  We modify Talagrand's generic chaining method to obtain upper bounds for all
p-th moments of the supremum of a stochastic process. These bounds lead to an
estimate for the upper tail of the supremum with optimal deviation parameters.
We apply our procedure to improve and extend some known deviation inequalities
for suprema of unbounded empirical processes and chaos processes. As an
application we give a significantly simplified proof of the restricted isometry
property of the subsampled discrete Fourier transform.
