A quick note on convergence in probability vs. convergence a.s.

Standard

Convergence in probability says \mathbb{P}( |X_n-X| > \epsilon) \to 0, which controls how a single random variable X_n is close to its putative limit X. Convergence almost surely controls how the entire tail approaches the limit simultaneously. I.e. we have that for all \epsilon > 0, \mathbb{P}(\lim \sup \{|X_n - X| > \epsilon \} = 1) \to 0. This is equivalent to saying that \mathbb{P} \{ w: X_n(w) \to X(w) \} = 1. This means that if you were simulating out the sequence \{X_n \} convergence a.s. to X would mean that with probability 1 after some finite number of steps all of your X_i wouldn’t deviate from X by more than \epsilon. Convergence in probability would only guarantee that the deviations happen with increasingly small frequency. With this intuition in mind it is much easier to visualize situations that converge in probability but do not converge almost surely.

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