Strong Approximations of the Quantile Process

Abstract

Let $q_n(y), 0 < y < 1,$ be a quantile process based on a sequence of i.i.d. rv with distribution function $F$ and density function $f.$ Given some regularity conditions on $F$ the distance of $q_n(y)$ and the uniform quantile process $u_n(y),$ respectively defined in terms of the order statistics $X_{k:n}$ and $U_{k:n} = F(X_{k:n}),$ is computed with rates. As a consequence we have an extension of Kiefer's result on the distance between the empirical and quantile processes, a law of iterated logarithm for $q_n(y)$ and, using similar results for the uniform quantile process $u_n(y),$ it is also shown that $q_n(y)$ can be approximated by a sequence of Brownian bridges as well as by a Kiefer process.