Almost sure convergence of certain slowly changing symmetric one- and multi-sample statistics
Consider real valued statistics \(S_ n=S_ n(X_ 1^{(1)},\dots,X^{(1)}_{n_ 1};\dots;X_ 1^{(k)},\dots,X^{(k)}_{n_ k})\) symmetric in each group \(X_ 1^{(i)},\dots,X^{(i)}_{n_ i}\), \(i=1,\dots,k\) where \(n=(n_ 1,\dots,n_ k)\in\mathbb{N}^ k\) and \(X_ j^{(i)}\), \(i=1,\dots,k\), \(j\in\mathbb{N}\), is an array of independent vectors in \(R^ d\) identically distributed for each \(i\). With the help of the Efron-Stein inequality [\textit{B. Efron} and \textit{C. Stein}, Ann. Stat. 9, 586-596 (1981; Zbl 0481.62035)] a normalized sequence \(S_{n(j)}\), \(j\in\mathbb{N}\), is shown to converge a.s. to a constant for appropriate \(n(j)\to\infty\) in \(\mathbb{N}^ k\) \((j\to\infty)\). The main assumption is a certain a.s. bound for all \(| S_{n_ 1,\dots,n_ k}-S_{n_ 1,\dots,n_{i-1},n_ i+1,n_{i+1},\dots,n_ k}|\), \(i-1,\dots,k\). Applications include strong limit theorems for some nearest neighbour graph statistics and for runs and empty blocks.
scientific article; zbMATH DE number 89803