Nonstandard strong laws for local quantile processes (Q5928934)

Let \(\{U_n:n\geq 1\}\) be a sequence of independent and identically distributed uniform \((0,1)\) random variables, and for every integer \(n\geq 1\), let \(V_n(x), -\infty<x<\infty\), denote the empirical quantile function pertaining to \(U_1,\dots,U_n\) (with \(V_n(x)=0\) for \(x<0\) and \(V_n(x)=V_n(1)\) for \(x\geq 1\)). Let \(\beta_n(x)=n^{1/2}(V_n(x)-x)\) denote the corresponding uniform empirical quantile process, and consider the increment functions NEWLINE\[NEWLINE\zeta_n(a,t;s)=\beta_n(t+as)-\beta_n(t) \quad \text{and}\quad v_n(a,t;s)=V_n(t+as)-V_n(t)NEWLINE\]NEWLINE for \(a\geq 0\) and \(-\infty<s,t<\infty\). The present paper describes in detail the asymptotic almost sure behavior of \(\zeta_n(h_n,t_0;1)\), \(\sup_{0\leq s\leq 1}\zeta_n(h_n,t_0;s)\) and \(v_n(h_n,t_0;1)\) for \(h_n\sim Bn^{-1}\log n\) with \(0<B<\infty\) and \(0<t_0<1\), both in a one-dimensional and functional form. As turns out, the limit laws for \(0<t_0<1\) are different from those valid for \(t_0=0\).