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It was shown by Burchard and Fortier that the expected $L^1$ distance between$f^*$ and $n$ random polarizations of an essentially bounded function $f$ withsupport in a ball of radius $L$ is bounded by$2dm(B_{2L})||f||_{\infty}n^{-1}$. This article complements and extends thisresult. The expected $L^1$ distance is bounded by $c_nn^{-1}$ with$\limsup_{n\rightarrow \infty}c_n \leq 2^{d+1}||\nabla f||_1$ for every $f \inW_{1,1}(B_L) \cap L^{\infty}(B_L)$. Furthermore, the expected $L^1$ distance is$O(n^{-1/q})$ for $f \in L^p(B_L)$ with $p>1$ and $\frac{1}{p} + \frac{1}{q} =1$. The rate $n^{-1}$ is best possible: $n$ times the measure of the symmetricdifference between the random polarizations of a ball and its correspondingSchwarz symmetrization converges in distribution to a random variable withmoments that are derived. It is also shown that the expected symmetricdifference between the random polarizations of a measurable set in $B_L$ andits corresponding Schwarz symmetrization is slower than $n^{-r}$ for any $r>2d$and if the rate is $n^{-1}$ then $n$ times the measure of the symmetricdifference between the random polarizations of the set and its correspondingSchwarz symmetrization converges in distribution. A new sequence of randompolarizations is introduced such that the transition probability depends on thestate of the underlying Markov chain. For compact sets with finite perimeter,the rate of convergence is $O(n^{-3/2})$ when $d=1$ and $O(n^{-(1 +\frac{d-1}{d(d+1)})})$ for $d>1$. Finally it is shown that for every compactset $A$ in $\mathbb{R}$ with finite perimeter there exists a sequence ofpolarizations $A_n$ of $A$ converging exponentially to its Schwarzsymmetrization.

A note on the rate of convergence for a sequence of random polarizations