Using the fact that tr (A ® B) = tr (A) tr (B) and the Cauchy-Schwarz inequal-
ity, we have
T12 XX tr(
Gx,T (xit xi) (χis xi) Gx,T
®Gx,T (xit xi) (xis xi) Gx,T
⅛∑ E©tr £Gx,T (Xit
xi) (xis
Xi)0 Gχ,τ]}
T X ∣∣Gχ,τ (xit
t
xi)k
Similarly,
T12 XXtr(
Gx,T (xit xi) (xis xi) Gx,T
®Gx,T (xis xi) (xit xi) Gx,T
T X ∣G∙,T (xi.
t
- xi )k2
and
1 XX tr f Gx,T (xit — xi) (xit — cci)° GX,T
T2 tl s' × ®Gx,T (xis - xi) (xis - xi) Gx,T
= (T X
tr Gx,T (xit
- xi) (xit
T X ∣∣Gχ,T (xit
t
- xi)k2
Thus, the right hand side of (84) is less than or equal to
3M1 κ4v sup sup E
i,T 1≤t≤T
T X ∣∣Gχ,T (xit
t
— xi)k2
Note that
sup sup E
i,T 1≤t≤T
2
T X ∣∣Gχ,t (xit — xi)k2
t
≤ sup sup E
i,T 1≤t≤T
2
T X l∣Gχ,Txitk2
t
≤ M supi,T sup1≤t≤T E ∣D1Tx1,it ∣4 + supi,T sup1≤t≤T E ∣x2,it ∣4
+supi,Tsup1≤t≤TE∣x3,it∣4
53
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