Let X1,..., Xn be i.i.d. with density p0 or p1, and consider testing the null hypothesis H that p0 is true. The MP level-α test rejects when n i=1r(Xi) ≥

Cn, where r(Xi) = pi(Xi)/p0(Xi), or equivalently when 1

√n log r(Xi) − E0[log r(Xi)]



≥ kn

(i) Show that, under H, the left side of (11.41) converges in distribution to N(0, σ2)
with σ2 = Var0[log r(Xi)], provided σ < ∞.
(ii) From (i) it follows that kn → σz1−α, where zα is the α quantile of N(0, 1).
(iii) The power of the test (11.41) against p1 tends to 1 as n → ∞. Hint: Use Problem 3.41(iv).