随机极化
Random polarizations
摘要：
我们推导出随机序列的极化（两点对称化）几乎必然收敛到对称递减重排的条件。极化的参数是独立的随机变量，其分布不需要是均匀的。收敛性的证明取决于对与极限的预期距离的估计，该估计也产生收敛收敛速率的界。在i.i.d.的特殊情况下。序列，我们得到几乎必然收敛，甚至从适当的小集合中随机选择的极化。作为推论，我们发现界的收敛速度收敛速率的Steiner对称化，不需要凸性假设，并表明，充分的旋转对称性可以实现随机交替Steiner对称化在有限数量的方向，满足明确的非退化条件。我们还提出了一些负面的结果的收敛速率，并给出例子，收敛失败。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
