# Lindeberg CLT condition on Discrete Uniform independent sequence of random variables

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:

$$E[X_i]=\frac{i-1}{2}$$

$$\operatorname{Var}[X_i]=\frac{i^2-1}{12}$$

$$B_n^2=\sum_{i=1}^{n}\operatorname{Var}[X_i]=\frac{2n^3+3n^2-5n}{72}\approx\frac{n^3}{36}$$, 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 $$n$$.

## 1 Answer

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.