1. Substitute Eq. (3) under expectation sign and use addiditve property of expectation
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1. Substitute Eq. (3) under expectation sign and use addiditve property of expectation. 2. All expectations are equal. (3). 4. Use the results of the first paragraph. 5. Algebraic rearrangements. 6. Used triangle inequality |a + b| ≤ |a| + |b|. 7. (8). (9). - PowerPoint PPT Presentation
Transcript of 1. Substitute Eq. (3) under expectation sign and use addiditve property of expectation
1. Substitute Eq. (3) under expectation signand use addiditve property of expectation
2. All expectations are equal
(3)
4. Use the results of the first paragraph
5. Algebraic rearrangements
6. Used triangle inequality|a + b| ≤ |a| + |b|
TT SdxSxk
dxxk
)/(
1)(
7.
(8)
(9)
2/1
2
2/1
2 )()()()( b
a
b
a
b
a
dxxgdxxfdxxgxf
10. Discrete version of Schwartz inequality below+
Equation (9) and
dxxk )(2
21
1
1
1
22 |)(||||)(|||Tj
Tj
Tj
Tj
jjjj
)1(|)(||||)(||)(|||||||
ojjT
SjT
T
SjS
Tj
qT
Tj
qT
Tj
qT
11.
Schwartz inequality:
See the next page
)0()!2(
1)(1lim
...)0(!
1...)0(
!2
1)0(
!1
11)(
)2(20
)(
0
2
nnx
knx
xk
xkq
xkxkxk
12.
13. Apply definition of kq from paragraph 4 to this expression
14. First expression multiplied by is o(1) and thus, this limit exists
because all previous expressions are algebraic transformations. Lastlimit is finite (see paragraph 6).
T
S qT
15. Consequence of paragraph 5 and
0|)(|||
)1(|)(|||
T
SCjj
T
k
OjjSk
qT
j
j
j
j
qqTq
Because in second Equation (15), the sumdoes not depend on T, the sum is finite.Here and below C is some constant.
See page 5
16. By the Eq. (15) )1()(
oT
fk qq
Because kqf(q) does not depend on T,1)( q
q CkfFinite, by paragraph 6Finite by
construction
17. The expressionis equal to q
TCS
18. Putting together the results of paragraph5, expression for the bias from the introductionand the definition of from paragraph 7. we obtain this result.
)(qf
