  Let $N_n(a, b)$ denote the number of real zeros of Gaussian elliptic
polynomials of degree $n$ on the interval $(a, b)$, where $a$ and $b$ may vary
with $n$. We obtain a precise formula for the variance of $N_n(a, b)$ and
utilize this expression to derive an asymptotic expansion for large values of
$n$. Furthermore, we provide sharp estimates for the cumulants and central
moments of $N_n(a, b)$. These estimates are instrumental in establishing
sufficient conditions on the interval $(a, b)$ for $N_n(a, b)$ to satisfy both
a central limit theorem and a strong law of large numbers. In the second part
of the paper, we extend our analysis to nondegenerate Gaussian analytic
functions, including well-known examples such as the Gaussian Weyl series and
Weyl polynomials.

Probability