I have to solve $$ n \int_{\mathbb{R}}{\left|\frac{1}{n}\sum_{j=1}^n{e^{(itY_j)}}-e^{-\frac{1}{2}t^2}\right|^2}\psi(t)dt $$ where $\psi(t)=(2\pi)^{-\frac{1}{2}}e^{-\frac{1}{2}t^2}$ is the density function of the standard normal distribution.
I tried to do it like this:
$$ T_n= n \int_{\mathbb{R}}{\left|\hat{\varphi_n}(t)-e^{-\frac{1}{2}t^2}\right|^2}\psi(t)dt\\ = n \int_{\mathbb{R}}{\left|\frac{1}{n}\sum_{j=1}^n{e^{(itY_j)}}-e^{-\frac{1}{2}t^2}\right|^2}\psi(t)dt\\ = n \int_{\mathbb{R}}{\left|\frac{1}{n}\sum_{j=1}^n{\cos{(tY_j)}}+i\frac{1}{n}\sum_{j=1}^n{\sin{(tY_j)}}-e^{-\frac{1}{2}t^2}\right|^2}\psi(t)dt\\ = n \int_{\mathbb{R}}{\left[\left(\frac{1}{n}\sum_{j=1}^n{\cos{(tY_j)}}-e^{-\frac{1}{2}t^2}\right)^2+\left(\frac{1}{n}\sum_{j=1}^n{\sin{(tY_j)}}\right)^2\right]}\psi(t)dt\\ = n \int_{\mathbb{R}}\Bigg[\frac{1}{n^2}\sum_{j=1}^n{\sum_{k=1}^n{\cos{(tY_j)}\cos{tY_k}}}-2\cdot\sum_{j=1}^n{\cos{(tY_j)}}e^{-\frac{1}{2}t^2}+e^{-t^2}\\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+\frac{1}{n^2}\sum_{j=1}^n{\sum_{k=1}^n{\sin{(tY_j)}\sin{(tY_k)}}}\Bigg]\psi(t)dt\\ = n \int_{\mathbb{R}}\Bigg[\frac{1}{n^2}\sum_{j,k=1}^n{\cos{(t(Y_j-Y_k))}}-2\cdot\sum_{j=1}^n{\cos{(tY_j)}}e^{-\frac{1}{2}t^2}+e^{-t^2}\Bigg]\psi(t)dt\\ = \frac{1}{n}\sum_{j,k=1}^n{\int_{\mathbb{R}}{(2\pi)^{-\frac{1}{2}}\cos{(t(Y_j-Y_k))}e^{-\frac{1}{2}t^2}}dt}-2\sum_{j=1}^n{\int_{\mathbb{R}}{(2\pi)^{-\frac{1}{2}}\cos{(tY_j)}e^{-t^2}}dt}\\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+n\int_{\mathbb{R}}{(2\pi)^{-\frac{1}{2}}e^{-\frac{3}{2}t^2}}dt\\ $$
I dont know how to solve the integrals to reach the expected form $$ T_n= \frac{1}{n}\sum_{j,k=1}^n{e^{-\frac{1}{2}\left|Y_j-Y_k\right|^2}}-2^{\frac{1}{2}}\sum_{j=1}^n{e^{-\frac{1}{4}\left|Y_j\right|^2}}+n3^{-\frac{1}{2}} $$