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I would like some help with the following problem. Thanks for any help in advance.

Does there exist a sequence of ${L^1}$ random variables $(X_n)_{n=1}^\infty$ such that

$P(X_n\neq n i.o.(n)) = 0$ and $\lim_{n\to\infty} EX_n = -\infty$

If your answer is yes, provide an example. If your answer is no, prove that no such example exists.

Community
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shmiggens
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1 Answers1

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Your notation is slightly confusing, but I presume you mean the probability that $X_n \ne n$ for infinitely many $n$ is $0$.

Take $X_n = n$ with probability $1-1/n^2$ and $-n^5$ with probability $1/n^2$.

Robert Israel
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