There is a worked exercise in my book, however there is a line that I am not sure sure about. I understand all of the work before and after this line to finish the proof.

Here is what we are given:

Let $X_i\sim \mathrm{Uniform}(0,1,\ldots,i-1)$, a sequence of independent discrete uniform random variables.

For reference it also uses:



$B_n^2=\sum_{i=1}^{n}\operatorname{Var}[X_i]=\frac{2n^3+3n^2-5n}{72}\approx\frac{n^3}{36}$, noting this approximation seems to be used somewhere.

For clarification Lindebergs condition is as follows in this case:

$\displaystyle\lim_{n\rightarrow\infty}\frac{1}{B_n^2}\sum_{k=1}^{n}E\big[Y_i^2\mathbb{I}_{(|Y_i|>\epsilon\cdot B_n)}\big]=0$ for all $\epsilon >0$ where $Y_i=X_i-\frac{i-1}{2}$

It then goes on to say the Lindeberg condition holds due to the following, stripping several bit's of Lindebergs condition away:

$E\big(|X_i-\frac{i-1}{2}|^2 \mathbb{I}_{\big(|X_i-\frac{i-1}{2}|>\epsilon\cdot B_n\big)}\big)=0$ for all $n\geq\frac{9}{\epsilon^2}$.

This line skipped some steps that I just do not see, and I am hoping to better understand applying Lindebergs CLT off of this example. I particularly am lost on how to deduce this value of $n$.


Observe that the random variable $\left\lvert Y_i\right\rvert$ is always smaller than $i$ (hence than $n$) hence the event $ \left\{\left\lvert Y_i\right\rvert\gt \varepsilon B_n\right\} $ is empty if $n\gt \varepsilon B_n$. Since $B_n$ is of order $n^3$, this certainly happen for $n$ large enough. A ranked depending on $\varepsilon$ where this starts to be true can be made explicit.


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