On the concentration phenomenon for ϕ-subgaussian random elements

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