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The present work provides an original framework for random matrix analysisbased on revisiting the concentration of measure theory from a probabilisticpoint of view. By providing various notions of vector concentration($q$-exponential, linear, Lipschitz, convex), a set of elementary tools is laidout that allows for the immediate extension of classical results from randommatrix theory involving random concentrated vectors in place of vectors withindependent entries. These findings are exemplified here in the context ofsample covariance matrices but find a large range of applications instatistical learning and beyond, thanks to the broad adaptability of ourhypotheses.

Concentration of Measure and Large Random Matrices with an application to Sample Covariance Matrices