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Motivated by new examples of functional Banach spaces over the unit disk, arising as the symbol spaces in the study of random analytic functions, for which the monomials 
$\{z^n\}_{n\geq 0}$ exhibit features of an unconditional basis yet they often don’t even form a Schauder basis, we introduce a notion called solid basis for Banach spaces and p-Banach spaces and study its properties. Besides justifying the rich existence of solid bases, we study their relationship with unconditional bases, the weak-star convergence of Taylor polynomials, the problem of a solid span and the curious roles played by c0. The two features of this work are as follows: (1) during the process, we are led to revisit the axioms satisfied by a typical Banach space of analytic functions over the unit disk, leading to a notion of 
$\mathcal{X}^\mathrm{max}$ (and 
$\mathcal{X}^\mathrm{min}$), as well as a number of related functorial constructions, which are of independent interests; (2) the main interests of solid basis lie in the case of non-separable (p-)Banach spaces, such as BMOA and the Bloch space instead of VMOA and the little Bloch space.
Let 
$\mathcal {R}f$ be the randomization of an analytic function over the unit disk in the complex plane where 
$$ \begin{align*}\mathcal{R} f(z) =\sum_{n=0}^\infty a_n X_n z^n \in H({\mathbb D}), \end{align*} $$
$f(z)=\sum _{n=0}^\infty a_n z^n \in H({\mathbb D})$ and 
$(X_n)_{n \geq 0}$ is a standard sequence of independent Bernoulli, Steinhaus, or complex Gaussian random variables. In this paper, we demonstrate that prescribing a polynomial growth rate for random analytic functions over the unit disk leads to rather satisfactory characterizations of those 
$f \in H({\mathbb D})$ such that 
${\mathcal R} f$ admits a given rate almost surely. In particular, we show that the growth rate of the random functions, the growth rate of their Taylor coefficients, and the asymptotic distribution of their zero sets can mutually, completely determine each other. Although the problem is purely complex analytic, the key strategy in the proofs is to introduce a class of auxiliary Banach spaces, which facilitate quantitative estimates.
Let 
$f(z)=\sum _{n=0}^\infty a_n z^n$ be an entire function on the complex plane, and let 
${\mathcal R} f(z) = \sum _{n=0}^\infty a_n X_n z^n$ be its randomization induced by a standard sequence 
$(X_n)_n$ of independent Bernoulli, Steinhaus, or Gaussian random variables. In this paper, we characterize those functions 
$f(z)$ such that 
${\mathcal R} f(z)$ is almost surely in the Fock space 
${\mathcal F}_{\alpha }^p$ for any 
$p, \alpha \in (0,\infty )$. Then such a characterization, together with embedding theorems which are of independent interests, is used to obtain a Littlewood-type theorem, also known as regularity improvement under randomization within the scale of Fock spaces. Other results obtained in this paper include: (a) a characterization of random analytic functions in the mixed-norm space 
${\mathcal F}(\infty , q, \alpha )$, an endpoint version of Fock spaces, via entropy integrals; (b) a complete description of random lacunary elements in Fock spaces; and (c) a complete description of random multipliers between different Fock spaces.
