Let $\bar{X}_n$ denote the sample mean of n iid random variables. Let $\bar{X}^*_n$ be the bootstrap sample mean. Does $\left|\mathbb{P}\left(n^{1/2}\bar{X}_n\leq x\right)-\mathbb{P}\left(n^{1/2}\bar{X}^*_n\leq x\right)\right|$ converge to $0$ in probability?
I am asking this question because I know from [Bickel, P.J. and D.A. Freedman (1981), Some asymptotic theory for the bootstrap] that the bootstrap can be used to approximate the distribution of the sample mean in the sense that $\left|\mathbb{P}\left(n^{1/2}\left(\bar{X}_n-\mu\right)\leq x\right)-\mathbb{P}\left(n^{1/2}\left(\bar{X}^*_n-\bar{X}_n\right)\leq x\right)\right|\overset{\mathbb{P}}{\rightarrow}0$ and I was wondering whether the centering is needed. Maybe to be able to use the Central Limit Theorem?
Thanks a lot!