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

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 bn(βn) = log(βn + 1)
δn +
δn 2 .
Show the left side of (13.67) is equal to + ∞
bn (βn )
[exp(δn x − δ2 n 2 ) − 1]φ(x)dx = (bn(βn)) − (bn(βn) − δn) , and show this last expression tends to zero by appropriate choice of βn.