Central limit theorems for empirical product densities of stationary point processes

We prove the asymptotic normality of kernel estimators of second- and higher-order product densities (and of the pair correlation function) for spatially homogeneous (and isotropic) point processes observed on a sampling window <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$W_n$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi>W</mi> <mi>n</mi> </msub> </math> </EquationSource> </InlineEquation>, which is assumed to expand unboundedly in all directions as <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$n \rightarrow \infty \,$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi>∞</mi> <mspace width="0.166667em"/> </mrow> </math> </EquationSource> </InlineEquation>. We first study the asymptotic behavior of the covariances of the empirical product densities under minimal moment and weak dependence assumptions. The proof of the main results is based on the Brillinger-mixing property of the underlying point process and certain smoothness conditions on the higher-order reduced cumulant measures. Finally, the obtained limit theorems enable us to construct <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$\chi ^2$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msup> <mi mathvariant="italic">χ</mi> <mn>2</mn> </msup> </math> </EquationSource> </InlineEquation>-goodness-of-fit tests for hypothetical product densities. Copyright Springer Science+Business Media Dordrecht 2014