Almost sure convergence of sample range
An almost sure limit theorem is derived for the sample range statistics \( R_n=\max_{1\leq i\leq n} X_i-\min_{1\leq i\leq n} X_i \), where \(X_i\) are i.i.d. r.v.s. It states that if for some nonrandom \(\alpha_n\), \(\beta_n\) and some nondegenerate CDF \(G\) \(\zeta_n=\alpha_n(R_n-\beta_n)\) converges in distribution to \(G\) then at any point \(x\) of continuity of \(G\) \[ \lim_{n\to\infty}{1\over\log n}\sum_{k=1}^n {1\over k}{\mathbf 1}_{\{\zeta_k\leq x\}}=G, \text{a.s.} \] and \[ \lim_{n\to\infty}{1\over\log n}\sum_{k=1}^n {1\over k} f(\zeta_k)=\int_{-\infty}^{+\infty} f(x)dG(x) \] for any bounded Lipschitz function \(f\).
scientific article; zbMATH DE number 5520591