# Does minimizing KL imply boundedness of density ratios

Suppose

$$q^*(\theta) = \underset{q \in \mathcal{Q}}{argmin} \,\, KL[q(\theta)||p(\theta)] = \min_{q\in\mathcal{Q}} \int_{\theta\in\Theta}q(\theta)\ln\Big( \frac{q(\theta)}{p(\theta)} \Big)d\theta$$

is an approximation to $$p(\theta)$$ where $$Q$$ is a family of densities. Does a solution imply boundedness of $$q^*/p$$ (if such a $$q^*$$ is allowed within the family $$Q$$)?