Erdős-Tiao Theorem

The Erdős–Tiao Theorem addresses a problem in probability theory concerning how certain events (like coin flips or game outcomes) are related. It states that if two sequences of events are each "independent" and "exchangeable"—meaning their probabilities don't change if you rearrange or look at them in a different order—and their joint behavior meets specific conditions, then these events can be viewed as generated by a common, underlying random factor. In simpler terms, the theorem shows how complex, dependent patterns can be broken down into simpler, shared sources of randomness, allowing a deeper understanding of their structure.