Proof for rate of convergence in CLT

Question

I am struggling to understand the following proof in this post. Indeed, to prove the claim i write that \begin{align*} | \mathbb{E}[Y_n] - \mathbb{E}[|W|] | & = |\mathbb{E}[Y_n\mathbb{1}\{Y_n < R\} + \mathbb{E}[Y_n\mathbb{1}\{Y_n > R\}] - \mathbb{E}[|W|\mathbb{1}\{|W| < R\}] - \mathbb{E}[|W|\mathbb{1}\{|W| > R\}] | \\ & \leq |\mathbb{E}[Y_n\mathbb{1}\{Y_n > R\} - \mathbb{E}[|W|\mathbb{1}\{|W| > R\}| + |\mathbb{E}[Y_n\mathbb{1}\{Y_n < R\} - \mathbb{E}[|W|\mathbb{1}\{|W| < R\}| \\ & \leq 2R^{-1} + |\mathbb{E}[Y_n\mathbb{1}\{Y_n < R\} - \mathbb{E}[|W|\mathbb{1}\{|W| < R\}| \end{align*}

Here, i'm not sure how to apply Berry-Esseen. Indeed, using the expectation identity we get that $$ \mathbb{E}[Y_n\mathbb{1}\{Y_n < R\}] = \int_{0}^R \mathbb{P}(Y_n \mathbb{1}\{Y_n < R\} > t) \mathrm{d}t$$ and similarly for |W|. But this is not $\mathbb{P}(Y_n > t)$ so how can i apply the theorem?

Also I do not know why the theorem holds for $Y_n$ and $|W|$.

Thank you in advance.

Answer 1

One thing you can notice is that for all $t\ge 0$,

$$\mathsf{P}\left(Y_n\mathbf{1}_{Y_n\le R}>t\right)=\mathsf{P}\left(t < Y_n \le R\right)=\mathsf{P}(Y_n\le R)-\mathsf{P}(Y_n\le t)$$

and similarly for $|W|$. Also, from the "expectation identity",

\begin{align*} \left|\mathsf{E} \left[Y_n\mathbf{1}_{Y_n\le R}\right]-\mathsf{E} \left[|W|\mathbf{1}_{|W|<R}\right]\right|&\le\int_0^\infty \left|\mathsf{P}\left(Y_n\mathbf{1}_{Y_n\le R}>t\right)-\mathsf{P}\left(|W|\mathbf{1}_{|W|\le R}>t\right)\right|dt\\ &=\int_0^R \left|\mathsf{P}\left(Y_n\mathbf{1}_{Y_n\le R}>t\right)-\mathsf{P}\left(|W|\mathbf{1}_{|W|\le R}>t\right)\right|dt \end{align*}

since when $t\ge R$, both probabilities are zero. Combining our observations and the triangle inequality,

\begin{align*} \left|\mathsf{E} \left[Y_n\mathbf{1}_{Y_n\le R}\right]-\mathsf{E} \left[|W|\mathbf{1}_{|W|\le R}\right]\right|&\le\int_0^R|\mathsf{P}(Y_n\le R)-\mathsf{P}(|W|\le R)|+|\mathsf{P}(Y_n\le t)-\mathsf{P}(|W|\le t)| \, dt. \end{align*}

Now how do we apply Berry-Esseen theorem? Note that

$$\mathsf{P}(Y_n\le z)=\mathsf{P}\left(\frac{1}{\sqrt{n}}\sum_{i=1}^n Z_i\le z\right)-\mathsf{P}\left(\frac{1}{\sqrt{n}}\sum_{i=1}^n Z_i<-z\right)$$

and

$$\mathsf{P}(|W|\le z)=\mathsf{P}(W\le z)-\mathsf{P}(W<-z)$$

so that by the triangle inequality,

$$|\mathsf{P}(Y_n\le z)-\mathsf{P}(|W|\le z)|\le \left|\mathsf{P}\left(\frac{1}{\sqrt{n}}\sum_{i=1}^n Z_i\le z\right)-\mathsf{P}(W\le z)\right|+\left|\mathsf{P}\left(\frac{1}{\sqrt{n}}\sum_{i=1}^n Z_i< -z\right)-\mathsf{P}(W< -z)\right|.$$

You should be able to conclude from here.

Answer 2

What @DavideGiraudo did was to consider $$\begin{align} \Big|E[|Y_n|]-E[|W|]\Big|&=\Big|\int^\infty_0 P[Y_n>t]-P[|W|>t)\,dt\Big|\leq \int^\infty_0|P[Y_n>t]-P[|W|>t)|\,dt\\ &=\int^R_0|P[Y_n>t]-P[|W|>t)|\,dt +\int^\infty_R|P[Y_n>t]-P[|W|>t)|\,dt\\ &\leq \frac{CR}{\sqrt{n}} +\int^\infty_R|P[Y_n>t]-P[|W|>t)|\,dt \end{align}$$ On the other hand, $$\int^\infty_R P[|Y_n|>t]\,dt\leq \int Y_n\mathbb{1}_{\{Y_n>R\}}\,dP= E[Y_n\mathbb{1}_{\{Y_n>R\}}]\leq\frac1R$$ Similar for $\int^\infty_RP[|W|>t]\,dt$.