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.
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.
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 $0$.