随机比特正交与在希尔伯特空间上分布的近似
Random Bit Quadrature and Approximation of Distributions on Hilbert Spaces
AI总结：
研究在分离希尔伯特空间上的高斯随机元素期望近似，提出受限蒙特卡洛算法，分析其误差衰减与协方差算子特征值衰减的关系，证明随机比特多级算法最优。
摘要：
我们研究高斯随机元素X在分离希尔伯特空间H上的期望E(f(X))的近似，其中f是Lipschitz连续函数。考虑仅使用随机比特而非随机数的受限蒙特卡洛算法，确定n阶最小误差的渐进行为，与协方差算子特征值衰减相关。分析表明，受限蒙特卡洛算法不逊于任意蒙特卡洛算法，合适随机比特多级算法最优。此问题分析导致量化问题的变种，即用有限点均匀分布近似希尔伯特空间上的概率测度。确定标准正态分布、上述高斯测度及标量自主SDE的最优近似误差渐进行为（至乘法常数）。
英文摘要：
We study the approximation of expectations $\E(f(X))$ for Gaussian random elements $X$ with values in a separable Hilbert space $H$ and Lipschitz continuous functionals $f \colon H \to \R$. We consider restricted Monte Carlo algorithms, which may only use random bits instead of random numbers. We determine the asymptotics (in some cases sharp up to multiplicative constants, in the other cases sharp up to logarithmic factors) of the corresponding $n$-th minimal error in terms of the decay of the eigenvalues of the covariance operator of $X$. It turns out that, within the margins from above, restricted Monte Carlo algorithms are not inferior to arbitrary Monte Carlo algorithms, and suitable random bit multilevel algorithms are optimal. The analysis of this problem leads to a variant of the quantization problem, namely, the optimal approximation of probability measures on $H$ by uniform distributions supported by a given, finite number of points. We determine the asymptotics (up to multiplicative constants) of the error of the best approximation for the one-dimensional standard normal distribution, for Gaussian measures as above, and for scalar autonomous SDEs.
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