On the Existence of the Moments of the Asymptotic Trace Statistic



as the inverse of a tridiagonal Minkowski matrix (see Neumann, 2000, and Yueh, 2006), i.e.

/

T

1

T

2

T

3 ...

2

1

/

1

1

0    ..

.0

0

ʌ

T1

T

2

T

2

T

3 ...

2

1

1

2

1     ..

.0

0

-*⅛

T

3

T

3

T

3 ...

2

1

0

1

2 ..

.0

0

F1 =

.

.

.

.

.

.

.

.

.

.

.

.

.

= ( -1)

.

.

.

.

..

.

.

. ..

.

.

.

2

.

2

.

2

...

.

2

.

1

.

0

.

0

0      ..

. 2

.

1

1V

1

1

1

...

1

1

1V

0

0

0    ..

.1

2

Using Theorem 2 of Yueh (2005), the positive (ordered) eigenvalues of F can
as

be represented


λt=

21


-----7------^-τ for t = 1,..., T 1.
cos (%⅛)          , ,

The series expansion of the cosine function provides, for a fixed m {1, . . . , T 1} and as
T→∞,


1


(2m

cos   2T


11)π =


(2m 1)2π2


2(2T - 1)2


+ o(T-3)


and therefore


T2 __(2m

λm T→∞


1)2π2

=: c1 .


(11)


Note that, for fixed m and T → ∞, λm is of the same order T 2 as the sum of all eigenvalues
of
F , since PtT=1 λt = tr(F) = T (T - 1)/2.

With the notation εt = (εt1, . . . , εtd)0, the last term in inequality (7) may be written as

tr(BT BT0 )


1          dd

T2 tr( ε,D,εε,) = ΣΣαi2j , where


(12)


TT

αij


T∑ ∑ εjti<-
=1 t=s+1

To prove (i), we first take the second power in (7) and apply the Cauchy-Schwarz in-
equality, which gives

E(ZT,d) E £tr(AT 1)tr(BtBt)]2 ©E[tr(AT 1)]4 E[tr(BtBt)]4}1 /2 .        (13)

Consequently, it suffices to verify that both expectations on the right-hand side of inequality
(13) are uniformly bounded in
T > d.

In view of (12), E[tr(BT BT0 )]4 is uniformly bounded in T if, for i, j {1, . . . , d},
sup
T E(α8j) . But this follows from εt ~ N(0,Id) i.i.d. and

E ( α8 ¢ = ɪ ∑...∑ ∑ ... ∑ E ( εs ι j ...εs 8 j εt ι i ...εt 8 i )

1 =1


8=1t1=s1+1 t8=s8+1


because E (εs1j . . . εs8jεt1i . . .εt8i ) = 0 if more than eight of the subscripts s1, . . . , s8, t1, . . . , t8

differ.



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