integral theorems for Monte Carlo

Nhat Ho and Stephen G. Walker have just arXived a paper on the use of (Fourier) integral theorems for Monte Carlo estimators, following the earlier entry of Parzen: namely that for any integrable function,

m(y)=\frac{1}{(2\pi)^d}\int_{\mathbb R^d}\int_{\mathbb R^d}\cos(s^\text{T}(y-x))m(x)\text dx\text ds

which can be turned into an estimator of a density m based on a sample from m. This identity can be rewritten as

m(y)=\lim_{R\to\infty}\frac{1}{\pi^d}\int_{\mathbb R^d}\prod_{i=1}^d\dfrac{\sin(R(y_i-x_i))}{y_i-x_i}\;m(x)\,\text dx

and the paper generalises this identity to all cyclic functions. Even though it establishes that sin is the optimal choice. After reading this neat result, I however remain uncertain on how this could help with Monte Carlo integration.

One Response to “integral theorems for Monte Carlo”

  1. Ho and Walker shows how Fourier integrals lead to natural Monte Carlo estimators in an earlier paper:

    Ho, N. and Walker, S.G., 2020. Multivariate smoothing via the Fourier integral theorem and Fourier kernel. arXiv preprint arXiv:2012.14482.

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