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Is there any pair of random variables (X,Y) such that Expected value of X goes to infinity, Expected value of Y goes to minus infinity but expected value of X+Y goes again to infinity?

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Just as in the deterministic case: find some real numbers $(x_n,y_n)$ such that $x_n$ goes to infinity, $y_n$ goes to minus infinity but $x_n+y_n$ goes to infinity. –  Did Oct 26 '12 at 14:08

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Let $H$ take positive values with density $\dfrac{2}{\pi (1+x^2)}$, i.e. a half Cauchy distribution with positive infinite expectation.

Let $X=2H$ and $Y=-H$ to meet your requirements.

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Try this: $$ \mathbf{E}X = 3 n \log n \\ \mathbf{E}Y=-2n \log n\\ \mathbf{E}[X+Y]=n \log n $$ The first expectation tends to $\infty$ as $n \to \infty$, the second one to $-\infty$, and the third one to $\infty$ again.

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