Limit theorems for short distances in \(\mathbb{R}^m\) (Q1124991)

Let \({\mathbf X}_1, {\mathbf X}_2,\dots\) be independent identically distributed random vectors in \(\mathbb{R}^m\) with continuous distribution function. Let \(S_n(t)= \sum_{1\leq i< j\leq n}I(d({\mathbf X}_i,{\mathbf X_j})\leq ta(n))\), where \(d({\mathbf x},{\mathbf y})\) is a nonnegative function which is not necessarily a distance in \(\mathbb{R}^m\). The authors prove the weak convergence theorems of \((S_n(t)- ES_N(t))/ (n^r a(n))\) \((r= \frac 32,1)\) which show that the limit process is always Gaussian but its form depends on how fast the sequence \(a(n)\) goes to zero. The results obtained generalize those results of the second author [Stochastic Processes Appl. 39, No. 1, 65-80 (1991; Zbl 0743.60034)].