On the ranked excursion heights of a Kiefer process (Q1827466)

The authors study the path properties of the process \(t \rightarrow M^{*}_{j}(t)\), where \(M^{*}_{1}(t) \geq M^{*}_{2}(t) \geq \cdots \geq M^{*}_{j}(t) \geq \cdots \geq 0 \) are the ranked excursion heights of \(K( \cdot, t)\) and \((K(s,t), 0 \leq s \leq 1, t \geq 1)\) is a Kiefer process. A Kiefer process is a continuous two-parameter centered Gaussian process indexed by \( [0,1]\times \mathbb{R}_{+} \) whose covariance function is given by \(\mathbb{E}(K(s_{1}, t_{1}),K(s_{2}, t_{2}))= (s_{1} \wedge s_{2} -s_{1}s_{2})t_{1} \wedge t_{2},\) \(0 \leq s_{1}, s_{2} \leq 1,\) \(t_{1}, t_{2} \geq 0.\) Kiefer introduced this process \(K\) to approximate the empirical process. The authors show that two laws of the iterated logarithm are established to describe the asymptotic behaviors of \(M^{*}_{j}(t)\) as \(t\) goes to infinity. The results are as follows: Fix \(j \geq 1\). We have \[ \lim\sup_{t\rightarrow\infty}\frac{M^{*}_{j}(t)}{\sqrt{t\log\log t}} = \frac{1}{j\sqrt{2}} \;\text{ a.s.} \] Fix \(j \geq 2.\) We have \[ \lim\inf_{t\rightarrow\infty}\frac{(\log t)^\chi}{\sqrt{t}}M^{*}_{j}(t)= \begin{cases} 0, & \text{if } \chi \leq \frac{1}{2}, \\ \infty, & \text{if } \chi > \frac{1}{2}\;\text{ a.s.} \end{cases} \]