When can limit of a sequence of characteristic be also a characteristic function?

Solution 1:

$X_{n}\xrightarrow{d} X\iff \phi_{X_{n}}(t)\xrightarrow{n\to\infty}\phi_{X}(t)$ .

So now suppose you construct $X_{1},X_{2},..$ such that they have their cf as $\phi_{X_{n}}$.

It can be shown that this sequence of rv's is a tight sequence.

Then using Tightness you get a subsequence and a random variable $X$ such that $X_{n_{k}}\xrightarrow{d} X$.

So $\phi=\phi_{X}$ ( using pointwise convergence).

Which in particular implies that $\phi$ is a cf as it agrees with $\phi_{X}$ which is a cf .

For the proof of Tightness you can refer to Rick Durrett as suggested in the comments on page 132-133. Here is a link for the pdf