Let \((\mathbb{Y},{\mathcal Y})\) be a standard Borel space with a non-atomic \(\sigma\)-finite measure \(\lambda\). Let \(\eta_t\) be a Poisson point process on \(\mathbb{Y}\) with intensity measure \(\lambda_t= t\lambda\). \(\eta^k_{t,\neq}\;(k\geq 1)\) stands for the set of all \(k\)-tuples of distinct points of \(\lambda_t\) and \(y\in\eta_t\) means that \(y\in\mathbb{Y}\) is charged by the random measure \(\eta_t\). Let \(f: \mathbb{Y}^k\to\mathbb{R}\) be a nonnegative measurable symmetric function which satisfies \(\lambda^k(f^{-1}([0, x]))<\infty\) for all \(x>0\). Let \(\xi_t= \{f(y_1,\dots, y_k):(y_1,\dots, y_k)\in \eta^k_{t,\neq}\}\) be a collection of points on the positive real-half axis \(\mathbb{R}_+\) and \(F^{(m)}_t\) the \(m\)-th order statistic of \(f\) applied to \(\eta^k_{t,\neq}\). Let \(\alpha_t(x)\) be the mean number of \(k\)-tuples \((y_1,\dots, y_k)\) of \(\eta^k_{t,\neq}\) for which \(f(y_1,\dots, y_k)\leq xt^{-\gamma}\) \((y> 0\), \(t\geq 1\), \(x> 0)\). Further, put \[ r_t(t)= \sup_{\substack{ \widehat y_1,\dots,\widehat y_j\in\mathbb{Y}^j\\ 1\leq j\leq k-1}} \lambda^j_t(\{(\widehat y_1,\dots, \widehat y_j)\in\mathbb{Y}^j: f(\widehat y_1,\dots,\widehat y_j, y_1,\dots, y_{k-j})\leq xt^{-\gamma}\}). \] In this paper, the authors prove that, under some conditions on \(\alpha(x)\) and \(r_t(x)\), the resealed point process \(t^\gamma\xi_t\) converges as \(t\to\infty\) in distribution to some Poisson point process \(\xi\) and \[ \Biggl| P(t^\gamma F^{(m)}_t> r)- e^{-\beta x^r} \sum^{m-1}_{i=0} {(\beta x^r)^i\over i!}\Biggr|\leq |\beta x^\tau-\alpha_t(x)|+ C_{f,x}\sqrt{r_t(x)} \] for all \(t\geq 1\), where \(C_{f,x}\) is a positive constant depending on \(f\) and \(x>0\).
The result is applied to various problems having a geometric flavour.