7.18 This problem will outline the argument needed to prove Theorem 7.19: (a) Show that mg(x) = mg(x), that is, g( )e|x |

7.18 This problem will outline the argument needed to prove Theorem 7.19:

(a) Show that ∇mg(x) = m∇g(x), that is,

∇



g(θ )e−|x−θ |

2 dθ =



[∇g(θ )] e−|x−θ |

2 dθ .

(b) Using part (a), show that r(π, δg) − r(gn, δgn ) =  

∇ log mg(x) − ∇ log mgn (x)





2 mgn (x) dx

=

 







∇mg(x)

mg(x) − ∇mgn (x)

mgn (x)









2 mgn (x) dx

≤ 2

 







∇mg(x)

mg(x) − mh2 n∇g(x)

mgn (x)









2 mgn (x) dx

+2  







mg∇h2 n (x)

mgn (x)









2 mgn (x) dx

= Bn + An.

(c) Show that An = 4  









mghn∇hn (x)

mgh2 n (x)











2 mgn (x) dx ≤ 4



mg(∇hn)2 (x) dx and this last bound → 0 by condition (a).

(d) Show that the integrand of Bn → 0 as n → ∞, and use condition

(b) together with the dominated convergence theorem to show Bn → 0, proving the theorem.