This is a cross-post from stats.stackexchange.com. No answer has appeared there. Since this is a theoretical question, mathoverflow.net seems to be a more appropriate venue for it.

What is the analog of the central limit theorem or concentration theorem for resampling, say, an i.i.d. samples? Are there any references for this topic?

Here is a simple example. Suppose there are $n$ i.i.d. random variables $\{x_1,x_2,\cdots,x_n\}$ with mean $0$ and standard deviation $1$. We sample uniformly randomly with replacement from this set $n$ times and obtain random variables ${y_1,y_2,\cdots,y_n}$. What is the distribution of the mean $\displaystyle y=\frac1n\sum_{i=1}^ny_i$ as $n\to\infty$?

This is a cross-post from stats.stackexchange.com. No answer has appeared there. Since this is a theoretical question, mathoverflow.net seems to be a more appropriate venue for it.

What is the analog of the central limit theorem or concentration theorem for resampling, say, an i.i.d. samples? Are there any references for this topic?

Here is a simple example. Suppose there are $n$ i.i.d. random variables $\{x_1,x_2,\cdots,x_n\}$ with mean $0$ and standard deviation $1$. We sample uniformly randomly with replacement from this set $n$ times and obtain random variables ${y_1,y_2,\cdots,y_n}$. What is the distribution of the mean $\displaystyle y=\frac1{\sqrt n}\sum_{i=1}^ny_i$ as $n\to\infty$?

This is a cross-post from stats.stackexchange.com. There has not been an answer there. Since this is a theoretical question, mathoverflow.net seems to be a more appropriate venue for it.

What is the analog of the central limit theorem or concentration theorem for resampling, say, an i.i.d. samples? Are there any references for this topic?

Here is a simple example. Suppose there are $n$ i.i.d. random variables $\{x_1,x_2,\cdots,x_n\}$ with mean $0$ and standard deviation $1$. We sample uniformly randomly with replacement from this set $n$ times and obtain random variables ${y_1,y_2,\cdots,y_n}$. What is the distribution of the mean $\displaystyle y=\frac1n\sum_{i=1}^ny_i$ as $n\to\infty$?

This is a cross-post from stats.stackexchange.com. No answer has appeared there. Since this is a theoretical question, mathoverflow.net seems to be a more appropriate venue for it.

What is the analog of the central limit theorem or concentration theorem for resampling, say, an i.i.d. samples? Are there any references for this topic?

Here is a simple example. Suppose there are $n$ i.i.d. random variables $\{x_1,x_2,\cdots,x_n\}$ with mean $0$ and standard deviation $1$. We sample uniformly randomly with replacement from this set $n$ times and obtain random variables ${y_1,y_2,\cdots,y_n}$. What is the distribution of the mean $\displaystyle y=\frac1n\sum_{i=1}^ny_i$ as $n\to\infty$?