Let X1,X2,...,Xn
iid∼ F, a distribution with finite mean µ, and suppose that the momentgenerating function m(t) of X exists for t in an open interval containing 0. Use Theorem 5.2.2 to show that X¯
D
→ δµ, the distribution degenerate at µ. (Hint: Write the mgf of X¯
as displayed below, and apply Lemma 5.3.1.
mX¯(t) = (mX (t/n))n =
1+ (t/n) EX + (t
2
/2!n
2
) EX2 + (t
3
/3!n
3
) EX3 +...n
.