On the asymptotic distribution of the number of random variables exceeding a given level
Let \(X\) be a non-degenerate r.v. in \([0,\infty)\) and let \(X_1, \dots, X_n\) be independent copies of \(X\). Denote the number of r.v.'s among \(X_1,X_2, \dots, X_{\mu(t)}\) exceeding a given level \(x\) by \(N_x(t)\) where \(\mu(t)\) is a stopping time. As usual \(\Pi(\cdot, \lambda)\) signifies the Poisson distribution with parameter \(\lambda\). The author establishes uniform (in \(x)\) estimates of the rate of convergence of \(P(N_x(t)=k)\) to \(\Pi(k, \lambda_k)\) and asymptotic expansions in the limit theorem. As consequences, there are given estimates of the rate of convergence for the number of ``long'' head-runs and the number of ``long'' intervals between consecutive jumps of a Poisson process. The main results of the paper may be used in studying some spatial extremal processes \(Z(t)\) for which \(\{Z(t) <x\} {\overset{d}=} \{N_x(t)=0\}\).
scientific article; zbMATH DE number 918007