A more particular characterisation of the normal distribution among the class of [infinitely divisible](http://en.wikipedia.org/wiki/Infinite_divisibility_%28probability%29) distributions is presented in [Steutel and Van Harn (2004)](http://www.amazon.com/Infinite-Divisibility-Probability-Distributions-Mathematics/dp/0824707249).

> A non-degenerate infinitely divisible random variable $X$ has a normal distribution if and only if it satisfies
$$-\limsup_{x\rightarrow\infty}\dfrac{\log{\mathbb P}(\vert X\vert>x)}{x\log(x)}=\infty.$$ 

This result characterises the normal distribution in terms of its tail behaviour.