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- W2920789816 abstract "The coordinates along any fixed direction(s), of points on the sphere $S^{n-1}(sqrt{n})$, roughly follow a standard Gaussian distribution as $n$ approaches infinity. We revisit this classical result from a nonstandard analysis perspective, providing a new proof by working with hyperfinite dimensional spheres. We also set up a nonstandard theory for the asymptotic behavior of integrals over varying domains in general. We obtain a new proof of the Riemann--Lebesgue lemma as a by-product of this theory. We finally show that for any function $f co mathbb{R}^k to mathbb{R}$ with finite Gaussian moment of an order larger than one, its expectation is given by a Loeb integral integral over a hyperfinite dimensional sphere. Some useful inequalities between high-dimensional spherical means of $f$ and its Gaussian mean are obtained in order to complete the above proof. A review of the requisite nonstandard analysis is provided." @default.
- W2920789816 created "2019-03-11" @default.
- W2920789816 creator A5065051750 @default.
- W2920789816 date "2019-01-29" @default.
- W2920789816 modified "2023-09-23" @default.
- W2920789816 title "Limiting Probability Measures" @default.
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