A Bernstein-type inequality for stochastic processes of quadratic forms of Gaussian variables

The concentration phenomenon of stochastic processes around their mean is of key importance in statistical estimation by model selection for getting nonasymptotic bounds for some statistics. For example in model selection via penalization, for devising sharp penalties and proving useful upper bounds for the risk of an estimator, one needs generally to control uniformly the statistic of the risk of an estimator by means of a sharp concentration inequality. This topic has received since recently (late nineties) a considerable interest among the statistical community above all further to the amazing series of works of Michel Talagrand which can be seen as the infinite dimensional analogue of the Bernstein’s inequality (see in particular (7) for an overview and (8) for later advances). Their application in non-asymptotic model selection has first been discovered by Birgé and Massart (e.g. (2)), then refined and popularized by the same authors (e.g. (3; 4)). For beautiful lectures on the topic, we refer the dear reader to (5; 6). In this small body of work, we establish a new Berstein-type inequality which serves to control (e.g. uniformly) quadratic forms of Gaussian variables and which happens to be useful for controlling, for example, uniformly the quadratic risk of a finite (or a countable) set of linear estimators in linear regression and linear inverse problems (see (1) for an application). In the remainder, we will give both the uncorrelated form and the correlated form of such an inequality.