Empirical Quantile CLTs for Time Dependent Data

Details Empirical Quantile CLTs for Time Dependent Data Empirical Quantile CLTs for Time Dependent Data

2011-11-19

arxiv.org/abs/1111.4591v1

Mathematics

Probability

52 pages

Scientific paper

We establish empirical quantile process CLTs based on $n$ independent copies of a stochastic process $\{X_t: t \in E\}$ that are uniform in $t \in E$ and quantile levels $\alpha \in I$, where $I$ is a closed sub-interval of $(0,1)$. Typically $E=[0,T]$, or a finite product of such intervals. Also included are CLT's for the empirical process based on $\{I_{X_t \le y} - \rm {Pr}(X_t \le y): t \in E, y \in R \}$ that are uniform in $t \in E, y \in R$. The process $\{X_t: t \in E\}$ may be chosen from a broad collection of Gaussian processes, compound Poisson processes, stationary independent increment stable processes, and martingales.

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