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I'm reading through some notes on Probability, and the statement is made that:

If random variables $X_1, \ldots, X_n$ converge to $X$ in mean square, then they also converge in probability.

Can someone please explain why this is the case? Regards.

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Fix $\delta>0$. Then $\delta^2 P(|X_n-X|\geq \delta)=\delta^2 P(|X_n-X|^2\geq \delta^2)\leq \int_{\Omega}|X_n-X|^2dP,$ so $P(|X_n-X|\geq \delta)\leq \frac 1{\delta^2}\int_{\Omega}|X_n-X|^2dP$ and we can conclude since the las integral converges to $0$.

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    Excellent, thanks for that.2012-02-09