3. Convergence in distribution of a class of random matrices and their
generalized eigenvalues
Our tests will be based on the following pair of random matrices:
m
 = Va χιTk,
m n, k n, k,
k= 1
m
Bn, ≈ Σ bn, fink ■
k= 1
depending on a natural number m ≥ q, where
m;(F„(•))/fin b = fin M(F1(•))
an, k .-------------------------------------------------------- , n, k .---------------------
y ʃʃFk(x )Fk (y )min(x, y ) dxdy y ʃFk(x )2 dx
with
nn
M(Fk) = fi∑F((t/n)z,, Mfiz(F.) - ɪ Fk( t/n )ʌzt •
t= 1 t= 1
where {Fk } is a class of differentiable real functions on the unit interval [0,1]. As will be
shown below, the functions Fk can be chosen such that
( ʌ
D . .,,ɔ m _ . .,,ɔ
→ C (1) C (1) T £ XkXkτ C (1) C (1) T) ,
(5)
→C(1) C(1) 1/2 ∑ Y1Y1T C(1) C(1) T) 1/2,
yk= 1
where the Xk ’s and Yk ’s are independent q-variate standard normal random vectors, and →D
indicates convergence in distribution. In order to apply the result of Andersen, Brons and
Jensen (1983), saying:
if for a pair of square random matrices Pn , Qn ,(Pn,Qn ) converges in distribution to
(P,Q), where Q is a.s. nonsingular, then the ordered solutions of the generalized
eigenvalue problem det(Pn -λQn ) = 0 converge in distribution to the ordered solutions of