Define for some density $q_{\kappa_n}$ (with respect to the Lebesgue measure) on $\Theta$, parameterized by $\kappa_n \in \mathbb{R}^d$. Next, define the sequence of measures $\mu_n$ as
$$\mu_n(A) = \int_{A}q_{\kappa_n}(\theta)d\theta, \text{ for all measurable } A\subset \Theta.$$
Suppose that (i) $\mu_n \overset{w}{\to} \delta_{\theta^{\ast}}$, i.e. the sequence of measures $\mu_n$ weakly converges to a Dirac measure (point mass) at $\theta^{\ast}$ and that (ii) $\kappa_n \to \kappa^{\ast}$, where the latter convergence occurs in the standard Euclidean topology.
Does this imply strong convergence? (I.e., does it hold that $\|\mu_n – \delta_{\theta^{\ast}}\|_{\text{TVD}} \to 0$?)
Best Answer
Clearly not. The point mass measures are all at TV distance 1 from all measures with density functions, and the set $A=\{\theta^*\}$ attains the maximum of $|\mu_n(A)-\delta_{\theta^*}(A)|$, which evaluates to $1$.