L2 convergence and probability

Topic: Expectation and convergence · more questions on this topic
A spike that keeps its mean ›

Use Markov’s inequality to show that L^2 convergence implies convergence in probability.

Worked answer (click to show)

Markov’s inequality applied to |X_n-X|^2 gives, for every \varepsilon>0,

\mathrm{P}(|X_n-X|>\varepsilon)\le\frac{\mathbb{E}|X_n-X|^2}{\varepsilon^2}.

If the numerator tends to 0, the probability tends to 0.