Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 210 / 3 / d / Solution

With the usual variance-proxy convention for a sub-Gaussian random vector, every column inner product is a sub-Gaussian random variable with proxy at most ${\tau }^2$. The two-sided Chernoff bound and union bound implywith probability at least $1-\delta$. No independence between these inner products is required.

The printed coefficient-one bound is false under this convention. Take $k=m=1$, $A=[1]$, and $g=\pm \tau$ with equal probability. This Rademacher random variable has the stipulated variance proxy, but for $\delta =0.99$ the printed threshold is less than $\tau$, whereas $|g|=\tau$ surely. Even a general sub-Gaussian assumption does not imply Gaussian concentration inequality for arbitrary Lipschitz functions. Under the stronger convention $\mathbb{E}{e}^{⟨a,g⟩}\le e^{{\tau }^2\Vert a{\Vert }_2^2/4}$, the same union bound proves the printed constants. That convention, however, differs from the binomial variance proxy used in 1(b).