  In this paper, we study the convergence properties of a randomized
block-coordinate descent algorithm for the minimization of a composite convex
objective function, where the block-coordinates are updated asynchronously and
randomly according to an arbitrary probability distribution. We prove that the
iterates generated by the algorithm form a stochastic quasi-Fej\'er sequence
and thus converge almost surely to a minimizer of the objective function.
Moreover, we prove a general sublinear rate of convergence in expectation for
the function values and a linear rate of convergence in expectation under an
error bound condition of Tseng type.
