On the Existence of the Moments of the Asymptotic Trace Statistic

Note that inequality (6) cannot be used to bound the moments of ZT_,d uniformly in T,
because the moments of a χ²-distributed random variable depend on the degrees of freedom.

Lemma 2. Let ZT,d be defined as in (3). Then there exist some constants a and b such
that, for all T > d,

(i) E (ZTd´ < a,

(ii) E (ZT,d´ < b.

Proof. Using an inequality of Coope (1994), we get, on account of (5),

ZT,d = tr(A_T^-1BTBT⁰ ) ≤ tr(A_T^-1)tr(BTBT⁰ ),                      (7)

since A_T^-1 and BT B_T⁰ are symmetric and nonnegative definite matrices of the same order.

To deal with AT, let λ1 ≥ ... ≥ λT-1 ≥ λT ≥ 0 and v1, . . . , vT be the eigenvalues and the
associated orthonormal eigenvectors, respectively, of the symmetric and nonnegative definite
(T × T) matrix F = D⁰D. Then, for any m ∈ {1, . . . , T - 1},

Tm

F = X λtvtv'_t ° λm X v_tv'0 =: Fm ,                      (8)

t=1               t=1

where ° denotes the Lowner partial ordering for symmetric matrices. Because of the or-
thonormality of the matrix V = (v1, . . . , vT), V⁰ε has the same distribution as ε, that is, with
the notation of Muirhead (1982), V⁰ε ~ N(0, IT ® I_d). This implies

m

ε^,F_m,ε = λmU, with U := ∑ε'vtv'tε ~ Wd(m,I_d),

t=1

and thus, in view of (4) and (8),

AT = 712^εF^ε ° T2^εFm^ε = Tm U =^: AT,m.                 ⁽⁹⁾

Clearly, AT,_m is almost surely positive definite if m ≥ d. Then (9) leads to A^-_m ° AT¹, so
that we arrive at

T²

tr(AT¹) ≤ tr(ATm) = — tr(U^-1).                       (10)

λ_m

Observing

it follows TT = 0, and λ ₁ ,...,TT_-1 are the eigenvalues of the positive definite matrix F
obtained from F by deleting the last column and the last row. This matrix can be represented

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