Game-Theoretic Foundations for Probability and Finance

2Bernoulli's and De Moivre's Theorems

In the preceding chapter, we studied the convergence of a frequency or an average, an event that depends on infinitely many successive outcomes. We turn now to related approximations involving only finitely many successive outcomes. Such approximations were first established by Jacob Bernoulli and Abraham De Moivre for independent trials of an event, such as a tossed coin falling heads, that happens on each trial with probability . Bernoulli, in work published in 1713 [33], showed that the frequency that the coin will come up heads will be close to with high probability in a large number of trials; this was the first law of large numbers. De Moivre, in 1733 [108], showed that the probabilities for deviations of the frequency from can be described by the Gaussian probability distribution; this was the first central limit theorem.

Bernoulli's and De Moivre's theorems can be stated as follows, where is the fraction of the first tosses in which the ...