Suppose that for each $n\geq1$ there is a given sequence of independent random variables $\xi_{n1},\xi_{n2},\ldots,\xi_{nn}$ with $E\xi_{nk}=0$, $V\xi_{nk}=\sigma_{nk}^2$, $\sum_{k=1}^n\sigma_{nk}^2=1$. Let $S_n=\xi_{n1}+\ldots+\xi_{nn},$ $F_{nk}(x)=P\{\xi_{nk}\leq x\},\ \Phi(x)=(2\pi)^{-1/2}\int_{-\infty}^x e^{-y^2/2}dy,\ \Phi_{nk}(x)=\Phi\left(\frac{x}{\sigma_{nk}}\right)$ On page 339 there is an inequality: $\sum_{k=1}^n\left| t\int_{-\infty}^{\infty}(e^{itx}-1-itx)(F_{nk}(x)-\Phi_{nk}(x))dx\right|\leq\frac{|t|^3}{2}\varepsilon\sum_{k=1}^n\int_{|x|\leq\varepsilon}|x||F_{nk}(x)-\Phi_{nk}(x)|dx+2t^2\sum_{k=1}^n\int_{|x|>\varepsilon}|x||F_{nk}(x)-\Phi_{nk}(x)|dx$ How can I get it?
Then the book says we can use $E|\xi|^n=\int_{-\infty}^{\infty}|x|^ndF(x)=n\int_0^\infty x^{n-1}[1-F(x)+F(-x)]dx$ to prove $\int_{|x|\leq\varepsilon}|x||F_{nk}(x)-\Phi_{nk}(x)|dx\leq2\sigma_{nk}^2$ But I can't figure why.
Thank you!