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.



More intriguing information

1. The Role of Immigration in Sustaining the Social Security System: A Political Economy Approach
2. PRIORITIES IN THE CHANGING WORLD OF AGRICULTURE
3. Wounds and reinscriptions: schools, sexualities and performative subjects
4. On the Relation between Robust and Bayesian Decision Making
5. Heavy Hero or Digital Dummy: multimodal player-avatar relations in FINAL FANTASY 7
6. ‘Goodwill is not enough’
7. Accurate and robust image superresolution by neural processing of local image representations
8. Evolving robust and specialized car racing skills
9. Menarchial Age of Secondary School Girls in Urban and Rural Areas of Rivers State, Nigeria
10. The name is absent