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Mathematics

The Law of the Iterated Logarithm and Its Implications

Quick fact

For a fair coin toss, the maximum deviation from the mean, scaled by sqrt(n), almost surely has limsup equal to sqrt(2 log log n). This means the largest fluctuations grow like sqrt(2 n log log n), a result that holds almost surely and is not captured by the central limit theorem or the strong law of large numbers.

Why this is interesting

You flip a fair coin 1,000,000 times and record the number of heads minus tails. The average drift is tiny, but occasionally the path wanders far. How far can it possibly go? The answer is surprisingly precise.