A spike that keeps its mean

Topic: Expectation and convergence · more questions on this topic
‹ L2 convergence and probability A bound for a sample proportion ›

For X_n=n\,1_{\{U\le 1/n\}} with U uniform on (0,1), compute \mathrm{P}(X_n>\varepsilon), \mathbb{E} X_n, and \mathbb{E} X_n^2. Which modes of convergence to 0 hold?

Worked answer (click to show)

For n>\varepsilon, the event \{X_n>\varepsilon\} is \{U\le 1/n\}, so \mathrm{P}(X_n>\varepsilon)=1/n\to 0. Also \mathbb{E} X_n=1 and \mathbb{E} X_n^2=n^2\cdot(1/n)=n. The sequence converges to 0 in probability and almost surely. It does not converge to 0 in L^1 or in L^2, because those expectations do not tend to 0.