Central limit theorem of the perturbed sample quantile for a sequence of $m$-dependent nonstationary random process

Given a sequence $X_i$, $i\ge1$, of $m$-dependent nonstationary random variables, the usual perturbed empirical distribution function is $\widehat F_n(x)=n^{-1}\sum_{i=1}^nK_n(x-X_i)$, where $K_n$, $n\ge1$, is a sequence of continuous distribution functions converging weakly to the distribution function with a unit mass at zero. In this paper, we study the perturbed sample quantile estimator $\hat\xi_{np}=\inf\{x\in\mathbf{R},\widehat F_n(x)\ge p\}$, $0<p<1$, based on a kernel $k$ associated with $K_n$ and a sequence of window-widths $a_n>0$. Under suitable assumptions, we prove the weak as well as the strong consistency of $\hat\xi_{np}$ and also provide sufficient conditions for the asymptotic normality of $\hat\xi_{np}$. Our central limit theorem for $\hat\xi_{np}$ generalizes a result of Sen [15] and also extends the results of Nadarya [8] and Ralescu and Sun [12].