Normal Limits, Nonnormal Limits, and the Bootstrap for Quantiles of   Dependent Data
  We will show under very weak conditions on differentiability 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 differentiability condition, quantiles might have a non-normal asymptotic distribution and the bootstrap might fail. 
O. Sh. Sharipov, M. Wendler
math.ST math.PR stat.TH