  For a set of $p$-variate data points $\boldsymbol y_1,\ldots,\boldsymbol
y_n$, there are several versions of multivariate median and related
multivariate sign test proposed and studied in the literature. In this paper we
consider the asymptotic properties of the multivariate extension of the
Hodges-Lehmann (HL) estimator, the spatial HL-estimator, and the related test
statistic. The asymptotic behavior of the spatial HL-estimator and the related
test statistic when $n$ tends to infinity are collected, reviewed, and proved,
some for the first time though being used already for a longer time. We also
derive the limiting behavior of the HL-estimator when both the sample size $n$
and the dimension $p$ tend to infinity.
