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On the Bahadur representation of sample quantiles for dependent sequences. (English) Zbl 1080.62024
From the paper: Let $$(\varepsilon_k)_{k\in\mathbb{Z}}$$ be independent and identically distributed (i.i.d.) random variables and let $$G$$ be a measurable function such that $X_n=G(\dots,\varepsilon_{n-1}, \varepsilon_n)$ is a well-defined random variable. Clearly $$X_n$$ represents a huge class of stationary processes. Let $$F(x)=\mathbb{P}(X_n\leq x)$$ be the marginal distribution function of $$X_n$$ and let $$f$$ be its density. For $$0<p<1$$, denote by $$\xi_p=\inf\{x:F(x)\geq p\}$$ the $$p$$-th quantile of $$F$$. Given a sample $$X_1,\dots,X_n$$, let $$\xi_{n,p}$$ be the $$p$$ th $$(0<p<1)$$ sample quantile and define the empirical distribution function $F_n(x)=n^{-1}\sum^n_{i=1}{\mathbf 1}_{X_i\leq x}.$ For simplicity we also refer to $$\xi_{n,p}$$ as the $$p$$ th quantile of $$F_n$$. We are interested in finding asymptotic representations of $$\xi_{n,p}$$.
We establish the Bahadur representation of sample quantiles for linear and some widely used nonlinear processes. Local fluctuations of empirical processes are discussed. Applications to the trimmed and Winsorized means are given. Our results extend previous ones by establishing sharper bounds under milder conditions and thus provide new insight into the theory of empirical processes for dependent random variables.

##### MSC:
 62G20 Asymptotic properties of nonparametric inference 62G30 Order statistics; empirical distribution functions 60F05 Central limit and other weak theorems 60F17 Functional limit theorems; invariance principles 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
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