0
$\begingroup$

Background

I have come across a problem where I want to derive a concentration inequality for sub-gaussian random variables. More precisely, I want to bound the spectrum of a certain empirical covariance matrix, where $(x_i)$ are sub-gaussian random vectors forming the rows of a matrix $X \in \mathbb{R}^{T\times n}$. Without specififying exactly what the dependence structure of the $x_i$ are, let us just say I want to bound the probability of an event of the form $$ \left\|\frac{1}{T}X^*X-\Sigma \right \|_\infty \geq f(T,\dots). $$

My argument relies on a fact which holds (more or less) uniquely for Gaussian random variables (I have a quotient of moment generating functions somewhere in my argument) and I have verified it to be true in the Gaussian case, but let us not dwell on this.

Question

Now, somehow the notion of sub-Gaussianity is meant to capture the fact the distribution is less wild than a Gaussian (in particular dispalyed by the moment generating definition so in the scalar case my question is trivial).

So, this got me thinking, is there a generic way to prove such an inequality for the Gaussian case and then use some sort of "extension/domination lemma" to give the result for the sub-Gaussian case?

I can already see one problem with such an approach since the entries of $X^*X$ really are sub-exponential and not sub-Gaussian anymore (although one can imagine them being dominated by $\chi$-squareds).

I would be very interested in any links, references, ideas, partial results that could help me better understand this problem. I am not interested in finding a different way to prove my bound, but really would like to explore this sort of "domination" argument!

$\endgroup$
1
$\begingroup$

You can use covering number arguments to get similar results for sub-gaussian distributions sometimes. The result that you are interested in is Theorem 4.6.1 (see Eq 4.22) in the book[1]. The book is available online and the proof uses covering number argument.

Also, Lemma 6.2.3 in [1] shows an example where the moment generating function of sub-gaussian random vector was bounded by mgf of a gaussian vector.

[1]: High-Dimensional Probability, Roman Vershynin. https://www.math.uci.edu/~rvershyn/papers/HDP-book/HDP-book.html

$\endgroup$
  • $\begingroup$ @Anktip Thank you. I'm aware of the reference and the theorem you cite is precisely the one I am playing around with (without the independence assumption). I have at this point kind of realized that the question might be a bit too imprecise for this exchange so I will accept your answer as sufficient. $\endgroup$ – sortofamathematician Feb 16 at 17:20
  • $\begingroup$ It will be useful for others, including me, if you could share other examples that you may have found where "domination arguments" was used. $\endgroup$ – Ankitp Feb 16 at 19:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.