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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 $u \ge u_1$.
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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 $B_n$ 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 $u\downarrow0$.

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  • $\begingroup$ Thanks for the input. I was hoping there would be broad classes distributions strictly between atomic and bounded density, which satisfy the conditions. $\endgroup$
    – dohmatob
    Commented Sep 20, 2020 at 22:22
  • $\begingroup$ @dohmatob : In your question, you mentioned symmetry and subggaussianity as prospective conditions for (1). Did you also mention there (non-)atomicity of the distribution and boundedness of the density? Anyhow, I have now also added that the boundedness of the density cannot be relaxed by allowing the density to explode to $\infty$ near $0$. So, the relevant conditions are shown to be in terms of boundedness of the density. $\endgroup$
    – Iosif Pinelis
    Commented Sep 21, 2020 at 0:01
  • $\begingroup$ Thanks for the update. As a side note, I didn't mention those because I thought the were examples and non-examples (Indeed, in my question, I used boundedness of density to prove that $N(0,1)$ was a member of the sought-for family). $\endgroup$
    – dohmatob
    Commented Sep 21, 2020 at 0:05

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