  Gaussian noise is an irreducible component of the background in gravitational
wave (GW) detectors. Although stationary Gaussian noise is uncorrelated in
frequencies, we show that there is an important correlation in time when
looking at the matched filter signal to noise ratio (SNR) of a template, with a
typical autocorrelation time that depends on the template and the shape of the
noise power spectral density (PSD). Taking this correlation into account, we
compute from first principles the false alarm rate (FAR) of a template in
Gaussian noise, defined as the number of occurrences per unit time that the
template's matched filter SNR goes over a threshold $\rho$. We find that the
Gaussian FAR can be well approximated by the usual expression for uncorrelated
noise, if we replace the sampling rate by an effective sampling rate that
depends on the parameters of the template, the noise PSD and the threshold
$\rho$. This results in a minimum SNR threshold that has to be demanded to a
given GW trigger, if we want to keep events generated from Gaussian noise below
a certain FAR. We extend the formalism to multiple detectors and to the
analysis of GW events. We apply our method to the GW candidates added in the
GWTC-3 catalog, and discuss the possibility that GW200308\_173609 and
GW200322\_091133 could be generated by Gaussian noise fluctuations.
