that there are others ways of using bootstrap that do not use sampling from the empirical distribution. In those cases the same explanation still hold if $$\hat F$$ is a consistent estimator of $$F$$.