I am occasionally asked whether there is a lower tail bound that is comparable to the upper tail bound from "A tail inequality for quadratic forms of subgaussian random vectors".

The answer is NO, at least without making some additional assumption.

- Suppose, for example, that n=1 and x=x_1 is a random variable that is 0 with probability 0.5 and otherwise -sqrt(2) and +sqrt(2) with probability 0.25 each. Then although x has mean zero and variance 1, Pr(x^2 = 0) = 0.5. So quadratic forms in x do not concentrate around the mean.

- An example of an additional assumption is that x has IID subgaussian components. In this case, one can obtain a lower tail bound. See Roman Vershynin's textbook for a very nice exposition: <https://www.math.uci.edu/~rvershyn/papers/HDP-book/HDP-2.pdf>