It has long been known that if \(\sum | a_ n|^ 2<\infty\), then for almost every choice of signs \(\epsilon_ n=\pm 1\), the function \(f(z)=\sum \epsilon_ na_ nz^ n\) belongs to \(H^ p\) for every \(p<\infty\). It was recently discovered that if \(\sum | a_ n|^ 2\log n<\infty,\) then \(f\in BMOA\) almost surely. The author explores the effect of the intermediate condition \(\sum | a_ n|^ 2(\log n)^{\beta}<\infty,\) where \(0\leq \beta \leq 1\). The main result is that this implies \[ \int^{1}_{0}(1-r)| \log (1-r)|^{\beta - 1}[M_{\infty}(r,f')]^ 2dr<\infty \quad a.s. \]