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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
lim νn,k (A) = γk (A) =
e−x /2 dx
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
Fn (c) =
e dνn,k (x) and G(c) =
Rk Rk By deﬁnition
Fn (c) = ˜
S n−1 1
eic·x e−x /2 dx = e−c /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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