The large-sample lower bounds for the tails of the distributions of PP Kolmogorov-Smirnov statistics (Q1200699)

Let \(\xi_ 1,\xi_ 2,\dots,\xi_ n\) be i.i.d. and \(R^ p\)-valued random vectors defined on a probability space \((\Omega,\tau,P)\) with common law \(P\). We construct PP Kolmogorov-Smirnov statistics as follows by the projection pursuit (abbreviated PP) method: \[ K^{(1)}_ n=\sup\left\{n^{-1}\biggl|\sum^ n_{i=1} \bigl[I_{[a^ \tau X_ i\leq t]}-P[a^ \tau X_ i\leq t]\bigr]\biggr|: | a|=1,\quad a\in R^ p,\quad t\in R\right\}, \] where \(I_{[\cdot]}\) is the characteristic function of a set and \(|\cdot|\) is the Euclidean norm. It is shown, for some general \(p\)-dimensional distributions (including the \(p\)-dimensional elliptically contoured distributions with positive definite dispersion matrix, the \(p\)- dimensional regular distributions with bounded support, and so on), that the best large-sample lower bound for \(P\bigl\{\sqrt n K^{(1)}_ n>\lambda\bigr\}\) is \(C(P)\lambda^{2(p-1)}\exp(-2\lambda^ 2)\) with a constant \(C(P)\) independent of \(\lambda\). Let \({\mathcal F}=\{[\underset\widetilde{} s,\underset\widetilde{} t]: \underset\widetilde{} s, \underset\widetilde{} t\in R^ p\), \(\underset\widetilde{} s\leq\underset\widetilde{} t\}\) and \(\underset\widetilde{} s\leq\underset\widetilde{} t\) denotes \(s_ i\leq t_ i\), \(1\leq i\leq p\). Let \(P\) be a continuous distribution and \(\{W_ P(f): f\in {\mathcal F}\}\) be the Kuiper empirical processes. \textit{R. J. Adler} and \textit{G. Samorodnitsky} [Ann. Probab. 15, 1339-1351 (1987; Zbl 0638.60059)] gave the best lower bound \(C\lambda^{2(2p-1)}\exp(- 2\lambda^ 2)\) for \(P\{\sup_{\mathcal F} W_ P(\cdot)>\lambda\}\) provided \(P\) is the uniform distribution on \([0,1]^ p\). In this note, under a weak regularity condition, we show that this best lower bound still holds for the general \(p\)-dimensional distribution \(P\). Similarly, some \textit{R. J. Adler} and \textit{L. D. Brown's} results [ibid. 14, 1-30 (1986; Zbl 0596.62053)] are improved.