I have been looking into bootstrapping lately and although I believe to have understood the basic process somewhat, I am fuzzy on the mathematical details. I will begin with my understanding of what bootstrapping is and then my understanding of the mathematics going on in the background. I might very well be mistaken on either one.

**Bootstrapping** (as I understand it):

The idea behind (most) bootstrapping is to study a random variable $X$ and its (unknown) distribution by resampling a sample $x=(x_1, ..., x_n)$ of $X$. Meaning, we choose randomly $n$ times an entry of $x$ to create a new sample $x^{(1)}=(x_1^{(1)}, ..., x_n^{(1)})$. This way, we create a bunch of samples $(x^{(i)})_i$. For each of these samples, we can now compute, for example, the mean and plot these means into a histogram. This histogram can be normalized to give us a distribution of the mean across all new samples. This process could of course be applied to any other statistic we might want to compute for any of the samples, giving us some histogram each time.

**Mathematical model** (as I understand it):

Let $X$ be a random variable with unknown distribution $f_X$. We have a sample $x=(x_1, ..., x_n)$ of $X$. This sample defines a (discrete) empirical distribution $f_x$. We can now define $n$ independent variables $Y_1, ..., Y_n \sim f_x$. Calculating the mean of a resample of $x$ would now be equivalent to sampling the random variable $X' = \frac{1}{n}(Y_1 + ... + Y_n)$. Sampling $X'$ a lot, producing and normalizing a histogram would tend towards the distribution of $\bar X = \frac{1}{m}(X'_1 + ... + X'_m)$ as $m$ tends to infinity where $X'_1, ..., X'_m$ are independent and distributed as $X'$. The CLT now tells us that this would be a Gaussian distribution.

Barring any mistakes I have made here, I have two questions:

- Am I correct here? Does the histogram of the mean of bootstrapped samples really resemble a Gaussian distribution? (In the sense that we can normalize the histogram and take the limit.)
- What if we replace the mean by any function $g$ on $X_1, ..., X_n$ that produces different $X'$, e.g. $g=\min$? Do we still get something Gaussian? My instinct says no but I cannot see why the CLT would not work the same.