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 $N\to \infty$ to the zero process. This gives rise to a polynomial decomposition for Brownian motion. We further study the fluctuation processes obtained through scaling by $\sqrt{N}$ and show that they converge in finite‐dimensional distributions as $N\to \infty$ 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.

中文翻译：

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迭代Kolmogorov环的半圆定律和解相关现象

我们考虑时间间隔[0,1]上的标准一维布朗运动，条件是其迭代时间积分消失到阶 $ñ$。我们证明了可以用移位的勒让德多项式和原始布朗运动明确地表示所得过程，并使用这些表示来证明这些过程弱收敛于$ñ\to \infty$归零过程。这引起布朗运动的多项式分解。我们进一步研究了通过缩放获得的波动过程。$\sqrt{ñ}$ 并证明它们收敛于有限维分布 $ñ\to \infty$到独立的零均值高斯随机变量的集合，其方差遵循缩放的半圆。波动结果是勒让德多项式极限定理的结果，该定理量化了它们的完整性和正交性。在后者的证明中，我们遇到了加泰罗尼亚三角。