  We investigate distributional properties of a class of spectral spatial
statistics under irregular sampling of a random field that is defined on
$\mathbb{R}^d$, and use this to obtain a test for isotropy. Within this
context, edge effects are well-known to create a bias in classical estimators
commonly encountered in the analysis of spatial data. This bias increases with
dimension $d$ and, for $d>1$, can become non-negligible in the limiting
distribution of such statistics to the extent that a nondegenerate distribution
does not exist. We provide a general theory for a class of (integrated)
spectral statistics that enables to 1) significantly reduce this bias and 2)
that ensures that asymptotically Gaussian limits can be derived for $d \le 3$
for appropriately tapered versions of such statistics. We use this to address
some crucial gaps in the literature, and demonstrate that tapering with a
sufficiently smooth function is necessary to achieve such results. Our findings
specifically shed a new light on a recent result in Subba Rao (2018a). Our
theory then is used to propose a novel test for isotropy. In contrast to most
of the literature, which validates this assumption on a finite number of
spatial locations (or a finite number of Fourier frequencies), we develop a
test for isotropy on the full spatial domain by means of its characterization
in the frequency domain. More precisely, we derive an explicit expression for
the minimum $L^2$-distance between the spectral density of the random field and
its best approximation by a spectral density of an isotropic process. We prove
asymptotic normality of an estimator of this quantity in the mixed increasing
domain framework and use this result to derive an asymptotic level
$\alpha$-test.
