Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 3 / 24J (Probability and Measure)

Convergence in probability means $\mathbb{P}(|X_n-X|>ϵ)\to 0$ for every $ϵ>0$. Convergence in distribution means $F_{X_n}(t)\to F_X(t)$ at every continuity point $t$ of $F_X$. The inclusions obtained by separating the event $|X_n-X|>ϵ$ giveTake lower and upper limits as $n\to \infty$, then $ϵ\downarrow 0$ at a continuity point. Both bounds tend to $F_X(t)$, proving that convergence in probability implies convergence in distribution.

Uniform integrability means ${\sup }_n\mathbb{E}[|X_n|1_{\{|X_n|>K\}}]\to 0$ as $K\to \infty$. In particular the first absolute moments are uniformly bounded. By the Portmanteau theorem, $\mathbb{E}|X|\le {\mathrm{liminf}}_n\mathbb{E}|X_n|<\infty$. Let $T_K(x)=\max (-K,\min (x,K))$, a bounded continuous truncation. Convergence in distribution gives $\mathbb{E}{T}_K(X_n)\to \mathbb{E}{T}_K(X)$. ThereforeThe first term tends to zero by uniform integrability, the second by integrability of $X$. Hence $\boxed{\mathbb{E}{X}_n\to \mathbb{E}X}$.