  Let $\pa{X_{t}}_{t\in T}$ be a family of real-valued centered random
variables indexed by a countable set $T$. In the first part of this paper, we
establish exponential bounds for the deviation probabilities of the supremum
$Z=\sup_{t\in T}X_{t}$ by using the generic chaining device introduced in
Talagrand (2005). Compared to concentration-type inequalities, these bounds
offer the advantage to hold under weaker conditions on the family
$\pa{X_{t}}_{t\in T}$. The second part of the paper is oriented towards
statistics. We consider the regression setting $Y=f+\eps$ where $f$ is an
unknown vector of $\R^{n}$ and $\eps$ is a random vector the components of
which are independent, centered and admit finite Laplace transforms in a
neighborhood of 0. Our aim is to estimate $f$ from the observation of $Y$ by
mean of a model selection approach among a collection of linear subspaces of
$\R^{n}$. The selection procedure we propose is based on the minimization of a
penalized criterion the penalty of which is calibrated by using the deviation
bounds established in the first part of this paper. More precisely, we study
suprema of random variables of the form $X_{t}=\sum_{i=1}^{n}t_{i}\eps_{i}$
when $t$ varies among the unit ball of a linear subspace of $\R^{n}$. We
finally show that our estimator satisfies some oracle-type inequality under
suitable assumptions on the metric structures of the linear spaces of the
collection.
