Proving consistency of quantile regression estimators I have asked the question on the statistic section on stack exchange, but no one was able to give me an answer. I think this is actually is a question that touches econometrics so I am going to ask it here (I would like to use the answer to this in an econometric context): 

I am a bit confused. The assumptions that have to be fulfilled so that OLS estimators are consistent (and efficient) are fairly straightforward.

I am currently trying to prove consistency of quantile regression (QR) estimators.

I have found the following lecture notes: https://eml.berkeley.edu/~powell/e241a_sp10/qrnotes.pdf

There are 4 assumptions listed (page 3) for the proof of consistency of the QR estimators. The first one being that the data $x_{t},y_{t}$ given $t=1,2,..,n$ has to be i.i.d. (independent and indentically distributed). 

In the case of OLS the data only has to be covariance-stationary. To my understanding the above mentioned assumption of i.i.d. data rules out the possibility of for example autoregressive processes, since in that case the data points are not independent from one another. This in turn puts quite restrictive boundaries on the possible applications of QR.

Am I missing something here? Could someone clarify the assumptions for consistency of the QR estimators for me? 

Thank you!
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.