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Suppose $\mu$ is a finite positive rotation invariant Borel measure on the open unit disc $\Delta$, and that the unit circle lies in the closed support of $\mu$. For $0\<p\<\infty$ the $A_{\mu }^p$ is the collection of functions in $L^p\left(\mu \right)$ holomorphic on $\Delta$. We show that whenever a Gaussian power series $f\left(z\right)=\Sigma {\zeta }_n{a}_n{z}^n$ almost surely lies in $A_{\mu }^p$ but not in ${\bigcup }_{q\>p}{A}_{\mu }^p$, then almost surely: a) the zero set $Z\left(f\right)$ of $f$ is not contained in any $A_{\mu }^q$ zero set ($q\>p$, and b) $Z(f+1)\cup Z(f-1)$ is not contained in any $A_{\mu }^q$ zero set.