Suppose we have a sequence of characteristic functions $\phi_n$ such that all of these functions are absolutely integrable and this sequence of functions converges point wise to $~1~$ . Now consider any distribution function $~F~$ with $a < b{}$ as continuity points , that has characteristic function $\phi{}$ . Then , can we write :
$$F(b)-F(a) = \lim_{n\rightarrow \infty} \frac{1}{2 \pi} \displaystyle \int_{\mathbb{R}}\frac{e^{-itb}-e^{-ita}}{-it}\phi_n(t)\phi(t)dt$$
I have observed that if we can swap the limit and integral , the result holds true . I suspect that some form of DCT can be used to justify this swapping . However , I am unable to use the following form of DCT ( I know only this version of DCT ) :
If $\{X_n\}_{~n ~\in~ \mathbb{N}} $ is a sequence of random variables such that $X_n(\omega) \rightarrow X(\omega)$ for all $\omega \in \Omega$ , such that $|X_n| \leq Y$ where $\mathbb{E}[Y] < \infty$ , then $\mathbb{E}[X_n] \rightarrow \mathbb{E}[X]$
The map $t \mapsto \frac{e^{-itb}-e^{-ita}}{-it}$ is continuous on $\mathbb R$ and converges to zero at $\pm \infty$. Therefore it is bounded, let say by $C>0$.
Then the sequence of maps $$\psi_n(t) =\frac{e^{-itb}-e^{-ita}}{-it}\phi_n(t)\phi(t)$$ is bounded by $C\vert \phi \vert$ and converges pointwise to $\frac{e^{-itb}-e^{-ita}}{-it}\phi(t)$. We can apply DCT and swap limit and integral.