Proving consistency of quantile regression estimators
I can give you literature. An authoritative book on the subject is Koenker, R. (2005). Quantile regression. Cambridge university press.. In there we read (p.126)


  4.6 QUANTILE REGRESSION ASYMPTOTICS UNDER DEPENDENT CONDITIONS    Relaxation of the independence condition on the observations is
  obviously possible and offers considerable scope for continuing
  research. Bloomfield and Steiger (1983) showed the asymptotic
  normality of a median autoregression estimator for a model in which
  the observations were assumed to be stationary and ergodic martingale
  differences. Weiss (1991) considers models under α-mixing conditions.
  Portnoy (1991) considers a considerably more general class of models
  with “m-decomposible” errors, a condition that subsumes m-dependent
  cases with *m* tending to infinity sufficiently slowly... More
  recently, Koul and Mukherjee (1994) and Mukherjee (2000) have
  considered the asymptotic behavior of both the primal and dual
  quantile regression process under conditions of long-range dependence
  of the error process. This work relies heavily on the theory of
  weighted empirical processes developed by Koul (1992).


The section continues with a presentation of the properties of Quantile regression for AR, ARMA, and ARCH models.