= Normal approximation to a compound Poisson aggregate
{title2=$S\approx\mathcal N(\lambda m_1,\lambda m_2)$}
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 $\lambda[\varphi_X(t/\sqrt{\lambda m_2})-1]-it\lambda m_1/\sqrt{\lambda m_2}$; the second-order expansion of $\varphi_X$ makes this tend to $-t^2/2$. The approximation loses the zero atom and assigns positive <probability> to negative losses.
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