Pointwise and uniform asymptotics of the Vervaat error process (Q1866068)

Let \(\alpha_n(t), 0\leq t\leq 1,\) and \(\beta_n(t), 0\leq t\leq 1,\) denote the uniform empirical and quantile processes of sample size \(n\geq 1\), respectively. The sum \(R_n=\alpha_n+\beta_n\) is known in the literature as the Bahadur-Kiefer process. For the integrated Bahadur-Kiefer process \(V_n(t)=2\sqrt{n}\int_0^t R_n(s) ds, 0\leq t\leq 1,\) called uniform Vervaat process, set \(Q_n(t)=V_n(t)-\alpha_n^2(t), 0\leq t\leq 1\). It is known that \(\sup_{0\leq t\leq 1}|Q_n(t)|\to 0\) in probability as \(n\to\infty\) so that \(Q_n\) can be interpreted as the ``Vervaat error process'' in the representation \(V_n=\alpha_n^2+Q_n\). The asymptotic behaviour of \(Q_n\), both for fixed \(t\) and uniformly in \(0\leq t\leq 1\), is investigated, including a strong approximation via a Kiefer process.