On central limit theorems in the random connection model

Consider a sequence of Poisson random connection models (Xn,λn,gn) on , where λn/nd→λ>0 and gn(x)=g(nx) for some non-increasing, integrable connection function g. Let In(g) be the number of isolated vertices of (Xn,λn,gn) in some bounded Borel set K, where K has non-empty interior and boundary of Lebesgue measure zero. Roy and Sarkar (Physica A 318 (2003) 1047) claim that where denotes convergence in distribution. However, their proof has errors. We correct their proof and extend the result to larger components when the connection function g has bounded support.