6 Appendix A: Preliminary Results
We here provide some preliminary lemmas that are useful to prove the main
results in Section 4. From now on, we use the notation M to denote a generic
constant, if no explanation follows.
Lemma 10 Let fi,T ,fi, and gi are integrable functions in probability space
(Ω, F,P). If fi,τ → fi a.s. uniformly in i as T → ∞, and there exists gi
such that |fi,T | ≤ gi for all i and T with E supi gi < ∞, then Efi,T → Efi
uniformly in i as T →∞.
Proof
Let hi,T = |fi,T
and hi,T → 0 uniformly in i
fi | . Under given assumptions, 0 ≤ supi hi,T ≤ 2 supi gi
as T →∞. Then, by Fatou’s Lemma
2E sup gi
i
E lim→i∞nf 2 sup gi
lim inf E 2 sup gi -
sup hi,T
sup hi,T
2E sup gi- lim sup E sup hi,T ,
i T→∞ i
from which we can deduce
lim sup E sup hi,T ≤ 0.
T→∞ i
Then, since
0 ≤lim sup sup |E (fi,T - fi)| ≤lim sup sup Ehi,T ≤lim sup E sup hi,T ≤ 0,
T→∞ i T→∞ i T→∞ i
we have the required result: Efi,T → Efi uniformly in i as T →∞. ¥
The following lemma is a uniform version of the Toeplitz lemma.
Lemma 11 Let ait be a sequence of real numbers such that ait → ai uniformly
in i as t → ∞ with supi |ai | < M. Then, (a) TT ɪ^t ait → ai uniformly in i, and
(b) TT Pt a2 → a2 uniformly in i.
Proof
From the uniform convergence of ait , for a given ε > 0, we can choose t0
such that t ≥ t0 implies that
sup |ait - ai | < ε.
i
Then, Part (a) follows because t ≥ t0 implies
sup 1 X (ait - ai)
iT
≤ 1 X sup |ait
Tt i
- ai | < ε.
31
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