A semicircle law and decorrelation phenomena for iterated Kolmogorov loops
A semicircle law and decorrelation phenomena for iterated Kolmogorov loops
复制标题
迭代柯尔莫哥洛夫环的半圆定律和去相关现象
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发表时间：
2019
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通讯作者：
Karen Habermann
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作者：
Karen Habermann
我们考虑时间区间[0，1]上的标准一维布朗运动，条件是具有直到N阶的消失迭代时间积分。我们证明了所得到的过程可以用移位勒让德多项式和原始布朗运动来明确表示，并且我们使用这些表示来证明过程当N→∞时弱收敛到零过程。这就产生了布朗运动的多项式分解。我们进一步研究了通过N尺度化得到的涨落过程，并证明了它们在有限维分布中当N→∞时收敛于一个独立的零均值高斯随机变量的集合，这些变量的方差遵循一个尺度圆。波动的结果是勒让德多项式的极限定理，量化其完整性和正交性的后果。在证明后者时，我们遇到了一个加泰罗尼亚三角形。
We consider a standard one‐dimensional Brownian motion on the time interval [0,1] conditioned to have vanishing iterated time integrals up to order N . We show that the resulting processes can be expressed explicitly in terms of shifted Legendre polynomials and the original Brownian motion, and we use these representations to prove that the processes converge weakly as N→∞ to the zero process. This gives rise to a polynomial decomposition for Brownian motion. We further study the fluctuation processes obtained through scaling by N and show that they converge in finite‐dimensional distributions as N→∞ to a collection of independent zero‐mean Gaussian random variables whose variances follow a scaled semicircle. The fluctuation result is a consequence of a limit theorem for Legendre polynomials which quantifies their completeness and orthogonality property. In the proof of the latter, we encounter a Catalan triangle.