(iii) Uniformly in i and r ∈ [0, 1],
D31T EFzi x31,it - Ex31,it → 0k31×1 a.s.;
D32τ (EFzix32,it — Ex32,it) → √1rIk32g32,i (zi) a.s.;
D33T EFzi x33,it - Ex33,it → τ33 (r) g33,i (zi) a.s.,
where
g32,i = (g1,32,i, ..., gk32,32,i)0 ; g33,i = (g1,33,i, ..., gk33,33,i)0 ,
and g32,i (zi) and g33,i (zi) are zero-mean functions of zi with
0 <Esup kg3k,i (zi)k4q < ∞, for some q>1,
i
and g3k,i 6= g3k,j for i 6= j, and τ33 (r) = diag (r-m1,33, ...r-mk33,33).
(iv) There exist τ (r) and Gi (zi) such that
∖∖D3τ (Er,.x3,it — Ex3,it) k ≤ e(r)Gei(zi),
where R τ (r)4q dr < ∞ and E supi (Gi (zi)4q < ∞ for some q > 1.
(v) Uniformly in (i, j) and r ∈ [0, 1];
D31T (Ex31,it — Ex31,jt) → 0k31 ×1,
D32T (Ex32,it — Ex32,jt) → √rlk32 μgg3∙2i — μfl32j) ,
D33T (Ex33,it — Ex33,jt) → τ33 (r) (μg33i — Mg33j) ,
with suPi kμg32ik, suPi ∖∖μg33ik < ∞.
Some remarks would be useful to understand Assumption 6. First, to have
an intuition about what the assumPtion imPlies, we consider, as an illustrative
example, the simple model in CASE 3 in Section 2.2, in which x3,it = ∖∖zz∕lm +
eit, where eit is indePendent of zi and i.i.d. across i. For this case,
D3τ (EFzi X3,it — Exз,it¢ = D3τ∏i (zi — Ezi) /^tm;
D3T (Ex3,it — Ex3,jt) = D3T (ΠiEzi — ΠjEzj) /tm.
Thus,
g3k,i (zi) = Πi (zi — Ezi);
μg3k,i = niEzi.
Second, Assumption 6(iii) makes the restriction that E supi ∖g3k,i (zi)∖4q is
strictly positive, for k =2, 3. This restriction is made to warrant that g3k,i (zi) 6=
0 a.s. If g3k,i (zi) = 0 a.s.,15then
D3kTEFzi (x3k,it — Ex3k,it) ~ τ3k (r) g32,i (zi) = 0 a.s.,
15An example is the case in which x3,it = eit Πizi /tm ,where eit is independent of zi with
mean zero.
22