Some asymptotic properties between smooth empirical and quantile processes for dependent random variables

Let $\widehat F_n$ be the smooth empirical estimator obtained by integrating a kernel type density estimator based on a random sample of size $n$ from continuous distribution function $F$. The almost sure deviation between smooth empirical and smooth quantile processes is investigated under $\phi$-mixing and strong mixing conditions. We derive a pointwise as well as a uniform Bahadur–Kieffer type representation for smooth quantiles under cases of $\phi$-mixing and strong mixing. These results extend those of Babu and Singh [J. Multivariate Anal., 8 (1978), pp. 532–549] and Ralescu [J. Statist. Plann. Inference, 32 (1992), pp. 243–258].