Let c = (c n ) n∈N be an arbitrary sequence of l 2 (N ) and let F c (ω) be a random series of the type where (g n ) n∈N * is a sequence of independent N C (0, 1) Gaussian random variables and (e n ) n∈N an orthonormal basis of L 2 (Y, M, µ) (the finite measure space (Y, M, µ) being arbitrary). By using the equivalence of Gaussian moments and an integrability theorem due to Fernique, we show that a necessary and sufficient condition for F c (ω) to belong to L p (Y, M, µ), p ∈ [2, ∞) for any c ∈

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... 2 (N ) almost surely is that sup n∈N e n L p (Y,M,µ) < ∞. One of the main motivations behind this result is the construction of a nontrivial Gibbs measure invariant under the flow of the cubic defocusing nonlinear Schrödinger equation posed on the open unit disc of R 2 . When σ = 0, then X ≡ 0, and its p.d.f. does not exist. However, for convenience, we suppose that 0 is a Gaussian random variable.