Concentration of quadratic forms under a Bernstein moment assumption

A similar concentration result is available for subgaussian random variables. It is known as the Hanson-Wright inequality and is given in Proposition 2 below. First versions of this inequality can be found in Hanson and Wright [5] and Wright [9], although with a weaker statement than Proposition 2 below since these results involve ||| (|aij |) |||2 instead of |||A|||2. Recent proofs of this concentration inequality with |||A|||2 instead of ||| (|aij |) |||2 can be found in Rudelson and Vershynin [6] or Barthe and Milman [2, Theorem A.5].