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I'm having trouble with the following inequality, I am not sure whether it is correct:

$$ P((X_n-X)^2+(Y_n-Y)^2>\epsilon^2) \stackrel{?}{\le} P(|X_n-X|>\epsilon)+P(|Y_n-Y|>\epsilon) $$

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    I think this doesn't always hold. It will be correct if we replace the left hand side by $P((X_n-X)^2+(Y_n-Y)^2 > 2\epsilon^2)$. To prove that inequality, note that $(X_n-X)^2+(Y_n-Y)^2 > 2\epsilon^2$ implies that either $|X-X_n| > \epsilon$ or $|Y-Y_n| > \epsilon$.2011-10-26

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