\(L_ 1\) nonuniform central limit bounds for generalized rank statistics (Q1120925)

Let \(X_ 1,X_ 2,...,X_ n\) be independent random variables. A generalized rank statistic (g.r.s.) is an expression of the form \[ T=\sum^{n}_{i=1}c_ iJ(R^ g_ i/(n+1))h(X_ i), \] where \(c_ 1,c_ 2,...,c_ n\) are known constants, \(R^ g_ i\) is the rank of \(g(X_ i)\) among \(\{g(X_ j)\), \(1\leq j\leq n\}\) and J, h, g are Borel measurable functions. Simple and signed linear rank statistics, L- statistics and rank combinatorial statistics are included as special cases. Let \(T^*_ n\), \(n\geq 1\), be a sequence of standardized g.r.s. with distribution functions \(F_ n\), \(n\geq 1\). It is proved, under certain conditions, that \[ d_ s(F_ n,\Phi):=\int^{+\infty}_{- \infty}(1+| x|^ s)| F_ n(x)-\Phi (x)| dx\quad =\quad O(n^{-1/2}) \] for all positive s, where \(\Phi\) denotes the standard normal distribution function.