On central limit theorems in the random connection model

Abstract: Consider a sequence of Poisson random connection models (X_n,lambda_n,g_n) on R^d, where lambda_n / n^d \to lambda > 0 and g_n(x) = g(nx) for some non-increasing, integrable connection function g. Let I_n(g) be the number of isolated vertices of (X_n,lambda_n,g_n) in some bounded Borel set K, where K has non-empty interior and boundary of Lebesgue measure zero. Roy and Sarkar [Phys. A 318 (2003), no. 1-2, 230-242] claim that (I_n(g) - E I_n(g)) / \sqrt Var I_n(g) converges in distribution to a standard normal random variable. However, their proof has errors. We correct their proof and extend the result to larger components when the connection function g has bounded support.