What can we say about the concentration around 0 linear transformation of Gaussian random variables? I have a matrix $X \in \mathbb{R}^{n \times m}$ such that each $A_{ij}$ is a Gaussian with mean $0$ and variance $1$. We have $m > n$.
I also have a vector $v \in \mathbb{R}^m$ such that $||v||_2 \le \epsilon$.
What can we say about the concentration of $Xv$ around the 0 vector?
Clearly, $E[Xv] = 0$ and the variance should be rather small because $||v||_2 \le \epsilon$. Easy to analyze when $m=n=1$, of course.
But what do I do in the general case? Can I have a statement of the following form:
$$p(||Xv||_2 \ge \delta) \le f(\delta,\epsilon,m,n)$$ etc. ?
 A: Are they also independent Gaussians? (If, say, they are all actually the same Gaussian, then you have a very different situation.)
Assuming they are independent, there is such a statement. The key fact is that all norms on finite dimensional spaces are equivalent. That means that there exist constants $c,C>0$ such that 
$$c \| A \|_F \leq \| A \|_2 \leq C \| A \|_F$$ 
where $\| \cdot \|_2$ is the Euclidean operator norm and $\| \cdot \|_F$ is the Frobenius norm. This will be useful for your statement because the expected value of the square of the Frobenius norm will be $mn$. The linear algebraic part of your problem will be finding $c$ and/or $C$. Once you've done that, I think the problem will be essentially just the Markov/Chebyshev inequality.
Here is one (blunt) way to estimate $C$. Note that $\| Ax \|_2$ is at most $m$ times the norm of the largest column of $A$ times the size of the largest entry of $x$. (This is just the triangle inequality.) The largest entry of $x$ is at most equal to the Euclidean norm, while $m$ times the largest column is at most $m$ times the Frobenius norm. Hence $\| Ax \|_2 \leq m \| A \|_F \| x \|_2$, so $C \leq m$.
Hence if $\| A \|_F \leq M$ and $\| x \|_2 \leq \epsilon$ then $\| Ax \|_2 \leq mM \epsilon$. Now you know $E[\| A \|_F^2]=mn$, so you should be able to finish from here.
