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Lecture #10 Notes

# Lecture #10 Notes - High-Dimensional Measures and Geometry...

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High-Dimensional Measures and Geometry Lecture Notes from Feb 18, 2010 taken by ALI S. KAVRUK 6.2.1 Theorem. Let ˜ S n 1 = { x R n : ° x ° = n } and ˜ μ n be the rotation invariant Borel probability measure on ˜ S n 1 .Con s ide r Φ: ˜ S n 1 R given by Φ( x 1 ,...,x n )= x 1 . Let ν n be the the probability measure on R induced by Φ .I f γ 1 is the standard Gaussian measure on R then for any Borel set A lim n →∞ ν n ( A γ 1 ( A ° A 1 2 π e x 2 / 2 dx and the density of ν n converges uniformly on compact subsets to γ 1 . Proof. It is enough to show the frst part oF the claim For an open interval A =( a,b ) . Assume n is large. Note that Φ 1 ( A { x R n : a<x 1 <b } . So For a fxed 1 , Φ 1 ( { x 1 } ) is the n 2 sphere oF radius ± n x 2 1 which we denote by ˜ S n 2 ( ± n x 2 1 ) .Hence ν n ( A )=˜ μ n 1 ( A )) = 1 n ( n 1) / 2 | S n 1 | ² ³´ µ normalization ° b a ˜ S n 2 · ¸ n x 2 1 ¹¶ ² ³´ µ ( n x 2 1 ) n 2 | ˜ S n 2 | º 1+(

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Lecture #10 Notes - High-Dimensional Measures and Geometry...

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