Normal approximation to a compound Poisson aggregate ($S\approx \mathcal{N}(\lambda m_1,\lambda m_2)$)

For a compound Poisson distribution of count parameter $\lambda$ and severity moments $m_1,m_2$, the expected value is $\lambda m_1$ and the variance is $\lambda m_2$. As $\lambda \to \infty$ with the severity law fixed and $m_2<\infty$, the standardized aggregate converges in distribution to the standard normal distribution. Indeed, the logarithm of its characteristic function is $\lambda [{\phi }_X(t/\sqrt{\lambda m_2})-1]-it\lambda m_1/\sqrt{\lambda m_2}$; the second-order expansion of ${\phi }_X$ makes this tend to $-t^2/2$. The approximation loses the zero atom and assigns positive probability to negative losses.