I've seen several examples when a sequence of r.v. converge in probability but not almost surely, yet none of them had the sequence to be independent. Would additional conditions of independence and convergence to a constant be sufficient to ensure almost surely convergence?
2026-03-28 15:20:27.1774711227
If independent r.v. converge in probability to a constant, do they converge almost surely?
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Let $A_n$ be independent events with $\mathbb{P}(A_n)=1/n$, and define $X_n=1_{A_n}$. Then $X_n\to0$ in probability, but $X_n$ does not converge almost everywhere.
Apply the second Borel-Cantelli lemma twice; once to the sequence $A_n$ and also to the sequence $A_n^c$, to conclude that
$$P([X_n=1\mbox{ infinitely often}] \cap [X_n=0\mbox{ infinitely often}])=1.$$