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n
S  a
n
n âˆ’ x1
ï¿¿ nâˆ’2 ï¿¿
ï¿¿1
ï¿¿ï¿¿
2
1 Î“(n/2) b
x2 n
x2
1
1
=âˆš
1âˆ’
1+
dx1
nâˆ’1
n
n âˆ’ x2
Ï€ n Î“( 2 ) a
1
ï¿¿ nâˆ’2
ï¿¿ bï¿¿
1 Î“(n/2)
x2 n
1
1 /2
(by M V T ) = âˆš
(1 + Î·n )
1âˆ’
dx1
n
Ï€ n Î“( nâˆ’1 )
a
2 for all large enough n for some constant
0 â‰¤ Î·n â‰¤ max(a, b)2
n âˆ’ max(a, b)2
1 so consequently limnâ†’âˆž Î·n = 0. Note that the integrand
ï¿¿ x2
1âˆ’ 1
n ï¿¿ nâˆ’2
n 2 â†’ eâˆ’x1 /2 not only pointwise but also uniformly over the closed interval [a, b]. (Which can be deduced by
taking logarithm of both sides and using the inequality
ï¿¿ï¿¿
ï¿¿
ï¿¿
2
2ï¿¿
ï¿¿
ï¿¿ln 1 âˆ’ x1 âˆ’ x1 ï¿¿ â‰¤ C
ï¿¿
n
2 ï¿¿ n2 where C depends on a and b.) Cosequently we can use the dominated convergence theorem and
the result follows.
Similirlarly this results holds for projections onto higher dimensional subspaces.
âˆš
Ëœ
6.2.2 Corollary. Let S nâˆ’1 = {x âˆˆ Rn : ï¿...
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This note was uploaded on 01/16/2014 for the course MATH 6397 taught by Professor Staff during the Spring '08 term at University of Houston.
 Spring '08
 Staff
 Geometry, Probability

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