Lecture #10 Notes

Proof we will use the fourier transform consider icx

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Unformatted text preview: x￿ = n} and µn be the rotation invariant Borel ˜ ˜ probability measure on S n−1 . Consider ˜ Φ : S n−1 → Rk given by Φ(x1 , ..., xn ) = (x1 , ..., xk , 0, ..., 0) Let νn,k be the the probability measure on Rk induced by Φ. If γk is the standard Gaussian measure on Rk then for any Borel set A ⊆ Rk ￿ 1 2 lim νn,k (A) = γk (A) = e−￿x￿ /2 dx k/2 n→∞ A (2π ) and the density of νn,k converges uniformly on compact subsets to density of γk . Proof. We will use the Fourier Transform. Consider ￿ ￿ ic·x Fn (c) = e dνn,k (x) and G(c) = Rk Rk By definition Fn (c) = ￿ ˜ S n−1 1 2 2 eic·x e−￿x￿ /2 dx = e−￿c￿ /2 . k/2 (2π ) eic·x dµn (x). ˜ Since µn is rotation invariant, by a suitable rotation, we may assume that c = (￿c￿, 0, ..., 0). So ˜ by using the previous theorem ￿ Fn (c) = eic·x dνn (x1 ) . R because only the projection of x on the first coordinate enters in c · x. Thi...
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