Biomedical Engineering Reference
In-Depth Information
It follows from (9.22) that
P
2
(; b
n
;
b
n
) 2P[`(; b
n
;
0
)(; b
n
;
b
n
)] + o
p
(1):
(9.23)
Because 2ab 2a
2
+ 2
1
b
2
for any a;b 2 R, then by (9.23) we have
P
2
(; b
n
;
b
n
) 2P`
2
(; b
n
;
0
) + 2
1
P
2
(; b
n
;
b
n
) + o
p
(1):
This implies
P
2
(; b
n
;
b
n
) 4P`
2
(; b
n
;
0
) + o
p
(1) = 4P`
2
(;
0
;
0
) + o
p
(1):
(9.24)
Therefore, P
2
(; b
n
;
b
n
) = O
p
(1). Consequently,
P
2
(;
0
;
b
n
) = O
p
(1):
(9.25)
By the definition of
0
, `(x;
0
;
0
) and (x;
0
;) are orthogonal in L
2
(P), that
is, P[`(;
0
;
0
)(;
0
;)] = 0 for any 2H. We have
P[`(; b
n
;
0
)(; b
n
;
b
n
)]
Pf[`(; b
n
;
0
) `(;
0
;
0
)][(; b
n
;
b
n
) (;
0
;
b
n
)]g
=
+Pf`(;
0
;
0
)[(; b
n
;
b
n
) (;
0
;
b
n
)]g
+Pf(;
0
;
b
n
)[`(; b
n
;
0
) `(;
0
;
0
)]g:
This equation, (9.25), assumption (A2), and the Cauchy{Schwarz inequality
imply that
P[`(; b
n
;
0
)(; b
n
;
b
n
)] !
p
0:
(9.26)
Combining (9.23) and (9.26), we obtain
P
2
(; b
n
;
b
n
) !
p
0:
(9.27)
Now,
P`
2
(b
n
;
b
n
; x)
P`
2
(;
0
;
0
) + 2P[`(;
0
;
0
)f`(; b
n
;
0
) `(;
0
;
0
)g]
+P[`(; b
n
;
0
) `(;
0
;
0
)]
2
+2P[`(; b
n
;
0
)(; b
n
;
b
n
)] + P
2
(; b
n
;
b
n
):
=
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