  This paper consists of two parts. In the first part, we focus on the average
of a functional over shifted Gaussian homogeneous noise and as the averaging
domain covers the whole space, we establish a Breuer-Major type Gaussian
fluctuation based on various assumptions on the covariance kernel and/or the
spectral measure. Our methodology for the first part begins with the
application of Malliavin calculus around Nualart-Peccati's Fourth Moment
Theorem, and in addition we apply the Fourier techniques as well as a soft
approximation argument based on Bessel functions of first kind.
  The same methodology leads us to investigate a closely related problem in the
second part. We study the spatial average of a linear stochastic heat equation
driven by space-time Gaussian colored noise. The temporal covariance kernel
$\gamma_0$ is assumed to be locally integrable in this paper. If the spatial
covariance kernel is nonnegative and integrable on the whole space, then the
spatial average admits Gaussian fluctuation; with some extra mild integrability
condition on $\gamma_0$, we are able to provide a functional central limit
theorem. These results complement recent studies on the spatial average for
SPDEs. Our analysis also allows us to consider the case where the spatial
covariance kernel is not integrable: For example, in the case of the Riesz
kernel, the first chaotic component of the spatial average is dominant so that
the Gaussian fluctuation also holds true.
