For Part (b), notice that
sup ∣a2t — a2∣ ≤ sup ∣αit + αi∣ sup ∣αit — αi∣
≤ M sup |ait - ai | → 0
as t →∞, where the last inequality holds because supi ∣ai ∣ < ∞ and sup ∣ait — ai ∣
→ 0. Then, by Part (a), we can obtain the desired result. ¥
Lemma 12 Suppose that Xi,T and Xi are sequences of random vectors. Sup-
pose that Xi,T → Xi in probability (or almost surely) uniformly in i as T →∞,
and NN Pi Xi → X in probability (or almost surely) as N → ∞. Then, as
(N,T→∞),
-1 X xiT → x
N ʌ i,T
i
in probability (or almost surely).
Proof
We only prove the lemma for the case of convergence in probability, because
the almost sure convergence case can be proven by the similar fashion. Since
NI Pi Xi →p X as N → ∞ and X →p Xi uniformly in i as T → ∞, for given
ε, δ > 0, we can choose N0 and T0 such that
P I 1 X Xi — X > εj ≤ δ ;
I Nγ i 2 ∫ ≤ 2;
P {sup ∣∣Xi,T - Xik > 2 j∙ ≤ 2,
whenever N ≥ N0 and T ≥ T0 . Now, suppose that N ≥ N0 and T ≥ T0 . Then,
Xi,T
—X
>ε
{∣ N XX (Xi
i,T
Xi) > 2
)+P {∣ N XX
Xi
P {sup k (Xi,T - Xi)k > 2J, + 2= δ ¥
Lemma 13 Suppose that a sequence of random vectors Zi is independently dis-
tributed across i. Let Fzi = σ (Zi) . Assume that QiT (k × k) is a sequence of
independent random matrices across i satisfying
sup EFzi kQiT k 1 {kQiT k >M} → 0 a.s., (36)
32
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