# Trouble understanding the Lindeberg-Feller Central Limit Theorem

I am having trouble understanding the notation in the following Lindeberg-Feller CLT. The notation isn't defined anywhere else in the text so I would like to clarify a few things.

Firstly, what is $y_{ni}$? Is $i$ indexing the observation? E.g., if there are 5 observations in the sample, then we have $y_{n1}, \cdots, y_{n5}$? What does the subscript $n$ mean? How is the sample average $\overline{y}$ defined? Is it $\frac{1}{m}\sum_{i=1}^{m} y_{ni}$ where $m$ is the total number of observations?

• $\overline y = \frac1n\sum_{i=1}^n y_{ni}$. For every $n\geq 1$ you have $n$ independent random variables $y_{n1},\ldots,y_{nn}$. And $\overline y$ depends on $n$. – NCh Jan 28 '18 at 5:01
• Thanks, but what is the point of indexing $y$ with $n$? Asymptotically, we only care about what happens when the sample size $n \rightarrow \infty$, so isn't indexing by $i$ enough? – elbarto Jan 28 '18 at 5:21
• This is more general settings: we have one observation for $n=1$, two - for $n=2$, and each series observations are independent, but vary vith $n$. Say, look at math.stackexchange.com/q/2601463/413376 Here the summands in CLT are $Y_{ni}=\frac{(n+1-i)X_i}{n^{3/2}}$, where $X_i$ are i.i.d.r.v. And CLT holds for this summands. – NCh Jan 28 '18 at 6:38
• Thanks @NCh, makes a lot of sense now! I can see that it is just a more general version of the version I saw on wikipedia: en.wikipedia.org/wiki/Central_limit_theorem#Lindeberg_CLT – elbarto Jan 28 '18 at 6:48