  We derive conditions under which random sequences of polarizations (two-point
symmetrizations) converge almost surely to the symmetric decreasing
rearrangement. The parameters for the polarizations are independent random
variables whose distributions need not be uniform. The proof of convergence
hinges on an estimate for the expected distance from the limit that also yields
a bound on the rate of convergence. In the special case of i.i.d. sequences, we
obtain almost sure convergence even for polarizations chosen at random from
suitable small sets. As corollaries, we find bounds on the rate of convergence
of Steiner symmetrizations that require no convexity assumptions, and show that
full rotational symmetry can be achieved by randomly alternating Steiner
symmetrization in a finite number of directions that satisfy an explicit
non-degeneracy condition. We also present some negative results on the rate of
convergence and give examples where convergence fails.
