Is there a succinct characterization of all random variables $\zeta$ on $\mathbb R$ with the following properties

• Symmetry: $\zeta \overset{d}{=} - \zeta$.
• Small-ball probability: there exists a constant $\alpha > 0$ such that $P(\|\zeta^\otimes\| \le u\sqrt{n}) \le (\alpha u)^n$, for all $u \ge 0$ and positive integer $n$. Note that $\zeta^\otimes$ is a random vector on $\mathbb R^n$ with iid coordinates having the same distribution as $\zeta$.

Of course, $\zeta \sim N(0,1)$ fits the bill. I'm looking for general characterization.

Is there a succinct characterization of all random variables $\zeta$ on $\mathbb R$ with the following properties

• 1. Symmetry: $\zeta \overset{d}{=} - \zeta$.
• 2. Small-ball probability: there exists a constants $\alpha > 0$ and $u_0 \in (0,\infty]$ such that $P(\|\zeta^\otimes\| \le u\sqrt{n}) \le (\alpha u)^n$, for all $u \in [0,u_0)$ and positive integer $n$. Note that $\zeta^\otimes$ is a random vector on $\mathbb R^n$ with iid coordinates having the same distribution as $\zeta$.

Of course, $\zeta \sim N(0,1)$ fits the bill. I'm looking for general characterization.

In case (and only in case) it turns out that the space of all such distributions is too "vast and unstructured", consider the following 3rd axiom

• 3. Sub-Gaussianity: There exists $\sigma>0$ and $u_1 \ge 0$ such that $P(|\zeta| \ge u) \le 2e^{-u^2/(2\sigma^2)}$, for all $u \ge u_1$.