1
$\begingroup$

In this paper, Adamczak defines, for $m>0$ and $\sigma\geq 0$, the class of probability distributions $\mathcal{M}(m,\sigma)$ over $\mathbb{R}$ as those $\mu$ satisfying the tail conditions $$\nu^+([x,\infty)) \leq \mu([x,\infty)),\quad \nu^-((-\infty,-x]) \leq \mu((-\infty,-x])$$ for every $x\geq m$, where $\nu^+,\nu^-$ are the measures with densities $g^+(x) = x\mu([x,\infty)$ and $g^-(x) = -x\mu((-\infty,-x])$, respectively. (See Definition 4, or the followup work by Adamczak and Strzelecki which generalizes the results of the first paper)

The paper also proves (Lemma 2) that the condition $\mathcal{M}(m,\sigma)$ implies $C(m,\sigma)$-subgaussianity for some constant $C(m,\sigma)$. Some more examples are given, but I am not fully seeing how this class $\mathcal{M}(m,\sigma)$ is related to subgaussianity:

How much stricter is the condition $\mathcal{M}(m,\sigma)$ than subgaussianity? Are there "natural" examples of subgaussian distributions which are not in $\mathcal{M}(m,\sigma)$ for any $m,\sigma$?

Moreover, in view of one of the main results of the paper, Theorem 3 (the log-Sobolev inequality for convex functions holds for measures in $\mathcal{M}(m,\sigma)$):

Is it known whether the subgaussian concentration inequality in Theorem 3 fails for some random variables which are sugaussian but not in any $\mathcal{M}(m,\sigma)$?

$\endgroup$
5
  • $\begingroup$ Such an inequality (the 2nd one in Theorem 3 of the first paper) holds for $\varphi(X)$ when $\varphi:\mathbb R^N \to \mathbb R$ is Lipschitz and $X=(X_1,\ldots,X_N)$ has iid sub-Gaussian coordinates. This follows from Lévy's inequality combined sub-G concentration (to further bound the Lévy concentration function of the situation). The details can be found in standard text like Boucheron et al Concentration inequalities. So my guess is that the authors in your referenced paper only need those conditions to bound the Lévy concentration function, which can be done via other assumptions. $\endgroup$
    – dohmatob
    Feb 11, 2021 at 11:31
  • $\begingroup$ @dohmatob interesting -- thanks. I guess what I'm missing in your comment is the bound on the concentration function for subgaussian (looking at the Boucheron et al. book, I couldn't find it for subgaussian, just for Gaussian or Hamming wrt weighted Hamming). I should probably go back to that book... $\endgroup$
    – Clement C.
    Feb 17, 2021 at 0:17
  • 1
    $\begingroup$ I was referring to Theorem 7.1 of the book, which holds for any metric space (including Hamming cube), but the real question is bounding $\alpha(t)$. To this end, If you take as definition of sub-Gaussian vector to mean the distribution satisfies a $T_1$ inequality (T for Talagrand), then you can use Theorem 3.1 of (S. Bobkov) hse.ru/data/2016/11/24/1113029206/…. Finally, otherwise, if $\psi$ is $1$-Lipschitz w.r.t $L_1$ distance, you can used results from this paper (A. Kantorovich ) arxiv.org/pdf/1309.1007.pdf. I hope any of this helps :) $\endgroup$
    – dohmatob
    Feb 18, 2021 at 19:51
  • $\begingroup$ Thanks, I'll look into those asap! $\endgroup$
    – Clement C.
    Feb 18, 2021 at 22:51
  • 1
    $\begingroup$ Erratum: The correct link to S. Bobkov's paper (Theorem 3.1) is this one citeseerx.ist.psu.edu/viewdoc/… $\endgroup$
    – dohmatob
    Feb 18, 2021 at 22:55

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.