Central limit theorems for \(k\)-nearest neighbour distances
Let \(X_1,X_2,\dots \) be i.i.d. \(d\)-dimensional random vectors with a density function \(f.\) Let \(R_{i,k,n}\) denote the distance from \(X_i\) to its \(k\)th nearest neighbour in the set \(\{X_1,\dots , X_n\}, n>k\), and similarly, let \(R_{i,k, N_n}\) be the distance from \(X_i\) to its \(k\)th nearest neighbour in \(\{X_1,\dots , X_{N_n} \},\) where for each \(n =1, 2,\dots\), \(\;N_n \) is a Poisson process with mean \(n\). Weak convergence of centered and scaled empirical distributions of \(\{f(X_i)R^d_{i, k_n, n}, 1\leq i\leq n\}\), respectively of \(\{f(X_i)R^d_{i, k_n, N_n}, 1\leq i\leq N_n\}\) is studied as \(n\to \infty \) and \(k_n = o(n^{2/(2+d)})\).
scientific article; zbMATH DE number 1792505
scientific article