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This is a follow-up to this question.

Now let $(X_n)$ be a sequence of positive random variables. Suppose that the limit of expectation of this sequence $\lim_{n\rightarrow\infty}\mathbb{E}[X_n]=\mu$. Does that imply that $(X_n)$ converges to $\mu$ in mean, i.e., that $\lim_{n\rightarrow\infty}\mathbb{E}[|X_n-\mu|]=0$?

My previous question was for a general sequence of random variables, and Deven Ware showed that the above implication does not hold in the general case...

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  • $\begingroup$ Similarly with your previous question, take $X_n$ iid, $X_n=1$ with probability $1/2$ and $X_n=3$ with probability $1/2$. $\endgroup$ Dec 5, 2013 at 7:08

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Of course not. Try $X_n=X_1$ for every $n$, with $X_1\geqslant0$.

The WP page on the convergence of random variables might help you delineate some plausible implications in this context.

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