## 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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