  This paper studies the inference problem in quantile regression (QR) for a
large sample size $n$ but under a limited memory constraint, where the memory
can only store a small batch of data of size $m$. A natural method is the
na\"ive divide-and-conquer approach, which splits data into batches of size
$m$, computes the local QR estimator for each batch, and then aggregates the
estimators via averaging. However, this method only works when $n=o(m^2)$ and
is computationally expensive. This paper proposes a computationally efficient
method, which only requires an initial QR estimator on a small batch of data
and then successively refines the estimator via multiple rounds of
aggregations. Theoretically, as long as $n$ grows polynomially in $m$, we
establish the asymptotic normality for the obtained estimator and show that our
estimator with only a few rounds of aggregations achieves the same efficiency
as the QR estimator computed on all the data. Moreover, our result allows the
case that the dimensionality $p$ goes to infinity. The proposed method can also
be applied to address the QR problem under distributed computing environment
(e.g., in a large-scale sensor network) or for real-time streaming data.
