Strong laws for the k-th order statistic when k\(\leq c\,\log _ 2\,n\) (Q1075678)

Let \(U_ 1,U_ 2,..\). be an i.i.d. sequence of random variables uniformly distributed over (0,1) and let for each n, \(U_{1,n}<U_{2,n}<...<U_{n,n}\) denote the order statistics of \(U_ 1,U_ 2,...,U_ n\). Let \((k_ n)\) be a non decreasing integer sequence such that \(1\leq k_ n\leq n\) for each n. Given a sequence \((k_ n)\), a sequence \((c_ n)\) of positive numbers is said to belong to (i) upper-upper class (lower-upper class) according as \(P(nU_{k_ n,n}>c_ n\quad i.o.)=0\) \((=1)\) and (ii) lower-lower class (upper-lower class) according as \(P(nU_{k_ n,n}<c_ n\quad i.o.)=0\) \((=1).\) In this paper the author obtains under certain regularity conditions on \((k_ n)\) and \((c_ n)\), criteria in terms of the convergence/divergence of an infinite series, for classifying \((c_ n)\) into upper-upper class, lower-upper class and lower-lower class. In obtaining the criteria for classifying \((c_ n)\) into upper-upper class or lower-lower class, \((k_ n)\) is just assumed to satisfy \(\lim \sup n^{-1}k_ n<\infty.\) However, in the criterion for classifying \((c_ n)\) into lower-upper class \(k_ n\) is assumed to be of \(O(\log \log n).\) No criterion is presented as regards upper-lower class sequences. When \(k_ n=k\), a constant, \textit{J. Kiefer} [Proc. 6th Berkeley Sympos. math. Statist. Probab., Univ. Calif. 1970, 1, 227-244 (1972; Zbl 0264.62015)] gives a criterion for \((c_ n)\) to belong to lower-lower class or upper-lower class while \textit{G. R. Shorack} and \textit{J. A. Wellner} [Ann. Probab. 6, 349-353 (1978; Zbl 0376.60034)] give a criterion for \((c_ n)\) to belong to upper-upper class or lower-upper class. When \(k_ n\uparrow \infty\) Kiefer has obtained iterated logarithm laws for \((U_{k_ n,n})\).