A note on supremum of a Kiefer process (Q1365189)

A Kiefer process \(K_F(x,t)\) on \(\mathbb{R}^d\times[0,\infty)\) associated with the df \(F\) is a Gaussian process with zero mean and covariance function \[ EK_F(x,t)\cdot K_F(y,s)=(F(x\land y)-F(x)\cdot F(y)). \] \textit{M. Csörgö, L. Horvath} and \textit{B. Szyszkowicz} (1994) characterized the sample paths of a Kiefer process by proving that \[ \limsup_{t\rightarrow\infty}\sup_{x\in \mathbb{R}^d}\frac{|K_F(x,t)|}{t^{1/2}\psi(t)}<\infty \quad\text{a.s.} \tag{*} \] if and only if \(\psi\) is a positive function on \([1,\infty)\) with \(I(\psi,c)=\int_1^\infty t^{-1}\exp\{-c\psi^2(t)\}dt<\infty\). The main result gives the precise value of the LHS of (*), namely it is a.s. equal to the constant \([2\tau c^*(\psi)]^{1/2}\), where \(\tau=\sup\{F(x)(1-F(x)):x\in \mathbb{R}^d\}\) and \(c^*(\psi)=\inf \{c>0:I(\psi,c)<\infty\}\).