Using betting strategy math to prove probability limits

This paper reframes the law of the iterated logarithm—a classical statistics result about how sample averages behave—through the lens of online betting. Instead of traditional probability arguments, the author uses wealth guarantees from betting strategies to derive empirical Bernstein inequalities. The connection is elegant: a bettor's accumulated winnings under adversarial conditions provide the same concentration bounds that statisticians use to quantify estimation error. The approach offers a concrete alternative perspective on convergence rates.