Let $(\epsilon_t)_t$ be a sequence of iid random variables, distributed according to the density $f:\mathbb{R}\to (0,\infty)$ and $$ x_t = q( \theta^\star, x_1,x_2, \ldots, x_{t-1}) + \epsilon_t \,. $$ Assume that the derivative of $q$ with respect to $\theta$ is bounded from below, i.e. for all $\theta, x$ $$ \frac{\partial q(\theta, (x_r)_{r<t})}{\partial \theta} \geq c > 0\,. $$ The posterior likelihood is defined as $$ L_t(\theta) = \sum_{s=1}^t \log f \Big(x_s-q(\theta, (x_r)_{r<t})\Big) + \log \pi_0(\theta)\,, $$ where the prior density $\pi_0:\mathbb{R} \to (0,\infty)$ is strictly positive.

**Question:** Are those assumptions sufficient to conclude that the posterior
$$
\pi_t(\theta) = \frac{e^{L_t(\theta)}}{\int e^{L_t(z)} dz}
$$
converges almost surely to a dirac measure on $\theta^\star$? How could I show this? What would be a good reference?

I know that Bernstein von-Mises Theorem yields this type of result, but I could not find a version which is general enough to cover this case.

Thanks a lot!