Introduction to Imprecise Probabilities

6.6 Limit theorems of probability

The main goal of this section is to give a constructive proof of Theorem 6.1, which says that event (6.1) holds strictly almost surely. The simple proof given here is inspired by [412] (proof of Theorem 3.1).

We say that a non-negative supermartingale forces an event if for every not in . We are required to construct a supermartingale forcing (6.1), where we can set , as we mentioned in Section 6.2. This is not difficult given Theorem 6.2.

Lemma 6.5

In Protocol 6.2, there is a game-theoretic supermartingale such that the supermartingale constructed in the proof of Theorem 6.2 forces

6.9

Proof.

Set . The protocol specifies that the initial capital is ...