# Bounding sub-Gaussian tail events by Gaussian tail events?

## 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!

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

• @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. – sortofamathematician Feb 16 at 17:20
• It will be useful for others, including me, if you could share other examples that you may have found where "domination arguments" was used. – Ankitp Feb 16 at 19:38