We reformulate the Littlewood theorem on random analytic functions in the Hardy spaces as a problem of determining the random symbol spaces, and we show that, under a general randomization scheme, the symbol space is always a subspace of H²(D) (Corollary 7.6). We then characterize completely when the symbol space is precisely H²(D) (Theorem 1.1). This result extends Littlewood's theorem and can also be considered a converse of the theorem, since previous literature has focused solely on the sufficiency part of the results. We establish an analog of the Fernique theorem by determining the optimal integrability exponent within the L^p-scale (Theorem 1.2), and we propose a conjecture concerning general Young functions. The issue of determining which vector spaces can emerge as symbol spaces is exemplified through examples.

G. Cheng is supported by NSFC (12371126). X. Fang is supported by NSTC of Taiwan (112-2115-M-008-010-MY2). C. Liu is supported by NSFC (12461028) and Yunnan Fundamental Research Projects (202501CF070076).