On the concentration phenomenon for [formula omitted]-subgaussian random elements

Abstract

We study the deviation probability P { | ∥ X ∥ - E ∥ X ∥ | > t } where X is a ϕ -subgaussian random element taking values in the Hilbert space l 2 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 ) = x 2 / 2 ) is given.