I am trying to apply CLT for multinomial distribution in the following way.
Suppose $v=(v_{1}\ldots v_{N})$ is multinomial random vector, so $v\sim M(n,p), p=(p_{1}\ldots p_{N})$.
Its characteristic function has the form:
$\psi_{v}(t)=(\sum_{j=1}^Np_{j}e^{it_{j}})^n$
Then we consider:
$v^{*}=\frac{v}{\sqrt{n}}-\sqrt{n}p$
Under this linear transformation the characteristic function becomes:
$\psi_{v^*}(t)=(\sum_{j=1}^Np_{j}e^{\frac{it_{j}}{\sqrt{n}}})^ne^{-\sqrt{n}i\sum_{j=1}^Np_{j}t_{j}}$
Taking log yields:
$\ln(\psi_{v^*}(t))=n\cdot \ln(\sum_{j=1}^Np_{j}e^{\frac{it_{j}}{\sqrt{n}}})-\sqrt{n}i\sum_{j=1}^Np_{j}t_{j}$
Then taking the limit and transforming it to $\frac{0}{0}$ gives:
$\lim_\limits{{n\to\infty}} \ln(\psi_{v^*}(t))=lim_{n\to\infty}\frac{\ln(\sum_{j=1}^Np_{j}e^{\frac{it_{j}}{\sqrt{n}}})-\frac{i\sum_{j=1}^Np_{j}t_{j}}{\sqrt{n}}}{n^{-1}}$
Using l'Hospital rule and making some straightforward transformations I achieve:
$\lim_\limits{{n\to\infty}}\ln(\psi_{v^*}(t))=\frac{i\sqrt{n}}{2}\sum_{j=1}^Np_{j}t_{j}(e^{\frac{it_{j}}{\sqrt{n}}}-1)$
Transfering $\sqrt{n}$ to denominator and using l'Hospital rule again I finally get:
$\lim_\limits{{n\to\infty}}\psi_{v^*}(t)=e^{-\frac{1}{2}\sum_{j=1}^Np_{j}t_{j}^2}$
It is the characteristic function of a multivariate normal distribution with zero covariances and variances equal to $p_{j}$. However this answer is wrong as true covariance matrix has the form $\Sigma_{ij}=p_{i}(1-p_{i})$ if $i=j$ and $\Sigma_{ij}=-p_{i}p_{i}$ otherwise.
Please help me figure out where I am wrong.