I've encountered the following question: suppose $X_n$ is a sequence of positive random variables such that $\mathbb{E}(X_n)=1$ for all $n$. Does it follow that $\sum X_n$ diverges almost surely?

This doesn't look true in general, but I could not find a counter example.

Following my intuition, I have tried to soften the claim:

  • if we suppose in addition that $X_n$ all have the same distribution (that is, independent of $n$), does the claim follow?
  • if not - if we assume that $X_n$ are also independent, does the claim follow?

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