Strong approximation of the quantile processes and its applications under strong mixing properties (Q1340277)

Let \(\{X_ n : n \in \mathbb{Z}\}\) be a real-valued strictly stationary sequence of random variables taking values in a space \((\Omega, {\mathcal F},P)\) with the common distribution function \(F(x)\). The random variables satisfy a strong mixing condition. Furthermore some regularity conditions on the distribution function \(F(x)\) are given, where \(f=F'\). The \(n\)th empirical and quantile measures are given by \[ F_ n(x) = n^{-1} \sum^ n_{i=1} I_{(-\infty,x]} (X_ i) \quad \text{and} \quad F_ n^{-1} (s) = \inf \bigl \{x:F_ n (x) \geq s \bigr\}, \] respectively. It is shown that the sequence of quantile processes \(\{n^{1/2} f(F^{- 1} (s)) (F_ n^{-1} (s)-F^{-1} (s));\) \(0 < s < 1\}\) behaves like a sequence of Brownian bridges \(\{B_ n (s);\;0 < s < 1\}\). The latter is then utilized to construct (i) simultaneous bounds for the unknown quantile function \(F^{-1} (s)\), and (ii) a tolerance interval for predicting a future observation. -- Also some numerical investigations of the results are discussed.