I am looking for probability distributions with CDF $F$ which satisfy the following property: for all $\varepsilon > 0$, \begin{equation} \frac{F(h+\varepsilon) - F(h)}{\varepsilon}\frac{1}{F(h+\varepsilon)} \to g(h)~, \end{equation} where $g(h) \cdot \exp(-h^2/2) \to 0$.
With stronger smoothness assumption, this looks equivalent to getting $F'(h)/F(h) \to g(h)$, where $g(h)$ is not increasing too quickly.
This property is satisfied e.g. by the normal distribution, where $g(h)$ is approximately $h$.
Are there general classes of distributions which satisfy this property? I have come across regularly varying functions in my search but that is still somewhat too restrictive.
