  Given a sequence $(\xi_n)$ of standard i.i.d complex Gaussian random
variables, Peres and Vir\'ag (in the paper ``Zeros of the i.i.d. Gaussian power
series: a conformally invariant determinantal process'' {\it Acta Math.} (2005)
194, 1-35) discovered the striking fact that the zeros of the random power
series $f(z) = \sum_{n=1}^\infty \xi_n z^{n-1}$ in the complex unit disc
$\mathbb{D}$ constitute a determinantal point process. The study of the zeros
of the general random series
  $f(z)$ where the restriction of independence is relaxed upon the random
variables $(\xi_n)$ is an important open problem. This paper proves that if
$(\xi_n)$ is an infinite sequence of complex Gaussian random variables such
that their covariance matrix is invertible and its inverse is a Toeplitz
matrix, then the zero set of $f(z)$ constitutes a determinantal point process
with the same distribution as the case of i.i.d variables studied by Peres and
Vir\'ag. The arguments are based on some interplays between Hardy spaces and
reproducing kernels. Illustrative examples are constructed from classical
Toeplitz matrices and the classical fractional Gaussian noise.
