The lecture schedules mention "Moment generating functions and statement (no proof) of continuity theorem.''
The continuity theorem (which I forgot to name) is in fact described in Remark 1 on page 91 of the notes. It is the result that if a sequence of random variables X1,X2,… have moment generating functions mi(θ), i=1,2,…, and mi(θ)→m(θ) as i→∞, pointwise for every θ, then Xi tends in distribution to the random variable having m.g.f. m(θ).
We use this when we give a "sketch of proof" of the Central limit theorem on page 91.
Lévy's continuity theorem is the same thing, but for characteristic functions.
The continuity theorem (which I forgot to name) is in fact described in Remark 1 on page 91 of the notes. It is the result that if a sequence of random variables X1,X2,… have moment generating functions mi(θ), i=1,2,…, and mi(θ)→m(θ) as i→∞, pointwise for every θ, then Xi tends in distribution to the random variable having m.g.f. m(θ).
We use this when we give a "sketch of proof" of the Central limit theorem on page 91.
Lévy's continuity theorem is the same thing, but for characteristic functions.
