  The present work provides an original framework for random matrix analysis
based on revisiting the concentration of measure theory from a probabilistic
point of view. By providing various notions of vector concentration
($q$-exponential, linear, Lipschitz, convex), a set of elementary tools is laid
out that allows for the immediate extension of classical results from random
matrix theory involving random concentrated vectors in place of vectors with
independent entries. These findings are exemplified here in the context of
sample covariance matrices but find a large range of applications in
statistical learning and beyond, thanks to the broad adaptability of our
hypotheses.
