Show that (14.42) holds with B = if V ar[Tj (X1)] is uniformly bounded in .

Show that (14.42) holds with B = ∞ if V arθ[Tj (X1)] is uniformly bounded in θ. Hint: Argue by contradiction. Suppose there exists hn with

|hn| ≥ b such that Ehnn−1/2 (φ∗

n) →  , where  is less than the right side of (14.42). This is a contradiction if Ehnn−1/2 (φ∗

n) → 1 if |hn|→∞. By taking subsequences if necessary, assume the jth component hn,j of hn satisfies |hn,j |→∞. Then, Ehnn−1/2 (φ∗

n) ≥ Phnn−1/2 {Z2 n,j > ck,1−α} .

It now suffices to show |Zn,j |→∞ in probability under hnn−1/2. But

|Eθ[Tj (X1)]| increases in θ (using properties of exponential families) while the variance of Zn,j remains bounded.