Weak Convergence of a Convolution of measures

Question

Suppose I have two sequences of probability measures defined on the interval $[a,b]$, $\mu_{n}$ and $\nu_{n}$ where $\mu_{n}\to_{w}\mu$ and $\nu_{n}\to_{w}\nu$, is it true that $\mu_{n}*\nu_{n}\to_w\mu*\nu$? I can show that $\mu_n*\nu\to\mu*\nu$ but I'm not sure the argument goes through when we take both sequences

Answer 1

This can be shown very easily using the fact that weak convergence can be equivalently characterized by the convergence of the associated characteristic functions.

Denote by $\hat{\mu}$ the characteristic function of a probability measure $\mu$, then

$$\widehat{\mu_n \ast \nu_n} = \hat{\mu}_n \cdot \hat{\nu}_n$$

and the right-hand side converges, by assumption, to $\hat{\mu} \cdot \hat{\nu}$ (which is the characteristic function of $\widehat{\mu \ast \nu}$). This, in turn, implies by Lévy's continuity theorem that $\mu_n \ast \nu_n \to \mu \ast \nu$ weakly.