Prove Lemma 13.5.1 by using Problem 13.32. That is, if 1 < n = o(n) and Yn,i = exp(n Xi 2 n 2 )

Prove Lemma 13.5.1 by using Problem 13.32. That is, if 1 < βn =

o(n) and Yn,i = exp(δn Xi − δ2 n

2 ) , show that E[|Yn,i − 1|I{|Yn,i − 1| > βn}] → 0 . (13.66)

Since Yn,i > 0 and βn > 1, this is equivalent to showing E[(Yn,i − 1)I{Yn,i > βn + 1}] → 0 . (13.67)

The event {Yn,i > λ + 1} is equivalent to {Xi > bn(βn)}, where Prove the right-hand inequality.
(ii) Prove the left inequality in (13.64). Hint: Feller (1968) p.179 notes the negative of the derivative of the left side ( 1 t − 1 t 3 )φ(t) is equal to (1 − 3t−4)φ(t), which is certainly less than φ(t).
(iii) Use (13.64) to show that, for any fixed α, any δ > 0, and all large enough n:

(1 − δ)2 log n ≤ z1− α
n ≤ 
2 log n . (13.65)