How to prove that if the bootstrapped sample means are approximately normal, then the sample mean is approximately normal?

Question

I'm considering the scenario described in this question, namely:

• we have IID $X_i, i = 1, ..., n$ sampled from a population with mean $\mu$, variance $\sigma^2$ unknown
• the goal is to put a confidence interval on the sample mean, $\bar{X}_n$
• we believe the population is not normal, say it is trimodal
• set $S_n^2 = \frac{\sum (X_i - \bar{X}_n)^2}{n-1}$ (sample variance)
• if the sample size $n$ is sufficiently large, we know by the CLT that $\bar{X}_n$ is approximately sampled from $N(\mu, S_n/\sqrt{n}$, which provides confidence intervals form the normal distribution.

The previous question was how to know if $n$ is sufficiently large, and the answer was boostrapping to gain a diagnostic:

• for $i = 1, ..., 10,000$, sample $Y_j(i)$ for $j = 1, ..., n$ with replacement from $\{X_1, ..., X_n\}$, and take their sample mean $\bar{Y}(i,n)$.
• use a QQ plot or other method to investigate if the bootstrapped sample means $\{ \bar{Y}(i,n): i = 1, ..., 10,000\}$ is indeed normal.
• If so, then $n$ is large enough to use the confidence intervals from $N(\mu, S_n/\sqrt{n})$ on $\bar{X}_n$.

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Question:

I am trying to understand the math underpinning of using this boostrapping + QQ plot technique as a diagnostic for $n$. I believe the desired theorems for this algorithm is as follows:

1. If $n$ is large enough that the sample mean $\bar{X}_n$ is (approximately) $N(\mu, S_n/n)$, then the bootstrapped sample means $\{ \bar{Y}(i,n): i = 1, ..., 10,000\}$ are (approximately) too.
2. If $n$ is large enough that the bootstrapped sample means $\{ \bar{Y}(i,n): i = 1, ..., 10,000\}$ are (approximately) normal, then sample mean $\bar{X}_n$ is (approximately) from the same normal too.

I believe I can prove 1. via these two facts:

• $Y_j(i)$ are IID samples of $\{X_1, ..., X_n\}$
• $\bar{Y}(i,n) - \bar{X}_n \to 0$ as $n\to \infty$ by CLT for each $i$

How do we prove 2? Here is a start:

• Suppose the bootstrapped sample means $\{ \bar{Y}(i,n): i = 1, ..., 10,000\}$ are (approximately) normal.
• We know from CLT that $\bar{Y}(i,n)$ are approximately $N(\bar{X}_n, S_n/n)$.
• ??

I think the missing piece is (roughly speaking) this statement: "the sample mean's distribution is approximately the same as the bootstrapped sample means' empirical distribution" but that is not a rigorous statement, nor do I know how to prove it. Also, given the proliferation of bootstrapping, I think some more general version holds: "the sample statistic's distribution is approximately the same as the bootstrapped sample statistics' empirical distribution"

I have run a few simulations to verify that the former seems true.

If this is not the correct formulation of a needed math formulation to trust the bootstrapping diagnostic algorithm, what is?