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Mathematics

Almost Sure Convergence and Convergence in Probability

Quick fact

Almost sure convergence is strictly stronger than convergence in probability: if a sequence converges almost surely, it also converges in probability, but the converse is false. For example, a sequence of random variables can converge in probability without converging at any single point of the sample space.

Why this is interesting

Flip a fair coin repeatedly. Does the proportion of heads get closer to 1/2? Yes, but there are two very different ways to understand that 'closeness'. That difference is the heart of this card.