Convergence in probability implies convergence of characteristic functions If $X_n$ converge in probability to $X$, then they also converge weakly, and we can conclude by the continuity theorem that the characteristic functions $\phi_n$ of $X_n$ converge pointwise to the characteristic function $\phi$ of $X$. How can we reach the same conclusion without appealing to the continuity theorem?

An alternative way of seeing this is as follows (solely for my own recording purposes). Let $Y_n(\theta)=\exp(e^{i \theta X_n})$. Note that, $|Y_n(\theta)|\leq 1$, and thus for every  fixed $\theta$, the sequence $\{Y_n(\theta)\}_{n\geq 1}$ is uniformly integrable. This, together with $Y_n \to Y$ (using continuous mapping thm) yield the $Y_n\to Y$ in $L^1$ (using a result given in Williams' book).  
