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.

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

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

Note that, for fixed m and T → ∞, λm is of the same order T ² as the sum of all eigenvalues
of F , since ^P_t^T₌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

TT

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(BtB_t)]² ≤ ©E[tr(AT ¹)]4 E[tr(BtB_t)]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 B_T⁰ )]⁴ is uniformly bounded in T if, for i, j ∈ {1, . . . , d},
supT E(α⁸j) < ∞. But this follows from ε_t ~ N(0,I_d) i.i.d. and

E ( α8 ¢ = ɪ ∑...∑ ∑ ... ∑ E ( ε_s ι j ...ε_s 8 j ε_t ι i ...ε_t 8 i )

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

differ.

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