Random Bit Quadrature and Approximation of Distributions on Hilbert Spaces
For researchers in computational statistics and numerical analysis, this work provides tight error bounds for Monte Carlo methods using random bits, a practical constraint in hardware implementations.
The paper determines the asymptotics of the n-th minimal error for approximating expectations of Lipschitz functionals of Gaussian random elements using only random bits, showing that restricted Monte Carlo algorithms are not inferior to arbitrary ones and that random bit multilevel algorithms are optimal. It also solves related quantization problems for Gaussian measures and scalar SDEs.
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.