Noncentral Limit Theorem and the Bootstrap for Quantiles of Dependent Data

We will show under minimal conditions on di erentiability and dependence that the central limit theorem for quantiles holds and that the block bootstrap is weakly consistent. Under slightly stronger conditions, the bootstrap is strongly consistent. Without the di erentiability condition, quantiles might have a non-normal asymptotic distribution and the bootstrap might fail. 1. Limit Behaviour of Quantiles Let (Xn)n∈Z be a stationary sequence of real-valued random variables with distribution function F and p ∈ (0, 1). Then the p-quantile tp of F is de ned as tp := F −1 (p) := inf { t ∈ R ∣∣F (t) ≥ p} and can be estimated by the empirical p-quantile, i.e. the dnp e-th order statistic of the sample X1 . . . , Xn. This also can be expressed as the p-quantile F −1 n (p) of the empirical distribution function Fn (t) := 1 n ∑n i=1 1Xi≤t. It is clear that F−1 n (p) is greater than tp i Fn (tp) is smaller than p. In the case of independent random variables, this converse behaviour was exploited by Bahadur [3] to show that the asymptotic behaviour of the quantile F−1 n (p) and the empirical distribution function Fn at the point tp is the same under the condition that F is di erentiable twice in a neighborhood of tp. Ghosh [7] established a weak form of the Bahadur representation, only assuming that F is di erentiable once in tp. He showed that F−1 n (p)− tp = p− Fn (tp) f (tp) +Rn, where f = F ′ is the derivative of the distribution function and Rn = oP ( n− 1 2 ) . As noticed by Lahiri [12], the condition that F is di erentiable is also necessary for the central limit theorem for F−1 n (p). Ghosh and Sukhatme [9] and de Haan and Taconis-Haantjes [10] investigated the noncentral limit theorem for F−1 n (p), if F is not di erentiable, but regular varying. We will extend their results to strongly mixing random variables. There is a broad literature on the Bahadur representation for strongly mixing data. Babu and Singh [2] proved such a representation under an exponentially fast decay of the strong mixing coe cients, this was weakened by Yoshihara [21], Sun [18] and Wendler [20] to a polynomial decay of the strong mixing coe cients. All these articles deal with the case that F is di erentiable. 2000 Mathematics Subject Classi cation. 62G30; 62G09; 60G10.