A Bernstein-type inequality for suprema of random processes with an application to statistics

We use the generic chaining device proposed by Talagrand to establish exponential bounds on the deviation probability of some suprema of random processes. Then, given a random vector $\xi$ in ${\mathbb{R}}^n$ the components of which are independent and admit a suitable exponential moment, we deduce a deviation inequality for the squared Euclidean norm of the projection of $\xi$ onto a linear subspace of ${\mathbb{R}}^n$. Finally, we provide an application of such an inequality to statistics, performing model selection in the regression setting when the errors are possibly non-Gaussian and the collection of models possibly large.