Let $X$ a random variable (in $\mathbb{R}$ for simplicity) and $\varphi(t) = \mathbb{E}[e^{i tX}]$ be its characteristic function. If $X$ has a moment of order $2$, then $|\varphi(t)|^2 = 1 - \frac12 \mathbb{E}[X^2] t^2 + o(t^2)$ as $t \to 0$.
When $X$ is heavy tailed, $|\varphi(t)|^2$ can be non analytic near zero. For example, for stable distributions, $|\varphi(t)|^2 = e^{-2 |t|^{\alpha}} = 1 - |2t|^{\alpha} + o(|t|^{\alpha})$ where $\alpha \in (0, 2)$.
But, the behavior near zero can contain logarithmic term. For example if $X$ is discrete with for all $n \geq 1$, $\mathbb{P}[X = n] = c/n^3$ where $c > 0$ is a normalization constant, then one has $|\varphi(t)|^2 = 1 + c' \log(t)|t|^2 + o(\log(t)|t|^2)$, for some $c'$. Note that $\log$ is a slowly varying function (https://en.wikipedia.org/wiki/Slowly_varying_function).
I wonder if this last behavior is the "worst" possible. Is there any example of real random variable such that $|\varphi(t)|^2 -1 \sim L(1/t)|t|^{\alpha}$ as $t \to 0$ is not satisfied for any slowly varying function $L$ and any $\alpha \in (0, 2)$?
