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In a course we have been given the following proposition:

For a random variable $X\geq0$ with essup$(X)=\infty$

i) $F$ is heavy tailed if $\lim_{u\rightarrow\infty}e_F(u)=\infty $

ii) $F$ is light-tailed if $\limsup_{u\rightarrow\infty}e_F(u)<\infty $

$e_F$ refers to the mean excess function. It is defined $e_F(u):=\mathbb{E}[X-u|X>u].$ An explicit formula is given by $e_F(u)=\frac{1}{1-F(u)}\int_u^\infty 1-F(x)dx$

It was claimed that in the case of a finite $\liminf$ and indefinite $\limsup$, $F$ can be either light- or heavy-tailed. Does anybody know an example for such distributions? I tried to find an example and constructed various discrete distributions and didn't even find any distribution such that $\limsup_{u\rightarrow\infty}e_F(u)=\infty $ and $\liminf_{u\rightarrow\infty}e_F(u)<\infty $. Any help would be appreciated.

EDIT: added definition of $e_F$.

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    Could you define more of your notation? What is $e_F$ here? – 2017-01-17
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    @Brick thanks for the comment, of course I forgot that one. – 2017-01-17
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    Is i) meant to be lim or lim inf? – 2017-01-17
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    It is meant to be lim. The examples I'm looking for have limsup = inf and lim inf < inf, so there is no lim there. – 2017-01-17

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