  A recursive estimator of the conditional geometric median in Hilbert spaces
is studied. It is based on a stochastic gradient algorithm whose aim is to
minimize a weighted L1 criterion and is consequently well adapted for robust
online estimation. The weights are controlled by a kernel function and an
associated bandwidth. Almost sure convergence and L2 rates of convergence are
proved under general conditions on the conditional distribution as well as the
sequence of descent steps of the algorithm and the sequence of bandwidths.
Asymptotic normality is also proved for the averaged version of the algorithm
with an optimal rate of convergence. A simulation study confirms the interest
of this new and fast algorithm when the sample sizes are large. Finally, the
ability of these recursive algorithms to deal with very high-dimensional data
is illustrated on the robust estimation of television audience profiles
conditional on the total time spent watching television over a period of 24
hours.
