Hey I need some help with a problem:
Let $X_{n1},X_{n2},...,X_{nn}$ be independent random variables with a common distribution as follows: $$P(X_{nk}=0)=1-\frac{1}{n}-\frac{1}{n^2}, P(X_{nk}=1)=\frac{1}{n}, P(X_{nk}=2)=\frac{1}{n^2}, $$
where k=1,2,.. and n=2,3,... $S_{n}=X_{n1}+X_{n2}+...+X_{nn}$, $n>=2$
Show that
$S_{n}\stackrel{d}{\rightarrow}Po(1)$ as n $\rightarrow \infty$
I've started like this:
$$E[e^{tS_{n}}]=E[e^{tX_{n1}+tX_{n2}+...+tX_{nn}}]=\prod_{k=1}^{n}E[e^{tX_{nk}}]=\prod_{n=2}^{\infty}(1-\frac{1}{n}-\frac{1}{n^2}+\frac{1}{n}e^{t}+\frac{1}{n^2}e^{2t})$$
but I'm not sure I'm going with the right approach. Does someone have any tips for me?