A more particular characterisation of the normal distribution among the infinitely divisible distributions is presented in Steutel and Van Harn (2004).

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.

A more particular characterisation of the normal distribution among the class of infinitely divisible distributions is presented in Steutel and Van Harn (2004).

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.

A more particular characterisation of the normal distribution among the infinitely divisible distributions is presented in Steutel and Van Harn (2004).

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

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

A more particular characterisation of the normal distribution among the infinitely divisible distributions is presented in Steutel and Van Harn (2004).

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.