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Counterexample

Is this true?

Xn→0X_{n} \to 0 almost surely implies E[Xn]→0E[X_{n}] \to 0

No — it is false.

The counterexample

The object that kills this claim comes with Trap Analysis.

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86 counterexamples are free — including most of the ones Part C leans on hardest.

Go deeper· Explained in full in Modes of convergence and the limit theorems· Asked in CSIR NET December 2025· From Limit Theorems and Markov Chains › Modes of convergence, WLLN, SLLN, CLT

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