The law of the iterated logarithm in probability theory

Educação Superior::Ciências Exatas e da Terra::MatemáticaThe law of the iterated logarithm is a refinement of the strong law of large numbers, a fundamental result in probability theory. In the particular case of an unlimited sequence of Bernoulli trials with parameter p, the strong law asserts that with probability one, the relative frequency of successes converges to p as the number of trials grows. The relative frequency of successes is simulated for 1,000,000 trials, and is plotted against a log scale for the number of trials. As the number of trials increases the relative frequency is observed to remain within the funnel-shaped region described by the law of the iterated logarithm, and only in rare cases will it land outside the funnel. Move the slider to change the value of p and watch the behavior of the relative frequencies as a function of the number of trials. Open the control for the slider and use the "Play" animation control to repeat the process automatically for different values of