On the concentration phenomenon for [phi]-subgaussian random elements

We study the deviation probability P{[short parallel]X[short parallel]-E[short parallel]X[short parallel]>t} where X is a [phi]-subgaussian random element taking values in the Hilbert space l2 and [phi](x) is an N-function. It is shown that the order of this deviation is exp{-[phi]*(Ct)}, where C depends on the sum of [phi]-subgaussian standard of the coordinates of the random element X and [phi]*(x) is the Young-Fenchel transform of [phi](x). An application to the classically subgaussian random variables ([phi](x)=x2/2) is given.