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 $u \ge u\_1$.



Characterization of random variables whose tensor powers have subexponential "small-ball" probabilities
Let $X:=(X\_1,\dots,X\_n)$, where the $X\_j$'s are iid copies of $\zeta$. Then the problem is about conditions for
$$P(\|X\|\le u\sqrt n)\le C^n u^n\tag{1}$$
for some real $C>0$, all natural $n$, and all small enough real $u>0$ (and then for all real $u>0$, possibly with a different $C>0$).


If the distribution of $X\_1$ has a density bounded by some real $c>0$, then the distribution of $X$ has a density bounded by $c^n$, so that
$$P(\|X\|\le u\sqrt n)\le c^n (u\sqrt n)^n|B\_n|\le C^n u^n$$
for some real $C>0$, all real $u>0$, and all natural $n$, where $|B\_n|$ is the [volume of the unit ball](https://en.wikipedia.org/wiki/Volume_of_an_n-ball#The_volume) $B\_n$ in $\mathbb R^n$; so, (1) holds.


On the other hand, (1) cannot hold (for $C,u,n$ specified above) if the distribution of $X\_1$ has an atom at $0$. Moreover, (1) cannot hold for any real $C>0$, any natural $n$, and all small enough real $u>0$ if the distribution of $X\_1$ has a density $p$ such that $p(x)\to\infty$ as $x\to0$. Indeed, then for each natural $n$ we have
$$P(\|X\|\le u\sqrt n)/u^n=\int\_{u\sqrt n\,B\_n}\prod\_{j=1}^n (p(x\_j)\,dx\_j)/u^n\to\infty$$
as $u\downarrow0$.


