Prove Lemma 13.5.1 as follows. Let = 1 r. Let L n = 1 n

Prove Lemma 13.5.1 as follows. Let η = 1 − √r. Let L˜ n = 1 n

n i=1 exp(δn Xi − δ2 n

2 )I{Xi ≤ 

2 log n} .

First, show Ln − L˜ n P

→ 0 (using Problem 13.41). Then, show E(L˜ n) = (η



2 log(n)) → 1 .

The proof then follows by showing V ar(L˜ n) → 0. To this end, show V ar(L˜ n) ≤

1 n E[X2 i I{Xi ≤ 

2 log n}] =

1 n exp(δ2 n )((2η − 1)



2 log n)

≤

1 n exp(δ2 n )φ((1 − 2η)



2 log n) = 1

√2π exp[−η2 log n] → 0 .