## Wednesday, 14 May 2014

### Continuity theorem

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 $X_1,X_2,\dotsc$ have moment generating functions $m_i(\theta)$, $i=1,2,\dotsc$, and $m_i(\theta)\to m(\theta)$ as $i\to\infty$, pointwise for every $\theta$, then $X_i$ tends in distribution to the random variable having m.g.f. $m(\theta)$.

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.